Summary
The implications of the validity of chiralSU 3×SU3 current algebra and the assumption that the chiral-symmetry breaking term transforms according to the (3,3*)+(3*, 3) representation ofSU 3×SU3 are investigated. It is argued that approximateSU 2×SU2 symmetry is expected on the grounds of the smallness of the pion mass together with some reasonably smooth behavior for matrix elements which involve a pion. The PCAC hypothesis, in its pole-dominance form, is applied to the π, K and ν cases in a step-by-step manner. Some results are obtained for the Kℓ3 form factors, includingf +(0)=1. Except for the equal-time commutation relations,SU 3 symmetry is not invoked even approximately. Sum rules involvingf π,f K,f ν and the pseudoscalar-meson masses are derived, some of which differ significantly from those obtained by authors using approximateSU 3 symmetry. In particular, an expression forf K/f π is inferred which involves pseudoscalar-meson masses alone.
Riassunto
Si studiano le implicazioni della validità dell'algebra chirale delle correntiSU 3×SU3 e l'ipotesi che il termine di rottura della simmetria chirale si trasformi secondo la rappresentazione (3, 3*)+(3*, 3) diSU 3×SU3. Si deduce che la simmetria approssimata diSU 2×SU2 si postula sulla base della piccolezza della massa del pione assieme con un comportamento ragionevolmente omogeneo per gli elementi di matrice che coinvolgono un pione. Si applica, nei casi di π, K e ν, in passi successivi l'ipotesi di PCAC, nella sua forma di dominanza del polo. Si ottengono alcuni risultati per i fattori di forma Kℓ3 comprendentif +(0)=1. Tranne che per le relazioni di commutazione di tempo uguale, non si invoca neanche approssimativamente la simmetriaSU 3. Si ricavano alcune regole comprendentif π,f K,f ν e le masse del mesone pseudoscalare; alcune di esse differiscono significativamente da quelle ottenute da altri autori usando simmetrieSU 3 approssimate. In particolare si deduce un'espressione perf K/f π che coinvolge solo masse mesoniche pseudoscalari.
Резюме
Исследуются выводы справедливости алгебры токов чиральнойSU 3×SU3 симметрии и предположение, что член, нарушающий чиральную симметрию, преобразуется согласно (3, 3*)+(3*, 3), представлениюSU 3×SU3. Утверждается, что прибиженнаяSU 2×SU2 симметрия предполагается на основе малости пионной массы совместно с некоторым разумно плавным поведением для матричных элементов, которые включают пионы. Гипотеза РСАС, в ее форме полюсного преобладания, применяется в случаях π, К и ν по методу последовательных приближений. Получаются некоторые результаты для форм-факторов Кℓ3, включаяf +(0)=1. За исключением коммутационных соотношений при равных временах, симметрияSU 3 не используется, даже приближенно. Выводятся правила сумм, включающиеf π,f K,f ν и псевдоскалярные мезонные массы, некоторые из которых отличаются от правил, полученных авторами при использовании приближеннойSU 3 симметрии. В частности, выводится выеажение дляf K/f π, которое включает только массы псевдоскалярных мезонов.
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M. Gell-Mann:Phys. Rev.,125, 1067 (1962);Physics,1, 63 (1964).
M. Gell-Mann, R. Oakes andB. Renner:Phys. Rev.,175, 2195 (1968), hereafter referred to as GOR.
With\(c \approx - \sqrt 2 \) this is just the result ofC. G. Callan andS. B. Treiman:Phys. Rev. Lett.,16, 153 (1966).
N. H. Fuchs:Phys. Rev.,170, 1310 (1968);172, 1532 (1968).
N. T. Nieh:Phys. Rev. Lett.,21, 116 (1968).
H. Pagels: preprint, Rockefeller University (1968);R. A. Coleman andJ. Moffat: preprint, University of Toronto (1969).
For an analysis along these lines, seeT. D. Lee andC. S. Wu:Ann. Rev. Nucl. Sci.,15, 381 (1966).
W. J. Willis:Proceedings of the Heidelberg International Conference on Elementary Particles (1968), p. 284.
B. W. Lee:Phys. Rev. Lett.,20, 617 (1968).
While this paper was being written, we received a very interesting preprint fromP. R. Auvil andN. G. Deshpande (Northwestern University) which studies chiral-symmetry breaking for the pseudoscalar mesons along similar lines to those of the present work.Auvil andDeshpande include ν-ν′ mixing. Their paper differs in some details from ours in both hypothesis and conclusions.
Fayyazuddin andRiazuddin: preprint (1968) (unpublished).
Knowledge of the constantf ν is important theoretically, since it is involved in some calculations of ν decay. SeeRiazuddin andA. Q. Sarker:Phys. Rev. Lett.,20, 1455 (1968) for an example of a calculation which is extremely sensitive tof ν/f π.
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To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. This work was supported in part by the U.S. Atomic Energy Commission, Contract No. AT(11-1)-1428.
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Fuchs, N.H., Kuo, T.K. ChiralSU 3×SU3 algebra and approximateSU 2×SU2 symmetry. Nuovo Cimento A (1965-1970) 64, 382–396 (1969). https://doi.org/10.1007/BF02754900
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DOI: https://doi.org/10.1007/BF02754900