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Il Nuovo Cimento A (1965-1970)

, Volume 15, Issue 1, pp 127–135 | Cite as

The derivation of a parton model sum rule within the framework of Dashen-Gell-Mann’s program

  • M. I. Pavković
Article

Summary

The sum rule\(\int\limits_1^\infty {(vW_2 /\omega ){\mathbf{ }}d\omega {\mathbf{ }} = {\mathbf{ }}\sum\limits_{N = 1}^\infty {P(N){\mathbf{ }}\left\langle {\sum\limits_{\sigma = 1}^N {e_\sigma ^2 } } \right\rangle _N } } \) is derived within the framework of Dashen-Gell-Mann’s program, under mild technical assumptions concerning the existence of the infinite-momentum limit. The derivation is consistently relativistic and mathematically exact.

Вывод правила сумм картонной модели в рамках программы Дащена-Гелл-Манна

Реэюме

Выводится правило сумм\(\int\limits_1^\infty {(vW_2 /\omega ){\mathbf{ }}d\omega {\mathbf{ }} = {\mathbf{ }}\sum\limits_{N = 1}^\infty {P(N){\mathbf{ }}\left\langle {\sum\limits_{\sigma = 1}^N {e_\sigma ^2 } } \right\rangle _N } } \) в рамках программы Дащена-Гелл-Манна при слабых технических предположениях, каса-юшихся сушествования предела бесконечного импульса. Предложенный вывод является последовательно релятивистским и математически точным.

Riassunto

Si deduce la regola di somma\(\int\limits_1^\infty {(vW_2 /\omega ){\mathbf{ }}d\omega {\mathbf{ }} = {\mathbf{ }}\sum\limits_{N = 1}^\infty {P(N){\mathbf{ }}\left\langle {\sum\limits_{\sigma = 1}^N {e_\sigma ^2 } } \right\rangle _N } } \) nel contesto del programma di Dashen e Gell-Mann con moderate ipotesi tecniche che riguardano l’esistenza del limite dell’impulso infinito. La deduzione è consistentemente relativistica e matematicamente esatta.

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References

  1. (1).
    For a recent review of the field seeE. D. Bloom et al.: preprint SLAC-PUB-796 (presented at theXV International Conference on High-Energy Physics, Kiev). See alsoH. W. Kendall: inProceedings of the International Symposium on Electron and Photon Interactions at High Energies, 1971, edited byN. B. Mistry (Ithaca, N. Y., 1972).Google Scholar
  2. (2).
    R. F. Dashen andM. Gell-Mann:Phys. Rev. Lett.,17, 340 (1966).MathSciNetCrossRefADSGoogle Scholar
  3. (3).
    S. D. Drell andJ. D. Walecka:Ann. of Phys.,28, 18 (1964). Note thatDrell andWalecka use the metrica·b=a·ba 0 b 0, while we use the metrica·b=a 0 b 0a·b, consistent with the metric tensorg 00=−g ii=1,g 00=—g ii=1,g μv= 0 forμ≠ν. The normalization of our states is 〈λ′p′|λp〉=<λP′λ|P′>(P′−P).CrossRefADSGoogle Scholar
  4. (4).
    R. P. Feynman: unpublished material andPhys. Rev. Lett.,23, 1415 (1969);J. D. Bjorken andE. A. Paschos:Phys. Rev.,185, 1975 (1969).CrossRefADSGoogle Scholar
  5. (5).
    A simple of the representative literature is given below:S. D. Drell, D. J. Levy andT. M. Yan:Phys. Rev. Lett.,22, 744 (1969);Phys. Rev.,187, 2159 (1969);Phys. Rev. D,1, 1035, 1617 (1970). The sum rule (I) can already be found in the firstPhys. Rev. reference.CrossRefADSGoogle Scholar
  6. (6).
    M. I. Pavković:Ann. of Phys.,62, 1 (1971);64, 474 (1971); Erratum,Ann. of Phys.,71, 611 (1972).CrossRefADSGoogle Scholar
  7. (7).
    H. Bebié, V. Gorgé andH. Leutwyler:Ann. of Phys.,74, 524 (1972). The sum rule (1) can be found in this reference. Note that in the models based on infinite-component wave equationsP(1) = 1,P(N) = 0 forN ≠ 1 and 〈e1/2〉1 = 1.CrossRefADSGoogle Scholar
  8. (8).
    K. Gottfried:Phys. Rev. Lett.,18, 1174 (1967).CrossRefADSMATHGoogle Scholar
  9. (9).
    Ref. (2). See alsoE. Roffman:Journ. Math. Phys.,8, 1954 (1967). We assume that\(e_\sigma {\mathbf{ }} = {\mathbf{ }}\lambda _\sigma ^3 /2{\mathbf{ }} + {\mathbf{ }}(1/\sqrt 3 ){\mathbf{ }}(\lambda _\sigma ^8 /2)\), where\(\delta _{\sigma Q} [\lambda _Q^i /2,{\mathbf{ }}\lambda _Q^j /2]{\mathbf{ }} = {\mathbf{ }}if_k^{ij} (\lambda _\sigma ^k /2) \cdot {\mathbf{ }}\{ f_k^{ij} |i,j,k{\mathbf{ }} = {\mathbf{ }}1,{\mathbf{ }}2,{\mathbf{ }}...,{\mathbf{ }}8\} \) are the Lie algebra structure constants of theSU 3 group. In the usual quark model\(\lambda _\sigma ^i /2\) are the familiar 3×3 Gell-Mann matrices.MathSciNetCrossRefADSGoogle Scholar
  10. (10).
    Shan-Jin Chang, R. Dashen andL. O’Raifeartaigh:Phys. Rev.,182, 1805, 1819 (1969).MathSciNetCrossRefADSGoogle Scholar
  11. (11).
    The statement that the infinite-component wave equations necessarily lead to scaling (in the senseενW 2dω ≠ 0) was first made in ref. (7), but the argument was based on the light-cone considerations and the observation that the associated field theories are canonical.CrossRefADSGoogle Scholar
  12. (12).
    See, for example,F. E. Close andJ. F. Gunion:Phys. Rev. D,4, 742 (1971).CrossRefADSGoogle Scholar
  13. (13).
    L. D. Landau andE. M. Lifshitz:Quantum Mechanics, Chap. XV (New York, 1958).Google Scholar
  14. (14).
    T. D. Lee andC. N. Yang:Phys. Rev.,126, 2239 (1962);S. Adler:Phys. Rev.,143, 1144 (1966);J. D. Bjorken:Phys. Rev.,179, 1547 (1969);T. M. Yan andS. Drell:Phys. Rev. D,1, 2402 (1970).MathSciNetCrossRefADSMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1973

Authors and Affiliations

  • M. I. Pavković
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

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