Summary
Formulae for the structure functionsW 1 andW 2 are obtained by means of the statistical approach to the Veneziano model together with pomeron dominance. A form forW 2 is obtained that scales and agrees well with the data. The question of the scaling ofW 1 is discussed. Phase-space corrections and generalized Glauber corrections are then studied in connection with e-D scattering; they are found to be small. Hence, contributions from both the pomeron and lower trajectories are used to generalize the model. Satisfactory agreement is obtained.
Riassunto
Per mezzo dell’approccio statistico al modello di Veneziano e della predominanza del pomerone, si ottengono formule per le funzioni di strutturaW 1 eW 2. Si ottiene una forma diW 2 che varia di scala e concorda bene con i dati. Si discute la questione della variazione di scala diW 1. Si studiano poi le correzioni dello spazio delle fasi e le correzioni di Glauber generalizzate in rapporto allo scattering e-D; si trova che esse sono piccole. Per cui si usano i contributi sia del pomerone che delle traiettorie inferiori per generalizzare il modello. Si ottiene un accordo soddisfacente.
Реэюме
Испольэуя статистический подход к модели Венециано вместе с доминантностью померона, мы получаем формулы для структурных функцийW 1 иW 2. Определяется формаW 2, которая определяет масщтаб и хорощо согласуется с имеюшимися данными. Обсуждается вопрос масщтабаW 1. Затем в свяэи с е-D рассеянием исследуются поправки фаэового пространства и обобшенные глаибе-ровские поправки. Эти поправки окаэываются малы. Следовательно, для обобшения зтой модели испольэуются вклады и от померона и от инэщих траекторий. Получается удовлетворительное согласие.
Similar content being viewed by others
References
J. D. Bjorken:Phys. Rev.,179, 1547 (1969).
R. P. Feynman:Phys. Rev. Lett.,23, 1415 (1969).
For a detailed discussion of this calculation as well as for a thorough review of all the theories that have been mentioned, see:C. H. Llewellyn-Smith:Introduction to highly inelastic lepton scattering and related processes, CERN Pub. TH 1188;
L. S. Brown:Causality in electroproduction at high energy, inLectures Given at the Summer Institute for Theoretical Physics, is a slightly older reference.
The following references would constitute a reasonably complete listing for the experimental data mentioned in this paper:L. W. Mo:Inelastic electron-proton scattering, lecture delivered at theStony Brook Conference on High-Energy Collisions, edited byC. N. Yang (New York, 1969);
R. E. Taylor andF. J. Gilman: inProceedings of the IV International Symposium on Electron and Photon Interactions and High Energy (Liverpool, 1969);
E. D. Bloom, D. H. Coward, H. DeStaebler, J. Drees, G. Miller, L. W. Mo, R. E. Taylor, M. Breidenbach, J. I. Friedman, G. C. Hartmann andH. W. Kendall:Phys. Rev. Lett.,23, 930 (1968);
G. Miller, E. D. Bloom, G. Buschhorn, D. H. Coward, H. DeStaebler, J. Drees, C. L. Jordan, L. W. Mo, R. E. Taylor, J. I. Friedman, G. C. Hartmann, H. W. Kendall andR. Verdier:Inelastic electron-proton scattering at large momentum transfers, SLAC preprint;
E. D. Bloom, G. Buschhorn, R. L. Cottrell, D. H. Coward, H. DeStaebler, J. Drees, C. L. Jordan, G. Miller, L. Mo, H. Piel, R. E. Taylor, M. Breidenbach, W. R. Ditzler, J. I. Friedman, G. C. Hartmann, H. W. Kendall andJ. S. Poucher: contributed paper to theXV International Conference on High-Energy Physics (Kiev, 1970), SLAC Pub. 796.
J. W. Moffat andV. G. Snell:Phys. Rev. D,3, 2848 (1971) as well as various unpublished University of Toronto preprints. Reference (4) also contains information concerning this model.
G. Veneziano:Nuovo Cimento,57 A, 190 (1968).
Y. Nambu: inProceedings of the International Conference on Symmetries and Quark Models (Detroit, Mich., 1969).
Y. Nambu:Phys. Rev. D,4, 1193 (1971).
P. V. Landshoff andJ. C. Polkinghorne:Nucl. Phys.,19 B, 432 (1970); see also,Y. Nambu: ref. (8) above.
L. N. Chang, P. G. O. Freund andY. Nambu:Phys. Rev. Lett.,24, 628 (1970).
H. Harari:Phys. Rev. Lett.,22, 1078 (1969);P. G. O. Freund:Phys. Rev. Lett.,20, 235 (1968).
D. Bloom andF. J. Gilman:Phys. Rev. Lett.,25, 1140 (1970);Phys. Rev. D,4, 2901 (1971).
R. J. Glauber: inLectures in Theoretical Physics, Vol.1 (New York, 1959), p. 315; inHigh-Energy Physics and Nuclear Structure, edited byG. Alexander (Amsterdam, 1967), p. 311; inHigh-Energy Physics and Nuclear Structure, edited byS. Devons (New York, 1970), p. 207.
S. Drell andJ. D. Walecka:Ann. of Phys.,28, 18 (1964). Most standard QED textbooks also discuss this calculation, at least for the elastic case.
L. N. Hand:Phys. Rev.,129, 1834 (1963).
For details of this calculation, seeY. Nambu andA. Hacinliyan:Veneziano models and e-pscattering, EFI preprint 70-67, unpublished, orS. Matsuda andJ. Manassah:Phys. Rev. D,4, 882, 3062 (1971).
For example, we can writeW i=S α i (sg 2). The Pomeranchuk theorem says thatα i (s, t) → 1 forW 1 and −1 forW 2 ass → ∞. Now if we holds large and fixed and write a dispersion relation forα i inq 2, we should have at least one subtraction. This subtraction is done at the vector-meson mass (or any fixed point ≪s). If one now passes to the scaling limit, assuming that the results scale, form factors of the form (3.6) are obtained. The author would like to thank Dr.K. Fujikawa for pointing this out. As a general reference of the phase representation, see, for instance,Y. Nambu andM. Sugawara:Phys. Rev.,132, 2724 (1963).
J. I. Friedman, H. W. Kendall, E. D. Bloom, D. H. Coward, H. DeStaebler, J. Drees, C. L. Jordan, G. Miller andR. E. Taylor:Behavior of the electromagnetic inelastic structure functions of the proton, SLAC preprint.
Fong Ching Chen:Phys. Lett.,34 B, 625 (1971).
G. B. West: Stanford preprint ITP-397, and private communication to be published in theAnn. of Phys. Nearly all of the results in this Subsection have been independently obtained and generalized byWest.
T. Hamada andI. D. Johnston:Nucl. Phys.,34, 382 (1962).
I. J. McGee:Phys. Rev.,151, 772 (1966).
R. J. Glauber:Phys. Rev.,100, 242 (1955);R. J. Glauber andV. Franco:Phys. Rev.,142, 1195 (1966).
Blankenbecler andGunion claim that the standard Glauber treatment fails here since the momentum transfers are large. However, the only significant difference between their result and classical Glauber theory is a phase factor. Thus our calculation clearly over-estimates these corrections, except very close to the threshold. For further details,J. F. Gunion andR. Blankenbecler:Phys. Rev. D,3, 2125 (1971) could be consulted.
K. Gottfried: inProceedings of the Stony Brook Conference (ref. (5a)).
The fact that arather large contribution from the lower trajectories is required forW 1p is compatible with Compton scattering data (reported inR. L. Anderson, D. Gustavson, J. Johnson, I. Overman, D. Ritson, B. W. Wiik, R. Talman, J. K. Walker andD. Worcester:Phys. Rev. Lett.,25, 1218 (1970);A. M. Boyarski, D. H. Coward, S. Ecklund, B. Richter, D. Sherden, R. Siemann andC. Sinclair:Phys. Rev. Lett.,26, 1600 (1971)). It is well known that the deep-inelastic-scattering cross-section is related to the imaginary part of the forward Compton scattering cross-section. Our crucial assumption is that the latter is purely imaginary near the forward direction. Then our structure function can be extended tot≠0. A typical contribution will be\( \begin{gathered} \left\langle {P\left| {J_\mu ^h \left( q \right)\pi \left( s \right)J_\nu ^h \left( {q'} \right)} \right|P} \right\rangle \sim \exp \left[ {q \cdot q'\sum\limits_n {\left| {\Delta f_n } \right|{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 n}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$n$}}} } \right] \exp \left[ {\beta q \cdot q'\sum\limits_n {\left| {\Delta f_n } \right|^2 } } \right] \sim \hfill \\ \sim \exp \left[ {At} \right]\exp \left[ { - \lambda t/4M\nu } \right] \hfill \\ \end{gathered} \). The second term is in the same form as what we had in the case of deep inelastic scattering. Because of the inequality of the two momenta, we now have an additional zeroth-order term exp [At]. Unfortunately, the kinematics makes it very difficult to verify contributions from the second factor. Combining the first factor above with the Regge factor ν(α)t, we essentially have a formc(ν/ν0)(α)t, a typical Regge fit. These fits have previously appeared in the literature (e.g.,Moffat andSnell, ref. (5)). We have attempted to fit both the neutron and proton data. In either case, the slope of the pomeron comes out as 0.54/(GeV)2, the slope of the lower trajectory around 1/(GeV)2 with 20% uncertainty. The combined neutron and proton data require only a 30% contribution from lower trajectories. For theproton data only, an alternate fit requires a contribution from lower trajectories which is 3 times that from the pomeron. Thex-squared values are roughly 1 per d.f. in the first case and 0.54 per d.f. in the second fit. This may enhance our finding that theremay be substantial contributions to the protonW 1 form factor from lower trajectories.
Author information
Authors and Affiliations
Additional information
Supported in part by the U.S. Atomic Energy Commission. Submitted to the Department of Physics, The University of Chicago, in partial fulfilment of the requirements for the Ph. D. degree.
Rights and permissions
About this article
Cite this article
Hacinliyan, A. Deep inelastic electron-proton and electron-neutron scattering. Nuov Cim A 8, 541–569 (1972). https://doi.org/10.1007/BF02722725
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02722725