Asymptotic form factors of hadrons and nuclei and the continuity of particle and nuclear dynamics

Stanley J. Brodsky; Benson T. Chertok

doi:10.1103/PhysRevD.14.3003

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Asymptotic form factors of hadrons and nuclei and the continuity of particle and nuclear dynamics

Stanley J. Brodsky^*

Benson T. Chertok^†

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

American University, Washington, D. C. 20016

^*Research supported by the Energy Research and Development Administration.
^†Research supported by the National Science Foundation under Grant No. MPS75-07325.

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Phys. Rev. D 14, 3003 – Published 1 December, 1976

DOI: https://doi.org/10.1103/PhysRevD.14.3003

Abstract

The large- $q^{2}$ behavior of the elastic form factor of a hadron or nucleus is related by dimensional counting to the number of its elementary constituents. Using the framework of a scale-invariant quark model, dimensional-scaling predictions are derived for the $\frac{B (q^{2})}{A (q^{2})}$ ratio in the Rosenbluth formula, multiple-photonexchange corrections, and the mass parameters which control the onset of the asymptotic power law in the meson, nucleon, and deuteron form factors. A simple "democratic chain" model predicts that for large $q^{2}$ , $F (q^{2}) \propto {(1 - \frac{q^{2}}{{m_{n}}^{2}})}^{1 - n}$ , where ${m_{n}}^{2}$ is proportional to the number of constituents $n$ . In the case of nuclear targets (or systems with several scales of compositeness), we also define the "reduced" form factor $f_{A} (q^{2}) = \frac{F_{A} (q^{2})}{Π {i = 1}^{A} [F_{i} ({q_{i}}^{2})]}$ in order to remove the minimal falloff of $F_{A}$ due to the nucleon form factors at ${q_{i}}^{2} = (\frac{{m_{i}}^{2}}{{M_{A}}^{2}}) q^{2}$ . Dimensional counting predicts ${(q^{2})}^{A - 1} f_{A} (q^{2}) \to const$ . A systematic comparison of the data for $π$ , $p$ , $n$ , and deuteron form factors with the dimensional-scaling quark-model predictions is given. Predictions are made for the large-spacelike- $q^{2}$ ${}^{3}{He}$ and $α$ -particle form factors. We also relate the deuteron form factor to (off-shell) fixed-angle $n - p$ scattering, and show that the experimental results for $t^{5} F_{d} (t)$ are consistent with the magnitude of the $s$ -wave wave function $u^{'} (0)$ obtained from soft-core potentials. The relation of the dynamics of an underlying six-quark state of the deuteron to the nucleon-potential and meson-exchange-current contributions is discussed. The scaling of $q^{2} f_{d} (q^{2})$ implies that the nuclear potential (after removing the effects of nucleon structure) displays the scale-invariant behavior of a theory without a fundamental length scale. Predictions are also given for the structure functions, fragmentation, and large-angle scattering of a nucleus.

References (32)

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[c1] R. Arnold et al., Phys. Rev. Lett. 35, 776 (1975)

[c2] J. Stack, Phys. Rev. 164, 1904 (1967)

[c3] S. Brodsky and G. Farrar, Phys. Rev. Lett. 31, 1153 (1973) Phys. Rev. D 11, 1309 (1975)

[c4] V. Matveev, R. Muradyan, and A. Tavkhelidze, Lett. Nuovo Cimento 7, 719 (1973)

[c5] S. Brodsky, in Proceedings of the International Conference on Few Body Problems in Nuclear and Particle Physics, Laval Univ., Québec, 1974, edited by R. Slobodrian (Les Presses de L'Université Laval, Quebec, 1975), p. 676

[c6] J. Gunion and L. Stodolsky, Phys. Rev. Lett. 30, 345 (1973)

[c7] R. Taylor, in Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies, Stanford, California, edited by W. Kirk (SLAC, Stanford, 1975), p. 679

[c8] Omitted endnote

[c9] [3] J. Stone et al., Bull. Am. Phys. Soc. Ser II, 21, 70 (1976)

[c10] [P. V. Landshoff, Phys. Rev. D 10, 1024 (1974)] [3] D. Sivers, S. Brodsky, and R. Blankenbecler, Phys. Rep. 23C, 1 (1976) M. Goldberger, A. Guth, and D. Soper, Phys. Rev. D 14, 1117 (1976)

[c11] E. D. Bloom and F. Gilman, Phys. Rev. D 4, 2901 (1971)

[c12] M. Moravcsik and P. Ghose, Phys. Rev. Lett. 32, 321 (1974)

[c13] J. Harte, Phys. Rev. 165, 1557 (1969) J. Ball and F. Zachariasen, ibid. 170, 1541 (1968)

[c14] M. Chanowitz and S. Drell, Phys. Rev. D 9, 2078 (1974)

[c15] C. Bebek et al., Phys. Rev. D 13, 25 (1976)

[c16] de Franceschi, Palumbo, and Simonov ([32])

[c17] R. Woloshyn, Phys. Rev. Lett. 36, 220 (1976)

[c18] M. Chemtob, E. Moniz, and M. Rho, Phys. Rev. C 10, 334 (1974)

[c19] R. Blankenbecler and J. Gunion, Phys. Rev. D 4, 718 (1971)

[c20] M. Gari and H. Hyuga, Phys. Rev. Lett. 36, 345 (1976)

[c21] D. Bollini et al., Lett. Nuovo Cimento 14, 418 (1975)

[c22] Ch. Berger et al., Phys. Lett. 35B, 87 (1971) W. Bartel et al., Nucl. Phys. B58, 429 (1973) W. Albrecht et al., Phys. Rev. Lett. 17, 1192 (1966) P. N. Kirk et al., Phys. Rev. D 8, 63 (1973) W. Atwood, SLAC Report No. SLAC-185, 1975 (unpublished)

[c23] W. Albrecht et al., W. Bartel et al., and Ch. Berger et al. in [22] K. Hanson et al., Phys. Rev. D 8, 753 (1973) J. Litt et al., Phys. Lett. 31B, 40 (1970) K. Chen et al., Phys. Rev. 140, B1267 (1965)

[c24] M. Gourdin, Phys. Rep. 11, 29 (1974) [10]

[c25] W. Bartel et al. and K. Hanson et al. in [22,23] R. Felst, DESY Report No. DESY 73/56 (unpublished)

[c26] J. Elias et al., Phys. Rev. 177, 2075 (1969) [4] S. Galster et al., Nucl. Phys. B32, 221 (1971)

[c27] J. McCarthy et al., Phys. Rev. Lett. 25, 884 (1970) M. Bernheim et al., Lett. Nuovo Cimento 5, 431 (1972)

[c28] F. Martin (private communication)

[c29] S. Gurvitz, Y. Alexander, and A. Rinat, Weizmann Report No. WIS-75-19 Ph., 1975 (unpublished)

[c30] F. Iachello, A. Jackson, and A. Lande, Phys. Lett. 43B, 191 (1973)

[c31] [30] Bartel et al. ([22]) K. Hanson et al. ([23]) W. Atwood ([22])

[c32] W. Schütz et al., SLAC Report No. SLAC-Pub-1770, 1976 (unpublished) L. L. Frankfurt and M. I. Strikman, Leningrad Reports Nos. LNPI-227 and 238, 1976 (unpublished) G. de Franceschi, F. Palumbo, and Yu. A. Simonov, Report No. LNF-76(P), 1976 (unpublished) V. I. Zakharov, Report No. ITEP-127, Rapporteur talk at the XVIII International Conference on High Energy Physics, Tbilisi, 1976 (unpublished)

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Asymptotic form factors of hadrons and nuclei and the continuity of particle and nuclear dynamics

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