Skip to main content
SpringerLink
Log in

Nonlinear σ-model and nucleon structure

  • Published:
Zeitschrift für Physik C Particles and Fields

Abstract

The nonlinear σ-model with the Wess-Zumino action describes the nucleon as a soliton and incorporates the non-abelian chiral anomalies. Several studies have shown that the model works well except for the nucleon mass, which comes out consistently too large. We investigate this question beginning with the more general framework of the linear σ-model, which has besides a pseudoscalar meson sector, a fermion or quark sector, a scalar field and an interaction between the fermions via the scalar field. Using a path integral formulation, we express the fermion measure of the model as the product of a Jacobian and an invariant measure. Identifying this Jacobian as exp[iΓ wz] , we find that the model breaks up into two parts, when in the pseudoscalar meson sector the scalar field is replaced by its vacuum value. The pseudoscalar part of the model becomes the nonlinear σ-model with the Wess-Zumino actionΓ wz. The other part involves chiral fermions, the scalar field and their interaction. We continue this part back to the Minkowski space to determine its ground state and energy levels. We find that for a scalar field that vanishes at smallr, but rises sharply to its vacuum value at someR, the ground state energy of the interacting quark-scalar-field system can be lower than the ground state energy of the non-interacting quark system. This means the interaction between quarks and the scalar field can lead to a condensed ground state or vacuum and can reduce the overall energy of the system (a phase transition as in superconductivity). It is, therfore, not surprising that the nonlinear σ-model predicts too large a nucleon mass, since it implicitly assumes a normal non-interacting vacuum in the quark sector. Quarks are now quasiparticles that appear as excitations of the condensed vacuum. The nucleon structure that emerges from this investigation agrees fully with the phenomenological nucleon structure found from analysis of high energy elasticpp and\(\bar p\) p scattering at CERN ISR and SPS Collider.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. U.-G. Meissner, N. Kaiser, W. Weise: Nucl. Phys. A 466 (1987) 685

    Google Scholar 

  2. M. Lacombe, B. Loiseau, R. Vinh Mau, W.N. Cottingham: Phys. Rev. D 38 (1988) 1491

    Google Scholar 

  3. P. Jain et al.: Phys. Rev. D 37 (1988) 3252; J. Schechter: in: Proceedings of the Storrs Meeting, p. 587, K. Haller et al. (eds.) Singapore: World Scientific 1989

    Google Scholar 

  4. M.M. Islam, V. Innocente, T. Fearnley, G. Sanguinetti: Europhys. Lett. 4 (1987) 189

    Google Scholar 

  5. M.M. Islam: in: Proceedings of the Second Int. Conf. on Elastic and Diffractive Scattering p. 251 K. Goulianos. (ed.) Gif-sur-Yvette: Editions Frontieres, 1988

    Google Scholar 

  6. I. Zahed, G.E. Brown: Phys. Rep. 142 (1986 1

    Google Scholar 

  7. E. Witten: Nucl. Phys. B223 (1983) 422, 433

    Google Scholar 

  8. Ö. Kaymakcalan, S. Rajeev, J. Schechter: Phys. Rev. D30 (1984) 594

    Google Scholar 

  9. T. Fujiwara et al.: Prog. Theo. Phys. 73 (1985) 926

    Google Scholar 

  10. K. Fujikawa: Phys. Rev. D29 (1984) 285

    Google Scholar 

  11. K. Fujikawa: Phys. Rev. D21 (1980) 2848; 22 (1980) 1499 (E)

    Google Scholar 

  12. It is useful to keep in mind that a fermion field configuration is an element of a Grassmann algebra, and what we are considering is a linear transformation from one set of Grassmann integration variables to another set of Grassmann integration variables

  13. Following the work of Witten [7], the Wess-Zumino action was derived by many authors using a variety of approaches, such as trial and error gauging, direct integration of Bardeen anomaly, direct gauging and dropping gauge invariant terms, integration by differential geometric method: Ö. Kaymakcalan, S. Rajeev, J. Schechter: Phys. Rev. D30 (1984) 594; N.K. Pak, P. Rossi: Nucl. Phys. B 250 (1985) 279; K. Chou, H. Guo, K. Wu, X. Song: Phys. Lett. 134 B (1984) 67; J.L. Mañes: Nucl. Phys. B250 (1985) 369. Our approach has been to focus on the transformation properties of the full fermion measure written in terms of Grassmann variables

  14. R. Rajaraman: Solitons and instantons, ch. 9. Amsterdam: North-Holland 1982

    Google Scholar 

  15. W.A. Bardeen et al.: Phys. Rev. D11 (1975) 1094

    Google Scholar 

  16. To be more precise, the relavant vacuum is the product vacuum |O c 〉|Ψ0〉, where |O c > is the coherent vacuum (see [15]) of the ζ-field:\(|O_c \rangle = \exp ( - i\smallint d^3 x\zeta _c (x)\hat \pi (x))|0\rangle .\).\(\hat \pi (x)\) is the canonical momentum density of the ζ-field. The classical field ζ c (x) is the field that occurs in (4.8) and (4.12). We see that the two vacua are tied together and constitute, in fact, one vacuum, since the form of ζ c (x) determines |Ψ0〉 and vacuum expectation value of ψ†γ0ψ in |Ψ0〉 determines ζ c (x).

  17. Higgs fieldh(x) is defined by ζ(x)=f π+h(x), so that\(\hat \zeta (x) = \hat h(x)\) andm 2 h =8#x03BB;f 2π

  18. Y. Nambu, G. Jona-Lasinio: Phys. Rev. 122 (1961) 345

    Google Scholar 

  19. P.D. Mannheim: Phys. Rev. D 12 (1975) 1772

    Google Scholar 

  20. U.-G. Meissner: Phys. Reports 161 (1988) 213

    Google Scholar 

  21. R. Friedberg, T.D. Lee: Phys. Rev. D15 (1977) 1694; Phys. Rev. D 16 (1977) 1096

    Google Scholar 

  22. T.D. Lee: Particle physics and introduction to field theory. ch. 20 New York: Harwood Academic 1981

    Google Scholar 

  23. L. Wilets: in: Chiral solitons. p. 362. K.-F. Liu (ed.) Singapore: World Scientific 1987

    Google Scholar 

  24. L.S. Celenza, A. Rosenthal, C.M. Shakin: Phys. Rev. C31 (1985) 212

    Google Scholar 

  25. S. Kahana, G. Ripka, V. Soni: Nucl. Phys. A415 (1984) 351

    Google Scholar 

  26. M.C. Birse, M.K. Banerjee: Phys. Rev. D31 (1985) 118

    Google Scholar 

  27. M.K. Banerjee, W. Broniowski, T.D. Cohen: Chiral soliton, ibid.Phys. Rev. D31 (1985) p. 255

    Google Scholar 

  28. D. Ebert, H. Reinhard: Nucl. Phys. B271 (1986) 188

    Google Scholar 

  29. D. Ebert, H. Reinhardt: Phys. Lett. B173 (1986) 453, 459

    Google Scholar 

  30. A. Dhar, S.R. Wadia: Phys. Rev. Lett. 52 (1984) 959; A. Dhar, R. Shankar, S.R. Wadia: Phys. Rev. D 31 (1985) 3256

    Google Scholar 

  31. H. Gomm, P. Jain, R. Johnson, J. Schechter: Phys. Rev. D33 (1986) 3476; P. Jain, R. Johnson, J. Schechter: Phys. Rev. D 35 (1987) 2230

    Google Scholar 

  32. M. Lacombe, B. Loiseau, R. Vinh Mau, W.N. Cottingham: Phys. Rev. D40 (1989) 3012

    Google Scholar 

  33. A.P. Balachandran, V.P. Nair, S.C. Rajeev, A. Stern: Phys. Rev. D27 (1983) 1153

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Connecticut, 06269, Storrs, CT, USA

    M. M. Islam

Authors
  1. M. M. Islam
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Islam, M.M. Nonlinear σ-model and nucleon structure. Z. Phys. C - Particles and Fields 53, 253–262 (1992). https://doi.org/10.1007/BF01597561

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01597561

Keywords

Search

Navigation