Effective Field Theories

  • G. Mack
  • T. Kalkreuter
  • G. Palma
  • M. Speh
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 409)


Effective field theories encode the predictions of a quantum field theory at low energy. The effective theory has a fairly low ultraviolet cutoff. As a result, loop corrections are small, at least if the effective action contains a term which is quadratic in the fields, and physical predictions can be read straight from the effective Lagrangian. Methods will be discussed how to compute an effective low energy action from a given fundamental action, either analytically or numerically, or by a combination of both methods. Basically, the idea is to integrate out the high frequency components of fields. This requires the choice of a “blockspin”, i.e. the specification of a low frequency field as a function of the fundamental fields. These blockspins will be the fields of the effective field theory. The blockspin needs not be a field of the same type as one of the fundamental fields, and it may be composite. Special features of blockspins in nonabelian gauge theories will be discussed in some detail. In analytical work and in multigrid updating schemes one needs interpolation kernels A from coarse to fine grid in addition to the averaging kernels C which determines the blockspin. A neural net strategy for finding optimal kernels is presented. Numerical methods are applicable to obtain actions of effective theories on lattices of finite volume. The special case of a “lattice” with a single site (the constraint effective potential) is of particular interest. In a Higgs model, the effective action reduces in this case to the free energy, considered as a function of a gauge covariant magnetization. Its shape determines the phase structure of the theory. Its loop expansion with and without gauge fields can be used to determine finite size corrections to numerical data.


Effective Action Gauge Field Multigrid Method Higgs Model Effective Field Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Göpfert, M., Mack, G.: Commun. Math. Phys. 82 (1982) 545Google Scholar
  2. 2.
    Mack, G., Pordt, M.: Commun. Math. Phys. 97 (1985) 267Google Scholar
  3. 3.
    Karliner, M., Mack, G.: Nucl. Phys. B225 [FS9] (1983) 371Google Scholar
  4. 4.
    Gawgdzki, K., Kupiainen, A.: Commun. Math. Phys. 77 (1980) 31Google Scholar
  5. 5.
    Nielsen, H.B., Ninomiya, M.: Phys. Lett. B105 (1981) 219Google Scholar
  6. 6.
    O'Raifeartaigh, L., Wipf, A., Yoneyama, H.: Nucl. Phys. B271 (1986) 653Google Scholar
  7. 7.
    Wilson, K.: Phys. Rev. D3 (1971) 1818; Rev. Mod. Phys. 55 (1983) 583; Kogut, J., Wilson, K.: Phys. Rep. 12C (1974) 75Google Scholar
  8. 8.
    Polchinski, J.: Nucl. Phys. B231 (1984) 269Google Scholar
  9. 9.
    Mack, G.: Multigrid Methods in Quantum Field Theory, Cargèse Lectures July 1987, in: 't Hooft, G.: et al., Nonperturbative Quantum Field Theory (Plenum Press, New York, 1989)Google Scholar
  10. 10.
    Gawgdzki, K., Kupiainen, A.: Commun. Math. Phys. 99 (1985) 197Google Scholar
  11. 11.
    Ringwald, A., Wetterich, C.: Nucl. Phys. B 334 (1990) 506; Wetterich, C.: Nucl. Phys. 352 (1991) 529; Wetterich, C.: The average action for scalar fields near phase transitions, DESY 91–088 (August 91)Google Scholar
  12. 12.
    Palma, G.: Finite Size Scaling Analysis of the Constraint Effective Potential computed from Multigrid Monte Carlo, DESY 90-083 (July 1990)Google Scholar
  13. 13.
    Palma, G.: Renormalized Loop Expansion to Compute Finite Size Effects of the Constraint Effective Potential, to appear in Z. Phys.Google Scholar
  14. 14.
    Palma, G.: Renormalized Loop Expansion for the Gauge Covariant Constraint Effective Higgs Potential on the Lattice, PhD-thesis Hamburg, DESY 92-031 (February 1992)Google Scholar
  15. 15.
    Mack, G., Meyer, S.: The effective action from multigrid Monte Carlo, DESY 89-009 (January 1989) and Nucl. Phys. B ( Proc. Suppl.) 17 (1990) 293Google Scholar
  16. 16.
    Lin, L., Kuti, J., Shen, Y.: Nucl. Phys. B (Proc. Suppl.) 9 (1989) 26Google Scholar
  17. 17.
    Löscher, M., Weisz, P.: Nucl. Phys. B290 [FS20] (1987) 25; Lüscher, M., Weisz, P.: Nucl. Phys. B295 [FS21] (1988) 65Google Scholar
  18. 18.
    Kalkreuter, T.: Projective block spin transformations in lattice gauge theory, preprint, DESY 90-158 (December 1990) to appear in Nucl. Phys. BGoogle Scholar
  19. 19.
    Mack, G.: Nucl. Phys. B235 [FS11] (1984) 197Google Scholar
  20. 20.
    Wetterich, C., Reuter, M.: Average Action for the Higgs Model with Abelian Gauge Symmetry, preprint DESY 92-037 (March 1992)Google Scholar
  21. 21.
    Klauder, J.R.: lectures at this schoolGoogle Scholar
  22. 22.
    Montvay, L: Nucl. Phys. B293 (1987) 479Google Scholar
  23. 23.
    Hasenfratz, A., Hasenfratz, P.: Phys. Rev. D34 (1986) 3160Google Scholar
  24. 24.
    Coleman, S., Weinberg, E.: Phys. Rev. D7 (1973) 1888Google Scholar
  25. 25.
    DeGrand, T.: lectures at this schoolGoogle Scholar
  26. 26.
    Kalkreuter, T., Mack, G., Speh, M.: Int. J. Mod. Phys. C3 (1992) 121Google Scholar
  27. 27.
    Mai, G.: Ein Blockspin für 2d/2 Fermionen, diploma thesis, Hamburg (October 1989)Google Scholar
  28. 28.
    Joos, H., Schaefer, M.: Z. Phys. C34 (1987) 465; Golterman, M.F.L: Nucl. Phys. B278 (1986) 417Google Scholar
  29. 29.
    Smit, J.: Acta Phys. Polon. B 17 (1986) 531Google Scholar
  30. 30.
    Ben-Av, R., Brandt, A., Solomon, S.: Nucl. Phys. B329 (1990) 193; Ben-Av, R., Brandt, A., Harmatz, M., Katznelson, E., Lauwers, P.G., Solomon, S., Wolowesky, K.: Phys. Lett. B253 (1991) 185; Nucl. Phys. B (Proc. Suppl.) 20 (1991) 102; Ben-Av, R., Lauwers, P.G., Solomon, S.: Nucl. Phys. B 374 (1992) 249; Lauwers, P.G., Solomon, S.: Int. J. Mod. Phys. C3 (1992) 149Google Scholar
  31. 31.
    Hackbusch, W.: Multi-grid Methods and Applications (Springer-Verlag, Berlin, 1985)Google Scholar
  32. 32.
    Hulsebos, A., Smit, J., Vink, J.C.: Int. J. Mod. Phys. C3 (1992) 161 Vink, J.C.: Phys. Lett. B272 (1991) 81; Nucl. Phys. B (Proc. Suppl.), to appear (LATTICE '91)Google Scholar
  33. 33.
    Brower, R.C., Moriarity, K.J.M., Myers, E., Rebbi, C.: in: Multigrid Methods, ed. S.F. McCormick (Marcel Dekker, New York, 1988)Google Scholar
  34. 34.
    Brower, R.C., Rebbi, C., Vicari, E.: Phys. Rev. D43 (1991) 1965; Phys. Rev. Lett. 66 (1991) 1263; Brower, R.C., Moriarty, K.J.M., Rebbi, C., Vicari, E.: Nucl. Phys. B (Proc. Suppl.) 20 (1991) 89; Phys. Rev. D43 (1991) 1974Google Scholar
  35. 35.
    Hulsebos, A., Smit, J., Vink, J.C.: Nucl. Phys. B (Proc. Suppl.) 20 (1991) 94; Nucl. Phys. B368 (1992) 379Google Scholar
  36. 36.
    Brower, R.C., Edwards, R.G., Rebbi, C., Vicari, E.: Nucl. Phys. B366 (1991) 689Google Scholar
  37. 37.
    Vyas, V.: Int. J. Mod. Phys. C3 (1992) 169Google Scholar
  38. 38.
    Kalkreuter, T.: Phys. Lett. B276 (1992) 485Google Scholar
  39. 39.
    Brandt, A.: Multigrid techniques: 1984 guide, GMD-Studie No. 85Google Scholar
  40. 40.
    Kalkreuter, T.: in preparation Google Scholar
  41. 41.
    Hertz, J., Krogh, A., Palmer, R.G.: Introduction to the Theory of Neural Computation(Addison-Wesley, Redwood City CA, 1991)Google Scholar
  42. 42.
    Mack, G., Speh, M.: in preparation Google Scholar
  43. 43.
    Balaban, T.: Commun. Math. Phys. 89 (1983) 571Google Scholar
  44. 44.
    Pinn, K.: Z. Phys. C45 (1990) 453; Pinn, K.: Z. Phys. C47 (1990) 325; Computation of Effective Hamiltonians by Monte Carlo Simulations with Fixed Blockspins, preprint DESY 88–123 (August 1988)Google Scholar
  45. 45.
    Hasenbusch, M., Mack, G., Meyer, S.: Nucl. Phys. B (Proc. Suppl.) 20 (1991) 110Google Scholar
  46. 46.
    Meyer, S., Hasenbusch, M.: Testing accelerated algorithms in the lattice CP 3 model, preprint Kaiserslautern-TH-91-23 (January 1992)Google Scholar
  47. 47.
    Grabenstein, M., Pinn, K.: Acceptance Rates in Multigrid Monte Carlo, preprint DESY 91–157 (February 1992), to appear in Phys. Rev. DGoogle Scholar
  48. 48.
    Palma, G.: in preparation.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • G. Mack
    • 1
  • T. Kalkreuter
    • 1
  • G. Palma
    • 1
  • M. Speh
    • 1
  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburg 50

Personalised recommendations