Abstract
The formulation of the non linear σ-model in terms of flat connection allows the construction of a perturbative solution of a local functional equation by means of cohomological techniques which are implemented in gauge theories. In this paper we discuss some properties of the solution at the one-loop level in D = 4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of counterterms parameters have to be introduced in the effective action in order to make the theory finite at one loop, while respecting the functional equation (fully symmetric subtraction in the cohomological sense). The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counterterms are expressed in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. These latter amplitudes contain only insertions of the composite operators φ0 (the constraint of the non linear σ-model) and F μ (the flat connection). The structure of the functional equation suggests a hierarchy of the Green functions. In particular once the amplitudes for the composite operators φ0 and F μ are given all the others can be derived by functional derivatives. In this paper we show that at one loop the renormalization of the theory is achieved by the subtraction of divergences of the amplitudes at the top of the hierarchy. As an example we derive the counterterms for the four-point amplitudes.
Similar content being viewed by others
References
Appelquist, T., Bernard, C. W. (1981). Physics Review D 23, 425.
Barnich, G., Brandt, F., Henneaux, M. (2000). Physics Reports 338, 439
Becchi, C., Rouet, A., Stora, R. (1975). Communications in Mathematical Physics 42, 127.
Becchi, C., Rouet, A., Stora, R. (1976). Annals of Physics 98, 287.
Bijnens, J., Colangelo, G., Ecker, G. (2000). Annals of Physics 280, 100
Blasi, A., Collina, R. (1987). Nuclear Physics B 285, 204.
Charap, J. M. (1970). Physics Review D 2, 1554.
Ecker, G., Honerkamp, J. (1971). Nuclear Physics B 35, 481.
Faddeev, L. D., Slavnov, A. A. (1973). Lettere al Nuovo Cimento 8, 117.
Ferrari, R. (2005). Journal of High Energy Physics 0508, 048
Ferrari, R., Quadri, A. (2006). Journal of High Energy Physics 0601, 003
Gasser, J., Leutwyler, H. (1984). Annals of Physics 158, 142.
Gasser, J., Leutwyler, H. (1985). Nuclear Physics B 250, 465.
Gerstein, I. S., Jackiw, R., Weinberg, S., Lee, B. W. (1971). Physics Review D 3, 2486.
Gomis, J., Paris, J., Samuel, S. (1995). Physics Reports 259, 1
Henneaux, M., Wilch, A. (1998). Physical Review D 58, 025017
Honerkamp, J., Meetz, K. (1971). Physics Review D 3, 1996.
Piguet, O., Sorella, S. P. (1995). Lecture Notes in Physics M28, 1.
Tataru, L. (1975). Physics Review D 12, 3351.
Author information
Authors and Affiliations
Corresponding author
Additional information
PACS numbers: 11.10.Gh, 11.30.Rd
Rights and permissions
About this article
Cite this article
Ferrari, R., Quadri, A. Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions. Int J Theor Phys 45, 2497–2515 (2006). https://doi.org/10.1007/s10773-006-9217-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-006-9217-x