Instantons of two-dimensional fermionic effective actions by inverse scattering transformation. (English) Zbl 0594.35085

A method to find symmetric solutions of the nonlinear and nonlocal saddle-point equations for the effective actions, containing the logarithm of a functional Dirac determinant, which appear in 1/N expansions of fermionic theories is proposed. This method consists in using the scattering data of the rotationally symmetric Dirac equation in two dimensions with the angular momentum as a spectral parameter. The method is applied to fermionic models with quartic coupling. It is shown that the effective action that generates the 1/N expansion admits a closed form in terms of the scattering data only in the particular case of Gross-Neveu and Chiral Gross-Neveu models. No instanton solutions are present in these two models. This fact suggested that the 1/N expansion could be convergent.
Reviewer: B.Konopelchenko


35Q99 Partial differential equations of mathematical physics and other areas of application
81T08 Constructive quantum field theory
81U99 Quantum scattering theory
Full Text: DOI


[1] Schwinger, J.: On the Green’s functions of quantized fields. I. Proc. Nat. Acad. Sci. USA37, 452 (1951) · Zbl 0044.43001
[2] Jona-Lasinio, G.: Relativistic field theories with symmetry-breaking solutions. Nuov. Cimento34, 1790 (1964)
[3] Coleman, S., Weinberg, E.: Radiative corrections as the origin of spontaneous symmetry breaking. Phys. Rev. D 7, 1888 (1973)
[4] de Vega, H.: Large orders in the 1/N perturbation theory by inverse scattering in one dimension. Commun. Math. Phys.70, 29 (1979)
[5] Avan, J.: 1/N series for quantum anharmonic oscillator eigenvalues and Green functions. Nucl. Phys. B237, 159 (1984)
[6] Avan, J., de Vega, H.: 1/N expansion for invariant potentials in quantum mechanics. Nucl. Phys. B224, 61 (1983)
[7] Avan, J., de Vega, H.: Classical solutions by inverse scattering transformation in any number of dimensions. I. The gap equation and effective action. Phys. Rev. D29, 2891 (1984)
[8] Avan, J., de Vega, H.: Classical solutions by inverse scattering transformation in any number of dimensions. II. Instantons and large orders of the 1/N series for the (?2)2 theory inv dimensions (1?v?4). Phys. Rev. D29, 2904 (1984)
[9] de Vega, H.: The inverse scattering transformation in the angular momentum plane. Commun. Math. Phys.81, 313 (1981)
[10] de Vega, H., Schaposnik, F.: Nonuniform external fields and vacuum properties in a two-dimensional gauge theory. Phys. Rev. D26, 2814 (1982)
[11] de Vega, H.: The non-linear sigma model in the 1/N expansion and the inverse scattering tranformation in the angular momentum. Phys. Lett. B98, 280 (1981)
[12] Lipatov, L.N.: Divergence of the perturbation-theory series and pseudoparticles. JETP Lett.25, 104 (1977)
[13] Zinn-Justin, J.: Perturbation series at large orders in quantum mechanics and field theories: Application to the problem of resummation. Phys. Rep.70, 109 (1981)
[14] Brézin, E., Le Guillou, J.C., Zinn-Justin, J.: Perturbation theory at large order. I. The ?2N interaction. Phys. Rev. D15, 1544, 1558 (1978)
[15] Zinn-Justin, J.: Les Houches lectures. Recent advances in quantum field theory, Stora, R., Zuber, J.B. (eds.). Amsterdam: North-Holland 1983
[16] Köberlé, R., Kurak, V., Swieca, A.: Scattering theory and 1/N expansion in the Chiral Gross-Neveu model. Phys. Rev. D20, 897 (1979)
[17] Coleman, S., Glaser, V., Martin, A.: Commun. Math. Phys.58, 211 (1978)
[18] Neveu, A., Papanicolaou, N.: Integrability of the classical \(\left[ {\bar \psi _i \psi _i } \right]_2^2 \) and \(\left[ {\bar \psi _i \psi _i } \right]_2^2 - \left[ {\bar \psi _i \gamma _5 \psi _i } \right]_2^2 \) interactions. Commun. Math. Phys.58, 31 (1978)
[19] Karowski, M., Thun, H.J.: CompleteS-matrix of theO(2N) Gross-Neveu model. Nucl. Phys. B190, 61 (1981)
[20] Karowski, M., Weisz, P.: Exact form factors in (1 + 1)-dimensional field theoretic models with soliton behaviour. Nucl. Phys. B139, 455 (1978)
[21] Abdalla, E., Berg, B., Weisz, P.: More about theS-matrix of the chiralSU(N) Thirring model. Nucl. Phys. B157, 387 (1979)
[22] Mitter, P.K., Weisz, P.: Asymptotic scale invariance in a massive Thirring model withU(n) symmetry. Phys. Rev. D8, 4410 (1973)
[23] Gross, D.J., Neveu, A.: Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D10, 3235 (1974)
[24] t’Hooft, G., Veltman, M.: Regularization and renormalization of gauge fields. Nucl. Phys. B44, 189 (1972)
[25] Bollini, C.G., Giambiagi, J.J.: Lowest order ?divergent? graphs inv-dimensional space. Phys. Lett. B40, 566 (1972)
[26] Bollini, C.G., Giambiagi, J.J.: Dimensional renormalization: The number of dimensions as a regularizing parameter. Nuovo Cimento B12, 20 (1972)
[27] Boerner, H.: Representation of groups, pp. 269-273. Amsterdam: North-Holland 1963 · Zbl 0112.26301
[28] Stratonovitch, R.L.: Dokl. Akad. Nauk. SSSR115, 1097 (1957)
[29] Berg, B., Weisz, P.: ExactS-matrix of the chiral invariantSU(N) Thirring model. Nucl. Phys. B146, 205 (1978)
[30] Coleman, S.: There are no Goldstone bosons in two dimensions. Commun. Math. Phys.31, 259 (1973) · Zbl 1125.81321
[31] Roskies, R., Schaposnik, F.: Comment on Fujikawa’s analysis applied to the Schwinger model. Phys. Rev. D23, 558 (1981)
[32] Magnus, W., Oberhettinger, R., Soni, R.: Formulas and theorems for the special functions of mathematical physics. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0143.08502
[33] Spencer, T.: The Lipatov argument. Commun. Math. Phys.74, 273 (1980)
[34] Breen, S.: Leading large order asymptotics for (?4)2 perturbation theory. Commun. Math. Phys.92, 179 (1981) · Zbl 0568.46055
[35] Magnen, J., Rivasseau, V.: Preprint CPhT 652-0285
[36] Ablowitz, M., Kaup, D., Newell, A., Segur, H.: The inverse scattering transform ? Fourier analysis for nonlinear problems. Stud. Appl. Math., vol. III4, 249 (1974) · Zbl 0408.35068
[37] Birman, M.S., Krein, M.O.: Sov. Phys. Dokl.3, 740 (1962)
[38] Buslaev, V.S.: Topics in Math. Phys., Vol. I, p. 69, Birman, M.Sh. (ed.). New York: Consultants Bureau 1968 · Zbl 0181.49601
[39] Gradhshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. New York: Academic Press 1980, 8/362
[40] Zamolodchikov, A.B., Zamolodchikov, Al.B.: Ann. Phys. (NY)80, 253 (1979)
[41] The problems of large orders in the coupling constant for fermionic theories are discussed for instance in: a. Parisi, G.: Asymptotic estimates in perturbation theory with fermions. Phys. Lett. B66, 382 (1977)
[42] Fainberg, V., Pakshaev, I.: On the fermion contribution to instantons. Phys. Lett. B77, 208 (1978)
[43] Bogomolnyi, V., Kubishin, Y.: Asymptotic estimates for graphs with a fixed number of fermion loops in quantum electrodynamics. The choice of the form of the steepest descent solutions. Sov. J. Nucl. Phys.34, 853 (1981) and: Asymptotic estimates for diagrams with a fixed number of fermion loops in quantum electrodynamics. The extremal configurations with the symmetry groupO(2) ?O(3).35, 114 (1982)
[44] Itzykson, C., Parisi, G., Zuber, J.B.: Asymptotic estimates in quantum electrodynamics. Phys. Rev. D16, 996 (1978)
[45] Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Preprint CPhT 644-0185
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.