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

Avan, J.; de Vega, H. J.

Commun. Math. Phys. 102, 463-496 (1985).

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

MSC:

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

Keywords:

symmetric solutions; saddle-point equations; effective actions; fermionic theories; scattering data; rotationally symmetric Dirac equation; instanton solutions

References:

[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