Geodesic
Proof of the absence of multiplicative renormalizability of the Gross–Neveu model in dimensional regularization $d=2+2\varepsilon$
Teoretičeskaâ i matematičeskaâ fizika, Tome 113 (1997) no. 1, pp. 85-99 
A. N. Vasil'ev; M. I. Vyazovskii. Proof of the absence of multiplicative renormalizability of the Gross–Neveu model in dimensional regularization $d=2+2\varepsilon$. Teoretičeskaâ i matematičeskaâ fizika, Tome 113 (1997) no. 1, pp. 85-99. http://geodesic.mathdoc.fr/item/TMF_1997_113_1_a7/
@article{TMF_1997_113_1_a7,
     author = {A. N. Vasil'ev and M. I. Vyazovskii},
     title = {Proof of the absence of multiplicative renormalizability of the {Gross{\textendash}Neveu} model in dimensional regularization $d=2+2\varepsilon$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {85--99},
     year = {1997},
     volume = {113},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1997_113_1_a7/}
}
TY  - JOUR
AU  - A. N. Vasil'ev
AU  - M. I. Vyazovskii
TI  - Proof of the absence of multiplicative renormalizability of the Gross–Neveu model in dimensional regularization $d=2+2\varepsilon$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1997
SP  - 85
EP  - 99
VL  - 113
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1997_113_1_a7/
LA  - ru
ID  - TMF_1997_113_1_a7
ER  - 
%0 Journal Article
%A A. N. Vasil'ev
%A M. I. Vyazovskii
%T Proof of the absence of multiplicative renormalizability of the Gross–Neveu model in dimensional regularization $d=2+2\varepsilon$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1997
%P 85-99
%V 113
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1997_113_1_a7/
%G ru
%F TMF_1997_113_1_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that the simplest four-fermion Gross–Neveu model with dimensional regularization $d=2+2\varepsilon$ is not multiplicatively renormalizable due to the counterterm generated by the three-loop vertex diagrams that is proportional to the evanescent operator [1] $V_3=(\bar\psi\gamma_{ikl}^{(3)}\psi )^2/2$, where $\gamma_{i_1\dots i_n}^{(n)}$ is the fully antisymmetric product of $n$ $\gamma$-matrices and is not zero in arbitrary dimensions. Therefore, calculations of the $(2+\varepsilon)$-expansion of the critical indices $\eta$ and $\nu$ in the framework of the simple Gross–Neveu model are correct only to $\varepsilon^4$ for $\eta$ and to $\varepsilon^3$ for $\nu$. In higher orders, one must take into consideration the generation of other (not only $V_3$) evanescent operators.

[1] A. Bondi, G. Curci, G. Paffuti, P. Rossi, Ann. Phys., 199 (1990), 268 | DOI | MR

[2] D. J. Gross, A. Neveu, Phys. Rev. D, 10 (1974), 3235 | DOI

[3] J. A. Gracey, Nucl. Phys. B, 341 (1990), 403 | DOI

[4] J. A. Gracey, Nucl. Phys. B, 367 (1991), 657 | DOI | MR

[5] C. Luperini, P. Rossi, Ann. Phys., 212 (1991), 371 | DOI | MR

[6] N. A. Kivel, A. S. Stepanenko, A. N. Vasiliev, Nucl. Phys. B, 424 (1994), 619 | DOI

[7] J. Zinn-Justin, Nucl. Phys. B, 367 (1991), 105 | DOI | MR

[8] A. N. Vasilev, S. E. Derkachev, N. A. Kivel, A. S. Stepanenko, TMF, 94 (1993), 179

[9] A. N. Vasilev, A. S. Stepanenko, TMF, 97 (1993), 364

[10] J. A. Gracey, Int. J Mod. Phys. A, 9 (1994), 567 | DOI

[11] J. A. Gracey, Int. J Mod. Phys. A, 9 (1994), 727 | DOI

[12] A. Kennedy, J. Math. Phys., 22 (1981), 1330 ; Л. В. Авдеев, ТМФ, 58 (1984), 308 | DOI | MR

[13] A. N. Vasilev, M. I. Vyazovskii, S. E. Derkachev, N. A. Kivel, TMF, 107 (1996), 27 | DOI | Zbl

[14] A. N. Vasilev, M. I. Vyazovskii, S. E. Derkachev, N. A. Kivel, TMF, 107 (1996), 359 | DOI | Zbl

[15] A. N. Vasilev, S. E. Derkachev, N. A. Kivel, TMF, 103 (1995), 179 | Zbl

[16] D. Kollinz, Perenormirovka, Mir, M., 1988 | MR