From Slavnov--Taylor identities to the ZJ equation

Jean Zinn-Justin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of modern theoretical and mathematical physics: Gauge theories and superstrings, Tome 272 (2011), pp. 299-303.

Résumé

In their work devoted to the proof of the renormalizability of non-Abelian gauge theories in the broken phase, Lee and Zinn-Justin have directly benefited from Slavnov's important contributions. Later generalized in the form of the BRST symmetry, the Slavnov–Taylor identities eventually have led to a remarkable quadratic equation for the renormalized gauge action (sometimes called the Zinn-Justin (ZJ) equation), allowing a completely general proof of renormalizability of non-Abelian gauge theories.

Bibliographie

[1] Slavnov A.A., “Ward identities in gauge theories”, Theor. Math. Phys., 10:2 (1972), 99–104 | DOI | MR

[2] Taylor J.C., “Ward identities and charge renormalization of the Yang–Mills field”, Nucl. Phys. B, 33 (1971), 436–444 | DOI | MR

[3] Zinn-Justin J., Quantum field theory and critical phenomena, Intern. Ser. Monogr. Phys., 113, 4th ed., Oxford Univ. Press, Oxford, 1989, Ch. 16 | MR

[4] Faddeev L.D., Popov V.N., “Feynman diagrams for the Yang–Mills field”, Phys. Lett. B, 25 (1967), 29–30 | DOI

[5] Faddeev L.D., “Faddeev–Popov ghosts”, Scholarpedia, 4:4 (2009), Art. 7389 | DOI

[6] Faddeev L.D., Slavnov A.A., Gauge fields: Introduction to quantum theory, Benjamin, Reading, MA, 1980 | MR | Zbl

[7] Slavnov A.A., “Slavnov–Taylor identities”, Scholarpedia, 3:10 (2008), Art. 7119 | DOI

[8] Lee B.W., Zinn-Justin J., “Spontaneously broken gauge symmetries. I, II, III, IV”, Phys. Rev. D, 5 (1972), 3121–3137 ; 5 (1972), 3137–3155 ; 5 (1972), 3155–3160 ; 7 (1973), 1049–1056 | DOI | DOI | DOI | DOI

[9] Slavnov A.A., “Invariant regularization of gauge theories”, Theor. Math. Phys., 13:2 (1972), 1064–1066 | DOI | MR

[10] Zinn-Justin J., “Renormalization and stochastic quantization”, Nucl. Phys. B, 275 (1986), 135–159 | DOI | MR

[11] Gauge theory of weak and electromagnetic interactions, eds. C.H. Lai, World Sci., Singapore, 1981 | MR

[12] Berezin F.A., The method of second quantization, Acad. Press, New York, 1966 | MR | MR | Zbl

[13] Becchi C., Rouet A., Stora R., “Renormalization of the abelian Higgs–Kibble model”, Commun. Math. Phys., 42 (1975), 127–162 | DOI | MR

[14] Becchi C., Rouet A., Stora R., “Renormalization of gauge theories”, Ann. Phys., 98 (1976), 287–321 | DOI | MR

[15] Tyutin I.V., Kalibrovochnaya invariantnost v teorii polya i statisticheskoi fizike v operatornoi formulirovke, Preprint No 39, FIAN, M., 1975; Tyutin I.V., Gauge invariance in field theory and statistical physics in operator formalism, E-print, 2008, arXiv: 0812.0580

[16] Kluberg-Stern H., Zuber J.B., “Ward identities and some clues to the renormalization of gauge-invariant operators”, Phys. Rev. D, 12 (1975), 467–481 | DOI

[17] Barnich G., Henneaux M., “Renormalization of gauge invariant operators and anomalies in Yang–Mills theory”, Phys. Rev. Lett., 72 (1994), 1588–1591 | DOI | MR | Zbl

[18] Barnich G., Brandt F., Henneaux M., “Local BRST cohomology in the antifield formalism. I, II”, Commun. Math. Phys., 174 (1995), 57–91 | DOI | MR | Zbl

[19] Zinn-Justin J., “Renormalization of gauge theories”, Trends in elementary particle theory, Lect. Notes Phys., 37, eds. H. Rollnik, K. Dietz, Springer, Berlin, 1975, 2–39

[20] Zinn-Justin J., “Renormalization of gauge theories and master equation”, Mod. Phys. Lett. A., 14:19 (1999), 1227–1235 | DOI | MR

[21] Zinn-Justin J., “Zinn-Justin equation”, Scholarpedia, 4:1 (2009), Art. 7120 | DOI | MR