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Conformal four point functions and the operator product expansion. (English) Zbl 1097.81734
Summary: Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants \(u,v\). A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables \(z,x\) which are simply related to \(u,v\). The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
33C90 Applications of hypergeometric functions
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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References:
[1] Ferrara, S.; Grillo, A.F.; Gatto, R.; Parisi, G.; Ferrara, S.; Grillo, A.F.; Gatto, R.; Parisi, G., Nucl. phys. B, Nuovo cimento A, 19, 667, (1974)
[2] Dobrev, V.K.; Petkova, V.B.; Petrova, S.G.; Todorov, I.T., Phys. rev. D, 13, 887, (1976)
[3] Lang, K.; Rühl, W., Nucl. phys. B, 400, 597, (1993)
[4] Herzog, C.P., Jhep, 0102, 038, (2001)
[5] Arutyunov, G.; Frolov, S.; Petkou, A.C., Nucl. phys. B, 586, 547, (2000)
[6] Dolan, F.A.; Osborn, H., Implications of \(N=1\) superconformal symmetry for chiral fields, Nucl. phys. B, 593, 599, (2001) · Zbl 0971.81550
[7] D’Eramo, M.; Parisi, G.; Peliti, L., Lett. nuovo cimento, 2, 878, (1971)
[8] Gradshteyn, I.S.; Ryzhik, I.M., Tables of integrals, series, and products, (1994), Academic Press San Diego · Zbl 0918.65002
[9] Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B., Nucl. phys. B, 241, 333, (1984)
[10] Osborn, H.; Petkou, A., Ann. phys. (N.Y.), 231, 311, (1994)
[11] Freedman, D.Z.; Mathur, S.D.; Matsusis, A.; Rastelli, L.; Freedman, D.Z.; Mathur, S.D.; Matsusis, A.; Rastelli, L.; D’Hoker, E.; Freedman, D.Z.; D’Hoker, E.; Freedman, D.Z., Nucl. phys. B, Phys. lett. B, Nucl. phys. B, Nucl. phys. B, 544, 612, (1999)
[12] D’Hoker, E.; Freedman, D.Z.; Rastelli, L., Nucl. phys. B, 562, 395, (1999)
[13] D’Hoker, E.; Freedman, D.Z.; Mathur, S.D.; Matsusis, A.; Rastelli, L., Nucl. phys. B, 562, 353, (1999)
[14] Liu, H.; Tseytlin, A.A., Phys. rev. D, 59, 086002, (1999)
[15] Liu, H., Phys. rev. D, 60, 106005, (1999)
[16] Sanjay, S., Mod. phys. lett. A, 14, 1413, (1999)
[17] Brodie, J.H.; Gutperle, M., Phys. lett. B, 445, 296, (1999)
[18] Hoffmann, L.C.; Petkou, A.C.; Rühl, W.; Hoffmann, L.C.; Petkou, A.C.; Rühl, W., Phys. lett. B, Adv. theor. math. phys., 4, 320, (2000)
[19] Witten, E., Adv. theor. math. phys., 2, 253, (1998)
[20] Symanzik, K., Lett. nuovo cimento, 3, 734, (1972)
[21] Hoffmann, L.; Mesref, L.; Rühl, W., Nucl. phys. B, 589, 337, (2000)
[22] Eden, B.; Howe, P.S.; Pickering, A.; Sokatchev, E.; West, P.C.; Eden, B.; Petkou, A.C.; Schubert, C.; Sokatchev, E., Nucl. phys. B, 581, 71, (2000)
[23] D’Hoker, E.; Mathur, S.D.; Matsusis, A.; Rastelli, L., Nucl. phys. B, 589, 38, (2000)
[24] Lang, K.; Rühl, W., Nucl. phys. B, 402, 573, (1993)
[25] Ferrara, S.; Grillo, A.F.; Gatto, R., Ann. phys., 76, 161, (1973)
[26] Petkou, A.C., Ann. phys. (N.Y.), 249, 180, (1996)
[27] Ussyukina, N.I.; Davydychev, A.I.; Ussyukina, N.I.; Davydychev, A.I.; Davydychev, A.I.; Tausk, J.B., Phys. lett. B, Phys. lett. B, Nucl. phys. B, 397, 133, (1993)
[28] Arutyunov, G.; Frolov, S., Phys. rev. D, 62, 064016, (2000)
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.