## Globally conformal invariant gauge field theory with rational correlation functions.(English)Zbl 1058.81054

Summary: Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields $$V_{\kappa}(x_1,x_2)$$ of dimension $$(\kappa,\kappa)$$. For a globally conformal invariant (GCI) theory we write down the OPE of $$V_{\kappa}$$ into a series of twist (dimension minus rank) $$2\kappa$$ symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field.We argue that the theory of a GCI hermitian scalar field $$L(x)$$ of dimension 4 in $$D=4$$ Minkowski space such that the 3-point functions of a pair of $$L$$’s and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density $$L(x)$$.

### MSC:

 81T13 Yang-Mills and other gauge theories in quantum field theory 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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### References:

 [1] Arutyunov, G.; Eden, B.; Petkou, A.C.; Sokatchev, E., Exceptional non-renormalization properties and OPE analysis of chiral 4-point functions in $$N=4 SYM4$$, Nucl. phys. B, 620, 380-404, (2002) · Zbl 0982.81049 [2] Bargmann, V.; Todorov, I.T., Spaces of analytic functions on the complex cone as carriers for the symmetric tensor representations of SO(n), J. math. phys., 18, 1141-1148, (1977) · Zbl 0364.46016 [3] Beisert, N.; Kristensen, C.; Staudacher, M., The dilation operator of conformal N=4 super-yang – mills theory · Zbl 1051.81044 [4] Bianchi, M.; Kovacs, S.; Rossi, G.; Stanev, Ya.S., Properties of Konishi multiplet in N=4 SYM theory, Jhep, 0105, 042, (2001) [5] Bogolubov, N.N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T., General principles of quantum field theory, (1990), Kluwer Dordrecht, (Original Russian edition: Nauka, Moscow, 1987) · Zbl 0732.46040 [6] Cunningham, E., The principle of relativity in electrodynamics and an extension thereof, Proc. London math. soc., 8, 77-98, (1910) [7] Di Francesco, P.; Mathieu, P.; Senechal, D., Conformal field theories, (1996), Springer Berlin [8] Dirac, P.A.M., Wave equations in conformal space, Ann. math., 37, 429-442, (1936) · Zbl 0014.08004 [9] Dobrev, V.K.; Mack, G.; Petkova, V.B.; Petrova, S.G.; Todorov, I.T., Harmonic analysis of the n-dimensional Lorentz group and its application to conformal quantum field theory, (1977), Springer Berlin · Zbl 0407.43010 [10] Dolan, F.A.; Osborn, H., Conformal four point functions and operator product expansion, Nucl. phys. B, 599, 459-496, (2001) · Zbl 1097.81734 [11] Eden, B.; Petkou, A.C.; Schubert, C.; Sokatchev, E., Partial nonrenormalization of the stress-tensor four-point function in N=4 SYM and AdS/CFT, Nucl. phys. B, 607, 191-212, (2001) · Zbl 0969.81576 [12] Faddeev, L.D., Mass in quantum yang – mills theory (comment on a Clay millenium problem), Bull. braz. math. soc., 33, 201-212, (2002) · Zbl 1033.81056 [13] Ferrara, S.; Gatto, R.; Grillo, A.; Parisi, G., The shadow operator formalism for conformal algebra vacuum expectation values and operator products, Nuovo cimento lett., 4, 115-120, (1972) [14] Fradkin, E.S.; Palchik, M.Ya.; Fradkin, E.S.; Palchik, M.Ya., Conformal quantum field theory in D dimensions, Phys. rep., 300, 1-112, (1996), Kluwer Dordrecht · Zbl 0712.35081 [15] Furlan, P.; Sotkov, G.M.; Todorov, I.T., Two-dimensional conformal quantum field theory, Riv. nuovo cimento, 12, 6, 1-202, (1989) [16] Haag, R., Local quantum physics, fields, particles, algebras, (1992), Springer Berlin · Zbl 0777.46037 [17] Mack, G., All unitary representations of the conformal group SU(2,2) with positive energy, Commun. math. phys., 55, 1-28, (1977) · Zbl 0352.22012 [18] Mack, G.; Symanzik, K., Currents, stress tensor and generalized unitarity in conformal invariant quantum field theory, Commun. math. phys., 27, 247-281, (1972) [19] Nikolov, N.M.; Todorov, I.T., Rationality of conformally invariant local correlation functions on compactified Minkowski space, Commun. math. phys., 218, 417-436, (2001) · Zbl 0985.81055 [20] Nikolov, N.M.; Stanev, Ya.S.; Todorov, I.T., Four-dimensional CFT models with rational correlation functions, J. phys. A: math. gen., 35, 2985-3007, (2002) · Zbl 1041.81097 [21] Nikolov, N.M.; Stanev, Ya.S.; Todorov, I.T., Global conformal invariance and bilocal fields with rational correlation functions, (), 256-268 · Zbl 1041.81097 [22] Osborn, H.; Petkou, A., Implications of conformal invariance for field theories in general dimensions, Ann. phys. (N.Y.), 231, 311-362, (1994) · Zbl 0795.53073 [23] Segal, I.E., Causally oriented manifolds and groups, Bull. amer. math. soc., 77, 958-959, (1971) · Zbl 0242.53009 [24] Streater, R.F.; Wightman, A.S., PCT, spin and statistics, and all that, (2000), Princeton Univ. Press Princeton, NJ · Zbl 0135.44305 [25] Todorov, I.T., Local field representations of the conformal group and their applications, (), 195-338 [26] Todorov, I.T., Infinite-dimensional Lie algebras in conformal QFT models, (), 387-443 [27] Todorov, I.T.; Mintchev, M.C.; Petkova, V.B., Conformal invariance in quantum field theory, (1978), Scuola Normale Superiore Pisa · Zbl 0438.22011 [28] Uhlmann, A.; Uhlmann, A., The closure of Minkowski space, Acta phys. Pol., Acta phys. Pol., 24, 295-296, (1963) · Zbl 0115.42303 [29] Wilson, K.G., Operator product expansions and anomalous dimensions in the Thirring model, Phys. rev. D, 2, 1473-1477, (1970) [30] Witten, E., Physical laws and the quest of mathematical understanding, Bull. amer. math. soc., 40, 21-29, (2003) · Zbl 1014.81039
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