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Conserved currents, consistency relations, and operator product expansions in the conformally invariant \(O(N)\) vector model. (English) Zbl 0873.47044

Summary: We discuss conserved currents and operator product expansions (OPE’s) in the context of an \(O(N)\) invariant conformal field theory. Using OPE’s we find explicit expressions for the first few terms in suitable short-distance limits for various four-point functions involving the fundamental \(N\)-component scalar field \(\psi^\alpha(x)\), \(\alpha=1,2,\dots,N\). We propose an alternative evaluation of these four-point functions based on graphical expansions. Requiring consistency of the algebraic and graphical treatments of the four-point functions we obtain the values of the dynamical parameters in either a free theory of \(N\) massless fields or a non-trivial conformally invariant \(O(N)\) vector model in \(2<d<4\), up to next-to-leading order in a \(1/N\) expansion. Our approach suggests an interesting duality property of the critical \(O(N)\) invariant theory. Also, solving our consistency relations we obtain the next-to-leading order in \(1/N\) correction for \(C_T\) which corresponds to the normalization of the energy momentum tensor two-point function.

MSC:

47N50 Applications of operator theory in the physical sciences
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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