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Conserved Currents, Consistency Relations and Operator Product Expansions in the Conformally Invariant O(N) Vector Model

Anastasios Petkou

TL;DR

This work analyzes conserved currents, Ward identities, and operator product expansions in the conformally invariant O(N) vector model. By pairing algebraic OPE constraints with a skeleton graphical expansion built from φ propagators and an auxiliary scalar ilde{O}, the paper fixes critical dimensions η and η_o and the normalizations C_T and C_J through a 1/N expansion, reproducing known results in the free theory and at the non-trivial fixed point in 2<d<4. The approach reveals a shadow symmetry and suggests a duality between the non-trivial CFT and a related (potentially non-unitary) theory, while also providing a non-trivial check of OPE associativity via multiple evaluation routes for four-point functions. Overall, the results connect OPE data, conformal constraints, and large-N dynamics, offering a non-Lagrangian bootstrap perspective on the O(N) vector model.

Abstract

We discuss conserved currents and operator product expansions (OPE's) in the context of a $O(N)$ 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 $φ^α(x)$, $α=1,2,..,N$. 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 normalisation of the energy momentum tensor two-point function.

Conserved Currents, Consistency Relations and Operator Product Expansions in the Conformally Invariant O(N) Vector Model

TL;DR

This work analyzes conserved currents, Ward identities, and operator product expansions in the conformally invariant O(N) vector model. By pairing algebraic OPE constraints with a skeleton graphical expansion built from φ propagators and an auxiliary scalar ilde{O}, the paper fixes critical dimensions η and η_o and the normalizations C_T and C_J through a 1/N expansion, reproducing known results in the free theory and at the non-trivial fixed point in 2<d<4. The approach reveals a shadow symmetry and suggests a duality between the non-trivial CFT and a related (potentially non-unitary) theory, while also providing a non-trivial check of OPE associativity via multiple evaluation routes for four-point functions. Overall, the results connect OPE data, conformal constraints, and large-N dynamics, offering a non-Lagrangian bootstrap perspective on the O(N) vector model.

Abstract

We discuss conserved currents and operator product expansions (OPE's) in the context of a 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 -component scalar field , . 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 massless fields or a non-trivial conformally invariant vector model in , up to next-to-leading order in a expansion. Our approach suggests an interesting duality property of the critical invariant theory. Also, solving our consistency relations we obtain the next-to-leading order in correction for which corresponds to the normalisation of the energy momentum tensor two-point function.

Paper Structure

This paper contains 13 sections, 147 equations, 9 figures.

Table of Contents

  1. Conserved Currents and Ward Identities in CFT
  2. Conserved Currents and Operator Product Expansions in the Conformally Invariant $O(N)$ Vector Model
  3. General Remarks
  4. Operator Product Expansions and the Four-Point Function in the Conformally Invariant $O(N)$ Vector Model
  5. The Four-Point Function in the Free Field Theory
  6. The Graphical Expansion
  7. The Consistency Relations
  8. Four-Point Function Involving the Field $O(x)$
  9. Discussion and Concluding Remarks
  10. Scalar Field Contribution to the Operator Product Expansion
  11. The One-Particle Exchange Graphs ${\cal{G}}_{1}^{(\tilde{\eta_{o}})}$
  12. The Box-Graphs ${\cal{G}}_{2}^{(\tilde{\eta}_{o})}$
  13. The Graphs in the Four-Point Function $\langle\phi\phi OO\rangle$

Figures (9)

  • Figure 1: The Graphical Expansion for $\Phi_{f}(x_{1},x_{2},x_{3},x_{4})$
  • Figure 2: The Conformally Invariant Three-Point Function $\langle\phi\phi \tilde{O}\rangle$.
  • Figure 3: The Three-Point Function $\langle\phi\phi\tilde{O}\rangle$ with amputation on the $\tilde{O}$ leg.
  • Figure 4: The Skeleton Graph Expansion for $\Phi^{(\tilde{\eta}_{o})}(x_{1},x_{2},x_{3},x_{4})$
  • Figure 5: The Graphical Expansion corresponding to $\langle\phi\phi OO\rangle$.
  • ...and 4 more figures