Momentum-space conformal blocks on the light cone

Abstract

We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks are polynomials in the cosine of the scattering angle, with degree $\ell$ corresponding to the spin of the intermediate operator. The coefficients of these polynomials are obtained in a closed-form expression for arbitrary spacetime dimension $d>2$. If the scaling dimension of the intermediate operator is large, the conformal block reduces to a Gegenbauer polynomial ${\mathcal{C}}_{\ell }^{(d-2)/2}$. If on the contrary the scaling dimension saturates the unitarity bound, the block is different Gegenbauer polynomial ${\mathcal{C}}_{\ell }^{(d-3)/2}$. These results are then used as an inversion formula to compute OPE coefficients in a free theory example.

arXiv comment30 pages, 6 figures & 1 table. v2: references added. v3: relation to partial wave expansion added, matches published version. v4: typo corrected

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• Operator product expansion coefficients in the exact renormalization group formalism
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• General Freeze-in and Freeze-out
• Shift operators from the simplex representation in momentum-space CFT
• ANEC in $\lambda {\varphi }^4$ theory
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• Spinors and conformal correlators
• $n$-point functions in Conformal Quantum Mechanics: A Momentum Space Odyssey
• Conformal bootstrap in momentum space at finite volume
• Nonperturbative dynamics of (2+1)d ${\varphi }^4$-theory from Hamiltonian truncation
• The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space