Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory

TL;DR

The paper develops a momentum-space OPE for Wightman functions in 2D CFTs, showing that conformal blocks in momentum space are simple products of 3-point functions. It proves pointwise convergence in a diamond-shaped kinematic region and establishes distributional convergence elsewhere, complemented by explicit analyses of generalized free field theory, the Ising model, and the energy-momentum tensor. A momentum-space bootstrap equation based on microcausality is formulated, with discussion of smearing needed for numerical use and projection of the identity operator. The results pave the way for momentum-space bootstrap studies and potential links to scattering amplitudes, and set the stage for generalization to higher dimensions and tighter CFT data bounds.

Abstract

General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics. This convergence is guaranteed to hold in the sense of a distribution, meaning that it holds for correlation functions smeared by smooth test functions. The conformal blocks for this OPE are conceptually extremely simple: they are products of 3-point functions. We construct the conformal blocks in 2-dimensional conformal field theory and show that the OPE in fact converges pointwise to an ordinary function in a specific kinematic region. Using microcausality, we also formulate a bootstrap equation directly in terms of momentum space Wightman functions.

Figures (4)

• Figure 1: The series defined in Eq. (\ref{['eq:deltasum']}) converges to the delta function in a distributional sense, but at any given point $p \neq 0$ the partial sums of the series oscillate with an amplitude that is fixed by the value of $p$. The envelope of this oscillation is given by $\pm 1/\sin(p/2)$.
• Figure 2: Two possible configurations of momenta. The Wightman 4-point function is non-zero only if all three momenta $p_1$, $p$ and $-p_4$ lie in the forward light cone (shaded region), which is the case here. The criterion for pointwise OPE convergence is that $p$ lies in the diamond delimited by $p_1$ and $-p_4$ (green region). The OPE is therefore pointwise convergent for the configuration in the left panel, but not for the configuration in the right panel.
• Figure 3: Contribution to the conformal block expansion of the 4-point function $\langle\sigma\sigma\sigma\sigma\rangle$ in the Ising model, for two different configurations of momenta corresponding to those of Fig. \ref{['fig:diamond']}. Only the absolute value of the contributions is shown, not their sign. The horizontal axis indicates the scaling dimension of the operators and the color encodes the spin (from violet for spin $s = 0$ to red for spin $s = \Delta$). In the convergent case (left panel) the contributions decrease exponentially with the scaling dimension, while in the divergent case (right panel) operators of all scaling dimensions give contributions of similar size.
• Figure 4: The connected part of the 4-point correlation function $\langle TTTT \rangle$ given by Eq. (\ref{['eq:TTTT:W']}) as a function of the momentum $k = k_1 + k_2$ (red dashed line), compared with different truncations of the conformal block expansion (blue lines, with $h_\text{max} = 2, 4, 6, 8, 10$ from lightest to darkest). Two different kinematic configurations are shown, one in which $k_1 \ll -k_4$ (left panel) and the other in which $k_1 \simeq -k_4$ (right panel).