Momentum-space conformal blocks on the light cone

TL;DR

This work develops a momentum-space framework for momentum-space conformal blocks in the light-cone limit of the 4-point function of identical scalars, focusing on null momenta. The imaginary part of the time-ordered 4-point function admits a conformal-block expansion with blocks $G_{\Delta,\ell}(x)$ that are even polynomials in $x=\cos\theta$ of degree $\ell$, with closed-form coefficients valid for any spacetime dimension $d>2$; in the large-$\Delta$ limit they reduce to $\mathcal{C}_\ell^{(d-2)/2}(x)$ and at the unitarity bound $\Delta=d-2+\ell$ to $\mathcal{C}_\ell^{(d-3)/2}(x)$. The authors derive explicit expressions for the momentum-space 3-point functions, construct the conformal blocks from products of these 3-point functions, and show how to invert the OPE in a free-scalar theory example to obtain OPE coefficients. This momentum-space approach clarifies the role of shadow operators and yields a practical inversion formula, with potential applications to momentum-space bootstrap and to theories where light-cone kinematics are central.

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.

Figures (6)

• Figure 1: $2 \to 2$ scattering configuration in the center-of-mass frame, with incoming massless particles carrying momenta $p_1$ and $p_2$, and outgoing particles momenta $p_3$ and $p_4$.
• Figure 2: The coefficient $\mathcal{N}_{\Delta, \ell}$ of Eq. \ref{['eq:N']} as a function of $\Delta$ and for $\ell = 0, 2$ and 4 in two different cases: the left panel corresponds to $d = 4$ spacetime dimensions with a scalar operator of scaling dimension $\Delta_\phi = 3$; the right panel to $d = 5$ and $\Delta_\phi = 5.5$.
• Figure 3: Feynman diagram representation of Eq. \ref{['eq:triangleintegral']}, in which the time-ordered 2-point function is indicated with a single solid line, and the Wightman 2-point functions with double lines.
• Figure 4: The 3 connected Feynman diagrams that contribute to the 4-point function of the operator $\mathcal{O} \sim \phi^2$ in the free scalar theory. The dashed lines indicate the external operator $\mathcal{O}$, while the solid lines represent propagators of the free field $\phi$.
• Figure 5: The combined contribution of conformal blocks up to a maximal spin $\ell_\text{max} = 0, 2, 4, 6$ (from lighter to darker blue lines) to the imaginary part of $\mathcal{M}$ for the operator $\phi^2$ in the free scalar theory, in $d = 5$ (left) and $d = 6$ (right) spacetime dimensions. The red dotted line indicates the full result given in Eq. \ref{['eq:G:freescalar']}. The upper panels correspond to the actual value of the conformal blocks, while the lower panels show their relative contribution.