A Scattering Amplitude in Conformal Field Theory

TL;DR

This work defines a scattering-like observable for conformal field theories by constructing the form factor $F(s,t,u)$ and the amplitude $A(s,t,u)$ from the Fourier transform of four-point functions of scalar primaries, using an LSZ-inspired on-shell limit. It provides two complementary derivations—a direct Lorentzian LSZ reduction and a Mellin-space approach—that establish crossing symmetry, analyticity, and a conformal partial-wave expansion in momentum space, with explicit connections to Landau singularities. The authors compute and validate the framework across perturbative fixed points ($\phi^4$ in $4-\varepsilon$, $\phi^3$ in $6+\varepsilon$), the non-perturbative 3d Ising model, and holographic CFTs, obtaining concrete CFT data such as anomalous dimensions and OPE coefficients from the form-factor expansion. This momentum-space construction opens avenues for conformal bootstrap in a new setting, enables perturbative and holographic checks, and suggests dispersion-like tools and extensions to non-scalar operators, deepening the link between bulk-like scattering and CFT data.

Abstract

We define form factors and scattering amplitudes in Conformal Field Theory as the coefficient of the singularity of the Fourier transform of time-ordered correlation functions, as $p^2 \to 0$. In particular, we study a form factor $F(s,t,u)$ obtained from a four-point function of identical scalar primary operators. We show that $F$ is crossing symmetric, analytic and it has a partial wave expansion. We illustrate our findings in the 3d Ising model, perturbative fixed points and holographic CFTs.

Figures (15)

• Figure 1: Regions of convergence of the partial wave expansion for the form factor \ref{['def:FormFactor']} in terms of the Mandelstam invariants $s$, $t$ and $u$, in units where $s + t + u = -1$. The $s$-channel partial wave expansion \ref{['CBexpansionF']} is defined for $s > 0$ and $t, u < 0$, corresponding to the red region, and it can be analytically continued to the white triangle with $s \leq 0$, as long as $t$ and $u$ remain negative. In other words, the expansion converges as long as the scattering angle is physical, $\cos\theta \in (-1,1)$, which corresponds to the wedge delimited by the dashed lines. The $t$- and $u$-channel expansions are respectively defined in the green and blue regions that are disjoint from the red region. However, they can also be analytically continued to the white triangle, where all 3 channels converge and the form factor is real.
• Figure 2: Integration contour of eq. \ref{['eq:k0integral:p1']} in the complex $k^0$ plane. The integral is finite when $p_1$ is timelike ($p_1^0 > |\vec{p}_1|$, left-hand side) since the contour can be deformed away from the pole, and similarly finite when $p_1$ is spacelike. There is a singularity in the limit $p_1^0 \to |\vec{p}_1|$ (right-hand side) since the pole approaches the end point of the integration contour.
• Figure 3: Integration contour of eq. \ref{['eq:k0integral:p2']} in the complex $k^0$ plane. The integral is finite when $p_2$ is timelike ($p_2^0 > |\vec{p}_2|$, left-hand side) since the contour can be deformed away from the pole. When $p_2$ is spacelike ($p_2^0 < |\vec{p}_2|$) the contour can also be deformed; it enters the branch cut but the integral remains finite. The singularity happens in the limit $p_2^0 \to |\vec{p}_2|$ (right-hand side), where the contour is pinched between the branch point and the pole.
• Figure 4: The configuration corresponding to the Landau singularity at $z=\overline{z}$. The red dot is lightlike separated from all the insertions. Notice that we can freely move the $x_i$ along the light-rays without affecting the cross-ratios, which are invariant under independent rescalings $x_i \to \lambda_i x_i$.
• Figure 5: The paths in the $\rho$ and $\overline{\rho}$ planes needed to reach the Landau singularity starting from a Euclidean configuration. The path of $\rho = e^{i(t+ \theta)}$ is the continuous line, while the path of $\overline{\rho} = e^{i(t-\theta)}$ is dashed. The time coordinate $t$ varies from $t=i \pi$ in the Euclidean regime to $t=-\pi$ in the Lorentzian regime. The wavy line marks the position of the cut in each conformal block.