Three-point functions in N=4 Yang-Mills theory and pp-waves

TL;DR

The paper tests the BMN pp-wave/SYM correspondence by computing planar BMN three-point functions in N=4 SYM to first order in the effective coupling $\lambda'$, and by deriving the corresponding three-string amplitudes in the pp-wave background. On the gauge theory side, it determines the normalization of BMN operators at order $\lambda'$ and extracts the three-point coefficients with explicit phase sums, finding a universal $1 - \frac{\lambda' n^2}{2}$ correction. On the string theory side, it computes Neumann matrices to all orders in $1/(\mu p^+ \alpha')^2$ and shows that the resulting amplitudes match the gauge theory predictions at $O(\lambda')$, providing an all-orders (in small $\lambda'$) prediction for the field theory correlators. The results constitute a concrete, nontrivial check of the pp-wave/SYM holographic relation beyond leading order and furnish explicit links between CFT data and light-cone string interactions.

Abstract

Recently it has been proposed that the coefficient of the three-point function of the BMN operators in N=4 supersymmetric Yang-Mills theory is related to the three-string interactions in the pp-wave background. We calculate three-point functions of these operators to the first order in the effective Yang-Mills coupling lambda' = g_{YM}^2 N/J^2 in planar perturbation theory. On the string theory side, we derive the explicit expressions of the Neumann matrices to all orders in 1/(μp^+ α')^2. This allows us to compute the corresponding three-string scattering amplitudes. This provides an all orders prediction for the field theory three-point functions. We compare our field theory results with the string theory results to the subleading order in 1/(μp^+ α')^2 and find perfect agreement.

Figures (6)

• Figure 1: Free diagrams for $G_3$ contributing to $P_1$. The labels $k$ and $l$ count the $Z$-lines as indicated (for the diagram drawn above, $k=2$, $l=4$).
• Figure 2: Interacting diagrams for $G_3$ contributing to $P_2$. Diagrams 2a and 2c have positive signs. Diagrams 2b and 2d have negative signs.
• Figure 3: Interacting diagrams for $G_3$. All diagrams come with a positive sign.
• Figure 4: Interacting diagrams for $G_3$. All diagrams come with a negative sign and precisely cancel those in figure 3.
• Figure 5: Free diagrams for $\tilde{G}_3$ contributing to $P_5$