Four-point function of determinant operators in $\mathcal{N}=4$ SYM

We calculate the four-point function of $1/2$-BPS (Bogomolnyi–Prasad–Sommerfield) determinant operators in $\mathcal{N}=4$ SYM (super Yang-Mills) at next-to-leading order at weak coupling. We use two complementary methods recently developed for a class of determinant three-point functions: one is based on Feynman diagrams and it extracts perturbative data at finite $N$, while the other one expresses a generic correlator of determinants as the zero-dimensional integral over an auxiliary matrix field. We generalize the latter approach to calculate one-loop corrections and we solve the four-point function in a semiclassical approach at large $N$. The results allow us to comment on the order of the phase transition that the four-point function is expected to exhibit in an exact integrability-based description.

Figure 1

Saddle points of the effective theory at zero coupling. The diagram is plotted in the parameter $\rho =\rho (z)$ of [31] and fixed $\alpha \overline{\alpha }=10(1-\alpha )(1-\alpha )=1$. The region I (blue) is defined by $u_1,u_1/u_2<1$, the region II (yellow) by $1,u_1/u_2<u_1$ and the region III (green) by $1,u_1<u_1/u_2$. The region II vanishes in the limit $y_{13}=y_{24}=0$ [see above (13)]. Each saddle point in (34) and below saturates (31) in the homonymous region. The boundary curves (black) separating the regions are the locus of critical points $z=z_c$ in Sec. 5. Small values of $\rho \overline{\rho }$ favor the saddle point I, while the complex phase of $\rho$ plays a role to favor I, II or III at higher values.