All-loop heavy-heavy-light-light correlators in $\mathcal{N}=4$ super Yang-Mills theory

We study Heavy-Heavy-Light-Light (HHLL) correlators $\langle \mathcal{H} \mathcal{H} \mathcal{O}_2 \mathcal{O}_2 \rangle$ in $\mathcal{N}=4$ super Yang-Mills theory with $SU(N)$ gauge group at generic $N$. The light operator $\mathcal{O}_2$ is the dimension two superconformal primary in the stress tensor multiplet and $\mathcal{H}$ is a general half-BPS superconformal primary operator with dimension (or $R$-charge) $\Delta_{\mathcal{H}}$. We consider the large-charge 't Hooft limit, where $\Delta_{\mathcal{H}} \rightarrow \infty$ with fixed 't Hooft-like coupling $\lambda:=\Delta_{\mathcal{H}}\, g_{_{\rm YM}}^2$. We show that the $L$-loop contribution to the HHLL correlators in the leading large-charge limit is universal for any choice of the heavy operator $\mathcal{H}$, given as $\lambda^L \sum_{\ell=0}^{L} \Phi^{(\ell)} \Phi^{(L-\ell)}$ with an $SU(N)$ colour factor coefficient, where $\Phi^{(\ell)}$ is the ladder Feynman integral, which is known to all loops. The dependence on the explicit form of the heavy operator lies only in the colour factor coefficients. We determine such colour factors for several classes of heavy operators, and show that the large charge limit leads to minimal powers of $N$. For the special class of "canonical heavy operators", one can even resum the all-loop ladder integrals and determine the correlators at finite $\lambda$. Furthermore, upon integrating over the spacetime dependence, the resulting integrated HHLL correlators agree with the existing results derived from supersymmetric localisation. Finally, as an application of the all-loop analytic results, we derive exact expressions for the structure constants of two heavy operators and the Konishi operator, finding intriguing connections with the integrated HHLL correlators.