The thermal representation of conformal ladder integrals

We present the details of a recently discovered representation of conformal four-point ladder integrals as thermal one-point functions in scalar field theories. We show that the conformal ladder integrals can be constructed from the partition function of two harmonic oscillators twisted by an imaginary chemical potential and that for any even dimension $D$ and any loop order $L$ they satisfy a familiar second order differential equation. In our representation, thermal one-point functions of higher-spin operators correspond to linear combinations of multi-loop ladder graphs in $D=2$ and $D=4$ dimensions. Moreover, we give a simple derivation for the all-loop resummation of conformal ladder integrals for arbitrary $D$. We conclude by highlighting possible connections between our work and recent developments in the thermal bootstrap, multiloop calculations, integrability, AdS/CFT and string theory.