Virial expansion for the Tan contact and Beth-Uhlenbeck formula from 2D SO(2,1) anomalies

The relationship between 2D $SO(2,1)$ conformal anomalies and the virial expansion is explored, using recently developed path-integral methods. In the process, the Beth-Uhlenbeck formula for the shift of the second virial coefficient $\delta b_2$ is obtained, as well as a virial expansion for the Tan contact for these systems. A possible extension of these techniques for higher order shifts is discussed.