Rare probes such as jets and heavy flavor hadrons provide unique opportunities to study the short-distance scale properties of the QGP created in heavy ion collisions. For example, recent measurements at the LHC and RHIC indicate that jet energy loss is correlated with the distance traveled through the QGP medium, and as a result, the initial geometry of the QGP. While estimates of differential azimuthal anisotropies, which we label $v^\prime_n$, have been experimentally measured for various rare probes, the fluctuations in $v^\prime_n$ have not been systematically studied.
We present a statistical formalism for the study of $v^\prime_n$ and its correlations with the integrated azimuthal anisotropies (reference flow), $v_n$, by constructing correlators of azimuthal angles from multiple rare probe objects. We then construct multivariate cumulants and moments using these correlators, which can measure fluctuations to arbitrary order in $v'_n$. The experimental methods and procedures for computing these quantities are also discussed.
To demonstrate how the measurements of the moments and cumulants can constrain the event by event distribution of $v'_n$, we propose a bivariate copula model to parametrize $v^\prime_n$ and $v_n$, and their correlations. We are able to constrain parameters in the marginal distributions for $v'_n$ by sampling the distribution and evaluating joint moments of $v^\prime_n$ and $v_n$.