  We define a large new class of conformal primary operators in the ensemble of
Brownian loops in two dimensions known as the ``Brownian loop soup,'' and
compute their correlation functions analytically and in closed form. The loop
soup is a conformally invariant statistical ensemble with central charge $c = 2
\lambda$, where $\lambda > 0$ is the intensity of the soup. Previous work
identified exponentials of the layering operator $e^{i \beta N(z)}$ as primary
operators. Each Brownian loop was assigned $\pm 1$ randomly, and $N(z)$ was
defined to be the sum of these numbers over all loops that encircle the point
$z$. These exponential operators then have conformal dimension
${\frac{\lambda}{10}}(1 - \cos \beta)$. Here we generalize this procedure by
assigning a more general random value to each loop. The operator $e^{i \beta
N(z)}$ remains primary with conformal dimension $\frac {\lambda}{10}(1 -
\phi(\beta))$, where $\phi(\beta)$ is the characteristic function of the
probability distribution used to assign random values to each loop. Using
recent results we compute in closed form the exact two-point functions in the
upper half-plane and four-point functions in the full plane of this very
general class of operators. These correlation functions depend analytically on
the parameters $\lambda, \beta_i, z_i$, and on the characteristic function
$\phi(\beta)$. They satisfy the conformal Ward identities and are crossing
symmetric. As in previous work, the conformal block expansion of the four-point
function reveals the existence of additional and as-yet uncharacterized
conformal primary operators.
