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Scalar conformal primary fields in the Brownian loop soup

The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$, with central charge $c=2 λ$. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary "layering vertex operators" $\mathcal{O}_β$ with dimensions $(Δ, Δ)$, with $Δ= \fracλ{10}(1-\cosβ)$, that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions $(Δ+ k/3, Δ+ k'/3)$, for all non-negative integers $k, k'$ satisfying $|k-k'| = 0$ mod 3. In this paper we introduce the edge counting field $\mathcal E(z)$ that counts the number of loop boundaries that pass close to the point $z$. We rigorously prove that the $n$-point functions of $\mathcal E$ are well defined and behave as expected for a conformal primary field with dimensions $(1/3, 1/3)$. We analytically compute the four-point function $\langle \mathcal{O}_β(z_1) \mathcal{O}_{-β}(z_2) \mathcal{E}(z_3) \mathcal{E}(z_4) \rangle$ and analyze its conformal block expansion. The operator product expansions of $\mathcal{E} \times \mathcal{E}$ and $\mathcal{E} \times \mathcal{O}_β$ produce higher-order edge operators with "charge" $β$ and dimensions $(Δ+ k/3, Δ+ k/3)$. Hence, we have explicitly identified all scalar primary operators among the new set mentioned above. We also re-compute the central charge by an independent method based on the operator product expansion and find agreement with previous methods.