Exact Correlation Functions in the Brownian Loop Soup

We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.


I. INTRODUCTION
The Brownian Loop Soup (BLS) [1] is a conformally invariant system consisting of closed, self-intersection loops randomly distributed in the plane according to a conformally invariant measure µ loop . The measure has one free parameter, an overall normalization λ > 0, that is called the intensity of the soup and determines the density of the loops.
In earlier work [2], three of us established that the soup has a central charge c = 2λ.
Since c is continuous and can be less than one, this demonstrates that the model cannot be unitary for all values of λ (the minimal models are the only unitary conformal field theories with c < 1, and they come in a discrete series). Nevertheless, we were able to identify a set of conformal primary operators with positive conformal dimensions. These operators are exponentials of the form e iβN (z) , where β is a real number (the "charge") and N (z) is an integer-valued operator that counts some characteristics of the loops.
In this paper we will consider the case where N (z) counts the layering number -each loop that encircles the point z contributes ±1 to N (z), with the sign chosen uniformly at random (see Fig. 1). Note that these layering vertex operators are sensitive only to the outer boundary of each Brownian loop (Fig. 1b).
Because the BLS is a Poissonian ensemble of loops (that is, each loop is independent of the others), the probabilities of events in the BLS can be expressed in terms of µ loop .
Specifically, we can obtain certain correlation functions in the BLS with suitable cutoffs simply by taking the exponential of λ times the µ loop -weights of certain sets of loops.
In [2] we demonstrated that, after removing the cutoff, e iβN (z) is a scalar primary with conformal weights that are periodic functions of β, ∆ =∆ = λ 10 (1 − cos β), and that correlation functions of products of these operators vanish unless i β i = 0 mod 2π. We computed the two-and three-point functions, but only up to multiplicative constants.
In this paper we use the results of [3] to determine various expressions for the twoand three-point correlation functions. Together with a result of [4], we use these results to compute analytically and in closed form the full four-point correlation function of the layering vertex operators in the plane, as a function of the positions of the four points, the intensity λ, and the four charges β i .
The results of [4] were obtained by taking the limit n → 0 of the critical O(n) model, which is conformally invariant and known to describe self-avoiding loops in this limit. As just  [5] guarantees that there is a unique (up to an overall multiplicative constant) conformally invariant measure on self-avoiding loops (this is in fact the measure induced by µ loop on the outer boundaries of Brownian loops).
With the four-point function in hand, we can expand it in conformal blocks. This reveals a new set of previously unknown conformal primary operators and their three-point function coefficients with the layering vertex operators. The physical interpretation and meaning of these operators is left to future work.
Using the results of [3] we also compute the two-point function in the upper half-plane (subject to a certain boundary condition on the real axis), as a function of the positions of the two points, the intensity λ, and the two charges β i . The results of [3] are rigorous and based on SLE theory [6]. They do not rely on the n → 0 limit of the O(n) model used in [4]. Limits of this two-point function help determine various constants in the fourpoint function. In particular, we obtain the interesting result that the three-point function coefficient for three canonically normalized layering vertex operators is exactly 1, consistent with the conformal block expansion of the four-point function.
We also determine the weights for Brownian loops to wind around one point and not another in the upper half-plane and full plane, and for several other configurations.

II. SUMMARY AND RESULTS
Our main results are the derivation of new correlation functions of exponentials of the layering operators in the BLS. In [2], three of us showed that the conformal dimensions of the operators e iβN (z) are In this work we obtain the two-point function of these operators in the upper half-plane with the boundary condition that any loop intersecting the real axis is erased, and the four-point function in the full plane.
The four-point function of these operators in the full plane C (Sec. VI) is given by where the operators are normalized so that O β 1 (z 1 )O β 2 (z 2 ) C = |z 1 − z 2 | −2∆ 1 and x is the cross-ratio (34).
All n-point functions in the full plane vanish unless a (periodic) charge conservation condition is satisfied: The conformal block expansion of the four-point function (4) (Sec. VII) reveals the spectrum of conformal primaries and associated three-point function coefficients. We find an apparently infinite new set of primary operators of integer spin in the BLS, with conformal where p and p are non-negative integers satisfying p − p = 0 mod 3.
Weights for Brownian loops to encircle various subsets of points in the plane or upper half-plane can be found in (27), (29), and (37).

A. Motivation and previous work
In [7], Freivogel and Kleban considered a toy model intended to capture the late-time physics of cosmic bubble nucleation in eternally inflating or de Sitter spacetime. In a spacetime with one time and two space dimensions, these bubbles will be disks (with random fluctuations to their shape) that expand exponentially after their nucleation due to the expansion of the ambient spacetime. This turns out to imply that on a late time slice the distribution of disks will be invariant under translations, scale transformations, rotations, and special conformal transformations [8]. This "disk soup" has intensity λ FK that is equal to the bubble production rate per Hubble time per Hubble volume.
It is widely believed that theories with Poincaré and scale invariance are fully conformally invariant. However, the disk soup model of [7] appears to be an exception. Operators of the form e iβN (z) , where N (z) now counts the number of disks that overlap the point z (the "layering" operator in the parlance of this paper) exhibit the behavior of primary operators A primary motivation for [2] was to obtain a full-fledged conformal field theory by replacing the disk distribution of [7] with the Brownian Loop measure [5]. The analog of the disk model with the disk distribution replaced by the Brownian loop measure is precisely the BLS [1]. Since the BLS is known to be locally conformally invariant, a theory defined by it should be a full-fledged local conformal field theory. While [2] demonstrated that the exponentials of the (loop) layering operators are conformal primary operators, we were unable to compute the three-point function coefficients or four-point correlation functions.
In this work we take a major step beyond [2] by obtaining explicit results for the fourpoint function in the plane -which indeed is an analytic function of the cross ratio and has a conformal block expansion -as well as the three-point function coefficients, and the two-point function on the upper half-plane.

III. THE TWO-POINT FUNCTION IN THE UPPER HALF-PLANE
In this section we will use the results of [2,3] to derive the general two-point function of layering vertex operators in the upper half-plane H. In this section and everything that follows, we will make extensive use of a main result from [2] n j=1 e iβ j N (z j ) = Here the product is over all nonempty subsets S ⊂ {z 1 , . . . , z n } and I S denotes the set of indices corresponding to the points of {z 1 , . . . , z n } contained in S. The α(S|S c ) are the weights, according to the Brownian loop measure, of the sets of loops that encircle the points in S but not those in S c . The loops need to be contained in some domain D, which in this paper is either the upper half-plane or the full plane. We will denote weights in the upper half-plane by α H and weights in the full plane simply by α, and correlation functions by . . . H and . . . C , respectively. In words, (7) states that a general n-point function of layering vertex operators in the BLS equals the exponential of terms consisting of the weights for loops that encircle various subsets of the points times the associated conformal weights.
The two-point function in any simply connected domain of the plane can be obtained from H by a conformal transformation, so computing the two-point function in H in principle gives the two-point function in a domain of any shape. The boundary condition is that all loops must be confined entirely to the interior of H (that is, one could consider the BLS on the full plane and remove all loops that intersect the lower half-plane).
Our strategy is to first find the weights of loops that encircle one or both points in H. Once we have these weights we can immediately write down the two-point function for general β i using (7). We adopt a notation related to that of [2]. For two points z 1 = x 1 + iy 1 , z 2 = Here δ > 0 is a short-distance regulator that we will later take to zero, γ is a loop (left panel of Fig. 1), γ is the interior of γ (right panel of Fig. 1, shaded region), and diam(γ) is its diameter (the largest distance between any two points on the loop).
In general, the weights of loops that encircle only one point (such as (8), first line) diverge as δ → 0 due to contributions from arbitrarily small loops, infinitely many of which encircle any given point. Weights of loops that encircle two or more points (such as (8), second line) are finite as δ → 0 because only loops whose diameter is larger or equal to the distance between the two closest points encircle them. For where and . We can use (10) and properties of the weight of the loops around z 1 in the Brownian loop measure to get an expression for α H (z 1 |z 2 ). Let Then, by scale invariance of the Brownian loop measure, and Lemma A1 of [2], if δ ≤ y 1 where α = α 1,H (i), the weight of the loops around the point z = i with diameter greater than or equal to 1 and contained in H, is a constant of the model.
On the other hand, if |z 1 − z 2 | ≥ δ, then Using (7) and denoting the conformal dimensions by we have Notice that the equation suggests a specific ultraviolet cutoff in this "renormalization scheme": in addition to the normalizing factor δ 2∆ j used in [2], it is natural to include Note that the one-point function can be immediately obtained from (19) by setting one of the β i = 0: Using the equivalence (9), (19) can also be written as The results obtained above are the two-point functions of the field obtained after renormalization and the limit δ → 0, whose existence in Sobolev spaces H −α (H), α > 3/2, is shown in [9] for ∆ < 1/2.

IV. THE TWO-AND THREE-POINT FUNCTIONS IN THE FULL PLANE
In this section we compute the two-and three-point functions of layering vertex operators in the full plane. A major difference from the half-plane is that in the full plane all correlation functions go to zero because of the contribution from large loops, unless the charge conservation condition is satisfied [2,7]. This is reminiscent of momentum or charge conservation for the vertex operators of the free boson, where the condition arises from integration over the zero mode.
Note that this condition would be clearly necessary were we to define these correlation functions on the sphere rather than the plane, because on a sphere a loop that covers a subset of points can equally well be interpreted as a loop that covers the complement of that set (on a compact space there is no notion of the "inside" versus the "outside" of the loop).
Since the plane and the sphere are conformally equivalent, (22) is necessary for consistency (cf. Sec. VI).
The z i dependence of the two-and three-point functions in the full plane follow from conformal invariance and the fact that the layering vertex operators are conformal primaries [2]. However, this argument does not fix the constant prefactors, which were not computed in [2]. As we show in Appendix A, by taking the limit that the points are far from the boundary, we can use our results from the upper half-plane to determine the multiplicative prefactors left undetermined in [2]. We find that the most convenient normalization in the plane is With this definition the two-and three-point functions in the limit δ → 0 are and the calculation of (25) is possible since, when the three-point function of the fields is expressed in terms of α(z 1 , z 2 ) and α(z 1 , z 2 , z 3 ), the net contribution of the last of these two terms vanishes by charge conservation (Appendix A). Remarkably, the three-point function coefficients are precisely 1 for all values of the β i satisfying (22).
In the notation of [2], these results are equivalent to . (26)

V. NACU-WERNER THINNESS FUNCTION
Starting from (16) and (10) and a small ultraviolet cutoff constant δ, one can get an explicit expression of the Nacu-Werner thinness function α(z 1 |z 2 ) (the weight of loops that encircles one point but not another) in terms of α. The fact that this function is finite appeared in [10]. In fact, for two points z 1 , z 2 ∈ C, δ ≤ |z 1 − z 2 |, and z 1 (t), z 2 (t) as in the previous section, we have where we defined the constant and againᾱ ≡ α 1,H (i) is the weight of loops in the upper half-plane with a diameter larger than 1 and that encircle z = i. This is the Nacu-Werner function with ultraviolet cutoff δ ≤ |z 1 − z 2 |, which turns out to be the fundamental solution of the Laplacian in C, except for one multiplicative and one additive constant. Note that the analogous calculation of On the other hand, by combining (16) with (9), we get an explicit expression for the linear term of the O(n) expansion in [4], which confirms their calculation, except for a factor of 6π/5, to correct for different scaling, and the precise form of the constant. In fact equals the expression in [4], bottom of Page 18, multiplied by 6π/5.
This confirms the validity of the O(n) expansion in [4] for the two point function. In the next section we use the same expansion for the four-point function, this time without an independent verification.

VI. THE GENERAL FOUR-POINT FUNCTION IN THE PLANE
In this section we will compute the general four-point function for the layering operators in the whole plane. As for the two-point function in the upper half-plane, we will first derive the weights for loops covering various subsets of the points, and then with these in hand we can immediately write down the correlation function using (7). One difference is that to derive the weights we rely on a result of [4].
Consider four points z 1 , z 2 , z 4 , z 4 and assume in what follows that the letters i, j, k, ∈ {1, 2, 3, 4} are always different. Using (7), the four-point function is The weights of loops encircling points in the full plane are defined analogously to (8).
As previously mentioned, if we consider the BLS on a sphere rather than the plane, charge conservation is necessary for consistency because there is no distinction between the inside and outside of a loop on a sphere. Another implication of this fact is that both "sides" of the loop must contribute equally to the correlation functions. Under stereographic projection to the plane, the "outside" of the loop is the side that contains the point that projects to infinity of the plane, but it remains the case that both the inside and the outside must contribute. This explains why only the paired weights α S appear in (31).
There are a total of seven pairs α S that contribute. Six of these can be determined from the results we have already obtained for the two-point functions (we can also obtain relations from the three-point functions, but they are not independent). For the seventh relation we will use a result of [4].
Comparing these, we obtain the relation where we used (27)  The system of six equations we obtain from (33) and its permutations has rank six. An additional independent relation is necessary to solve for the seven α S , and is provided by [4].
Defining z jk = (z j − z k ), the cross-ratio and the function equations (21) and (22) of [4] imply that Cardy and Gamsa derived this result by solving a linear differential equation that does not fix the overall normalization or the additive constant, so we have included an overall coefficient P , and retained an additive constant that we denote 4(Q + R) for future convenience (we will see shortly that R = 0, and Q is defined by (28)).
We now have seven independent equations for the seven α S . The solutions are α S (z 1 |z 2 , z 3 , z 4 ) = 1 5 log z 12 z 14 z 24 We can now show that R = 0. Consider the four points z i arranged in a rectangle in cyclical order. If we let a pair of points approach the other pair by taking z 1 → z 4 and z 2 → z 3 it is clear that α(z 1 , z 3 |z 2 , z 4 ) → 0, since the probability of a loop passing between the pairs of points goes to zero. In the same limit we have that x → 1, A(x) → 0.
This allows us to write the fully general, normalized four-point function as where A(x) is defined by (35) and the ∆ i , ∆ ij by (17).
With some algebra and using the identity one can check that the four-point function is invariant under exchange of any pair of indices, establishing crossing invariance.

A. Free-field limit
There is a limit in which the correlators in the full plane become those of free field vertex operators (the same limit was considered and the same result obtained in [7], for the disk model studied there). Consider taking β i → 0 and λ → ∞ with the product λβ 2 i fixed. We define the field ψ by βN such that the conformal dimension of the operator e iβN (z) = e i √ 2γψ(z) becomes This is the correct dimension for a canonically normalized free-field vertex operator e i √ 2γψ .

Now consider (38) and note that
where we used the fact that 4 i=1 γ i = 0. Therefore the factor in (38) involving hypergeometric functions goes to 1 in this limit, and the remainder reduces immediately to the correct form for the four-point function of free-field vertex operators: This same limit should reduce the n-point function in the plane for all n to the free-field case.
Interestingly, the correlators in the upper half-plane do not reduce to those of free fields in the same limit. To see this, note that the coefficient of the hypergeometric function in

VII. EXPANSION IN CONFORMAL BLOCKS
The four-point function of a conformal field theory contains information about the threepoint function coefficients, as well as the spectrum of primary operators. To obtain this data, one makes use of the operator algebra by performing a conformal block expansion.
By a global conformal transformation, one can always map three of the points appearing in the four-point function to fixed values. The remaining dependence is only on the crossratio (34) and its conjugatex. Each cross-ratio is invariant under global conformal transformations.
We can now proceed to expand the four-point function in Virasoro conformal blocks The sum over P runs over all primary operators in the theory, and the C P ij are the threepoint function coefficients of the operators labeled by i, j with P. Each P with a non-zero C contributes a term consisting of a holomorphic function times an anti-holomorphic function of the cross-ratio. These functions -the Virasoro conformal blocks -depend only on x, the central charge c, and the conformal dimensions ∆ i , ∆ P of the five operators.
The conformal blocks are given perturbatively by a power series where coefficients F K are determined by the Virasoro algebra. The first 3 terms are given by F 0 = 1 (48) with (51) (see [11,12]).
We now take the limit in the four-point function (38) to obtain G: where A(x) is given by (35) and Any consistent four-point function of scalar primary operators must obey the crossing relations: These relations follow from the invariance of (38) under exchange of any pair of indices, which we have already verified. However as a check, we can verify (56) directly. By taking different limits of the four points we obtain It is easy to see that (54), (57) and (58) indeed satisfy (56) (again making use of the identity (39)).
Furthermore, the three-point function coefficient is equal to 1. Presumably, this operator is indeed e −i(β 3 +β 4 )N (z) = e i(β 1 +β 2 )N (z) , although we cannot be certain as we do not have complete knowledge of all its three-point function coefficients.
The next term in the expansion of G 21 34 (x) comes from the x 1/3 2 F 1 (2/3, 1; 4/3; x) = x 1/3 (1 + O(x)) term. Since there are no other terms with the power x 1/3 , there must be at least one primary operator with dimension ∆ 12 + 1/3. Similarly, expanding the exponential to quadratic order gives a term proportional to the square of the previous one, x 2/3 2 F 1 (2/3, 1; 4/3; x) 2 /2. This indicates the existence of a primary with dimension ∆ 12 +2/3. The question of whether there is a primary with ∆ 12 + 3/3 = ∆ 12 + 1 is more subtle, because this power of x also appears in the expansion of the ∆ 12 conformal block. To see that such an operator indeed exists, we could compute (the square of) its three-point function coefficient by subtracting the contribution from that of the ∆ 12 block, and see that the result is non-zero.
Let us now make this procedure systematic for the first few levels of operators. As noted before, it appears there are operators with dimensions for all non-negative integers p that couple to e iβ 3 N (z 3 ) e iβ 4 N (z 4 ) and e iβ 1 N (z 1 ) e iβ 2 N (z 2 ) . We need to compare the expansion of a mn x m/3xn/3 , to the conformal block expansion (46). In the expansion we must allow the operators to have non-zero spin s = ∆ −∆. To accommodate the previous results we write where we now sum over all non-negative integers m, n, p, p . F . Every non-zero combination shows the existence of an operator with dimensions H p in the operator spectrum to which two operators fuse on to.
We use a code developed by Matt Headrick [12] to generate the conformal block coefficients up to third order, which allows us to consider terms up to order O(x 11/3 ). The non-zero three point coefficients are marked in Fig. 2. We denote µ = Below, we give the first few three-point function coefficients from different blocks. These grow very rapidly in complexity with increasing p, p .

B. Interpretation
We leave the physical interpretation of these new primaries to future work. A hint is provided by [13], which considers the O(n) model as n → 0. There the four-point function is essentially the log of the one considered here for β i = π, and only a finite number of primaries appear in the fusion products. The primary corresponding (p = 1, p = 1) in our notation has dimension 1 3 , 1 3 when β i = π, and is identified as the leading order energy density operator of the O(n) model (in the limit n → 0).
One important caveat to our results in this section is that the three-point function coefficients obtained from the conformal block expansion do not entirely determine the spectrum of primaries. Clearly, there could be primaries in the theory with vanishing three-point coefficients with the layering vertex operators, and these would be invisible to us. A more subtle issue can also arise in the other direction if there are multiple operators with the same conformal dimensions that couple to the vertex operators. In that case one can only determine the sum of the squares of the three-point coefficients. Since these squared coefficients can evidently be negative, there could be cancellations. Therefore it is logically possible we are missing some operators in the theory that couple to the vertex operators, as there could be multiple degenerate primaries that couple with three-point coefficients with squares that sum to zero.

VIII. OUTLOOK
Our results for the correlation functions raise many interesting questions. First, it is possible that we can extend our techniques to compute n-point correlation functions for arbitrary n. This would provide new results for the winding probabilities of Brownian loops.
The spectrum of new primary operators we discovered needs investigation, as we do not know how to identify these operators either in terms of a previously known CFT, or in terms of the BLS.
Another interesting direction is to generalize the random variables assigned to the loops.
Here we considered the layering operator and assigned a random ±1 to each loop. In ongoing work to appear soon, two of us (Foit and Kleban) have considered more general distributions of random weights. This gives rise to infinite class of new conformally invariant systems for which we can compute exact four-point functions that depend on additional continuous parameters characterizing the distribution of weights.
where H t is the half-plane {(x, y) : y ≥ −t}. The weights α Ht (z j |z k ), for unequal j, k = 1, 2, are increasing in t and bounded above by α(z j |z k ), which is finite by thinness of the Brownian loop measure µ loop [10]; this implies that they have a finite limit as t → ∞. Moreover, where z 1 z 2 is the segment connecting z 1 and z 2 , as t → ∞, again by thinness. This shows that lim t→∞ α Ht (z i |z j ) = α(z i |z j ) and proves (A2). Notice also that It follows that From the equality of (9) and (10) we have that With charge conservation β 1 + β 2 = 2πZ, we have from (19) that where we used the fact that ∆ 1 = ∆ 2 . This gives an explicit expression of the constant appearing in the two-point function for the plane in [2]; with the normalization used in [2] (see the Summary and Results section, below (2.3)) the constant C 2 defined there is Absorbing the constants 2e It turns out we can also compute the three-point functions in the full plane using only the weights for the Brownian loop measure encircling one and two points in the upper halfplane. Given z i , z j , z k ∈ C, for distinct i, j, k ∈ {1, 2, 3} and δ ≤ min ij |z i − z j |, we have the six relations It follows from (7) With charge conservation β 1 + β 2 + β 3 = 2πZ it follows that cos(β i + β j ) = cos(β k ), and hence the coefficient of α H (z 1 , z 2 , z 3 ) is identically zero. This gives the three-point function e iβ 1 N (z 1 ) e iβ 2 N (z 2 ) e iβ 3 N (z 3 ) where σ jk = |z j −z k | 2 |z j −z k | 2 . Again, using charge conservation, and inserting the canonical normalization factors found for the two point functions, we derive the three-point function in the full plane by taking the limit (analogous to (A2)) which has the correct z i dependence for a three-point function. Surprisingly, the overall coefficient of the three-point function is simply 1, and does not depend on the β i (this was also the case for the three-point functions in the disk model of [7]).
From this result we obtain an explicit expression for the constant appearing in the threepoint function in [2]; with the normalization used there (see the Summary and Results section, below (2.3)) . (A18)