Logarithmic behaviour of connected correlation function in CFT

We study $(m)$-type connected correlation functions of OPE blocks with respect to one spatial region in two dimensional conformal field theory. We find logarithmic divergence for these correlation functions. We justify the logarithmic behaviour from three different approaches: massless free scalar theory, Selberg integral and conformal block. Cutoff independent coefficients are obtained from analytic continuation of conformal blocks. A UV/IR relation has been found in connected correlation functions. We could derive a formal ``first law of thermodynamics'' for a subsystem using deformed reduced density matrix. Area law of connected correlation function in higher dimensions is also discussed briefly.


Introduction
Divergent problem in continues quantum field theory is not a catastrophe. Usually, it can be cured by introducing an energy scale (UV cutoff) in the theory. This is still true in computing entanglement entropy in conformal field theory (CFT) [1][2][3][4]. One famous result is that one interval Rényi entropy in CFT 2 presents logarithmic behaviour and univeral [5]. In higher dimensions, it also diverges and obeys area law [6]. Rényi entropy, roughly speaking, is a generator of connected correlation functions of modular Hamiltonians On the other hand, motivated from holographic duality of modular Hamiltonian [7,8], we showed that connected correlation functions of modular Hamiltonians H m A H n B c , m, n ≥ 1 (1.2) are finite [9]. For CFT 2 , they are also universal due to conformal Ward identity [14]. More generally, the author claimed that (p, m − p)-type connected correlator is also finite [10], where Qs are OPE blocks [11][12][13]. Unlike (m, n)-type correlator, (m)-type correlator is divergent in general. There is no way to remove divergent terms in these correlators. Therefore, one nature problem is to understand the divergent behaviour of (m)-type connected correlation functions.
In this paper, we show that (m)-type correlator may present logarithmic behaviour like (1.1) in CFT 2 where L is the length of interval and is UV cutoff. Therefore one could still extract cutoff independent information from coefficient C[h i ]. We extract C[h i ] using a method of regularization developed by [9]. We could validate these coefficients using three approaches. Firstly, we prove the logarithmic behaviour in massless free scalar theory. Secondly, we obtain the same divergent behaviour by analytic continuation of Selberg integrals [16]. We also find that the logarithmic behaviour of (m)-type correlator can be obtained from analytic continuation of (p, m − p)-type correlator, which is equivalent to take the limit B → A of (1.3). Since (m − 1, 1)-type correlator is claimed to be conformal block [10], we could prove the logarithmic behaviour by analytic continuation of conformal block for general (m)-type correlators. As a consequence of this observation, we obtain a UV/IR relation between coefficients C[h i ] and D[h i ] where C[h i ] characterizes the divergent behaviour when B coincides A (UV) and D[h i ] characterizes the leading order behaviour when B and A are far away (IR). Our result shows that deformed reduced density matrix encodes rich information of CFT 2 , its vacuum expectation value also presents logarithmic divergence. We could derive a formal "first law of thermodynamics" associated deformed reduced density matrix for any subsystem where E A , Ω A , T, S A , µ and Q A are "energy","grand potential","temperature","Gibbs entropy", "chemical potential" and "charge" of the subsystem in parallel to statistical mechanics.
The structure of this paper is as follows. In Section 2 we will review known results on connected correlation function and indicate that (m)-type correlators obey logarithmic law in CFT 2 . We will check this logarithmic behaviour up to m ≤ 4 in Section 3. After that, we will work on massless free scalar theory to support our observation. In Section 5, analytic continuation of Selberg integral has been used to justify the logarithmic behaviour. We establish the relation between (p, m − p)-type correlator and (m)-type correlator in Section 6. We find more logarithmic behaviour by analytic continuation of conformal block to another region in the following section. In Section 8, we study the expectation value of deformed reduced density matrix. Motivated by the similarity between deformed reduced density matrix and density matrix of thermodynamic system, we establish a formal "first law of thermodynamics" for any subsystem. In section 9, we discuss the area law of connected correlation function in higher dimensions. We conclude with a discussion of future directions for this program in the last section. Technical details on conformal blocks, integrals and analytic continuation are collected in four appendices.

Review
Modular Hamiltonian [17], which is the logarithm of reduced density matrix, plays a central role in the evolution of a subsystem. The subsystem we are interested in is one interval A of a CFT 2 . The total system is in Minkowski spacetime and the state is in vacuum. We use (t, z) to denote spacetime coordinates. The domain of dependence of A is D(A). The endpoints of interval A are x 1 = −1, x 2 = 1. In other words, the radius of A is 1 and the center is at origin. Therefore the length of interval A is L = 2. (2.1)

Modular Hamiltonian for one interval is [18]
where stress tensor is evaluated at t = 0 slice. The stress tensor can be seperated into homomorphic and anti-holomorphic part. We will only consider chiral operators in this paper, the holomorphic part of modular Hamiltonian is where we used the convention T (z) = −2πT tt | hol , following [19]. Zero modes of modular Hamiltonian are operators in D(A) which are annihilated by modular Hamiltonian [20] [H A , Q A ] = 0. (2.4) They are invariant under modular flow and have a gravitational dual which is evaluated at minimal surface in the bulk. Since modular Hamiltonian commutes with itself, modular Hamiltonian is also a zero mode. All zero modes form a closed algebra in D(A). Among all of the zero modes, OPE blocks are most intriguing. OPE blocks are the most natural objects in operator product expansion of two primary operators 2 (2.5) For interval A, it may have a closed form where h is the conformal weight of primary operator O. The coefficient c h is an unspecified constant which depends on convention. OPE block is an eigenvector of Casimir operator L 2 of SL(2, R) with eigenvalue −h(h − 1), therefore its correlation function with other OPE blocks in a disjoint region is an eigenfunction of Casimir operator L 2 with the same eigenvalue −h(h−1). Such kind of eigenfunction is conformal block [21,22] In [9,10], the author studied connected correlation functions and concluded that these connected correlators are finite in general. When n = 1, the correlator is conformal block where N h i , C 123 are constants in two and three point correlation functions The conformal weight of two operators O 1 and O 2 are assumed to be equal, h 1 = h 2 . When h 1 = h 2 , there is also a natural generalization of OPE blocks, see [15]. We will not discuss those objects in this work.

The most general connected correlation function from OPE blocks is
where capital letters A, B, C denote distinct spatial regions. We assume they are disjoint. A correlator (2.14) with m 1 OPE blocks in region A, m 2 OPE blocks in region B, etc. will be called Y -type, following the convention [10]. We use a Young diagram to label correlator (2.14). Interestingly, the correlators of (m, n)-type with m ≥ n ≥ 1 have been studied extensively in [9,10] while correlators of (m)-type have not been explored systematically though they correspond to one row Young diagrams, which should be much simpler than two row Young diagrams. This is partly because (m)-type correlators are divergent in general. However, we will show that they may always obey logarithmic law where is a UV cutoff. Therefore the coefficients C[h i ] are cutoff independent and encode rich information of the theory. The logarithmic behaviour of (m)-type correlator is similar to Rényi entropy [5] S (n) for any CFT 2 . Therefore, one should not discard (2.16) just because they are divergent. Actually, Rényi entropy is a special combination of (2.16) when all zero modes are modular Hamiltonians. To see this point, we definẽ We conclude that Rényi entropy is a generator of connected correlation function of modular Hamiltonian, H m A c in (2.20) is a special example of (m)-type correlators (2.16). We may regard (2.16) as a generalization of logarithmic law of Rényi entropy.

Logarithmic behaviour of connected correlation function
In this section, we will calculate (m)-type correlators in CFT 2 order by order. The lowest order is m = 1. This is not fixed as one can always shift Q A [O] by a constant, The constant term does not affect correlators with m ≥ 2 since its effects are canceled in connected correlation functions. However, it indeed affect (1)-type correlators. To determine (1)-type correlators, one should specify the constant term by other requirement. When Q A is modular Hamiltonian H A , reduced density matrix is normalized which may be used to fix the constant term. However, for other OPE blocks, we don't find a nature condition to fix this constant term. One may choose this term to be 0. In the following, we will consider the cases m ≥ 2.

(2)-type
(2)-type correlator is fixed by conformal symmetry As [9], : · · · : means that one should regularize the integral since the integrand is divergent at z 1 = z 2 . We will omit : · · · : to simplify notation in the following. The method of regularization has been justified by [9,10], however, we'd like to evaluate it carefully since we will meet new features in the integral. The first step is to calculate z 2 integral as if there is no pole, then (3.5) The integral is still divergent once z 1 approaches the bondary of the interval, therefore, we insert a UV cutoff into the integral The system is symmetric by exchanging the two end points, therefore the insertion of is chosen to be symmetric. Now (3.6) is well defined, we find Matching it with formula (2.16) and using L = 2, we find . (3.8)

(3)-type
(3)-type correlator is also fixed by conformal symmetry up to a structure constant of three point function, with (3.10) The integral can be evaluated for integer conformal weight case by case. We could check that it always obeys logarithmic law (3.12)

(4)-type
Four point function of primary operators can only be fixed up to a function of cross ratio by global conformal invariance. However, it is much more simpler when four primary operators are identical and their conformal weight h is an integer [23],  (3.14) The function f (θ) only depends on θ follows from the discrete symmetry z 1 ↔ z 2 or z 3 ↔ z 4 . It should be a polynomial of θ since there is no branch cut. The maximum degree of the polynomial is 2h since the most singular behaviour of the four point function is z −2h 13 when z 1 → z 3 . Therefore the function f (θ) is fixed up to a finite number of unkown constants [24] f (θ) = 2h j=0 a[h, j]θ j . (3.15) The first and last coefficient are fixed to 1 by examing the limit z 1 → z 2 and Therefore we may consider the (4)-type correlator where The last three terms are inserted because the correlator is connected. The basic integral is The integrals from last three terms of It turns out that the general structure of j I 4 is In Table 2, we calculate each basic integral j I 4 up to h ≤ 6.
Several remarks: is non-zero only for j = 0 and 2h, they are proportional to each other Therefore their contribution to connected correlation function is canceled in

One can read
We will return to this formula in the following sections.

Massless free scalar theory
In Section 3, we test the formula (2.16) for 2 ≤ m ≤ 4 in general CFT 2 . In this section, we will extend these results to higher orders for massless free scalar theory. In this theory, modular Hamiltonian can be evaluated in momentum space [9]  where b v , b † v are annihilation and creation operators by quantizing the theory in subregion D(A). They obey standard commutation relations. The constant term is fixed by normalization of reduced density matrix Rényi entropy are found from momentum space 4 , The Dirac delta function δ(0) is divergent in momentum space The author [9] noticed that it matches with (2.17) if one regularizes the Dirac delta function by a UV cutoff , in real position space. For massless free scalar theory, there are infinite many primary conserved currents [25] We will study real scalar, therefore s is even, The conformal weight of J s is equal to s, therefore the corresponding OPE block is They have rather simple form in momentum space [10] The function S(s; a, b) is a generalized hypergeometric function For integer s, it is a polynomial of two variables a and b. The normalization constant c s is chosen such that Since OPE blocks are zero modes of modular Hamiltonian, one can define deformed reduced density matrix [10] ρ which is related to the generator of connected correlation function of OPE blocks by (4.14) The deformed reduced density matrix is not normalized in this work. The generator is similar to partition function of a thermodynamic systerm. Expanding T A (α i ) for small α i , we could find its relation to connected correlation function Dirac delta function has the same origin in momentum space as (4.3). Therefore using the rule (4.5), (4.18) should be regarded as a "Rényi entropy" corresponds to deformed reduced density matrix. It still obeys logarithmic law. For Several interesting properties appear in (4.18).
1. The formula (4.18) is equivalent to (2.16) for free scalar theory. Note the primary operator is quadratic in terms of scalar field in this example. The connected correlation function The function f s (v) and the particle number N (v) = 1 e 2πv −1 have the following asymptotic behaviour (4.23) Therefore one can integrate by parts, The coefficients up to m ≤ 4 and spin less than 6 are collected in Table 3.
In the table, m 1 , m 2 , m 3 are number of spin 2,4,6, respectively. In the following we will match them with corresonding results of Section 3.
(c) The coefficients a[h, j] in function (3.15) are given in Table 5. Using formula (3.22) and They are exactly the corresponding coefficients of Table 3.
It should be non-negative for all v, otherwise the integrand for F (α i ) is not real.
Therefore we find 3. Connected correlation functions are related to small α i expansion of generator T A (α i ). One may also regard T A (α i ) as an independent quantity. As has been pointed out in [10], one can keep α s (s = 2) finite while which is independent of α s .

Analytic continuation of Selberg integral
In Section 4, we use the example of free scalar to justify the logarithmic behaviour derived from conformal field theory in Section 3. In this section, we use a rather different approach, analytic continuation of Selberg integral [16], to validate our method of regularization. Selberg integral is a direct multi-dimensional generalization of integral representation of Beta function. The basic Selberg integral is It is convergent in the region The development of Selberg integral can be found in [26]. 1.
(2)-type. The integral (3.4) is a "Selberg integral" for integer conformal weight, However, γ is in the wrong region since (3.4) is divergent. Nevertheless, this provides a new way to regularize I 2 [h, h]. By transforming variables z i to t i , We insert a˜ 2 since the integral is divergent, we continue Selberg integral to˜ 2 → 0  which could also be found from Table 1. 3. (4)-type. One can use Selberg integral only for j = 2h 3 , the conformal weight h = 3k, k ∈ Z + , then the following basic integral We will comment on analytic continuation of Selberg integral. The advantage of analytic continuation of Selberg integral is obvious. However, the disadvantages are also serious . Firstly, one cannot obtain all (m)-type correlators by using Selberg integral, even for m = 3. A basic Selberg integral has only three parameters a, b, c for a fixed n. To use Selberg integral, the conformal weight should be equal. The degree of z i − z j should be the same. This restricts the applications of Selberg integral. One should develop Selberg integral further. A general correlation function is invariant under spacetime translation, then it will only depends on the distance of positions. Therefore, we propose a more general integral where α, β are vectors whose elements are The matrix γ is a n × n symmetric matrix whose diagonal terms are zero, The total number of parameters of S n ( α, β, γ) is Secondly, we should introduce a small paramter˜ i to control the divergent behaviour for analytic continuation of Selberg integral, the relationship between˜ i and UV cutoff is unclear without matching several examples.

Analytic continuation of conformal block
Motivated by the success of analytic continuation of Selberg integral, we develop another method to overcome several difficulties in previous sections. The method is similar to pointsplitting regularization in quantum field theory. For example, to extract the singular and finite part the expectation value of a quadratic operator (∂φ) 2 , one should split the operator to ∂φ(z + )∂φ(z) firstly, and then take the limit → 0, The (m)-type connected correlation function of OPE block is similar, we first split the region A to two disjoint regions A and B, The right hand side of (6.2) is a (m−1, 1)-type correlator, which has been shown to be finite [10] and proportional to conformal block, see equation (2.9). Formally, we take the limit B → A, then (m − 1, 1)-type correlator should be (m)-type correlator, The right hand side of (6.3) is only a function of cross ratio it approaches −1 when B → A. The way to approach A is subtle, we choose a way which is symmetric 5 . More explicitly, the end points of A and B are parameterized as The parameter characterizes the distance between point x 3 to x 1 , therefore it should be the same UV cutoff in Section 3. The cross ratio is The conformal block G h (η) presents logarithmic behaviour The terms in · · · are finite and suppressed in the limit → 0, therefore It is intriguing that logarithmic behaviour in Section 3 emerges as analytic continuation of conformal block while the latter is fixed by conformal symmetry. We read a neat relation between In the following we will discuss the consequences of (6.9). 1.
(2)-type. D[h i , h j ] can be found in (2.10), then It is indeed (3.8) for integer conformal weight.
which should be very hard to evaluate in Section 3. It reproduces results of Table 1.
. (6.14) It is the same as (5.10) which is obtained from analytic continuation of Selberg integral.
This identity can be checked for free scalar [10].

Coefficient D[h i ] characterizes the leading behaviour (m − 1, 1)-type correlator when A
and B are far away, therefore it is an IR constant. On the other hand, C[h i ] characterizes the logarithmic divergent behaviour of (m)-type correlator when A and B coincide, so it is a UV constant. In this sense, (6.9) is a UV/IR relation.
We will comment on the analytic continuation of conformal block further at the end of this section. There are different ways to seperate m OPE blocks, the general way is to uplift (m)-type correlator to (p, m − p)-type, As has been noticed, the right hand side only depends on cross ratio η and finite, though it is not necessary conformal block. Nevertheless, we should find This may put contraints on possible structure of (p, m − p)-type correlator since (m)-type is only logarithmically divergent. We can test (6.23) using the (2, 2)-type correlator of modular Hamiltonians of [9], we read This is consistent with (6.15). We can extend this observation further since one can uplift (m)-type correlator to (m 1 , · · · , m k )-type correlator with k i=1 m i = m. We hope to return to this topic in the future.

More logarithmic behaviour
The logarithmic behaviour in Section 3 happens when region B coincide with region A. When region B attaches to region A, (m, n)-type correlator may also be divergent. To check this point, we parameterize the two intervals as The cross ratio η = 1 + · · · (7.2) approches infinity when x 3 → x 2 . Then conformal block We use the notation to denote the corresponding correlator, then is divergent logarithmically when the two intervals attach to each other. This is another type of logarithmic behaviour. Interestingly, the coefficient is not independent. One can also consider general (m 1 , m 2 , · · · , m k )-type correlator, for example, (2, 2)-type correlator However, it is unclear whether these correlators always have logarithmic behaviour.

Deformed reduced density matrix and grand canonical ensemble of a subsystem
It has been noticed that connected correlation function of OPE blocks [9,10] can be generated by the expectation value of deformed reduced density matrix More explicitly, they are generated by The function Z(α i ) is similar to the partition function (grand potential) in equilibrium thermodynamic systems. For example, assume where Q A is an OPE block, then where is defined to be "grand potential" of subsystem. The parameter β is inverse "temperature" and µ is "chemical potential" associated with OPE block Q A . "Gibbs entropy" is One can easily show that "Gibbs entropy" is von Neumann entropy "Gibbs entropy" defined in this way is not usual entanglement entropy in general. Expectation value of "energy" H A and "charge" Q A are Then "grand potential", "Gibbs entropy", "energy" and "charge" satisfy the identity Note "charge" can also be generated by "grand potential" 14) differential of "grand potential" becomes Combining (8.13) and (8.15), we obtain an equation of "first law of theromodynamics" for any subsystem The "first law" should be distinguished with "first law of entanglement" found in [27]. The former is valid at all order while the latter is valid up to linear order of perturbation.
The direct consequence of (2.16) is that any deformed reduced density matrix 6 may be associated with a logrithmic behaviour It is obtained in (4.18) where the operator is quadratic for free scalar theory. In previous example, W = aH A + αQ A , we find

Area law in higher dimensions
In this section, we will briefly comment on connected correlation function of OPE blocks to higher dimensions. We assume region A and B are disjoint, their radius are R and R , respectively. The distance of their center is r, r > R + R . We assume they are located at the same time slice, therefore there is only one independent cross ratio OPE block corresponding to a primary operator O of dimension ∆ and spin J is [28] where vector K µ is the generator of modular Hamiltonian of spherical region of CFT d . Constant c ∆,J depends on conventions and we leave it free. The integral region is in the domain of dependence of A. When the operator is conserved, the integral can be reduced to an integral inside region A. One special example is modular Hamiltonian H A itself. One can construct (m, n)-type correlators As d = 2 case, the correlators are finite in general. For n = 1, (m, 1)-type correlator may be a conformal block Just as d = 2 case, we continue the function by perserving spherical symmetry, The cross ratio approaches −∞ by Now it is interesting to check the asymptotic behavaiour of conformal block in this limit. The general d dimensional conformal block in the diagonal limit (z = x)can be found in [31]. For any finite conformal weight ∆ and spin J, it is a finite sum of 3 F 2 functions. We collect the discussion on conformal block in Appendix D. Hypergeometric function behaves as where Conformal block is divergent The const. term can be obtained directly, however, we don't need them in this paper. (m)-type correlator becomes 7 We have inserted back the radius R. Terms in · · · are subleading, we will discuss them later. Note the area of the boundary of A is we conclude that (m)-type obeys area law (9.14) When the OPE block is modular Hamiltonian, (9.14) is equivalent to area law of Rényi entropy [2]. In Rényi entropy, one may read cutoff independent information from subleading terms. For example, for a system with gravitational dual, the typical behaviour of entanglement entropy is [32] q is cutoff independent and encodes useful information of the theory. In a similar way, we would like to understand the subleading behaviour in (m)-type correlator. It turns out that the subleading behaviour depends on dimension ∆ and J. We will check several examples in the following.
1. d = 3, J ≤ 10. The general behaviour is where is not necessary to be in (9.9), but in the small limit, they are of the same order = + o( ). (9.17) For general conformal weight, q 1 = 0, therefore q 1 is cutoff independent. Only for special operators, for example, stress tensor has J = 2, ∆ = 3, q 1 = 0, then in this case, q 0 is cutoff independent. This is consistent with holographic result (9.15). Interestingly, when the operator is a symmtric traceless conserved current, its dimension obeys unitary bound [36] ∆ = J + 1, for J ≥ 1, (9.18) we always find q 1 = 0.
2. d = 4, J ≤ 10. The general behaviour is Again, is (9.17), it can be chosen such that no linear term R in (9.19). In general, q 2 is cutoff independent. One simple example is ∆ = 4, J = 0, However, for special combination of ∆ and J, q 2 = 0. One example is stress tensor ∆ = 4, J = 2, it has q 2 = 0, q 1 = 0, therefore q 1 is cutoff independent. Again, this is consistent with holographic result [32]. We also observe that q 2 = 0 for which is exactly unitary bound for a symmetric traceless conserved current in four dimensions.

(9.22)
For general conformal weight ∆ and spin J, the logarithmic behaviour in (9.22) is not the same as (9.15). When the operator is a conserved current it recovers the same phenomenon (9.15).

Conclusion and discussion
Connected correlation functions of (m)-type in CFT 2 have been studied extensively. They are divergent and obey logarithmic law, which is summarized by formula (2.16). The coefficient C[h i ] is independent of UV cutoff and encodes details of CFT 2 . We could check this formula by direct regularization of (m)-type correlators. To justify our results, we also provide three different ways to deal with (m)-type correlators. Firstly, we calculate the logarithm of the expectation value of an exponential operator e −H A − s≥2 αsQ A [Js] , the result is presented in (4.18). Secondly, we use Selberg integral to regularize several correlators, we could also find logarithmic behaviour by carefully identifying the regularization parameter. Finally, we use analytical continuation of (p, m − p)-type correlators to reproduce the same results. Interestingly, (p, m − p)-type correlation functions considered in [9,10]  are not independent. Mathematically, this follows from the asymptotic behaviour of conformal block near η = 0, −1 and ∞, (10.2) respectively. Physically, it is a rather intriguing UV/IR relation in connected correlation functions.
There are many problems to be solved in this direction.
1. Deformed reduced density matrix (8.1) is claimed to be meaningful in [9,10] by its formal similarity to Wilson loop [29,30]. In this paper, its expectation value ρ A = tr A e −H A −W is shown to be controllable, in the sense that it obeys logarithmic law. We notice that it is analogous to grand canonical ensemble in equilibrium thermodynamic system. We define corresponding "grand potential", "Gibbs entropy", " energy" and " charge" for a subsystem. A "first law of thermodynamics" for a subsystem has been derived in parallel to statistic mechanics. However, the physical meaning of these quantities deserves further study.
2. We discussed the area law of (m)-type correlators in higher dimensions from analytic continuation of conformal block. Interestingly, this is equivalent to the area law associated with exponential of OPE block For conserved current J , the area law becomes This is a direct generalization of Rényi entanglement entropy (9.15). It is better to validate this behaviour in more details. The cutoff independent coefficients before log R or log 2 R encode rich CFT data, there is no work on how to find out them so far. In this paper, we find one, F (α i ), for 2d massless free scalar theory.
3. The logarithmic behaviour in higher dimensions (10.4) can be continued to d = 2, which is consistent with (1. The conformal weight is larger than spin in general. The OPE block for these operators are which is consistent with analytic continuation from (10.3) in higher dimensions to 2d. One can also understand the log 2 L divergence in another way. The holomorphic part contributes one log L and anti-holomorphic part contributes another logarithmic divergence. Therefore, the divergent behaviour is the product of holomorphic and anti-holomorphic part, which is exactly log 2 L .

We defined another type of correlators when region A attaches region
These correlators are also divergent. However, we showed in several examples that one can also obtain cutoff independent information from the divergent behaviour. For m, n ≥ 2, we can not use the trick of analytic continuation of conformal block. Therefore it is not guaranteed that it always obeys logrithm law in two dimensions (or area law in higher dimensions). One should develop other method to tackle this problem. In higher dimensions, a new divergent behaviour for Q m−1 A Q B c is also very interesting.
5. In AdS/CFT correspondence, one could extract OPE data from holographic correlation function using Witten diagrams [33]. "Geodesic Witten diagram" [34,35] is a bulk description of conformal block. On the other hand, (m, 1)-type correlators are also claimed to be conformal block. It would be better to establish a relationship between geodesic Witten diagram and (m, 1)-type correlator. OPE block has a natural gravitational dual in the bulk, for example, the dual object in large N limit is schematically γ φ, (10.11) where γ is minimal surface associated with boundary region A, φ is the bulk field which is dual to primary operator O. Our work suggests that the following exponential operator We expect that it obeys area law in higher dimensions and logarithmic law in two dimensions. For two disjoint regions, one may consider the following correlation functions where γ i , i = 1, 2 is minimal surface corresponds to region A and B. The CFT results in [9,10] indicate that I γ 1 ,γ 2 (φ 1 , φ 2 ) is finite in large N limit.
6. We introduced an integral (5.16) with n(n+3) 2 parameters to deal with analytic continuation of integrals in this paper. This integral is crucial for our understanding of general connected correlation functions. When n = 1, the integral is an integral representation of Beta function. When n = 2 and α 1 = α 2 , β 1 = β 2 , the function is standard Selberg integral. Unfortunately, a closed formula is lack for general cases, as far as we know. This integral may reveal deep connection between mathematics and physics.

Appendices
A (m, 1)-type correlator In this section, we show that (m, 1)-type correlator is conformal block (2.9). We use the differential method as [22]. Assume the end points of A are z 1 , z 2 and z 3 , z 4 for B, then since OPE blocks in this paper are invariant under conformal transformation, a general (m, 1)-type correlator is only a function of cross ratio Casimir operator of SL(2, R) is where L AB are generators of global SL(2, R). It is a summation SL(2, R) generators act on each end point of B, In coordinate representation, the holomorphic part of Casimir L 2 is Casimir operator acts on (m, 1)-type correlator, then Therefore, Note the leading order of H(η) is η h since Therefore the solution of (A.7) is which is exactly conformal block. Since Casimir operator is a second-order partial differential operator, (m, n)-type correlator with n ≥ 2 is not conformal block in general.

B Integral
The One should carefully seperate these two classes of divergent behaviour. We will use h = 1 as an example to show how to calculate such kind of integrals.
At the second step, we use integrate z 1 , z 2 by ignoring any poles in the integrand. We insert UV cutoff to regularize the integral. since the log 2 2 terms should be canceled in the connected correlation function. Therefore we may subtract a term in the integral.
Now we change variables to y 3 , y 4 by

C Analytic continuation
In Section 6, we obtain logarithmic behaviour by taking the limit B → A. We choose the way to approach the coincidence limit symmetrically, The center of two intervals are the same while the radius are not. However, since two regions are non-local, there are other ways. We could choose the center of two intervals are not coincide, The logarithmic behaviour is the same as (6.7), including the coefficient. The problem appears in the constant term, which is complex due to the term proportional to iπ. However, it doesn't affect the result in this paper.

D Conformal block in general dimensions
We will collect some basic facts about conformal block in general dimensional conformal field theory. The reader is referred to [21,22,37,38] for more details. The motivation to study conformal block is from four point function in CFT, commutes with all generators L AB conformal algebra. Therefore conformal block is an eigenfunction of the following second order partial differential operator . The boundary condition is Therefore one could fix the form of conformal block in general dimensions. For example, in four dimensions, In this paper, two spherical regions are at the same time slices, x = z (D. 13) this is the so called "diagonal limit" of conformal block [31]. In diagonal limit, conformal blocks satisfy ordinary differential equations, third order for = 0 and fourth order for general cases. For equal external operator dimensions, ∆ 12 = 0, the closed form of diagonal limit of conformal block is a finite sums of 3 F 2 functions 14) The variable Z is The parameter λ is λ = ∆ + 2 .
(D. 16) The general form of F λ,2n and F λ,2n+1 are This is exactly two dimensional conformal block we used in this work. Note for odd h, there is an extra minus sign. For general non-chiral operator, ∆ = h +h, J = h −h, the conformal block becomes a product of two (D.19) in the diagonal limit. This is the origin of log 2 L divergence for non-chiral OPE blocks.