Area law of connected correlation function in higher dimensional conformal field theory

We study $(m)$-type connected correlation function(CCF) of OPE blocks with respect to one spatial region in higher dimensional conformal field theory (CFT). The leading term of CCF obeys area law. In the sub-leading terms of CCF, we find logarithmic correction which is cutoff independent. The logarithmic behaviour is characterized by a parameter $q$ and a constant $p_q$, where $q$ is the maximal power of logarithmic divergence and $p_q$ is the coefficient before the logarithmic term. We derive a UV/IR formula which relates $(m)$-type CCF to $(m-1,1)$-type. A cyclic identity for the coefficient $p_q$ has also been checked carefully.


Introduction
There are diverse area laws in different branches of physics. The prototype is originated from black hole physics where the thermal entropy of a black hole is proportional to the area of event horizon [1,2]. This unusual property of black hole has stimulated varies modern idea of theoretical physics.
In the context of quantum field theory (QFT), people have already noticed a similar area law for geometric entanglement entropy [3][4][5][6] several decades ago. One could find the details in the review paper [7]. Its connection to gravity has been established by the work of Ryu and Takayanagi [8], in which they proposed that the entanglement entropy of a CFT is equal to the area of a minimal surface in the bulk AdS spacetime.
In this paper, we present a new area law in general higher dimensional CFTs (d > 2) following the work [9]. In that work, the author argued that (m)-type CCF [10,11] of OPE block may obey area law from the analytic continuation of (m − 1, 1)-type CCF. As entanglement entropy of continues QFT, it is divergent. The leading term obeys area law whose coefficient depends on the energy scale. In the sub-leading terms, cutoff independent information can be extracted, usually, this is encoded in a logarithmic divergent term. However, the logarithmic structure turns out to be much more richer than entanglement entropy. We summarize the area law and logarithmic behaviour schematically in the following formula (1.1)

In this equation, Q A [O]
is an OPE block associated with a primary operator O of CFT. We will review the definition of OPE block in the following section. The subscript A denotes the spacetime region where the OPE block lives in. The quantity R is the typical size of region A. The small positive parameter is a UV cutoff. The constant γ is cutoff dependent, therefore it is not physical. The integer q is the maximal power of logarithmic terms in the CCF whose coefficient p d is non-zero. The exact value of q may depend on the OPE block and the spacetime dimension. According to the value of q, we classify the logarithmic behaviour of (m)-type CCFs. We will detail its value in the following sections. When the positive value m ≤ 3, we find that q may be in the region 0 ≤ q ≤ 2. (1. 2) The · · · terms in the formula are the possible sub-leading terms which are cutoff dependent. Therefore we will not be careful about their exact forms. The physical information is encoded in the coefficient p q . We establish a UV/IR relation to extract the coefficient p q based on the analytic continuation of conformal block.
This paper is organised as follows. We begin by introducing OPE block and CCF used in this work in section 2. In section 3 we will derive the area law and logarithmic behaviour of (m)-type CCF. We classify different CCFs according to the maximal power q of the logarithmic term in the CCFs. At the same time, we obtain a UV/IR relation which is useful to extract the cutoff independent coefficient p d . We compute several examples in the following section. In section 5, we discuss a severe "inconsistency" found in section 3. Section 6 contains some concluding remarks in this work.

OPE block
In CFTs, operators are classified into (quasi-)primary operators O and their descendants ∂ µ ∂ ν · · · O. A general primary operator is characterized by two quantum numbers, conformal weight ∆ and spin J. Under a global conformal transformation x → x , a primary operator 2 transforms as where |∂x /∂x| is the Jacobian of the conformal transformation of the coordinates, ∆ is the conformal weight of the primary operator and d is the spacetime dimension. Operator product expansion(OPE) of two separated primary scalar operators O i (x 1 )O j (x 2 ) is to expand it in a local complete basis around at a suitable point where · · · are descendants of the primary operator O k . Its form is fixed by global conformal symmetry, therefore it just contains kinematic information of the CFT. The constants C ijk are called OPE coefficients which is related to the three point function of primary operators. They are the only dynamical parameters in a CFT. The constants ∆ i , ∆ j , ∆ k are conformal weights of the corresponding primary operators. The distance of two points x 1 and x 2 is denoted as |x 12 |. By collecting all kinematic terms in the summation, we can rewrite OPE (2.2) as 3) The objects Q ij k (x 1 , x 2 ) are called OPE blocks [12][13][14]. They are non-local operators in the CFT and depend on the position of external operators x 1 and x 2 . The upper index i and j show that it also depends on the quantum number of the external operators O i and O j . It is easy to see that OPE block has dimension zero. Under a global conformal transformation The explicit form of f (x 1 , x 2 ) is not important in this work. When the two external operators are the same, we have f (x 1 , x 2 ) = 1 and OPE block will be invariant under global conformal transformation. One can also show that the OPE block is independent of the external operator in this special case. We will relabel such kind of OPE block as The subscript A denotes the region determined by the two points x 1 and x 2 where the two external operators insert into. The operator in square bracket reflects the fact that OPE block is generated by a primary operator O k . We omit the information of i since OPE block is insensitive to the external operators in this case. We will classify the primary operator O k into conserved currents J and non-conserved operators O. A general primary operator obeys the following unitary bound [15]   All other primary operators are non-conserved operators. Correspondingly, the OPE block (2.5) generated by conserved currents J will be called type-J OPE block. On the other hand, the OPE block (2.5) generated by non-conserved operators O will be called type-O OPE block.
When two operators are time-like separated, region A is a causal diamond. The two operators are at the sharp corner of the diamond A. We can use conformal transformation to fix The center of the ball is x A . The boundary of Σ A is a unit sphere S. In the context of geometric entanglement entropy, the surface S is an entanglement surface which separates the ball Σ A and its complement. Leading term of entanglement entropy is proportional to the area of surface S in general higher dimensions (d > 2). There is a conformal Killing vector K which preserves the diamond A, where the primary operator O µ 1 ···µ J is non-conserved It has dimension ∆ and spin J. When the operator is a conserved current the corresponding type-J OPE block is It can be obtained from (2.9) by using conservation law (2.11) and reducing it to a lower d − 1 dimensional integral. The coefficient c Jµ 1 ···µ J is also redefined at the same time. In (2.9) and (2.12), the coefficients c Oµ 1 ···µ J and c Jµ 1 ···µ J are free parameters, we set them to be 1.
A very special type-J OPE block is modular Hamiltonian [17,18] of the region Σ A , Modular Hamiltonian is the logarithm of reduced density matrix ρ A It plays a central role in the context of entanglement entropy, More generally, Rényi entanglement entropy has been shown to satisfy an area law generally where A is the area of the entanglement surface S. and is a UV cutoff. The constant γ is cutoff dependent. The subleading terms · · · contain a logarithmic term in even dimensions The coefficientp We will introduce the definition of CCF in the following subsection.

Deformed reduced density matrix and connected correlation function
Reduced density matrix is a highly non-local operator of a subregion A by tracing out degree of freedom in the complement of A ρ A = trĀρ (2.22) where ρ is the density matrix of the system. It can also be written as an exponential operator formally (2.14) For the causal diamond A, H A is a type-J OPE block. Therefore it is natural to define a deformed reduced density matrix [11] by replacing modular Hamiltonian with a general OPE block where we still use ρ A to label deformed reduced density matrix. The constant µ is an independent constant. In the "first law of thermodynamics" [9] associated with deformed reduced density matrix, it may be interpreted as a chemical potential which is dual to OPE block Q A . The OPE block Q A can also be a linear superposition of multiple OPE blocks. We don't restrict the OPE block in (2.24) to be type-J. A subtle problem is that the spectrum of Q A is not always non-negative, therefore deformed reduced density matrix may not be well defined in general. However, as we will show below, it is still a useful formal tool to generate CCF.
We define a formal generator of (m)-type CCF through the logarithm of the vacuum expectation value of deformed reduced density matrix, The first few orders are When there are multiple space-like separated regions, one can define a general Y -type CCF with the Young diagram Y = (m 1 , m 2 , · · · ), m 1 ≥ m 2 ≥ · · · ≥ 1. (2.28) The OPE block generated from operator O is an eigenvector of Casimir operator of conformal group with the eigenvalue C = ∆(∆−d)−J(J +d−2). Combining with the boundary behaviour when x 1 → x 2 for OPE block, any (m, 1)-type CCF will be proportional to conformal block where B is another causal diamond, z denotes the cross ratios corresponding to two diamonds A and B. The OPE blocks can be different in (2.29), we write the general result as The coefficient D characterize the large distance behaviour of (m, 1)-type CCF. The references to discuss conformal block are [19,20]. In this work, we just need the diagonal limit of conformal block [21].

Area law and logarithmic behaviour
Motivated by the area law of Rényi entanglement entropy (2.18), or equivalently, area law of (m)-type CCF of modular Hamiltonian (2.20), we are interested in the divergent behaviour of the (m)-type CCF of OPE blocks When the OPE block is modular Hamiltonian, we should reproduce the area law of modular Hamiltonian (2.20). Therefore it is natural to conjecture that (3.1) also obeys area law for general OPE blocks. In the subleading terms, one may also read out cutoff independent information. It turns out that the structure is much more richer, As discussed in the introduction, the maximal power of log R is q. We will call q the degree of the (m)-type CCF (3.1). For example, the degree q is one for the CCF of modular Hamiltonian (2.20) or (2.18) in even dimensions. In this paper, we will restrict the integer m ≤ 3, then the degree may satisfy 0 ≤ q ≤ 2. More explicitly, We will use the degree q to distinguish CCFs (3.1). In the following, we will discuss the logarithmic behaviour in detail.

Logarithmic behaviour
In even dimensions, as (3.3), we could distinguish two classes according to the logarithmic behaviour in the subleading terms.
1. Class I. The degree of the (m)-type CCF is 1. We can write (3.2) more explicitly as where we detail the dependence of primary operator O i for the coefficients γ and p 1 . The upper index e in p 1 indicates that the spacetime dimension is even.
We can choose the region B as the causal diamond of a unit ball Σ B whose radius is R .
The center of the ball is origin. Therefore the unique cross ratio of Σ A and Σ B is 3 (3.7) The conformal block G 4,2 (z) is well defined for 0 < z < 1, which is exactly the case that A and B are space-like separated. Now we move the diamond B to A, then the (m − 1, 1)-type CCF becomes an (m)-type CCF. Roughly speaking The limit B → A is subtle, we first move x A → 0 and then take the limit R → 1, The cross ratio z approaches −∞ by In this limit, the conformal block G 4,2 (z) becomes divergent We have inserted back the radius R in the expression. The leading term is proportional to area of the surface S.
If the coefficient D is finite in (3.13), then (3.12) is exactly the same as (2.20). The equation (3.13) is a typical UV/IR relation for modular Hamiltonian in the sense of [9]. The left hand side is the cutoff independent coefficient as B and A coincides (UV) while D characterizes the leading order behaviour of CCF when two regions are far away to each other (IR). The constant −120 is from conformal block associated with stress tensor in four dimensions. Therefore it is a kinematic term which is totally fixed by conformal symmetry. Note the constant γ is cutoff dependent, therefore it may depend on the energy scale we choose.
The discussion on modular Hamiltonian may extend to other OPE blocks. Interestingly, we find that a conformal block G ∆,J (z) in even dimensions has either degree q = 1 or q = 2 (3.14) where the constant E[∆, J] is determined by quantum numbers of the primary operator. When all the primary operators are conserved currents, ∆ = J + d − 2, we conclude that the (m)-type CCF of type-J OPE blocks may has degree q = 1 with where we have replaced the quantum numbers in E function by the primary operator. Some remarks are shown as follows.
(a) Cyclic identity. For a general (m)-type CCF of type-J OPE block (3.4), we have different ways to uplift (m)-type to (m − 1, 1) type. However, the cutoff independent coefficient should be equal since they characterize the same CCF. For example, m = 3, the coefficients p e 1 should satisfy the following cyclic identity can be read out from the conformal block G ∆,J corresponding to the primary operator O. For conserved currents, we find The constant γ also depends on the way to uplift (m)-type CCF to (m − 1, 1) type.
Since it is cutoff dependent, we don't expect they are equal to each other, 2. Class II. For this class, the degree q = 2, (3.19) Therefore the coefficient p e 2 is cutoff independent while p e 1 is not. As Class I, we can read UV/IR relation The coefficient p e 2 should also satisfy cyclic property as (3.16), We read E[O] from conformal block G ∆,J (z) for non-conserved operators, several examples are shown below There are some constraints on the conformal weight. For scalar primary operator, the unitary bound in four dimensions will constrain ∆ ≥ 1. We notice that the function E[O] becomes divergent when ∆ = 1. On the other hand, when ∆ = 2, the function is zero. Therefore we should be careful with the two special points. Since the physical coefficient is the product of E and D, see (3.20), we cannot make the conclusion that p e 2 is divergent for ∆ = 1 and zero for ∆ = 2 since it also depends on the behaviour of function D near the two special points. When the non-conserved operators have spin J ≥ 1, the unitary bound constrains ∆ > J + 2 (3.23) for CFT 4 . This is the inequality at the second and third line of E [O]. We also note that as ∆ → J + 2, E[O] actually approaches zero. If the function D is finite in this limit, (3.20) implies that p e 2 is zero for ∆ = J + 2. Then p e 1 becomes cutoff independent, which is consistent with the conclusion in Class I.
In odd dimensions, the logarithmic behaviour is a bit different, however, we could still distinguish two classes according to the degree q. It turns out that the maximal degree q is 1 in odd dimensions. We discuss them briefly in the following as it is parallel to even dimensions.
1. Class O. In this class, the degree q = 0, There is no logarithmic divergence in this case. The upper index in p o 0 denotes that the spacetime dimension is odd.
2. Class I'. In this class, the degree q = 1, We can also find the corresponding UV/IR relations. For example, in three dimensions,

A puzzle
According to the discussion in previous subsection, we have examined the area law of (m)-type CCF when all the OPE blocks are the same type. However, we avoid the following CCF is a type-J OPE block. Then we will find a "puzzle" on the logarithmic behaviour. Let's set spacetime dimension to be even and m = 3. According to the method of analytic continuation, we could move either type-O OPE block or type-J OPE block to region B, in the first case, we find where ∆, J are the conformal weight and spin of primary operator O. In the second case, we find where ∆ , J are the conformal weight and spin of primary conserved current J . From analytic continuation of (3.28), we find a (3)-type CCF with degree q = 2, At the same time, from analytic continuation of (3.29), we find a (3)-type CCF with degree However, the cutoff independent structure should be the same while (3.30) contrasts with (3.31) since they predict rather different logarithmic behaviour. We will try to solve this puzzle in Section 5.

Examples
In this section, we will use several examples to check the results in the previous section. We will set spacetime dimension d = 4 from now on.

Class I
Type-J OPE block is We will consider conserved currents with lower spin J ≤ 2.
1. Spin 1 current. We will use two methods to compute the CCF The symbol : : means that one should remove the divergence from the two operators attach to each other [10]. This requires a way of regularization. In the following, we will omit the symbol : :.
(a) We transform the coordinates to spherical coordinates At the first step, we make use of the two point function of spin 1 current where the symmetric tensor is The constant C J defines the normalization of the current J µ . At the time slice t = 0, we have n 0 = 0 and I 00 = η 00 . At the third line, we define the angle θ between the two vectors ω and ω , ω · ω = cos θ. The factor S n is the area of the unit n-sphere S n , . (4.8) The integrand at the fourth line has poles at According to the regularization method in [10], we can just ignore those poles. These poles are from the two currents J µ (x) and J ν (x ) attach to each other. We expect they can be removed 4 . At the fifth line, the integrand is also divergent for r → 1. Therefore we insert a small positive into the upper bound of the integration. The small parameter characterizes the distance to the entanglement surface, therefore it is a UV cutoff. At the last step, we insert back the radius R = 1 to balance the dimension. The term in · · · is an unimportant constant. Now we can extract the cutoff independent coefficient (4.10) (b) Now we can also compute the same CCF (4.2) by uplifting the (2)-type CCF to (1, 1)-type, namely The (1, 1)-type CCF is easy to compute as we just need to fix the leading order coefficient D[J µ , J ν ] as A and B are far apart.
At the first step, we insert back the center of Σ A . The center of Σ B is assumed to be 0. At the second step, we use the assumption that A and B are far away to each other, x A → ∞. At the third step, we rewrite x A in terms of cross ratio (4.13) We read out value (4.14) Then we use the UV/IR relation (3.15) and the function E[J ] = 12 for spin 1 current, As we expect, the coefficients (4.10) and (4.15) are the same. It is also easy to check that the coefficient γ are not the same for the two methods. Since γ has no cutoff independent meaning, it depends on the regularization. One can redefine the cutoff such that they are the same.
2. Spin 2 current. As spin 1 current, we use two method to regularize the integral.
(a) The first method is to regularize the integral directly, we need the two point function for spin 2 current where At the time slice t = 0, we find We read We can also compute (1, 1)-type CCF firstly, Therefore we get Combining with E[T µν ] = −120 for stress tensor and UV/IR relation, Again, we find the cutoff independent coefficients (4.20) and (4.23) are equal. We note that Q A [T µν ] 2 c is related to the universal property of Rényi entanglement entropy by [22]. Transforming to the notation of that paper, we have In the equation above, we just include the cutoff independent term. It is consistent with (4.20) and (4.23). Note this is also an independent check for the method of regularization.
In the integral (4.19), there will be poles when the two stress tensors attach to each other, their effects have been discussed in Appendix A. Since they do not appear in the context of Rényi entanglement entropy, it is fine to remove these effects through our regularization.

(3)-type
We will consider the following two examples.
1. Spin 1-1-2. In this case, the three point function is [23] T and The tensors h 1 µν , h 2 µνσρ , h 3 µνσρ are traceless It is the consequence of traceless condition of stress tensor. The variable . (4.29) The Ward identity from conservation of currents or stress tensor leads to Only two constants are independent. In four dimensions, We only need the component Then , (4.34) where we defined a compact constant C T J J which is a linear combination of a and b. Now we can use three different methods to extract the logarithmic term in CCF of OPE blocks.
(a) We can regularize the integral directly At the second line, we have defined a surface integral I 3 (2, 2, 1) whose details are discussed in the Appendix B.1. Roughly speaking, the integral I 3 (2, 2, 1) has the structure I 3 (2, 2, 1) =f 2,2,1 +g 2,2,1 log r 1 + r 2 |r 1 − r 2 | +h 2,2,1 log where the functionsf ,h,g,ĩ are rational functions of r 1 , r 2 , r 3 . Therefore the definite integral becomes elementary. The integrand at the third line has pole r 1 = 1 therefore we insert a small positive UV cutoff . The logarithmic term is indeed has degree 1, We can compute the following (2, 1)-type CCF firstly, then we can extract the D function by taking the limit x A → ∞, Therefore, using the UV/IR relation we extract the logarithmic term (c) We can also compute another (2, 1)-type CCF, and read out the large x A behaviour Now we can extract the logarithmic term Interestingly, the three results (4.37),(4.40) and (4.43) are equal to each other. This is also the first example that the cyclic identity for p e 1 has been checked.

Spin 2-2-The three point function of stress tensor is
The structure of t µνσραβ (X) could be found in [24]. There are three independent coefficients A, B, C in the three point function of stress tensor. In this paper, we just need the component We can use two different methods to extract the logarithmic term.
(a) The first method is to compute the logarithmic term directly, (4.47) As previous example, we define the integral I 3 (2, 2, 2) in Appendix B.1. We also insert a small in the integral of r 1 at the last step. From the result, we read The second method is to use UV/IR relation. We first read the coefficient D in the large x A limit, The cutoff independent term is the same for different methods. We also check that the result can be mapped to the second derivative of Rényi entanglement entropy [25],

Class II
We change the coordinates to The metric of Minkowski spacetime becomes The new metric (4.54) covers the diamond A twice, then Then the Type-O OPE becomes The measure (4.57) The subscript J is used to label the spin J in the measure. The dimension is understood as d = 4 in this expression. The region D 2 is a square with −1 < ζ,ζ < 1.
(4.58) Some integrals used in the following has been discussed in Appendix B.2.
The angular between ω and ω is denoted as θ. The regularization of (4.59) is not easy for general ∆. However, we can compute several examples. For ∆ = 4, At the first step, we integrate the angular part. At the second step, we integrate ζ,ζ part, the integrand becomes singular for ζ = ±1 andζ = ±1, (4.63) therefore we insert a small UV cutoff into the integral. Then the final result obeys area law and there is a logarithmic term with degree 2. The · · · term includes a logarithmic term with power 1 and a constant. Therefore, the cutoff independent information is The method can be extended to other even conformal weight, for example, Now we'd like to use UV/IR relation to obtain this result. (4.67) Note at the second step, we use the approximation that A and B are far apart and only extract the leading order behaviour. At the third step, H 0 is defined in Appendix B.2. Therefore we can use UV/IR relation (4.70) We parameterize where the coefficients are e = ζ +ζ, f = −(ζ +ζ ), The coefficients a, b can be found in (4.61). When A and B are far away to each other, the leading term is Note at the second step, we define the unit vector in the direction of x A asˆ x A . After some efforts, we find Therefore We could check this formula (4.75) by computing for special values of ∆. For example, The cutoff independent term matches with the general formulae (4.75).
3. Spin 2. Like previous example, we find From the leading behaviour when A and B are far away, (4.79) The cutoff independent term is (4.80) The formula could be checked for special values of ∆. For example,
1. For general conformal weights, we have We don't find an obvious way to prove the cyclic property (3.21) from this expression. It would be quite interesting to check the cyclic property for (4.87).

Discussion
In this section, we will focus on the puzzle which is mentioned in Section 3.2. The incompatibility between (3.31) and (3.30) is from the (m)-type CCF (3.27). We are trying to tackle this puzzle using two examples, this will partly solve the puzzle.
1. The first CCF we'd like to discuss is where T µν is the stress tensor and O is a spinless primary operator. The three point function [23] T µν ( is fixed up to a theory dependent coefficient a. We just need the 00 component, it is easy to find The region ∆ > 2 is from the convergence of the integral. We obtain As we discussed, we can also compute another CCF We have defined The integral is not easy, therefore we just compute several examples. Let's set ∆ = 4. Interestingly, we find a logarithmic divergent coefficient D, where · · · is a cutoff dependent constant. Now if we take the limit B → A, the conformal block of stress tensor will contribute one logarithmic divergence as usual. However, since the coefficient D also has a logarithmic divergence with degree one, there will be a logarithmic term with degree 2 in the final result, we get We use a subscript "log" to denote the logarithmic term in D coefficient. This is consistent with (5.5) for ∆ = 4. Some interesting new properties appear now.
(b) The inconsistency between (3.30) and (3.31) disappears for the special example. The puzzle in Section 3.2 is superficially. The sacrifice is that CCF of (m − 1, 1)-type is not always convergent in higher dimensions. The convergence of (m − 1, 1)-type CCF has been checked for several examples in CFT 2 , where the OPE blocks are always type-J [11]. In this example, we have both type-J and type-O OPEs in the CCF.
(c) Though the (2, 1)-type CCF is not convergent in this example, we find that the coefficient D still contains cutoff independent information. The UV/IR relation (3.20) is still valid even for divergent D coefficient. Due to the divergence of D, we expect that (m)-type OPE block Q A [O] · · · Q A [J ] c belongs to Class II.
(d) The logarithmic divergence of D in (5.8) is the key to solve the puzzle. However, it also leads to the following leading behaviour We use cutoff , to distinguish the divergent behaviour since their origin are quite different. The R 2 2 term is from analytic continuation of conformal block for stress tensor, the log R term is from D coefficient, this breaks the area law using another (2, 1)-type CCF, The leading behaviour (5.12) looks different compare to (5.13). The mismatch can be cured by mapping the cutoff , to . However, a simple dilation transformation → Λ cannot cure this problem. This causes the problem whether the area law is broken or not. We don't have a clear answer to this problem yet. In the following example, we also meet similar problem while the cutoff independent coefficient p e 2 still matches with each other. We believe this is not a coincidence. The method of analytic continuation is good enough to obtain p e 2 , though it may be not true to get the correct leading order behaviour. If we insist that the area law is not broken, the analytic continuation (5.12) should be forbidden while (5.13) is allowed. We hope to return to this problem in the future.

The second CCF we'd like to discuss is
where T µν , T σρ are stress tensor and O is a spin 0 primary operator with conformal weight ∆. The three point function is [23] T µν ( The tensors h 1 µν , h 2 µνσρ , h 3 µνσρ can be found in (4.27). The conservation of stress tensor leads to two linear relations between a, b, c There is an overall constant for the three point function T T O . In four dimensions, we find We need the component where the function ϕ is Note the time component of x 1 and x 2 is 0 for (5.19). Therefore We could read (5.23) The integral is finite for general ∆ > 3. After some efforts, we find We can also compute the following CCF, The D constant is where the function y = (ζ −ζ) 2 4(r 2 2 + ζζ − r 2 (ζ +ζ) cos θ) . (5.27) The integral (5.26) is not easy, we choose ∆ = 4, then It is divergent, therefore the cutoff independent coefficient is p e 2 , The result is consistent with (5.24) for ∆ = 4. We can also calculate other examples, 46800 log R + · · · , ∆ = 6, π 3 a 403200 log R + · · · , ∆ = 12, · · · . (5.30) They are all divergent with a logarithmic term. Then The result (5.31) matches with (5.24), correspondingly.

Conclusion
In this paper, we calculate the divergent behaviour of (m)-type CCF of OPE blocks. Due to the complexity of the integrals, we only tackle the case for m = 2 and 3. We classify OPE blocks to type-J and type-O, according to the primary operator in the definition of OPE block. The logarithmic behaviour has been discussed for varies (m)-type CCFs. In even/odd dimensions, we could identify two classes of (m)-type CCF according to the degree q.
We establish a formula which is to relate (m)-type CCF to (m − 1, 1)-type CCF, we call it UV/IR relation. Schematically, it has the simple form where p is the cutoff independent coefficient in (m)-type CCF. The coefficient D is the coefficient before conformal block for (m − 1, 1)-type CCF. The coefficients p and D encode useful information of the CFT. On the other hand, the coefficient E is completely fixed by conformal symmetry, which is a kinematic term. We check the UV/IR relation (6.1) for various examples, in all cases, the cyclic property of p is always valid, see (3.16) or (3.21).
When the OPE blocks belong to different types in (m)-type CCF, the logarithmic behaviour is incompatible superficially using different continuation method. However, we could solve this puzzle partly in two explicit examples, the sacrifice is that one coefficient D should also have logarithmic divergence. Therefore (m − 1, 1)-type CCF is not always convergent. This is a generalization of the conclusion in [11] where the author considered (m − 1, 1)-type CCF of type-J OPE blocks in two dimensions. However, we could still obtain cutoff independent coefficient from the logarithmic term in D coefficient. The UV/IR relation is still valid after replacing D by its cutoff independent part.
In all the examples we compute in this work, we always find q ≤ 2. Since we just consider the cases m ≤ 3, it is not clear whether q could be larger than 2 or not for general m. If the coefficient D is always finite, then the degree q must be less than or equal to 2. However, since we find a (2, 1)-type CCF which shows logarithmic behaviour, it would be quite interesting to explore higher (m)-type CCFs.
Higher (m)-type CCF of OPE blocks is also very important to understand the deformed reduced density matrix ρ A = e −µQ A [O] , a formal exponential non-local operator defined in [11]. This operator is similar to "Wilson loop" [26,27]  A naive continuation from conformal block shows that this quantity (6.2) also obeys area law [9]. Since conformal block is fixed by conformal invariance, the area law of (6.2) is protected by conformal symmetry. We'd like to study this point in the future.

A Singularity
When two operators attach to each other, there could be singularities. In this Appendix, we will show that these singularities does not affect the cutoff independent coefficient using explicit examples. In (4.4), at the fourth line, the singularities are at r = r , we'd like to examine the singular behaviour carefully. The typical integral is I 1 = 1 0 dr 1 0 dr r 2 r 2 (r 2 + r 2 ) (r − r ) 4 (r + r ) 4 . (A.1) We could separate the singularity by replacing the integral by The integral I 1 is the one used at the fifth line of (4.4). The first term on the right hand side of (A.2) is the effect of the singularity, it has been removed from the regularization method in the context. It is easy to find there is no extra logarithmic term. Therefore we conclude that the terms that have been removed do not affect the cutoff indepdendent coefficient. In the same way, the singularity in (4.19) is also r = r , the relevant integral is The typical integrals used in this paper is where x i = r i ω i . The integrand only depends on the angle between vectors ω i and ω j . The constants α ij are assumed to be real. If some of them are positive, then the integral has poles. We assume r 1 > r 2 > r 3 to avoid the pole and this doesn't lose any information of the integral. For n = 2, the integral is elementary. In this paper, we need the result for n = 3. We expand the function | x i − x j | −2α ij in terms of Legendre function of the first kind f (r i , r j , α ij )P (cos ψ ij ) (B.2) with cos ψ ij = cos θ i cos θ j + sin θ i sin θ j cos(φ i − φ j ). (B. 3) The function f is f (r i , r j , α ij ) = 2 + 1 2 At the first line, we used the orthogonal relation of Legendre function of the first kind At the second step, we used the integral formula [28] 1 −1 The parameter z ij = While the infinite sum of the triple product of Legendre Polynomials of first kind has been found long time ago [29], we don't find a close formula for general α 12 , α 13 , α 23 . Fortunately, we just need the result for special value of α 12 , α 13 , α 23 . With some efforts, the general structure of J is as follows for positive integer α 12 , α 13 , α 23 J(α 12 , α 13 , α 23 ) = f α 12 ,α 13 ,α 23 +g α 12 ,α 13 ,α 23 log r 1 + r 2 |r 1 − r 2 | +h α 12 ,α 13 ,α 23 log r 1 + r 3 |r 1 − r 3 | +i α 12 ,α 13 ,α 23 log r 2 + r 3 |r 2 − r 3 | , (B.12) 5 In general, the three variables z 12 , z 13 , z 23 are not related to each other. In our case, they are constrainted by the identity Selberg integral is defined as [30,31] Sel n (α, β, γ) = Therefore .