Area law and OPE blocks in CFT

Jiang Long111e-mail<EMAIL_ADDRESS>

School of Physics, Huazhong University of Science and Technology,
Wuhan, Hubei 430074, China

This is an introduction to the relationship between area law and OPE blocks in
conformal field theory.

###### Contents

  1. 1 Introduction
  2. 2 Setup
    1. 2.1 Area law
    2. 2.2 OPE block
    3. 2.3 Modular Hamiltonian and area law
    4. 2.4 Deformed reduced density matrix and connected correlation function
  3. 3 Area law
    1. 3.1 Continuation
    2. 3.2 Kinematic information
    3. 3.3 UV/IR relation
    4. 3.4 Discussion
  4. 4 Generalizations
  5. 5 Summary and outlook

##  1 Introduction

This report consists a summary of our recent progress on the relationship
between area law and OPE blocks. Area law has been a continuous topic in
physics. The prototype of area law dates back to black hole physics in general
relativity. The unusual property that the thermal entropy of a black hole is
proportional to the event horizon of the black hole [1, 2] has stimulated
varies modern idea of theoretical physics, including the famous holographic
principle.

OPE block [3, 4], on the other hand, is a relatively new topic in conformal
field theory, though it has been noticed at the early stages of conformal
field theory[5, 6]. The operator product expansion of two primary operators is
equivalent to a summation of OPE blocks with corresponding three point
function coefficients. It is a smeared operator which is generated from the
so-called (quasi-)primary operator.

Modular Hamiltonian, the logarithm of the reduced density matrix [7], plays a
central role in the context of geometric entanglement entropy [8, 9, 10, 11].
Entanglement entropy is a von Neumann entropy generated from reduced density
matrix of a subregion of spacetime. An intriguing fact of entanglement entropy
is that it obeys area law in the leading order, though one should introduce a
cutoff to secure the divergent behaviour. Its connection to gravity has been
established by the work of Ryu and Takayanagi [12], 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.

Modular Hamiltonian is a special OPE block generated by the stress energy-
momentum tensor for a ball region. This leads to the conjecture that OPE block
may be related to area law as modular Hamiltonian. Indeed, in a series of
papers [13, 14, 16, 17], we have shown that the quantity which satisfies area
law is the type-$(m)$ connected correlation function (CCF). More explicitly,
the leading term of the type-$(m)$ CCF is proportional to the area of the
boundary of the ball. In the subleading terms, we find a logarithmic
divergence with degree $q$. The degree $q$ is a natural number which is no
larger than 2 in general dimensions. The coefficient $p_{q}$ for the
logarithmic term with degree $q$ is cutoff independent. We establish a
relationship between $p_{q}$ and the type-$(m-1,1)$ CCF of OPE blocks for two
balls which are far away to each other. The coefficient $p_{q}$ obeys a cyclic
identity which is independent of the order of the operators.

This paper is organised as follows. In section 2, we will introduce some basic
concepts and conventions used in this paper. Section 3 is devoted to the study
of the new area law which is related to the OPE blocks. Varies generalizations
have been given in section 4. We conclude in section 5 with a number of
general open problems that deserve, in our opinion, more work.

##  2 Setup

In this section, we introduce some basic concepts and conventions used in this
paper.

###  2.1 Area law

In any continues quantum field theory(QFT), physical degrees exist at each
point $(t,x^{i}),i=1,\cdots,d-1$ of spacetime $M$. At each time slice
$t=t_{0}$, the data on the Cauchy surface $\Sigma$ determines the evolution of
the fields. One can divide the surface $\Sigma$ into a spacelike subregion $A$
and its complement $\bar{A}$, $\Sigma=A\cup\bar{A}$. The boundary $\partial A$
is a codimension 2 surface whose area is $\mathcal{A}$. The causal development
of A is denoted by $\mathcal{D}(A)$. The physical data on $A$ can only
determine the evolution of the fields in $\mathcal{D}(A)$. The causal
development $\mathcal{D}(A)$ is an independent subsystem of the original
spacetime $M$. Operators in this subsystem are collected to form an algebra
$\bm{a}(A)$. Assume the QFT in the spacetime $M$ is described by a density
matrix $\rho$, then by integrating out the degree of freedom in the complement
of $\bar{A}$, one achieves a reduced density matrix $\rho_{A}$

$\rho_{A}=\text{tr}_{\bar{A}}\rho.$ (2.1)

The reduced density matrix $\rho_{A}$ is a special operator in $\bm{a}(A)$
since it describes the subsystem $\mathcal{D}(A)$ effectively. A general
quantity $\mathcal{Q}(A)$ in $\bm{a}(A)$ is said to obey area law if its
leading term is proportional to the area of the boundary $\partial A$,

$\mathcal{Q}(A)\propto\mathcal{A}+\cdots.$ (2.2)

One typical example is the black hole entropy in Einstein gravity. The black
hole entropy is proportional to the area of its event horizion,

$S_{bh}=\frac{\mathcal{A}}{4G}$ (2.3)

where $G$ is the Newton constant. At the loop level, black hole entropy
requires logarithmic corrections [19, 18, 20, 21, 22, 23]. Usually, the
logarithmic correction is in the form $C\log\mathcal{A}$ where the constant
$C$ may encode useful information of the black hole.

Sometimes the area law is divergent, one typical example is the geometric
entanglement entropy

$S_{A}=-\text{tr}_{A}\rho_{A}\log\rho_{A}.$ (2.4)

In this case, one should insert a cutoff $\epsilon>0$,

$S_{A}=\gamma\frac{\mathcal{A}}{\epsilon^{d-2}}+\cdots.$ (2.5)

In the subleading terms, there may be a logarithmic term whose coefficient is
independent of the cutoff,

$S_{A}=\gamma\frac{R^{d-2}}{\epsilon^{d-2}}+\cdots+p\log\frac{R}{\epsilon}+\cdots$
(2.6)

where the parameter $R$ is the characteristic length of the region $A$.

In this report, we will present a quantity $\mathcal{Q}(A)$ which has a
slightly different logarithmic behaviour

$\mathcal{Q}(A)=\gamma\frac{R^{d-2}}{\epsilon^{d-2}}+\cdots+p_{q}\log^{q}\frac{R}{\epsilon}+\cdots.$
(2.7)

The maximum power $q$ of the logarithmic terms is a natural number. We will
call it the degree of the quantity $\mathcal{Q}(A)$. The coefficient $p_{q}$
is cutoff independent and encodes useful information of the theory. In the
special case that the subregion $A$ is a ball, $R$ could be chosen as its
radius. The subregion $A$ and its causal development $\mathcal{D}(A)$ are in
one-to-one correspondence, we will not distinguish them in the following.

In two dimensions, there is no polynomial terms of $\frac{R}{\epsilon}$, the
modified “area law” is

$\mathcal{Q}(A)=p_{q}\log^{q}\frac{R}{\epsilon}+\cdots.$ (2.8)

###  2.2 OPE block

In any d dimensional CFT, operators are classified into (quasi-)primary
operators $\mathcal{O}$ and their descendants
$\partial_{\mu}\partial_{\nu}\cdots\mathcal{O}$. A general primary operator is
characterized by two quantum numbers, conformal weight $\Delta$ and $so(d-1)$
spin $J_{ij}$ with magnitude $J$. Under a global conformal transformation
$x\to x^{\prime}$, a primary spin 0 operator transforms as

$\mathcal{O}(x)\to|\frac{\partial x^{\prime}}{\partial
x}|^{-\Delta/d}\mathcal{O}(x).$ (2.9)

where $|\partial x^{\prime}/\partial x|$ is the Jacobian of the conformal
transformation of the coordinates, $\Delta$ is the conformal weight of the
primary operator. Operator product expansion(OPE) of two separated primary
scalar operators $\mathcal{O}_{i}(x_{1})\mathcal{O}_{j}(x_{2})$ is to expand
their product in a local orthogonal and complete basis around a suitable point

$\mathcal{O}_{i}(x_{1})\mathcal{O}_{j}(x_{2})=\sum_{k}C_{ijk}|x_{12}|^{\Delta_{k}-\Delta_{i}-\Delta_{j}}(\mathcal{O}_{k}(x_{2})+\cdots),$
(2.10)

where $\cdots$ are descendants of the primary operator $\mathcal{O}_{k}$. Its
form is fixed by global conformal symmetry, therefore it just contains
kinematic information of the CFT. The summation is over all possible primary
operators of the CFT. Here we expand the product around the point $x_{2}$. The
distance of any two points $x_{i},x_{j}$ is written as $|x_{ij}|$. The
constant $C_{ijk}$ is called the OPE coefficient which is related to the three
point function of primary operators

$\langle\mathcal{O}_{i}(x_{1})\mathcal{O}_{j}(x_{2})\mathcal{O}_{k}(x_{3})\rangle=\frac{C_{ijk}}{|x_{12}|^{\Delta_{12,3}}|x_{23}|^{\Delta_{23,1}}|x_{13}|^{\Delta_{13,2}}},\quad\Delta_{ij,k}=\Delta_{i}+\Delta_{j}-\Delta_{k}.$
(2.11)

They are the only dynamical parameters in the CFT. The constants
$\Delta_{i},\Delta_{j},\Delta_{k}$ are conformal weights of the corresponding
primary operators. By collecting all kinematic terms in the summation, we can
rewrite the OPE (2.10) as

$\mathcal{O}_{i}(x_{1})\mathcal{O}_{j}(x_{2})=|x_{12}|^{-\Delta_{i}-\Delta_{j}}\sum_{k}C_{ijk}Q^{ij}_{k}(x_{1},x_{2}).$
(2.12)

The objects $Q^{ij}_{k}(x_{1},x_{2})$ are called OPE blocks [5, 6, 3]. They
are non-local operators in the CFT and depend on the position $x_{1}$ and
$x_{2}$ of the external operators. The upper index $i$ and $j$ show that it
also depends on the quantum number of the external operators $\mathcal{O}_{i}$
and $\mathcal{O}_{j}$. It is easy to see that OPE block has dimension zero.
Under a global conformal transformation $x\to x^{\prime}$, an OPE block
$Q^{ij}_{k}(x_{1},x_{2})$ will transform as

$Q^{ij}_{k}(x_{1},x_{2})\to
f(x_{1}^{\prime},x_{2}^{\prime})Q^{ij}_{k}(x_{1}^{\prime},x_{2}^{\prime}).$
(2.13)

The explicit form of $f(x_{1}^{\prime},x_{2}^{\prime})$ is not important in
this work. When the two external operators are the same, we have
$f(x_{1}^{\prime},x_{2}^{\prime})=1$ and OPE block will be invariant under the
global conformal transformation. One can also show that the OPE block is
independent of the external operator in this special case. Due to this reason,
we relabel such kind of OPE block as

$Q_{A}[\mathcal{O}_{k}]=Q_{k}^{ii}(x_{1},x_{2}).$ (2.14)

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 the
square bracket reflects the fact that OPE block is generated by a primary
operator $\mathcal{O}_{k}$. We omit the information of $i$ since the OPE block
is insensitive to the external operators in this case. We will classify the
primary operators $\mathcal{O}_{k}$ into conserved currents $\mathcal{J}$ and
non-conserved operators $\mathcal{O}$. A general symmetric traceless primary
operator obeys the following unitary bound [24]

$\displaystyle\left\\{\begin{array}[]{l}\Delta\geq J+d-2,\quad J\geq 1,\\\
\Delta\geq\frac{d-2}{2},\quad J=0.\end{array}\right.$ (2.16)

A conserved current $\mathcal{J}$ with spin $J(J\geq 1)$ will satisfy
$\Delta=J+d-2$. All other primary operators are non-conserved operators.
Correspondingly, the OPE block (2.14) generated by a conserved current
$\mathcal{J}$ will be called a type-J OPE block. On the other hand, the OPE
block (2.14) generated by a non-conserved operator $\mathcal{O}$ will be
called a type-O OPE block.

When two operators are time-like separated, the region $A$ is a causal
diamond. The two operators are at the sharp corner of the diamond $A$. We can
use the conformal transformation to fix

$x_{1}=(1,\vec{x}_{0}),\quad x_{2}=(-1,\vec{x}_{0}),$ (2.17)

then the causal diamond $A$ intersects $t=0$ slice with a unit ball which we
will also denote it as $A$

$A=\\{(0,\vec{x})|(\vec{x}-\vec{x}_{0})^{2}\leq 1\\}.$ (2.18)

The center of the ball is $\vec{x}_{0}$. The boundary of the ball $A$ is a
unit sphere $\partial A$. In the context of geometric entanglement entropy,
the surface $\partial A$ is an entanglement surface which separates the ball
$A$ and its complement. The leading term of entanglement entropy is
proportional to the area of the surface $\partial A$ in general higher
dimensions ($d>2$). In two dimensions, the entanglement entropy is
logarithmically divergent with the logarithmic degree $q=1$. There is a
conformal Killing vector $K$ which preserves the diamond $A$,

$K^{\mu}=\frac{1}{2}(1-(\vec{x}-\vec{x}_{A})^{2}-t^{2},-2t\vec{x}).$ (2.19)

The conformal Killing vector $K$ is null on the boundary of the diamond $A$.
It generates a modular flow of the diamond $A$. A type-O OPE block corresponds
to point pair (2.17) or unit ball $A$ (2.18) is [4]

$Q_{A}[\mathcal{O}_{\mu_{1}\cdots\mu_{J}}]=c_{\mathcal{O}_{\mu_{1}\cdots\mu_{J}}}\int_{\mathcal{D}(A)}d^{d}xK^{\mu_{1}}\cdots
K^{\mu_{J}}|K|^{\Delta-d-J}\mathcal{O}_{\mu_{1}\cdots\mu_{J}},$ (2.20)

where the primary operator $\mathcal{O}_{\mu_{1}\cdots\mu_{J}}$ is non-
conserved

$\partial^{\mu_{1}}\mathcal{O}_{\mu_{1}\cdots\mu_{J}}\not=0.$ (2.21)

It has dimension $\Delta$ and spin $J$. When the operator is a conserved
current

$\partial^{\mu_{1}}\mathcal{J}_{\mu_{1}\cdots\mu_{J}}=0,$ (2.22)

the corresponding type-J OPE block is

$Q_{A}[\mathcal{J}_{\mu_{1}\cdots\mu_{J}}]=c_{\mathcal{J}_{\mu_{1}\cdots\mu_{J}}}\int_{A}d^{d-1}\vec{x}(K^{0})^{J-1}\mathcal{J}_{0\cdots
0}.$ (2.23)

It can be obtained from (2.20) by using the conservation law (2.22) and
reducing it to a lower $d-1$ dimensional integral. The coefficient
$c_{\mathcal{J}_{\mu_{1}\cdots\mu_{J}}}$ is also redefined at the same time.
In (2.20) and (2.23), the coefficients
$c_{\mathcal{O}_{\mu_{1}\cdots\mu_{J}}}$ and
$c_{\mathcal{J}_{\mu_{1}\cdots\mu_{J}}}$ are free parameters, we set them to
be 1.

###  2.3 Modular Hamiltonian and area law

A very special type-J OPE block is the modular Hamiltonian [7, 25] of the ball
$A$,

$H_{A}=2\pi\int_{A}d^{d-1}\vec{x}K^{0}T_{00}=2\pi\int_{A}d^{d-1}\vec{x}\frac{1-(\vec{x}-\vec{x}_{0})^{2}}{2}T_{00}(0,\vec{x}).$
(2.24)

Modular Hamiltonian is the logarithm of the reduced density matrix $\rho_{A}$

$H_{A}=-\log\rho_{A}.$ (2.25)

It plays a central role in the context of entanglement entropy,

$S_{A}=-\text{tr}_{A}\rho_{A}\log\rho_{A}=\text{tr}_{A}e^{-H_{A}}H_{A}.$
(2.26)

More generally, Rényi entanglement entropy

$S_{A}^{(n)}=\frac{1}{1-n}\log\text{tr}_{A}\rho_{A}^{n}$ (2.27)

has been shown to satisfy an area law generally

$S_{A}^{(n)}=\gamma\frac{\mathcal{A}}{\epsilon^{d-2}}+\cdots,$ (2.28)

where $\mathcal{A}$ is the area of the entanglement surface $\partial A$ and
$\epsilon$ is a UV cutoff. The constant $\gamma$ is cutoff dependent. The
subleading terms $\cdots$ contain a logarithmic term with degree $q=1$ in even
dimensions

$S_{A}^{(n)}=\gamma\frac{\mathcal{A}}{\epsilon^{d-2}}+\cdots+p_{1}(n)\log\frac{R}{\epsilon}+\cdots,$
(2.29)

where we have inserted back the radius $R=1$. The area $\mathcal{A}$ is
related to the radius $R$ through the power law

$\mathcal{A}\sim R^{d-2}.$ (2.30)

The coefficient $p_{1}(n)$ encodes useful information of the CFT. The relation
between modular Hamiltonian and area law motivates the conjecture that OPE
block maybe related to area law in a suitable way. We will give the framework
to discuss this problem in the following subsection.

###  2.4 Deformed reduced density matrix and connected correlation function

Given a primary operator $\mathcal{O}$ in a ball A, one can always define a
corresponding OPE block $Q_{A}[\mathcal{O}]$. We construct an exponential
operator formally [14]

$\rho_{A}=e^{-\mu Q_{A}}$ (2.31)

which is still in the subregion $A$. The constant $\mu$ is free. Operators of
the form (2.31) is called deformed reduced density matrix. Note we use the
same symbol $\rho_{A}$ to label deformed reduced density matrix. Recall that
the modular Hamiltonian is a special OPE block, if one replaces the OPE block
by the modular Hamiltonian (2.31) and set $\mu=1$, the deformed reduced
density matrix becomes the reduced density matrix exactly. We can relax the
definition, namely, $Q_{A}$ in (2.31) could be a linear superposition of
several OPE blocks. Note our definition of deformed reduced density matrix is
a direction extension of the generalized reduced density matrix in the context
of the so-called charged Rényi entropy [15]. In that work, $Q_{A}$ is a charge
which is generated by a $U(1)$ current. The corresponding charged Rényi
entropy is holographically dual to the thermal entropy of a charged black hole
with hyperbolic horizon. However, in our definition, $Q_{A}$ is just a general
OPE block or their linear superposition. As a naive generalization of Rényi
entanglement entropy, we construct the logarithm of the vacuum expectation
value of the deformed reduced density matrix,

$T_{A}(\mu)=\log\langle\rho_{A}\rangle=\log\langle e^{-\mu Q_{A}}\rangle.$
(2.32)

When $Q_{A}$ is modular Hamiltonian, the above quantity is related to the
Rényi entropy for the vacuum state.

However, a direct computation of $T_{A}(\mu)$ is hard in general. A much more
severe problem is that OPE block has no lower bound in general, therefore the
definition is not valid for general OPE blocks. To solve this problem, we
observe that $T_{A}(\mu)$ could be expanded for small $\mu$,

$T_{A}(\mu)=\sum_{m=1}^{\infty}\frac{(-\mu)^{m}}{m!}\langle
Q_{A}^{m}\rangle_{c}.$ (2.33)

The Tayler expansion coefficient

$\langle
Q_{A}^{m}\rangle_{c}=(-1)^{m}\frac{\partial^{m}}{\partial\mu^{m}}T_{A}(\mu)|_{\mu\to
0}$ (2.34)

is called Type-(m) connected correlation function (CCF) of the OPE block
$Q_{A}$. For each definite $m$, one can always calculate the corresponding CCF
without knowing $T_{A}(\mu)$. The first few CCFs are

$\displaystyle\langle Q_{A}^{2}\rangle_{c}$ $\displaystyle=$
$\displaystyle\langle Q_{A}^{2}\rangle-\langle Q_{A}\rangle^{2},$
$\displaystyle\langle Q_{A}^{3}\rangle_{c}$ $\displaystyle=$
$\displaystyle\langle Q_{A}^{3}\rangle-3\langle Q_{A}^{2}\rangle\langle
Q_{A}\rangle+2\langle Q_{A}\rangle^{3}.$ (2.35)

Using CCF, there is no issue of lower bound of the OPE block. As an
application of the concept of CCF, we choose the OPE block as the modular
Hamiltonian, then it is easy to show that CCF of modular Hamiltonian $H_{A}$
satisfies area law with logarithmic degree $q=1$ in even dimensions,

$\langle
H_{A}^{m}\rangle_{c}=\tilde{\gamma}\frac{\mathcal{A}}{\epsilon^{d-2}}+\cdots+\tilde{p}_{1}^{(m)}\log\frac{R}{\epsilon}+\cdots,\quad
m\geq 1.$ (2.36)

The coefficient $\tilde{p}_{1}^{(m)}$ is determined from $p_{1}(n)$ by

$\tilde{p}_{1}^{(m)}=(-1)^{m}\partial_{n}^{m}(1-n)p_{1}(n)|_{n\to 1}.$ (2.37)

There could be multiple spacelike-separated balls $A_{1},A_{2},\cdots$, each
region has associate OPE block $Q_{A_{i}}$. We insert $m_{i}$ OPE blocks into
region $A_{i}$, then we can define the corresponding type-Y CCF

$\langle Q_{A_{1}}^{m_{1}}Q_{A_{2}}^{m_{2}}\cdots\rangle_{c}$ (2.38)

where the Young diagram $Y$ is

$Y=(m_{1},m_{2},\cdots),\quad m_{1}\geq m_{2}\geq\cdots\geq 1.$ (2.39)

The generator of all type-Y CCFs is

$T_{\cup A_{i}}(\mu_{1},\mu_{2},\cdots)=\log\frac{\langle
e^{-\sum_{i}\mu_{i}Q_{A_{i}}}\rangle}{\prod_{i}\langle
e^{-\mu_{i}Q_{A_{i}}}\rangle}.$ (2.40)

When there are only two balls $A$ and $B$, the generator is

$T_{A\cup B}(\mu_{1},\mu_{2})=\log\frac{\langle
e^{-\mu_{1}Q_{A}-\mu_{2}Q_{B}}\rangle}{\langle e^{-\mu_{1}Q_{A}}\rangle\langle
e^{-\mu_{2}Q_{B}}\rangle}=\sum_{m_{1}\geq 1,m_{2}\geq
1}\frac{(-1)^{m_{1}+m_{2}}\mu_{1}^{m_{1}}\mu_{2}^{m_{2}}}{m_{1}!m_{2}!}\langle
Q_{A}^{m_{1}}Q_{B}^{m_{2}}\rangle_{c}.$ (2.41)

We parameterize $A$ and $B$ as

$A=\\{(0,\vec{x})|(\vec{x}-\vec{x}_{0})^{2}\leq 1\\},\quad
B=\\{(0,\vec{x})|\vec{x}\leq R^{\prime 2}\\}.$ (2.42)

There is only one cross ratio

$\xi=\frac{4R^{\prime}}{x_{0}^{2}-(1-R^{\prime})^{2}}.$ (2.43)

When the two regions $A$ and $B$ are spacelike-separated,
$|x_{0}|>1+R^{\prime}$, the cross ratio is between 0 and 1,

$0<\xi<1.$ (2.44)

In some cases, it is more convenient to use an equivalent cross ratio

$\eta=\frac{\xi}{1-\xi}=\frac{4R^{\prime}}{x_{0}^{2}-(1+R^{\prime})^{2}}.$
(2.45)

For spacelike-separated regions $A$ and $B$, the range of the cross ratio
$\eta$ is

$0<\eta<\infty.$ (2.46)

Since the OPE block $Q_{A}[\mathcal{O}]$ is invariant under conformal
transformation, any type-$(m_{1},m_{2})$ CCF should be a function of cross
ratio $\xi$ or $\eta$. Actually the OPE block is an eigenvector of the
conformal Casimir

$[L^{2},Q_{A}[\mathcal{O}]]=C_{\Delta,J}Q_{A}[\mathcal{O}]$ (2.47)

where $L^{2}$ is the Casimir operator of the global conformal group. The
eigenvalue $C_{\Delta,J}$ is

$C_{\Delta,J}=-\Delta(\Delta-d)-J(J+d-2).$ (2.48)

Therefore, any type-$(m-1,1)$ CCF should be a conformal block

$\langle Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{m-1}]Q_{B}[\mathcal{O}_{m}]\rangle_{c}=D^{(d)}[\mathcal{O}_{1},\cdots,\mathcal{O}_{m}]G^{(d)}_{\Delta_{m},J_{m}}(\xi).$
(2.49)

The subscript $\Delta_{m},J_{m}$ are the conformal weight and spin of the
primary operator $\mathcal{O}_{m}$. The index $(d)$ is used to label the
dimension of spacetime. The conformal block can be constructed explicitly in
even dimensions [26, 27]. In this paper, we just need the diagonal limit of
conformal block[28]. Any type-$(m_{1},m_{2})$ CCF with $m_{1}\geq m_{2}\geq 2$
is not a conformal block .

##  3 Area law

We conjecture that the type-$(m)$ CCF of OPE blocks obeys the following area
law

$\langle Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{m}]\rangle_{c}=\gamma\frac{R^{d-2}}{\epsilon^{d-2}}+\cdots+p_{q}\log^{q}\frac{R}{\epsilon}+\cdots.$
(3.1)

The leading term is proportional to the area of the boundary $\partial A$. We
inserted the radius $R=1$ into the formula to balance the dimension. The small
positive constant $\epsilon$ is the UV cutoff which is roughly the distance
from the cutoff to the boundary $\partial A$. The constant $\gamma$ depends on
the choice of the cutoff and the method of regularization, we will not be
interested in its explicit value. The $\cdots$ terms are subleading and cutoff
dependent. Therefore we omit their forms. The degree $q$ characterizes the
maximal power of the logarithmic terms. The coefficient $p_{q}$ is invariant
under the rescaling of the cutoff, therefore it encodes detail universal
information of the theory. When all the OPE blocks are equal to the modular
Hamiltonian, the degree $q=1$ for even dimensions according to (2.36).
However, as we will see, $q$ is not necessary equal to 1 in general. To
distinguish different type-$(m)$ CCFs in different dimensions, we write the
area law (3.1) more explicitly as

$\langle Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{m}]\rangle_{c}=\gamma[\mathcal{O}_{1},\cdots,\mathcal{O}_{m}]\frac{R^{d-2}}{\epsilon^{d-2}}+\cdots+p^{(d)}_{q}[\mathcal{O}_{1},\cdots,\mathcal{O}_{m}]\log^{q}\frac{R}{\epsilon}+\cdots.$
(3.2)

###  3.1 Continuation

The two formulas (2.49) and (3.2) are actually related to each other through
an analytic continuation. We use the example of the two dimensional modular
Hamiltonian to illustrate this relation. For any CFT2, the modular Hamiltonian
can be decomposed into the holomorphic and anti-holomorphic part, we focus on
the holomorphic part

$H_{A}=-\int_{-1}^{1}dz\frac{1-z^{2}}{2}T(z+x_{0})+c.$ (3.3)

The constant $c$ can be fixed by the normalization condition

$\text{tr}_{A}\rho_{A}=\text{tr}_{A}e^{-H_{A}}=1.$ (3.4)

Its value doesn’t affect the type-Y CCF with any $\sum_{i}m_{i}\geq 2$. We
also used the convention $T(z)=-2\pi T_{zz}$ where the subscript $z$ is the
holomorphic coordinate $z=t+x$. The radius of the interval $A$ is 1, we have
shifted the variable $z$ such that the dependence of the center $x_{0}$ is in
the stress tensor. The modular Hamiltonian of region $B$ can be obtained by
setting $x_{0}=0$ and restoring the radius $R^{\prime}$. The type-$(m-1,1)$
CCF of the modular Hamiltonian is

$\langle
H_{A}^{m-1}H_{B}\rangle_{c}=D^{(2)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}]G^{(2)}_{2}(\eta).$
(3.5)

The two dimensional conformal block for a chiral operator can be labeled by
the conformal weight $h$ of the operator

$G_{h}^{(2)}(\eta)=(-\eta)^{h}{}_{2}F_{1}(h,h,2h,-\eta).$ (3.6)

We can move the interval $A$ to $B$ such that they coincide. In this limit,
any type-$(m-1,1)$ CCF should approach a type-$(m)$ CCF . This is equivalent
to set $\eta\to-1$. We can set $x_{0}\to 0$ and then take the limit
$R^{\prime}\to 1$,

$x_{A}\to 0,\quad R^{\prime}=1-\epsilon,\quad\epsilon\to 0.$ (3.7)

The cross ratio $\xi\to-\infty$ or $\eta\to-1$ by

$\xi=-\frac{4(1-\epsilon)}{\epsilon^{2}}\approx-\frac{4}{\epsilon^{2}},\quad\eta=-\frac{4(1-\epsilon)}{(2-\epsilon)^{2}}\approx-1+\frac{\epsilon^{2}}{4}.$
(3.8)

On the right hand side of (3.5), we find a logarithmic divergent term in this
limit

$G^{(2)}_{2}(\eta)=12\log\frac{2}{\epsilon}+\cdots=12\log\frac{R}{\epsilon}+\cdots$
(3.9)

The left hand side of (3.5) approaches type-$(m)$ CCF, therefore

$\langle
H_{A}^{m}\rangle_{c}=12D^{(2)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}]\log\frac{R}{\epsilon}+\cdots.$
(3.10)

We read out the cutoff independent coefficient

$p_{1}^{(2)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}]=12D^{(2)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}].$
(3.11)

The relation (3.11) is a typical UV/IR relation for the modular Hamiltonian.
The left hand side is the universal coefficient for $B$ and $A$ coincides
(UV). On the right hand side, the $D$ coefficient characterizes the leading
order behaviour of CCF when $B$ and $A$ are far away to each other (IR). They
provide equivalent information of the CFT since the constant $12$ is
completely fixed by conformal symmetry. The continuation of the conformal
block can be generalized to higher dimensions. For example, in four
dimensions, the conformal block associated with stress tensor becomes
divergent as $A$ approaches $B$,

$G^{(4)}_{4,2}\approx\tilde{\gamma}\frac{R^{2}}{\epsilon^{2}}+\cdots-120\log\frac{R}{\epsilon}+\cdots.$
(3.12)

The leading term is exactly proportional to the area of the boundary and the
logarithmic divergent term also appears in the subleading terms. We can read
out the type-$(m)$ CCF of the modular Hamiltonian in four dimensions

$\langle
H_{A}^{m}\rangle_{c}=\gamma\frac{R^{2}}{\epsilon^{2}}+\cdots+p_{1}^{(4)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}]\log\frac{R}{\epsilon}+\cdots$
(3.13)

with

$p_{1}^{(4)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}]=-120D^{(4)}[T_{\mu_{1}\nu_{1}},\cdots,T_{\mu_{m}\nu_{m}}].$
(3.14)

Note we obtain the area law and the logarithmic behaviour of the type-$(m)$
CCF of the modular Hamiltonian without using any knowledge of Rényi
entanglement entropy. The method of analytic continuation can be applied to
general dimensions and OPE blocks. A conformal block $G_{\Delta,J}^{(d)}(\xi)$
obeys area law in the limit $\xi\to-\infty$ in even dimensions. It has degree
$q=1$ only for $\Delta=J+d-2$,

$G^{(d)}_{\Delta,J}(\xi)=\tilde{\gamma}\frac{R^{d-2}}{\epsilon^{d-2}}+\cdots+E^{(d)}[\Delta,J]\log\frac{R}{\epsilon}+\cdots,\quad\xi\to-\infty.$
(3.15)

This means that type-$(m)$ CCF of type-J OPE blocks may always obey area law
with degree $q=1$, the cutoff independent coefficient is

$p_{q}^{(d)}[\mathcal{O}_{1},\cdots,\mathcal{O}_{m}]=E^{(d)}[\mathcal{O}_{m}]\times
D^{(d)}[\mathcal{O}_{1},\cdots,\mathcal{O}_{m}].$ (3.16)

We have replaced the quantum numbers in E function by the corresponding
primary operator. For non-conserved operators, the conformal block
$G_{\Delta,J}^{(d)}$ also obeys area law in the limit $\xi\to-\infty$ in even
dimension, though it has degree $q=2$

$G^{(d)}_{\Delta,J}(\xi)=\tilde{\gamma}\frac{R^{d-2}}{\epsilon^{d-2}}+\cdots+E^{(d)}[\Delta,J]\log^{2}\log\frac{R}{\epsilon}+\cdots,\quad\xi\to-\infty.$
(3.17)

Therefore, type-$(m)$ CCF of type-O OPE blocks obeys area law with degree
$q=2$. We can obtain similar UV/IR relations as (3.16). In odd dimensions, the
story is the same. The degree $q$ is $0$ for type-$(m)$ CCF of type-J OPE
blocks and $1$ for type-O OPE blocks.

###  3.2 Kinematic information

The function $E^{(d)}[\mathcal{O}]$ is completely fixed by conformal symmetry.
It can be obtained by reading out the coefficient of the logarithmic term with
degree $q$. For each fixed quantum number $\Delta$ and $J$, there is a unique
number $E^{(d)}[\mathcal{O}]$. For any type-J OPE block in two dimensions, the
primary operator $\mathcal{O}$ has dimension $\Delta=J=h$. The conformal block
(3.6) has degree $q=1$ in the limit $\eta\to-1$. The function
$E^{(2)}[\mathcal{O}]$ is

$E^{(2)}[\mathcal{O}]=\frac{2\Gamma(2h)}{\Gamma(h)^{2}},\quad\Delta=J=h.$
(3.18)

For type-O OPE block, the primary operator $\mathcal{O}$ has dimension
$\Delta=h+\bar{h}$ and spin $J=h-\bar{h}$. The conformal block has degree
$q=2$ in the limit $\eta\to-1$. The function $E^{(2)}[\mathcal{O}]$ is

$\displaystyle
E^{(2)}[\mathcal{O}]=\left\\{\begin{array}[]{cc}\frac{2^{4h}\Gamma(h+\frac{1}{2})^{2}}{\pi\Gamma(h)^{2}}&\quad
J=0,\ h>0\\\
-\frac{4^{2h-1}\Gamma\left(h-\frac{1}{2}\right)\Gamma\left(h+\frac{1}{2}\right)}{\pi\Gamma(h-1)\Gamma(h)}&\quad
J=1,\ h>1\\\
\frac{4^{2h-3}(h-2)(h-1)(2h-3)(2h-1)\Gamma\left(h-\frac{3}{2}\right)^{2}}{\pi\Gamma(h)^{2}}&\quad
J=2,\ h>2\\\ \cdots&\end{array}\right.$ (3.23)

In four dimensions, we also find

$E^{(4)}[\mathcal{O}]=\left\\{\begin{array}[]{cc}12&\quad\Delta=3,\ J=1\\\
-120&\quad\Delta=4,\ J=2\\\ 840&\quad\Delta=5,\ J=3\\\
\cdots&\end{array}\right.$ (3.24)

for conserved currents and

$\displaystyle
E^{(4)}[\mathcal{O}]=\left\\{\begin{array}[]{cc}-\frac{2^{2\Delta-1}\Gamma(\frac{\Delta-1}{2})\Gamma(\frac{\Delta+1}{2})}{\pi\Gamma(\frac{\Delta-2}{2})^{2}}&\quad\Delta>1,\
J=0,\\\
\vspace{4pt}\frac{2^{2\Delta-1}\Gamma(\frac{\Delta}{2})\Gamma(\frac{\Delta+2}{2})}{\pi\Gamma(\frac{\Delta-3}{2})\Gamma(\frac{\Delta+1}{2})}&\quad\Delta>3,\
J=1,\\\
\vspace{4pt}-\frac{4^{\Delta-1}(\Delta-2)\Gamma(\frac{\Delta-3}{2})\Gamma(\frac{\Delta+3}{2})}{\pi\Gamma(\frac{\Delta-4}{2})\Gamma(\frac{\Delta+2}{2})}&\quad\Delta>4,\
J=2,\vspace{4pt}\\\ \cdots\end{array}\right.$ (3.29)

for non-conserved operators. In three dimensions, we find

$\displaystyle
E^{(3)}[\mathcal{O}]=\left\\{\begin{array}[]{cc}-\frac{2^{2\Delta-1}(\Delta-1)\Gamma(\Delta-\frac{1}{2})}{\sqrt{\pi}\Gamma(\Delta-1)}&\quad\Delta>\frac{1}{2},\
J=0.\\\
\vspace{4pt}\frac{2^{\Delta+1}\Delta\Gamma(\Delta-\frac{1}{2})}{\Gamma(\frac{\Delta-2}{2})\Gamma(\frac{\Delta+1}{2})}&\quad\Delta>2,\
J=1,\\\
\vspace{4pt}-\frac{2^{2\Delta-1}(\Delta^{2}-1)\Gamma(\Delta-\frac{1}{2})}{\sqrt{\pi}(\Delta-2)^{2}\Delta\Gamma(\Delta-3)}&\quad\Delta>3,\
J=2,\vspace{4pt}\\\ \cdots\end{array}\right.$ (3.34)

for non-conserved operators. Note for conserved currents in odd dimensions,
the function $E^{(3)}[\mathcal{O}]$ may depend on explicit choice of the
cutoff. For example, a transformation $\epsilon\to\epsilon(1+a\epsilon)$ may
shift its value. This is because the degree is $0$, there is no logarithmic
divergence at all.

###  3.3 UV/IR relation

The UV/IR relation (3.16) relates type-$(m)$ CCF to type-$(m-1,1)$ CCF. This
relation may simplify computation in many cases. To see this point, let’s
compute the following type-$(2)$ CCF in two dimensions

$\displaystyle\langle Q_{A}[\mathcal{O}]^{2}\rangle_{c}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dz_{1}\int_{-1}^{1}dz_{2}\frac{(1-z_{1}^{2})^{h-1}(1-z_{2}^{2})^{h-1}}{(z_{1}-z_{2})^{2h}}$
(3.35) $\displaystyle=$
$\displaystyle\frac{(-1)^{-h}\sqrt{\pi}\Gamma(h)}{\Gamma(h+\frac{1}{2})}\int_{-1}^{1}dz_{1}\frac{1}{1-z_{1}^{2}}$
$\displaystyle=$
$\displaystyle\frac{(-1)^{-h}\sqrt{\pi}\Gamma(h)}{\Gamma(h+\frac{1}{2})}\log\frac{2}{\epsilon}.$

This is a double integral with poles at $z_{1}=z_{2}$. We regularize the
integral by ignoring these poles at the second step. At the last step, we
insert a UV cutoff to regularize the integral. However, using UV/IR relation,
one just need to fix the coefficient $D$ which is related to the large
distance behaviour of the type-$(1,1)$ CCF,

$\langle
Q_{A}[\mathcal{O}]Q_{B}[\mathcal{O}]\rangle_{c}=\int_{-1}^{1}dz_{1}\int_{-1}^{1}dz_{2}\frac{(1-z_{1}^{2})^{h-1}(1-z_{2}^{2})^{h-1}}{(z_{1}-z_{2}+x_{0})^{2h}}.$
(3.36)

In the large distance limit, $x_{0}\to\infty$, the integral becomes simpler

$\displaystyle\langle Q_{A}[\mathcal{O}]Q_{B}[\mathcal{O}]\rangle_{c}$
$\displaystyle\approx$
$\displaystyle\int_{-1}^{1}dz_{1}\int_{-1}^{1}dz_{2}\frac{(1-z_{1}^{2})^{h-1}(1-z_{2}^{2})^{h-1}}{x_{0}^{2h}}$
(3.37) $\displaystyle=$ $\displaystyle
4^{-h}(\frac{\sqrt{\pi}\Gamma(h)}{\Gamma\left(h+\frac{1}{2}\right)})^{2}\eta^{h}.$

We have used the relation $\eta\approx\frac{4}{x_{0}^{2}}$ in the large
distance limit. Then we can read out

$D^{(2)}[\mathcal{O},\mathcal{O}]=(-1)^{-h}4^{-h}(\frac{\sqrt{\pi}\Gamma(h)}{\Gamma\left(h+\frac{1}{2}\right)})^{2}.$
(3.38)

Combining UV/IR relation and (3.18), we find

$p^{(2)}_{1}[\mathcal{O},\mathcal{O}]=E^{(2)}[\mathcal{O}]\times
D^{(2)}[\mathcal{O},\mathcal{O}]=\frac{(-1)^{-h}\sqrt{\pi}\Gamma(h)}{\Gamma(h+\frac{1}{2})}.$
(3.39)

The result is exactly the same as (3.35). We use the UV/IR relation to obtain
type-$(3)$ CCF for type-J OPE blocks in two dimensions, the cutoff independent
coefficient is

$p^{(2)}_{1}[\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}]=\frac{C_{123}\pi^{3/2}(-1)^{\frac{h_{1}+h_{2}+h_{3}}{2}}\Gamma(h_{1})\Gamma(h_{2})\Gamma(h_{3})\kappa}{\Gamma(\frac{1+h_{1}+h_{2}-h_{3}}{2})\Gamma(\frac{1+h_{1}+h_{3}-h_{2}}{2})\Gamma(\frac{1+h_{2}+h_{3}-h_{1}}{2})\Gamma(\frac{h_{1}+h_{2}+h_{3}}{2})},$
(3.40)

where the constant $\kappa=\frac{1}{2}[1+(-1)^{h_{1}+h_{2}+h_{3}}]$. We notice
that the result is totally symmetric under the exchange of any two conformal
weights. Since there are different ways to uplift type-$(m)$ to
type-$(m-1,1)$, the cutoff independent coefficient should be identical since
they characterize the same CCF after taking the limit $A\to B$. For $m=3$,
this is a cyclic identity

$p_{q}^{(d)}[\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}]=p_{q}^{(d)}[\mathcal{O}_{2},\mathcal{O}_{3},\mathcal{O}_{1}]=p_{q}^{(d)}[\mathcal{O}_{3},\mathcal{O}_{1},\mathcal{O}_{2}].$
(3.41)

The UV/IR relation and the cyclic identity have been checked for type-$(m)$
CCF (m=2,3) in four dimensions. We list the cutoff independent coefficients
below [17].

  * •

Type-(2). The normalization constants are set to 1.

    * –

Spin 1-1 conserved currents.

$\displaystyle
p_{1}^{(4)}[\mathcal{J}_{\mu},\mathcal{J}_{\nu}]=-\frac{\pi^{2}}{3}.$ (3.42)

    * –

Spin 2-2 conserved currents.

$\displaystyle p_{1}^{(4)}[T_{\mu\nu},T_{\rho\sigma}]=-\frac{\pi^{2}}{40}.$
(3.43)

    * –

Spin 0-0 non-conserved operators.

$\displaystyle
p_{2}^{(4)}[\mathcal{O},\mathcal{O}]=-\frac{4\pi^{2}(\Delta-1)\Gamma(\Delta-2)^{2}\Gamma(\frac{\Delta}{2})^{4}}{\Gamma(\Delta)^{2}\Gamma(\Delta-1)^{2}}.$
(3.44)

    * –

Spin 1-1 non-conserved operators.

$\displaystyle
p_{2}^{(4)}[\mathcal{O}_{\mu},\mathcal{O}_{\nu}]=-\frac{4^{1-\Delta}\pi^{3}\Delta\Gamma(\frac{\Delta-3}{2})\Gamma(\frac{\Delta+1}{2})}{\Gamma(\frac{\Delta}{2}+1)^{2}},\quad\Delta>3.$
(3.45)

    * –

Spin 2-2 non-conserved operators.

$\displaystyle
p_{2}^{(4)}[\mathcal{O}_{\mu\nu},\mathcal{O}_{\rho\sigma}]=-\frac{3\pi^{2}(\Delta-2)\Delta^{2}\Gamma(\frac{\Delta}{2}-2)^{2}\Gamma(\frac{\Delta}{2}-1)^{2}}{64\Gamma(\Delta-4)\Gamma(\Delta+2)},\quad\Delta>4.$
(3.46)

  * •

Type-$(3)$.

    * –

Spin 1-1-2 conserved currents. The three point function of zero components are
fixed by conformal symmetry

$\langle
T_{00}(x_{1})\mathcal{J}_{0}(x_{2})\mathcal{J}_{0}(x_{3})\rangle_{c}=\frac{C_{T\mathcal{J}\mathcal{J}}}{x_{12}^{4}x_{13}^{2}x_{23}^{2}}.$
(3.47)

Then the coefficient

$\displaystyle
p_{1}^{(4)}[\mathcal{J}_{\mu},\mathcal{J}_{\nu},T_{\rho\sigma}]=-\frac{\pi^{3}}{2}C_{T\mathcal{J}\mathcal{J}}.$
(3.48)

    * –

Spin 2-2-2 conserved currents. The three point function of zero components are
fixed by conformal symmetry

$\langle
T_{00}(x_{1})T_{00}(x_{2})T_{00}(x_{3})\rangle_{c}=\frac{C_{TTT}}{x_{12}^{4}x_{13}^{4}x_{23}^{4}}.$
(3.49)

Then the coefficient

$p_{1}^{(4)}[T_{\mu\nu},T_{\rho\sigma},T_{\alpha\beta}]=\frac{\pi^{3}}{12}C_{TTT}.$
(3.50)

    * –

Spin 0-0-0 non-conserved currents.

$\displaystyle p_{2}^{(4)}[\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}]$
$\displaystyle=$
$\displaystyle-2^{4-\Delta_{1}-\Delta_{2}-\Delta_{3}}\pi^{3}C_{123}\int_{\mathbb{D}^{2}}d\zeta
d\bar{\zeta}(\zeta+\bar{\zeta})^{2}\int_{\mathbb{D}^{2}}d\zeta^{\prime}d\bar{\zeta}^{\prime}(\zeta^{\prime}+\bar{\zeta}^{\prime})^{2}$
$\displaystyle\times(1-\zeta^{2})^{\frac{\Delta_{1}-4}{2}}(1-\bar{\zeta}^{2})^{\frac{\Delta_{1}-4}{2}}(1-\zeta^{\prime
2})^{\frac{\Delta_{2}-4}{2}}(1-\bar{\zeta}^{\prime
2})^{\frac{\Delta_{2}-4}{2}}\int_{0}^{\pi}d\theta\frac{\sin\theta}{(a+b\cos\theta)^{\frac{\Delta_{12,3}}{2}}},$

Though the expression (LABEL:cyc) is not symmetric superficially under the
exchange of any two conformal weights, we checked explicitly that it satisfies
the cyclic identity for integer conformal weights.

For $m=4$, the UV/IR relation and the cyclic identity are much more harder to
check. We considered type-$(4)$ CCF for massless free scalar theory [13, 14].
In this theory, one can construct an infinite tower of conserved currents with
even spin [29]. The four point functions can be calculated explicitly.
Therefore we can find type-$(3,1)$ and type-$(4)$ CCFs and read out the
corresponding coefficients. For example, for spin-2-2-2-4 conserved currents
[14],

$D[2,2,2,4]=\frac{3}{70}D[2,2,4,2].$ (3.52)

Both of them leads to the cutoff coefficients

$p_{1}^{(2)}[2,2,2,4]=\frac{2\Gamma(8)}{\Gamma(4)^{2}}D[2,2,2,4]=\frac{2\Gamma(4)}{\Gamma(2)^{2}}D[2,2,4,2]=p^{(2)}_{1}[2,2,4,2].$
(3.53)

The cyclic identity is obeyed.

###  3.4 Discussion

The UV/IR relation should be slightly modified when the CCF contains both
type-J and type-O OPE blocks. One simple example is the following type-$(3)$
CCF

$\langle
Q_{A}[\mathcal{J}]Q_{A}[\mathcal{O}]Q_{A}[\tilde{\mathcal{O}}]\rangle_{c}$
(3.54)

where $Q_{A}[\mathcal{J}]$ is a type-J OPE block while $Q_{A}[\mathcal{O}]$
and $Q_{A}[\tilde{\mathcal{O}}]$ are type-O OPE blocks. This CCF is related to
the following two type-$(2,1)$ CCFs

$\displaystyle\langle
Q_{A}[\tilde{\mathcal{O}}]Q_{A}[\mathcal{J}]Q_{B}[\mathcal{O}]\rangle_{c}$
$\displaystyle=$ $\displaystyle
D^{(d)}[\tilde{\mathcal{O}},\mathcal{J},\mathcal{O}]G^{(d)}_{\Delta,J}(\xi),$
(3.55) $\displaystyle\langle
Q_{A}[\mathcal{O}]Q_{A}[\tilde{\mathcal{O}}]Q_{B}[\mathcal{J}]\rangle_{c}$
$\displaystyle=$ $\displaystyle
D^{(d)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]G^{(d)}_{\Delta^{\prime},J^{\prime}}(\xi).$
(3.56)

We choose $d=4$. Taking the limit $A\to B$ from (3.55), we find a type-$(3)$
CCF with degree $q=2$, the UV/IR relation reads

$p_{2}^{(4)}[\tilde{\mathcal{O}},\mathcal{J},\mathcal{O}]=E^{(4)}[\mathcal{O}]\times
D^{(4)}[\tilde{\mathcal{O}},\mathcal{J},\mathcal{O}]$ (3.57)

We can also take the limit $A\to B$ from (3.56), then we will find a
type-$(3)$ CCF with degree $q=1$, the UV/IR relation reads

$p_{1}^{(4)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]=E^{(4)}[\mathcal{J}]\times
D^{(4)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}].$ (3.58)

The equations (3.57) and (3.58) are not identical superficially since the
subscript $q$ are not equal to each other. However, an explicit calculation
for spin 2-0-0 and spin 2-2-0 in four dimensions [17] shows that the
coefficient $D^{(4)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]$ is actually
divergent logarithmically,

$D^{(4)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]=D^{(4)}_{\text{log}}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]\log\frac{R}{\epsilon}+\cdots.$
(3.59)

The terms in $\cdots$ are finite and depends on cutoff scale. Due to the
logarithmic divergence behaviour of the coefficient
$D^{(4)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]$, the degree of
type-$(3)$ CCF from (3.56) increases 1, the modified UV/IR relation becomes

$p_{2}^{(4)}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}]=E^{(4)}[\mathcal{J}]\times
D^{(4)}_{\text{log}}[\mathcal{O},\tilde{\mathcal{O}},\mathcal{J}].$ (3.60)

We checked explicitly that the two constants (3.57) and (3.60) are equal to
each other. The cyclic identity is still satisfied after counting the
logarithmic divergence of the $D$ function.

##  4 Generalizations

The area law and logarithmic behaviour in the subleading terms can be extended
in different directions. In this section, we mention several extensions.

  * •

UV/IR relation. In general, one can uplift any type-$(m)$ CCF to a
type-$(p,m-p)$ CCF

$\langle Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{m}]\rangle_{c}\stackrel{{\scriptstyle
uplift}}{{\longrightarrow}}\langle Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{p}]Q_{B}[\mathcal{O}_{p+1}]\cdots
Q_{B}[\mathcal{O}_{m}]\rangle_{c},\quad 1\leq p\leq m-1.$ (4.1)

When $p$ is not $1$ and $m-1$, the type-$(p,m-p)$ CCF is not a conformal
block. It is still a function of cross ratio $\xi$, therefore it should
reproduce the type-$(m)$ CCF after taking the limit $A\to B$,

$\langle Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{m}]\rangle_{c}=\lim_{\xi\to-\infty}\langle
Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{p}]Q_{B}[\mathcal{O}_{p+1}]\cdots
Q_{B}[\mathcal{O}_{m}]\rangle_{c}.$ (4.2)

Obviously, this also defines a UV/IR relation between $p_{q}^{(d)}$ and
several coefficients in the type-$(p,m-p)$ CCF. Since the right hand side is
not proportional to any conformal block, it is not easy to write out an
explicit formula. Nevertheless, one may still check the relation (4.2) case by
case. One example is to consider the type-$(2,2)$ CCF of the modular
Hamiltonian in CFT2. By making use of the universal feature of the CCF of the
stress tensor, one can fix the generator of type-$(m_{1},m_{2})$ CCFs [14]

$T_{A\cup
B}(\mu_{1},\mu_{2})=-\frac{c}{2}\text{tr}\log[\bm{1}-\left(\begin{array}[]{cc}\mathcal{A}&\mathcal{C}\\\
\mathcal{D}&\mathcal{B}\end{array}\right)],$ (4.3)

where the matrices $\mathcal{A},\mathcal{B},\mathcal{C}$ and $\mathcal{D}$ are

$\displaystyle\mathcal{A}_{xx^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\eta^{2}}{4}\int_{0}^{\infty}dy\frac{\sqrt{xx^{\prime}}y\sinh\pi\mu_{1}x\
\sinh\pi\mu_{2}y}{\sinh\pi x^{\prime}\ \sinh\pi y\ \sinh\pi(1+\mu_{1})x\
\sinh\pi(1+\mu_{2})y}(\frac{x_{13}}{x_{23}})^{i(x-x^{\prime})}\mathcal{F}(x,x^{\prime},y),$
(4.4) $\displaystyle\mathcal{B}_{xx^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\eta^{2}}{4}\int_{0}^{\infty}dy\frac{\sqrt{xx^{\prime}}y\sinh\pi\mu_{1}x\
\sinh\pi\mu_{2}y}{\sinh\pi x^{\prime}\ \sinh\pi y\ \sinh\pi(1+\mu_{1})x\
\sinh\pi(1+\mu_{2})y}(\frac{x_{13}}{x_{23}})^{-i(x-x^{\prime})}\mathcal{F}(x^{\prime},x,y),$
(4.5) $\displaystyle\mathcal{C}_{xx^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\eta^{2}}{4}\int_{0}^{\infty}dy\frac{\sqrt{xx^{\prime}}y\sinh\pi\mu_{1}x\
\sinh\pi\mu_{2}y}{\sinh\pi x^{\prime}\ \sinh\pi y\ \sinh\pi(1+\mu_{1})x\
\sinh\pi(1+\mu_{2})y}(\frac{x_{13}}{x_{23}})^{i(x+x^{\prime})}\mathcal{F}(x,-x^{\prime},y),$
(4.6) $\displaystyle\mathcal{D}_{xx^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\eta^{2}}{4}\int_{0}^{\infty}dy\frac{\sqrt{xx^{\prime}}y\sinh\pi\mu_{1}x\
\sinh\pi\mu_{2}y}{\sinh\pi x^{\prime}\ \sinh\pi y\ \sinh\pi(1+\mu_{1})x\
\sinh\pi(1+\mu_{2})y}(\frac{x_{13}}{x_{23}})^{-i(x+x^{\prime})}\mathcal{F}(-x,x^{\prime},y).$
(4.7)

with

$\displaystyle\mathcal{F}(x,x^{\prime},y)$ $\displaystyle=$
$\displaystyle{}_{2}F_{1}(1+ix,1-iy,2,-\eta)\
{}_{2}F_{1}(1-ix^{\prime},1+iy,2,-\eta)$ (4.8)
$\displaystyle+{}_{2}F_{1}(1+ix,1+iy,2,-\eta)\
{}_{2}F_{1}(1-ix^{\prime},1-iy,2,-\eta).$

$\mathcal{F}$ and its complex conjugate obey

$\mathcal{F}^{*}(x,x^{\prime},y)=\mathcal{F}(x^{\prime},x,y),\quad\mathcal{F}^{*}(-x,-x^{\prime},y)=\mathcal{F}(x,x^{\prime},y).$
(4.9)

so

$\mathcal{A}=\mathcal{B}^{*},\quad\mathcal{C}=\mathcal{D}^{*}.$ (4.10)

We read out the first few CCFs

$\displaystyle\langle H_{A}^{m}\rangle_{c}$ $\displaystyle=$
$\displaystyle\frac{cm!}{12}\log\frac{2}{\epsilon},$ $\displaystyle\langle
H_{A}^{m-1}H_{B}\rangle_{c}$ $\displaystyle=$ $\displaystyle\frac{cm!}{144}\
G_{2}^{(2)}(\eta).$ $\displaystyle\langle H_{A}^{2}H_{B}^{2}\rangle_{c}$
$\displaystyle=$ $\displaystyle
c\\{\frac{1+\eta}{\eta^{2}}[4\text{Li}_{3}(1+\eta)-2\log(1+\eta)\text{Li}_{2}(1+\eta)+\frac{2\log(1+\eta)}{3}\text{Li}_{2}(-\eta)$
$\displaystyle+\frac{1+\eta}{3}\log^{2}(1+\eta)-\frac{\pi^{2}}{3}\log(1+\eta)-4\zeta(3)]+\frac{2+\eta}{3\eta}[2\text{Li}_{2}(-\eta)+3\log(1+\eta)]-\frac{4}{3}\\},$

where the polylogrithm $\text{Li}_{n}(z)$ is

$\text{Li}_{n}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}.$ (4.12)

The relation (4.2) can be checked for $p=2,m=4$. The right hand side is

$\lim_{\eta\to-1}\langle
H_{A}^{2}H_{B}^{2}\rangle_{c}=2c\log\frac{2}{\epsilon}+\cdots.$ (4.13)

The cutoff independent coefficient $2c$ matches with the one in $\langle
H_{A}^{4}\rangle_{c}$.

  * •

New power law. In the previous discussion, we focus on the case that $B$ and
$A$ coincide with each other. However, there are other cases that the CCFs are
still divergent. One can consider the limit that $A$ just attaches the edge of
$B$,

$R^{\prime}=1,\quad x_{0}=2+\epsilon,\quad\epsilon\to 0.$ (4.14)

The cross ratio $\xi$ does not approach $-\infty$ but $1$

$\xi=\frac{4}{(2+\epsilon)^{2}}=1-\epsilon+\cdots.$ (4.15)

We can define a new CCF which is also divergent from type-$(m-1,1)$ CCF

$\langle Q_{A}[\mathcal{O}_{1}]\cdots Q_{A}[\mathcal{O}_{m-1}]\odot
Q_{B}[\mathcal{O}_{m}]\rangle_{c}=\lim_{\xi\to 1}\langle
Q_{A}[\mathcal{O}_{1}]\cdots
Q_{A}[\mathcal{O}_{m-1}]Q_{B}[\mathcal{O}_{m}]\rangle_{c}$ (4.16)

The continuation of conformal block tells us that the new CCF obeys a new
power law

$\langle Q_{A}[\mathcal{O}_{1}]\cdots Q_{A}[\mathcal{O}_{m-1}]\odot
Q_{B}[\mathcal{O}_{m}]\rangle_{c}=\bar{\gamma}(\frac{R}{\epsilon})^{\frac{d-2}{2}}+\cdots+\bar{p}_{q}^{(d)}\log^{q}\frac{R}{\epsilon}+\cdots.$
(4.17)

The leading term is proportional to

$\mathcal{L}=R^{\frac{d-2}{2}}=\sqrt{\mathcal{A}}$ (4.18)

which is the characteristic length of the region $A$ in four dimensions. In
two dimensions, the leading term is a logarithmic term with power $q$. In this
case, there is a new UV/IR relation between $\bar{p}_{q}$ and $D$ coefficient
, we write it schematically

$\bar{p}_{q}=\bar{E}\times D.$ (4.19)

The function $\bar{E}^{(d)}[\mathcal{O}]$ is proportional to
$E^{(d)}[\mathcal{O}]$. The proportional constant is shown below.

    * –

$d$ is even.

      * *

For conserved current $\mathcal{O}$ with conformal weight $\Delta=J+d-2$,

$\bar{E}^{(d)}[\mathcal{O}]=\frac{(-1)^{J}}{2}E^{(d)}[\mathcal{O}].$ (4.20)

      * *

For non-conserved current $\mathcal{O}$ with conformal weight $\Delta$ and
spin $J$,

$\bar{E}^{(d)}[\mathcal{O}]=\frac{(-1)^{J}}{4}E^{(d)}[\mathcal{O}].$ (4.21)

We checked the relation for $d=2,4$ and spin $J\leq 2$.

    * –

$d$ is odd.

      * *

For non-conserved current $\mathcal{O}$ with conformal weight $\Delta$ and
spin $J$,

$\bar{E}^{(d)}[\mathcal{O}]=\frac{(-1)^{J}}{2}E^{(d)}[\mathcal{O}].$ (4.22)

      * *

For conserved current $\mathcal{O}$, there is no logarithmic divergent term in
the CCF.

We checked the relation for $d=3$ and spin $J\leq 2$.

Since $D$ function is the same, we find a relation between two cutoff
independent coefficients $p$ and $\bar{p}$,

$\frac{p}{E}=\frac{\bar{p}}{\bar{E}}.$ (4.23)

##  5 Summary and outlook

In this report, we have introduced the area law (3.1) of type-$(m)$ CCFs of
OPE blocks. It is a generalization of the area law of entanglement entropy. We
will list several open problems for future work.

  * •

Higher $m\geq 4$. In most of the work, we restrict to the region $m\geq 3$.
This is because the structure of $m$-point correlation function of primary
operators in CFT is fixed up to $m=3$. For $m\geq 4$, it is harder to extract
cutoff independent coefficient.

  * •

UV/IR relation. The UV/IR relation

$p=E\times D$ (5.1)

has been checked for several examples. A rigorous proof is still lacking.

  * •

Cyclic identity. The cyclic identity of $p$ reflects the fact that $p$ is
independent of the way to regularize the type-$(m)$ CCF. However, we feel that
a direct computation is impossible to check this identity.

  * •

New power law. We generalize the type-$(m_{1},m_{2})$ CCF to the case that $A$
and $B$ just attaches with each other. The corresponding CCF is divergent with
a new power law (4.17). The corresponding new UV/IR relation

$\bar{p}=\bar{E}\times D$ (5.2)

also needs understanding.

  * •

Deformed reduced density matrix. This exponential operator is similar to the
“Wilson loop” in gauge theories [30, 31] despite the fact that the OPE block
has no lower bound in general. When the OPE block has a lower bound, the
logarithm of the vacuum expectation value of the deformed reduced density
matrix

$\log\langle e^{-\mu Q_{A}}\rangle$ (5.3)

should also obey area law with logarithmic divergence. There may be a
gravitational dual for this quantity as [32, 33]. The similarity of the area
law between this program and black hole entropy implies that the classical
part contributes to the area term while quantum effects lead to logarithmic
corrections.

  * •

Multiple integrals. According to the method of continuation of conformal
block, area law of type-$(m)$ CCF is protected by conformal invariance.
However, the method of continuation itself cannot guarantee that it always
leads to the correct result. One has to develop other methods to deal with the
multiple integrals. In two dimensions, one should generalize Selberg integrals
[34, 35] to include more parameters [16].

## Acknowledgements

This work was supported by NSFC Grant No. 12005069.

## References

  * [1] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7 (1973), 2333-2346 doi:10.1103/PhysRevD.7.2333
  * [2] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975), 199-220 doi:10.1007/BF02345020
  * [3] B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, “A Stereoscopic Look into the Bulk,” JHEP 1607, 129 (2016) doi:10.1007/JHEP07(2016)129 [arXiv:1604.03110 [hep-th]].
  * [4] J. de Boer, F. M. Haehl, M. P. Heller and R. C. Myers, “Entanglement, holography and causal diamonds,” JHEP 1608, 162 (2016) doi:10.1007/JHEP08(2016)162 [arXiv:1606.03307 [hep-th]].
  * [5] S. Ferrara, A. F. Grillo and R. Gatto, “Manifestly conformal covariant operator-product expansion,” Lett. Nuovo Cim. 2S2, 1363 (1971) [Lett. Nuovo Cim. 2, 1363 (1971)]. doi:10.1007/BF02770435
  * [6] S. Ferrara, A. F. Grillo and R. Gatto, “Manifestly conformal-covariant expansion on the light cone,” Phys. Rev. D 5, 3102 (1972). doi:10.1103/PhysRevD.5.3102
  * [7] R.Haag, “ Local quantum physics: fields, particles, algebras ,” Springer, Berlin, Germany (1992).
  * [8] L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, “A Quantum Source of Entropy for Black Holes,” Phys. Rev. D 34, 373 (1986).
  * [9] M. Srednicki, “Entropy and area,” Phys. Rev. Lett. 71, 666 (1993) doi:10.1103/PhysRevLett.71.666 [hep-th/9303048].
  * [10] C. G. Callan, Jr. and F. Wilczek, “On geometric entropy,” Phys. Lett. B 333, 55 (1994) doi:10.1016/0370-2693(94)91007-3 [hep-th/9401072].
  * [11] H. Araki, “Relative Entropy of States of Von Neumann Algebras,” Publ. Res. Inst. Math. Sci. Kyoto 1976, 809 (1976).
  * [12] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) doi:10.1103/PhysRevLett.96.181602 [hep-th/0603001].
  * [13] J. Long, “Correlation function of modular Hamiltonians,” JHEP 11 (2019), 163 doi:10.1007/JHEP11(2019)163 [arXiv:1907.00646 [hep-th]].
  * [14] J. Long, “Correlation function with the insertion of zero modes of modular Hamiltonians,” JHEP 01 (2020), 173 doi:10.1007/JHEP01(2020)173 [arXiv:1911.11487 [hep-th]].
  * [15] A. Belin, L. Y. Hung, A. Maloney, S. Matsuura, R. C. Myers and T. Sierens, “Holographic Charged Renyi Entropies,” JHEP 12 (2013), 059 doi:10.1007/JHEP12(2013)059 [arXiv:1310.4180 [hep-th]].
  * [16] J. Long, “Logarithmic behaviour of connected correlation function in CFT,” [arXiv:2001.05129 [hep-th]].
  * [17] J. Long, “Area law of connected correlation function in higher dimensional conformal field theory,” [arXiv:2007.15380 [hep-th]].
  * [18] S. N. Solodukhin, “The Conical singularity and quantum corrections to entropy of black hole,” Phys. Rev. D 51 (1995), 609-617 doi:10.1103/PhysRevD.51.609 [arXiv:hep-th/9407001 [hep-th]].
  * [19] S. N. Solodukhin, “On ‘Nongeometric’ contribution to the entropy of black hole due to quantum corrections,” Phys. Rev. D 51 (1995), 618-621 doi:10.1103/PhysRevD.51.618 [arXiv:hep-th/9408068 [hep-th]].
  * [20] R. K. Kaul and P. Majumdar, “Logarithmic correction to the Bekenstein-Hawking entropy,” Phys. Rev. Lett. 84 (2000), 5255-5257 doi:10.1103/PhysRevLett.84.5255 [arXiv:gr-qc/0002040 [gr-qc]].
  * [21] S. Carlip, “Logarithmic corrections to black hole entropy from the Cardy formula,” Class. Quant. Grav. 17 (2000), 4175-4186 doi:10.1088/0264-9381/17/20/302 [arXiv:gr-qc/0005017 [gr-qc]].
  * [22] T. R. Govindarajan, R. K. Kaul and V. Suneeta, “Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole,” Class. Quant. Grav. 18 (2001), 2877-2886 doi:10.1088/0264-9381/18/15/303 [arXiv:gr-qc/0104010 [gr-qc]].
  * [23] A. Sen, “Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions,” JHEP 04 (2013), 156 doi:10.1007/JHEP04(2013)156 [arXiv:1205.0971 [hep-th]].
  * [24] S. Minwalla, “Restrictions imposed by superconformal invariance on quantum field theories,” Adv. Theor. Math. Phys. 2, 783 (1998) doi:10.4310/ATMP.1998.v2.n4.a4 [hep-th/9712074].
  * [25] H. Casini, M. Huerta and R. C. Myers, “Towards a derivation of holographic entanglement entropy,” JHEP 1105, 036 (2011) doi:10.1007/JHEP05(2011)036 [arXiv:1102.0440 [hep-th]].
  * [26] F. A. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,” Nucl. Phys. B 599, 459 (2001) doi:10.1016/S0550-3213(01)00013-X [hep-th/0011040].
  * [27] F. A. Dolan and H. Osborn, “Conformal partial waves and the operator product expansion,” Nucl. Phys. B 678, 491 (2004) doi:10.1016/j.nuclphysb.2003.11.016 [hep-th/0309180].
  * [28] M. Hogervorst, H. Osborn and S. Rychkov, “Diagonal Limit for Conformal Blocks in $d$ Dimensions,” JHEP 1308, 014 (2013) doi:10.1007/JHEP08(2013)014 [arXiv:1305.1321 [hep-th]].
  * [29] I. Bakas and E. Kiritsis, Nucl. Phys. B 343 (1990), 185-204 [erratum: Nucl. Phys. B 350 (1991), 512-512] doi:10.1016/0550-3213(90)90600-I
  * [30] J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. 80, 4859 (1998) doi:10.1103/PhysRevLett.80.4859 [hep-th/9803002].
  * [31] S. J. Rey and J. T. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C 22, 379 (2001) doi:10.1007/s100520100799 [hep-th/9803001].
  * [32] D. L. Jafferis and S. J. Suh, “The Gravity Duals of Modular Hamiltonians,” JHEP 09 (2016), 068 doi:10.1007/JHEP09(2016)068 [arXiv:1412.8465 [hep-th]].
  * [33] D. L. Jafferis, A. Lewkowycz, J. Maldacena and S. J. Suh, “Relative entropy equals bulk relative entropy,” JHEP 06 (2016), 004 doi:10.1007/JHEP06(2016)004 [arXiv:1512.06431 [hep-th]].
  * [34] A. Selberg, “Bemerkninger om et multipelt integral,”Norsk.Mat.Tidsskr.24(1944)71-78.
  * [35] P. Forrester and S. Warnaar, “The importance of the Selberg integral,”Bull. Amer. Math. Soc. (N.S.) 45 (2008), 489-534.