More on Heavy-Light Bootstrap up to Double-Stress-Tensor

Yue-Zhou Li$\,{}^{1\Delta}$, Hao-Yu Zhang$\,{}^{2J}$

$\,{}^{1}$Center for Joint Quantum Studies and Department of Physics,
School of Science, Tianjin University, Tianjin 300350, China
$\,{}^{2}$ George P. & Cynthia Woods Mitchell Institute for Fundamental
Physics and Astronomy,
Texas A&M University, College Station, TX 77843, USA

ABSTRACT

We investigate the heavy-light four-point function up to double-stress-tensor,
supplementing 1910.06357. By using the OPE coefficients of lowest-twist
double-stress-tensor in the literature, we find the Regge behavior for lowest-
twist double-stress-tensor in general even dimension within the large impact
parameter regime. In the next, we perform the Lorentzian inversion formula to
obtain both the OPE coefficients and anomalous dimensions of double-twist
operators $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin $J$ in
$d=4$. We also extract the anomalous dimensions of double-twist operators with
finite spin in general dimension, which allows us to address the cases that
$\Delta_{L}$ is specified to the poles in lowest-twist double-stress-tensors
where certain double-trace operators $[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$
mix with lowest-twist double-stress-tensors. In particular, we verify and
discuss the Residue relation that determines the product of the mixed
anomalous dimension and the mixed OPE. We also present the double-trace and
mixed OPE coefficients associated with $\Delta_{L}$ poles in $d=6,8$. In the
end, we turn to discuss CFT2, we verify the uniqueness of double-stress-tensor
that is consistent with Virasoso symmetry.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Review of the heavy-light bootstrap
    1. 2.1 The heavy-light four-point function
    2. 2.2 Comment on the mixing problem
  3. 3 The Regge pole of double-stress-tensor
    1. 3.1 The Regge limit of conformal blocks
    2. 3.2 From Sommerfeld-Watson resummation
  4. 4 $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin
    1. 4.1 Warm-up: $d=4$
    2. 4.2 General dimensions
      1. 4.2.1 Explicit examples in $d=6,8,10$
  5. 5 On the $\Delta_{L}$ poles
    1. 5.1 Summation instead of integration
    2. 5.2 Verify the Residue relation
      1. 5.2.1 $d=4$
      2. 5.2.2 $d=6$
    3. 5.3 The universality of the Residue relation
  6. 6 Discussion on CFT2
    1. 6.1 Finite spin double-twist operators
    2. 6.2 Uniqueness of double-stress-tensor
  7. 7 Conclusion
  8. A Lorentzian inversion formula
  9. B Series expansion of conformal blocks
  10. C Stress-tensor block from geodesic Witten diagram
  11. D $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ from holography
  12. E Double-trace and mixed OPE coeffcieints
    1. E.1 $d=6$
      1. E.1.1 $\Delta_{L}=4$
      2. E.1.2 $\Delta_{L}=3$
    2. E.2 $d=8$
      1. E.2.1 $\Delta_{L}=6$
      2. E.2.2 $\Delta_{L}=5$
      3. E.2.3 $\Delta_{L}=4$

##  1 Introduction

A conformal field theory (CFT) is characterized by the conformal dimensions,
spin and operator product expansion (OPE) coefficients of all existed local
primary operators. Remarkably, by taking advantage of the conformal symmetry
combined with the general consistency conditions such as the unitarity and the
crossing equation, conformal bootstrap enables us to explore the conformal
dimensions and OPE coefficients in a powerful and efficient way. For example,
the numerical bootstrap sets the crossing equation as the semi-definite-
programming, providing the powerful ability to narrow down the possible value
of conformal dimensions and OPE coefficients up to high precession, e.g. see
[1] for review.

In parallel to the numerical bootstrap, the analytic bootstrap, in particular,
the lightcone bootstrap makes use of the singularity near the lightcone limit
(in the Lorentzian signature) of the crossing equation, from which the large
spin operators [2] arises naturally and were analyzed extensively, e.g. [3, 4,
5, 6]. Generally speaking, the CFT data of the large spin operators can be
explored by asymptotically expanding them in terms of the inverse powers of
spin $1/J$ and subsequently solving the singular part of crossing equation
order by order algebraically [7, 8]. The large spin perturbation theory was
then developed to extract the large spin data up to all orders of $1/J$ [9].
The CFT data bootstrapped from this analytic procedure are actually valid to
all spin except for few low spins $J=0,1$ [10, 11], which can be explained by
the analyticity in spin explicitly from Lorentzian inversion formula [12, 13,
14]. By employing these highly-developed techniques of lightcone bootstrap,
considerable progress was made in numerous subjects, for example, Wilson-
Fisher and $O(N)$ models [15, 17, 16] and AdS supergravity [18, 19, 20].

Recently, the four-point function with heavy states in large
$N\sim\sqrt{C_{T}}$ CFT where the heavy conformal dimension is comparable to
$C_{T}$ charge, known as heavy-light four-point function, draws a lot of
attentions. In $d=2$, the heavy-light four-point function is universally
constructed by the Virasoro identity block [21] and enjoys plentiful
applications in the context of AdS/CFT [22, 23, 24], e.g. [21, 25, 26, 27,
28]. It is natural to expect that higher dimensional CFT may have similarity
to CFT2 within a certain kinematic limit, for example, the Virasoro-like
structure can be observed near the lightcone limit [29, 30]. The heavy-light
four-point function would be an appropriate window to probe the similarity. It
is indeed found from holographic set-ups that the universal piece of heavy-
light four-point function is the lowest-twist multi-stress-tensors [31, 32],
similar to the universality in CFT2 imposed by Virasoro symmetry. Thus it
would be interesting and important to utilize the developed lightcone
bootstrap to study heavy-light four-point function from which the underlying
CFT data can be extracted [33, 34, 35, 36] and the evidence of the
universality can be provided and understood [35]. This paper is a supplement
to the previous paper [35] that bootstraps the heavy-light four-point function
by using the Lorentzian inversion formula. We discuss the large impact
parameter regime of the Regge limit [37] for lowest-twist double-stress-tensor
in the heavy-light four-point function, and more importantly, we extend the
universality of the double-twist operators
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ at the leading order
$\mathcal{O}(\mu)$ from large spin to finite spin and then tackle the
$\Delta_{L}$ poles referred in [31, 32, 38, 35]. We also discuss the case in
$d=2$, showing the consistency with Virasoro identity block [21].

The paper is organized as follows: in section 2, we review the heavy-light
four-point function and the recent progress on this subject and then we argue
that there is no mixing problem in the heavy-light bootstrap; in section 3, we
analyze the Regge behavior of lowest-twist double-stress-tensor where a
general even dimensional expression in the large impact parameter regime is
provided; in section 4, we employ the Lorentzian inversion formula to extract
the OPE coefficients and anomalous dimension of double-twist operators
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin in $d=4$, and
moreover, we also manage to find the anomalous dimension in any dimension $d$
by using $\bar{z}\rightarrow 1$ expansion of the logarithmic part of the
stress-tensor conformal block we found; in section 5, we consider the case
that $\Delta_{L}$ approaches the poles in the lowest-twist double-stress-
tensor OPE and verify the Residue relation proposed in [38], the mixed OPE
coefficients in $d=4$ is obtained and the universality of the Residue relation
is discussed; in section 6, we consider $d=2$ and obtain the double-twist data
with finite spin. By using these data, we then find the uniqueness of double-
stress-tensor which is consistent with Virasoro symmetry; the paper is
summarised in section 7; the Lorentzian inversion formula is reviewed in
Appendix A and the series expansion of conformal blocks is reviewed in
Appendix B; in Appendix C, we briefly explain how we derive
$\bar{z}\rightarrow 1$ expansion of the logarithmic part of the stress-tensor
conformal block from geodesic Witten diagram; in Appendix D, we show how to
derive the answer of anomalous dimension of
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin from holography by
using the holographic Hamiltonian perturbation theory; in Appendix E, we
consider all examples of $\Delta_{L}$ poles in $d=6,8$ and work out the OPE
coefficients of double-trace operators
$[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$ up to the mixed twist.

##  2 Review of the heavy-light bootstrap

In this section, we review the set-up and the recent progress of bootstrapping
heavy-light four-point function, moreover, we would like to comment on the
mixing problem of heavy-light bootstrap mentioned in [35] and argue that we do
not need to worry about the mixing problem.

###  2.1 The heavy-light four-point function

We consider a generic large $N\sim\sqrt{C_{T}}$ CFT with a large gap
$\Delta_{\rm gap}$ such that it admits holographic gravity dual [39]. The
objects we are interested in bootstrapping is the heavy-light four-point
function
$\langle\mathcal{O}_{H}\mathcal{O}_{L}\mathcal{O}_{L}\mathcal{O}_{H}\rangle$
in such a theory, which consist of two heavy operators with conformal
dimension $\Delta_{H}\sim C_{T}$ and two light operators with conformal
dimension $\Delta_{L}\ll C_{T}$. The heavy-light four-point function can be
studied from both $s$-channel
($\mathcal{O}_{H}\mathcal{O}_{L}\rightarrow\mathcal{O}_{L}\mathcal{O}_{H}$)
and $t$-channel
($\mathcal{O}_{H}\mathcal{O}_{H}\rightarrow\mathcal{O}_{L}\mathcal{O}_{L}$),
and the crossing equation can be established within an appropriate conformal
frame

$(z\bar{z})^{{\frac{\Delta_{H}+\Delta_{L}}{2}}}\langle\mathcal{O}_{H}\mathcal{O}_{L}\mathcal{O}_{L}\mathcal{O}_{H}\rangle=\mathcal{G}^{s}(z,\bar{z})={\frac{(z\bar{z})^{{\frac{\Delta_{H}+\Delta_{L}}{2}}}}{((1-z)(1-\bar{z}))^{\Delta_{L}}}}\mathcal{G}^{t}(1-\bar{z},1-z)\,,$
(2.1)

where both $s$ and $t$-channel correlators $\mathcal{G}^{s,t}$ shall be
expanded in terms of $\mu\sim\Delta_{H}/C_{T}$ around the degenerate point
$\mu\rightarrow 0$111Note this is not contradict to the heavy-limit
$\Delta_{L}\ll\Delta_{H}\sim C_{T}$ we consider. We can take the heavy-limit
$\Delta_{L}\ll\Delta_{H}\sim C_{T}$ for correlators at first such that the
correlators are functions of $\mu$ and in general they are not polynomials,
for example, see examples in CFT2 [21]. Then we can expand the resulting
correlators in terms of $\mu$ around $\mu\rightarrow 0$. Another perspective
is that we can start with large $N$ expansion where the degenerate point is
located at $C_{T}\rightarrow\infty$ and then we collect $\Delta_{H}$ to
reorganize the expansions in terms of $\mu$ [35].. To be more precise, we
adopt the convention in [41]

$\mu=\frac{4\Gamma(d+2)}{(d-1)^{2}\Gamma(\frac{d}{2})^{2}}\frac{\Delta_{H}}{C_{T}}\,.$
(2.2)

At the degenerate point, the heavy-light four-point function is the four-point
function of generalized free field theory, thus $\mathcal{G}^{t}=1$ and the
degenerate double-twist operators
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}=\mathcal{O}_{L}\partial^{2n}\partial_{\mu_{1}}\cdots\partial_{\mu_{J}}\mathcal{O}_{L}$
with conformal dimensions $\Delta=\Delta_{H}+\Delta_{L}+J+2n$ are exchanged in
$s$-channel weighted by the free OPE coefficients [40], where heavy-limit
should be imposed

$\tilde{c}_{n,J}^{\textrm{free}}=\frac{(J+1)\Gamma(\Delta_{L}+n-1)\Gamma(\Delta_{L}+n+J)}{\Gamma(n+1)\Gamma(n+J+2)\Gamma(\Delta_{L}-1)\Gamma(\Delta_{L})}+\mathcal{O}(\frac{1}{\Delta_{H}})\,.$
(2.3)

In general, the degeneracy of the double-twist operators breaks down via
acquiring anomalous dimensions
$\Delta_{L}=\Delta_{H}+\Delta_{L}+J+2n+\tilde{\gamma}_{n,J}$ and the
corresponding OPE coefficients are corrected to be $\tilde{c}_{n,J}$ where

$\displaystyle\tilde{c}_{n,J}=\tilde{c}_{n,J}^{\textrm{free}}\sum_{k=0}\mu^{k}\tilde{c}_{n,J}^{(k)},\qquad\tilde{\gamma}_{n,J}=\tilde{c}_{n,J}^{\textrm{free}}\sum_{k=1}\mu^{k}\tilde{\gamma}_{n,J}^{(k)}\,,$
(2.4)

where it turns out $\tilde{\gamma}^{(k)}$ and $\tilde{c}^{(k)}$ are universal
at large spin limit and behave as [41, 42, 33, 34, 35]

$\tilde{c}_{n,J}^{(k)}\,\,\textrm{and}\,\,\tilde{\gamma}_{n,J}^{(k)}\sim(J^{\prime})^{-\frac{k(d-2)}{2}}$
(2.5)

By dimensional analysis, the operators exchanged in $t$-channel shall be
multi-stress-tensors $T^{k}$ (the multi-twist operators constructed from
stress-tensor) living at the order $\mathcal{O}(\mu^{k})$

$T\partial\cdots T\partial\cdots T\cdots\,,$ (2.6)

for example, at $\mathcal{O}(\mu)$ it is assumed that there is only single-
stress-tensor with OPE coefficient completely determined by Ward identity [43,
44]

$c_{T}=\frac{\Delta_{L}\Gamma(\frac{d}{2}+1)^{2}}{4\Gamma(d+2)}\mu\,,$ (2.7)

which is universal by depending only on $\Delta_{H},\Delta_{L},C_{T}$. In
general, as the convention used for $s$-channel, we usually factorize $\mu$
out and denote

$c_{T^{k}(\Delta,J)}=\mu^{k}c_{T^{k}}\,,\qquad\Delta-J=kd-J_{T}+2n\,,\qquad
J_{T}\leq 2k\,,\qquad J\geq J_{T}\,,$ (2.8)

where $J_{T}$ is the spin of stress-tensors.

Recently, investigating heavy-light four-point function in both $s$ and
$t$-channel becomes increasingly attractive, driven by the motivation of
looking for the Virasoro-like structures in higher dimensional CFT. In fact,
both the holography and the bootstrap allow us to explore
$\tilde{c}^{(k)}_{n,J}$, $\tilde{\gamma}^{(k)}_{n,J}$ and $c_{T^{k}}$. The
holography treats $s$ and $t$-channel separately by using different
holographic techniques. In $s$-channel, one can use the bulk phase shift [45,
46, 47, 48] or holographic Hamiltonian perturbation theory [6, 21] to study
the large spin limit of $\tilde{c}^{(k)}_{n,J},\tilde{\gamma}^{(k)}_{n,J}$
[41, 42]; in $t$-channel, the formalism developed in [31] deals with heavy
operators by assuming they form a black hole and consequently one is allowed
to compute $c_{T^{k}}$ by analyzing two-point functions under black holes [31,
32, 38]. The holographic treatments surprisingly suggest that the large spin
double-twist operators $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ and the
lowest-twist multi-stress-tensors (i.e. $n=0$ and $J_{T}=2k$) are all
universal where it turns out the higher derivative gravity terms play no role.
In addition, the universality in Regge limit can also be investigated from
holography [32, 49].

From pure CFT perspective, the bootstrap handles $s$ and $t$-channel within
the same framework: the crossing equation (2.1) enables us to understand both
$s$ and $t$-channel. By establishing the ansatz of multi-stress-tensor
conformal blocks and assuming large spin expansion of double-twist operators
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$, the crossing equation (2.1) can then
be solved algebraically order by order in $\mu$ from which the large spin
limit of $\tilde{c}^{(k)}_{n,J},\tilde{\gamma}^{(k)}_{n,J}$ and lowest-twist
parts of $c_{T^{k}}$ can be obtained [34]. With more general ansatz around the
lightcone limit, both universal parts and non-universal parts of $1/J$
corrections in $\tilde{c}^{(k)}_{n,J},\tilde{\gamma}^{(k)}_{n,J}$ and OPE
coefficients of higher twist multi-stress-tensors $c_{T^{k}}$ can also be
investigated [36]. In fact, it turns out even the ansatz of stress-tensor
conformal blocks and the assumption of large spin expansion are not necessary
in bootstrapping heavy-light four-point function and understanding the hidden
universality (though, the efficiency is largely promoted provided with those
ansatz and assumption, see [36]), the Lorentzian inversion formula can used
back and forth to provide the strong evidence that the double-twist operators
at large spin limit and lowest-twist multi-stress-tensors show universality,
strikingly bootstrapping the large spin limit of
$\tilde{c}^{(k)}_{n,J},\tilde{\gamma}^{(k)}_{n,J}$ and the lowest-twist OPE
coefficients $c_{T^{k}}$ from nothing more than single-stress-tensor OPE (2.7)
[35]. The extracted OPE coefficients and anomalous dimensions achieve the
exact agreements with holographic set-ups [41, 42, 32]. However, some
questions quoted in [35], e.g. $\Delta_{L}$ poles, mixing problems, etc, still
remain unclear.

To go ahead, in this paper, we mainly follow [35], using the Lorentzian
inversion formula to compute the finite spin results of $\mathcal{O}(\mu)$
double-twist operators and picking these finite spin results up to clarify the
points about $\Delta_{L}$ poles observed in [31, 32, 38, 35] up to double-
stress-tensor. In fact, the whole of this paper should be viewed as the
supplement to [35]: we also study the Regge behavior of lowest-twist double-
stress-tensor in the large impact regime and discuss the case in $d=2$,
supplementing [35]. In the next subsection, we comment on the mixing problem
raised up in [35] and argue that heavy-light bootstrap does not suffer from
the mixing problem.

###  2.2 Comment on the mixing problem

There are two aspects as referred in [35] that literatures have rarely taken
into account. The first one is that the multi-stress-tensor sector has some
poles in $\Delta_{L}$ as observed in [31, 32, 38, 35]. As analyzed in [35],
these $\Delta_{L}$ poles are by-products of multi-stress-tensor trajectories
and reflect the fact that multi-stress-tensor $T^{k}$ could mix with double-
trace operators $[\mathcal{O}_{L}\mathcal{O}_{L}]$ for those $\Delta_{L}$ that
coincides with the poles. The situation when $\Delta_{L}$ is specified on the
poles is the main topic undertaken in this paper and we leave the details in
section 5.

The second aspect is about the situation that there are additional light
operators $\Delta^{(k)}_{L}=\Delta_{L}+2k$ where $k\in\mathbb{N}$. In this
case, the mixing appears among a family of double-twist operators
$[\mathcal{O}_{H}\mathcal{O}^{(k)}_{L}]_{n-k,J}$ which shares the same twist
and dimension at the leading order. The consequence is that the OPE and
anomalous dimension at the order $\mathcal{O}(\mu)$ should be interpreted as
the average over the whole family, i.e.

$\tilde{c}^{(1)}_{n,J}={\frac{\langle\tilde{c}^{(0)}_{\Delta,J}\tilde{c}^{(1)}_{\Delta,J}\rangle}{\langle\tilde{c}^{(0)}_{\Delta,J}\rangle}}\,,\qquad\tilde{\gamma}^{(1)}_{n,J}={\frac{\langle\tilde{c}^{(0)}_{\Delta,J}\tilde{\gamma}^{(1)}_{\Delta,J}\rangle}{\langle\tilde{c}^{(0)}_{\Delta,J}\rangle}}\,,\qquad\Delta-J=\Delta_{H}+\Delta_{L}+2n\,.$
(2.9)

The double-stress-tensor at $\mathcal{O}(\mu^{2})$ order thus requires us to
evaluate
$\langle\tilde{c}^{(0)}_{\Delta,J}\big{(}\tilde{\gamma}^{(1)}_{\Delta,J}\big{)}^{2}\rangle/\langle\tilde{c}^{(0)}_{\Delta,J}\rangle$
instead of simple $\big{(}\tilde{\gamma}^{(1)}_{n,J}\big{)}^{2}$. The similar
problem appears in bootstrapping loop corrections of supergravity in
AdS${}_{5}\times$S5 [18]. Nevertheless, in this subsection, we would like to
provide an argument to state that there is no mixing in the heavy-light
bootstrap.

The key point to resolve the mixing problem, as suggested in [18], is to
consider the complete set of (mixed) correlators, in our case
$\langle\mathcal{O}_{H}\mathcal{O}^{(p)}_{L}\mathcal{O}^{(q)}_{L}\mathcal{O}_{H}\rangle$.
The averaged anomalous dimension now can be organized as a matrix. Let us take
a simple example to show this. We consider $n=1$ and there are two light
operators $(\Delta_{L},\Delta_{L}^{(1)})$, then the averaged anomalous
dimension forms a matrix

$\Gamma=\left(\begin{array}[]{cc}\tilde{\gamma}_{00}&\tilde{\gamma}_{01}\\\
\tilde{\gamma}_{10}&\tilde{\gamma}_{11}\\\ \end{array}\right)\,,$ (2.10)

where $\tilde{\gamma}_{nm}$ stands for the anomalous dimension associated with
$\langle\mathcal{O}_{H}\mathcal{O}^{(m)}_{L}\mathcal{O}^{(n)}_{L}\mathcal{O}_{H}\rangle$
which is contributed by the single stress-tensor sector exchanged in
$\langle\mathcal{O}_{H}\mathcal{O}_{H}\mathcal{O}^{(m)}_{L}\mathcal{O}^{(n)}_{L}\rangle$.
The squared anomalous dimension at the next order
$\langle\tilde{\gamma}^{(1)2}\rangle$ can thus be read off as the element of
the matrix $\Gamma^{2}$ [18]. It is now obvious why normally we have
$\langle\tilde{\gamma}^{(1)2}\rangle\neq\langle\tilde{\gamma}^{(1)}\rangle^{2}$.
However, in our case, it follows that
$\tilde{\gamma}_{01}=\tilde{\gamma}_{10}=0$! This is stemming from a simple
fact that the structure constant of scalar-scalar-stress-tensor three-point
function $\langle\mathcal{O}_{1}\mathcal{O}_{2}T\rangle$ is identically
vanishing if the two scalars are different, in other words

$\lambda_{\mathcal{O}_{1}\mathcal{O}_{2}T}\equiv 0\,,\qquad\text{if
$\mathcal{O}_{1}\neq\mathcal{O}_{2}$}.$ (2.11)

One can easily observe this fact from Ward identity associated with stress-
tensor which relates $\langle\mathcal{O}_{1}\mathcal{O}_{2}T\rangle$ to two-
point function $\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle$. As the result,
the four-point function
$\langle\mathcal{O}_{H}\mathcal{O}_{H}\mathcal{O}^{(m)}_{L}\mathcal{O}^{(n)}_{L}\rangle$
contains no single-stress-tensor and thus gives zero anomalous dimension to
its cross-channel double-twist operators. Thus the anomalous dimension matrix
in the heavy-light context is automatically diagonal and hence does not admit
the mixing.

##  3 The Regge pole of double-stress-tensor

In this section, we study the Regge behavior of the lowest-twist double-
stress-tensor, especially in the large impact parameter regime. Normally, the
Regge limit of stress-tensors can be used to obtain the cross-channel double-
twist anomalous dimensions at large spin limit by means of the impact
parameter representation [41], and the results are consistent with ones
extracted from using Lorentzian inversion formula back and forth [35]. Thus it
is concluded that we should construct the correct Regge limit from OPE
coefficients of lowest-twist multi-stress-tensors. Indeed, it can be observed
that the results of lowest-twist double-stress-tensor OPE contain poles in $J$
which are presumably the Regge pole. We would use the Regge pole to find the
Regge behavior in an efficient way as [32] did, generalizing what was worked
out in $d=4$ [32, 49] to general dimensions. However, we can only perform the
analysis for double-stress-tensor due to the lack of symbolic $J$ expression
for triple and higher stress-tensors. Nevertheless, the Regge behavior for any
lowest-twist $k$-stress-tensors in $d=4$ was achieved recently using
holography [49], where the singular behavior turns out to be ${\rm
leading}\,\sigma^{-k}+\text{next-to-leading}\,\sigma^{1-k}$, and we verify
this is indeed the case for $k=2$ in general even dimensions.

###  3.1 The Regge limit of conformal blocks

In this subsection, we would like to present the Regge limit of conformal
blocks up to next-to-leading singular order in general dimension.

Consider a four-point function decomposed into conformal blocks

$\langle\mathcal{O}_{1}(0)\mathcal{O}_{2}(z,\bar{z})\mathcal{O}_{3}(1)\mathcal{O}_{4}(\infty)\rangle={\frac{1}{(z\bar{z})^{{\frac{\Delta_{1}+\Delta_{2}}{2}}}}}\sum_{\Delta,J}c_{\Delta,J}G_{\Delta,J}(z,\bar{z})\,,$
(3.1)

the Regge limit is defined as the kinematic limit $z\rightarrow
0,\bar{z}\rightarrow 0$ with fixed $z/\bar{z}$ on the second sheet of
$\bar{z}$. To be more precise, we can reach the Regge limit by keeping $z$
fixed from the start, taking $\bar{z}$ to go around $1$ and then sending
$z,\bar{z}\rightarrow 0$. Thus we are allowed to go through the following
procedure to obtain the Regge conformal block in general dimension order by
order:

  * •

we start with the series expansion of conformal blocks (see Appendix B), pick
up the dominant terms in the limit $z\ll\bar{z}$ and then analytically
continue $\bar{z}$ by going around $1$, i.e. $(\bar{z}-1)\rightarrow
e^{2i\pi}(\bar{z}-1)$, in the end we define $z=\sigma e^{\rho},\bar{z}=\sigma
e^{-\rho}$ and take $\sigma\rightarrow 0$.

As we will see, the Regge behavior of conformal block is singular in $\sigma$
for higher spin $J>2$, and what we are interested is the most two singular
terms, i.e. leading and next-to-leading terms. The leading Regge conformal
block is well-known as the hyperbolic space propagator [50, 48] and we would
like to follow the mentioned procedure to rederive this. Typically, a certain
order of Regge conformal block comes from the same order of series expansion.
We thus consider the leading term of series expansion (B.1) in Appdenxi B. By
expanding the Gegenbauer polynomial as the series of
$z/\bar{z}\rightarrow\infty$ and picking up the dominant terms, we find the
Gegenbauer becomes

$\tilde{C}_{J}({\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}})\rightarrow{\frac{(-1)^{J}\Gamma\big{(}{\frac{d-1}{2}}\big{)}\Gamma\big{(}2J+d-2\big{)}}{\Gamma\big{(}J+d-2\big{)}\Gamma\big{(}{\frac{1}{2}}(2J+d-1)\big{)}}}\xi^{J}\,_{2}F_{1}\Big{(}{\frac{3-d-2J}{2}},-J,3-d-2J,\xi^{-1}\Big{)}\,,$
(3.2)

where $\xi$ is

$\xi={\frac{1}{2}}(1-{\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}})\,.$ (3.3)

Going to the second sheet simply flip $(\Delta\rightarrow 1-J,J\rightarrow
1-\Delta)$ with an overall factor
$-{\frac{i}{\pi}}e^{-i\pi(a+b)}(\kappa_{\Delta+J}^{a,b})^{-1}$ [12], we thus
find

$\displaystyle i\pi
e^{i\pi(a+b)}\kappa_{\Delta+J}^{a,b}G^{(0)\circlearrowleft}_{\Delta,J}$
$\displaystyle=$
$\displaystyle{\frac{2^{1-d+2\Delta}\sqrt{\pi}\Gamma\big{(}d-2\Delta\big{)}}{\Gamma\big{(}{\frac{1}{2}}(1+d-2\Delta)\big{)}\Gamma\big{(}{\frac{d}{2}}-\Delta\big{)}}}\sigma^{1-J}e^{-(1-\Delta)\rho}(1-e^{\rho})^{2(1-\Delta)}\times$
(3.6)
$\,{}_{2}F_{1}\Big{(}\Delta-1,{\frac{1-d+2\Delta}{2}},1-d+2\Delta,-{\frac{4e^{\rho}}{(1-e^{\rho})^{2}}}\Big{)}\,.$

The next-to-leading order is actually very straightforward from the recursion
(B.12) in Appendix B

$\sigma^{-1}G^{(1)\circlearrowleft}_{\Delta,J}=-\frac{1}{4}(\Delta+J-2)G^{(0)\circlearrowleft}_{\Delta,J}\big{|}_{\Delta\rightarrow\Delta-1}+\frac{(\Delta-1)(\Delta-d+2)(\Delta-
d-J+2)}{(2\Delta-d+2)(2\Delta-d)}G^{(0)\circlearrowleft}_{\Delta,J}\big{|}_{\Delta\rightarrow\Delta+1}\,.$
(3.7)

###  3.2 From Sommerfeld-Watson resummation

With the Regge conformal block in hand, we are now ready to discuss the Regge
behavior up to the next-to-leading order of the full lowest-twist double-
stress-tensor

$\mathcal{G}_{T^{2}}^{\circlearrowleft}=\sum_{J=4}c_{\Delta,J}G^{\circlearrowleft}_{\Delta,J}\,,\qquad\Delta=2(d-2)\,,$
(3.8)

where the OPE coefficient can be easily computed in even dimensions by
following the algorithm proposed in [35], and the explicit expressions in
$d=4,6,8,10$ can be found in [35], see also [34] for $d=4,6$. It can be
observed that the double-stress-tensor OPE in even dimensions has a Regge pole
located at $J=3$ that would allow us to compute (3.8) without really trying
[32]. The trick is to consider the Sommerfeld-Watson transform, writing (3.8)
as

$\mathcal{G}_{T^{2}}^{\circlearrowleft}\sim\int_{C}dJ{\frac{c_{\Delta,J}}{\sin(\pi
J)}}(1+(-1)^{J})G^{\circlearrowleft}_{\Delta,J}\,,$ (3.9)

where the integration contour $C$ is depicted with blue line in Figure 1. Then
we just deform the contour to the opposite direction as the dotted line in
Figure 1 and pick up those Regge poles $J<4$ encoded in
$(1+(-1)^{J})c_{\Delta,J}/\sin(\pi J)$.

Figure 1: Contour and the deformed contour in the complex $J$ plane.

Since the leading and the next-to-leading Regge behavior goes like
$\sigma^{1-J}$ and $\sigma^{2-J}$ respectively, the leading Regge behavior of
lowest-twist single-stress-tensor is completely determined by the Regge pole
at $J=3$ from the leading Regge conformal block (3.6) and it behaves as
$\sigma^{-2}$. In addition, the next-to-leading Regge behavior is shaped by
the Regge pole $J=3$ from the next-to-leading Regge conformal block (3.7) and
$J=2$ from leading one (3.6).

Although no pattern is found for lowest-twist double-stress-tensor OPE in
general even dimensions [35], we find that the Regge limit of lowest-twist
double-stress-tensor sector in general even dimension with large impact
parameter limit $\rho\rightarrow\infty$ is generally given by

$\displaystyle\mathcal{G}_{T^{2}}^{\circlearrowleft}=\sigma^{-2}\mathcal{F}_{0}+\sigma^{-1}(-{\frac{d}{2}}e^{\rho}\mathcal{F}_{0}+\mathcal{F}_{1})\,,$
(3.10) (3.11) $\displaystyle\mathcal{F}_{0}=-\frac{2^{2d-3}\pi
e^{-2(d-1)\rho}\Gamma(\frac{d+1}{2})^{2}\Gamma(-d+\Delta_{L}+2)\Gamma(\frac{d}{2}+\Delta_{L}+1)}{\Gamma(1+\frac{d}{2})^{2}\Gamma(\Delta_{L})\Gamma(-\frac{d}{2}+\Delta_{L}+1)}\,,$
(3.12) (3.13)
$\displaystyle\mathcal{F}_{1}=\frac{-i\pi^{{\frac{1}{2}}}2^{2d-4}e^{-(2d-3)\rho}\Gamma({\frac{1}{2}}+d)\Gamma\left(-d+\Delta_{L}+2\right)\Gamma\left(\frac{d}{2}+\Delta_{L}\right)}{\Gamma(d)\Gamma\left(\Delta_{L}\right)\Gamma\left(-\frac{d}{2}+\Delta_{L}+1\right)}\,.$
(3.14)

Specifying $d=4$, our result (3.14) agrees with the one found in [49, 32].

##  4 $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin

In [35], by using Lorentzian inversion formula, the author shows that the
large spin limit of the double-twist operators
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ at any order of
$\mathcal{O}(\mu^{k})$ are universal and behave as $J^{-k(d-2)/2}$. In fact,
this conclusion can be put forward for leading order $\mathcal{O}(\mu)$: the
double-twist operators $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ at the order
$\mathcal{O}(\mu)$ are universal for any $J$. Just notice that if our starting
point is the universal single-stress-tensor that is assumed as the only thing
contribute to linear $\mathcal{O}(\mu)$ in
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$, then exactly performing the
inversion integral can actually lead us to find the universal double-twist
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin support. This
conclusion can also be verified from holography as demonstrated in the
Appendix D: the higher derivative curvature terms start to show up at
$\mathcal{O}(\mu^{2})$. As the completion to [35], we would like to extract
the universal $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ at the order
$\mathcal{O}(\mu)$ with finite spin support by using Lorentzian inversion
formula in this section.

There is one other thing motivates us to know double-twist operators with
finite spin at the linear order. Typically, we wish we can handle the
situation when $\Delta_{L}$ approaches the corresponding poles in lowest-twist
multi-stress-tensors, especially for double-stress-tensors. It turns out the
pole arises from the integration over large spin [35], thus the naive
expectation is that the summation over finite spin can resolve this problem.
We will discuss this shortly in the next section. For this reason, we are more
interested in the anomalous dimension rather than OPE, since OPE part will not
survive under dDisc [35].

###  4.1 Warm-up: $d=4$

We first discuss $d=4$ since the full expression of conformal block in $d=4$
is known and rather simple

$\displaystyle
G^{a,b}_{\Delta,J}(z,\bar{z})={\frac{z\bar{z}}{z-\bar{z}}}(k^{a,b}_{\Delta+J}(z)k^{a,b}_{\Delta-J-2}(\bar{z})-k^{a,b}_{\Delta+J}(\bar{z})k^{a,b}_{\Delta-J-2}(z))\,,$
(4.1) (4.2) $\displaystyle
k^{a,b}_{\beta}(x)=x^{{\frac{\beta}{2}}}\,_{2}F_{1}\Big{(}a+{\frac{\beta}{2}},b+{\frac{\beta}{2}},\beta,x\Big{)}\,.$
(4.3)

The conformal block is invariant by interchanging $z$ and $\bar{z}$, thus we
can only use a half of it for simplicity and put a $2$ factor on the result.
Then we have the single-stress-tensor correlator

$\displaystyle\mathcal{G}_{T}(z,\bar{z})=-{\frac{\Delta_{L}(z-1)^{-1-\Delta_{L}}(\bar{z}-1)^{1-\Delta_{L}}(z\bar{z})^{{\textstyle{\frac{\scriptstyle\Delta_{H}+\Delta_{L}}{\scriptstyle
2}}}}\big{(}3(1-z^{2})+(z^{2}+4z+1)\log z\big{)}}{40(z-\bar{z})}}\,,$ (4.4)

where we slip off the overall $\mu$.

Now there are some differences from large spin limit in [35]. At the large
spin limit, one can simply separated (4.4) into logarithmic part and non-
logarithmic part, obtaining anomalous dimension and OPE coefficient
respectively. However, this is not true for the data with finite spin, since
the logarithmic part can also contribute to OPE coefficient at sub-leading
large spin level. In addition, the lightcone expansion of heavy inverted
conformal block $G_{J+d-1,\Delta-d+1}$ would be different from [35] due to
finite spin effects. Analyzing the Casimir recursion equation (see, e.g.
Appendix A in [12] and Appendix A in [35]) at the heavy limit but arbitrary
spin, we find

$\displaystyle\kappa^{a,b}_{\beta}\mu^{a,b}(z,\bar{z})G_{J+d-1,\Delta-d+1}(z,\bar{z})=\sum_{p=0}\tilde{B}^{a,b}_{p}\kappa^{a,b}_{\beta+2p}z^{{\frac{J-\Delta}{2}}+p-1}k^{a,b}_{\beta+2p}(\bar{z})\,,$
(4.6) (4.7)
$\displaystyle\tilde{B}^{a,b}_{p}=\frac{(-1)^{p}\Gamma\left(\frac{d}{2}\right)(J+2p)\Gamma\left(\frac{d}{2}+J\right)\Gamma(J+p)}{\Gamma(J+1)\Gamma(p+1)\Gamma\left(\frac{d}{2}-p\right)\Gamma\left(\frac{d}{2}+J+p\right)}\,,$
(4.8)

where $\kappa^{a,b}_{\beta}$ can be found in Appendix A and
$a=b=1/2(\Delta_{L}-\Delta_{H})$. To perform the inversion integral

$\int
dzd\bar{z}\sum_{p=0}\tilde{B}^{a,b}_{p}\kappa^{a,b}_{\beta+2p}z^{{\frac{J-\Delta}{2}}+p-1}k^{a,b}_{\beta+2p}(\bar{z}){\rm
dDisc}[\mathcal{G}_{T}]\,,$ (4.9)

we expand $\mathcal{G}_{T}$ in terms of $z$ and $(1-\bar{z})/\bar{z}$ and then
use the formula (A.9) termwise, in the end, we just need to work out the
remaining integral over $z$. We still work on logarithmic part and non-
logarithmic part separately just to see clearly how $\log z$ contributes to
OPE coefficient. With the unit of free OPE coefficient at the heavy limit
(2.3), we find the non-logarithmic part gives us a rather simple expression

$\tilde{c}^{(1){\rm non-
log}}_{n,J}=-{\frac{3\Delta_{L}(\Delta_{L}+2n-1)}{4(J+1)}}\,.$ (4.10)

The logarithmic part is a bit complicated, typically, the integration with
$\log z$ will provide us the factor

${\frac{1}{\big{(}\Delta-J-(\Delta_{H}+\Delta_{L}+2n)\big{)}^{2}}}\,.$ (4.11)

We parameterize $\Delta-J=\Delta_{H}+\Delta_{L}+2n^{\prime}$ and then the
inversion integral is evaluated to give us

$\displaystyle\tilde{c}(n^{\prime},J)=\frac{\left(\Delta_{L}^{2}-\Delta_{L}+6n\Delta_{L}+6n^{2}-6n\right)\Gamma(J+n+2)\Gamma\left(J+n^{\prime}+\Delta_{L}\right)}{4(J+1)\Gamma(J+n^{\prime}+2)\Gamma\left(J+n+\Delta_{L}\right)(n-n^{\prime})^{2}}\,.$
(4.12)

Now it is clear that why logarithmic part would also contribute a part of OPE
coefficient: the OPE data (4.12) contains $n^{\prime}$ in the numerator, then
evaluating the OPE data around the physical twist $n$ would result in
additional $n^{\prime}-n$ and thus give us a part of OPE coefficient.
Specifically, by ignoring the regular terms we have

$\tilde{c}(n^{\prime},J)=\frac{\left(\Delta_{L}^{2}-\Delta_{L}+6n\Delta_{L}+6n^{2}-6n\right)}{4(J+1)(n-n^{\prime})^{2}}\Big{(}1+(\psi_{n+J+2}-\psi_{n+J+\Delta_{L}})(n-n^{\prime})\Big{)}\,.$
(4.13)

Thus the OPE and anomalous dimensions can be readily read off from the pole
structures

(4.14)
$\displaystyle\tilde{\gamma}^{(1)}_{n,J}=-\frac{\left(\Delta_{L}^{2}-\Delta_{L}+6n\Delta_{L}+6n^{2}-6n\right)}{2(J+1)}\,,\qquad\tilde{c}^{(1){\rm
log}}_{n,J}=-{\frac{1}{2}}\tilde{\gamma}^{(1)}_{n,J}(\psi_{n+J+2}-\psi_{n+J+\Delta_{L}})\,.$
(4.15)

In summary, in $d=4$, we find the anomalous dimension of double-twist
operators $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin is given
by the first formula in (4.15) and the OPE coefficient is given by

$\tilde{c}^{(1)}_{n,J}=\tilde{c}^{(1){\rm non-log}}_{n,J}+\tilde{c}^{(1){\rm
log}}_{n,J}\,,$ (4.17)

where the specific details can be found in (4.10) and the second formula in
(4.15). Whenever our results are expanded in terms of $1/J$, the agreements
with [36] can be observed.

We should state that our results are valid to any $J$, including $J=0$.
Standardly, Lorentzian inversion formula may not applicable for $J<2$ [12,
13]. A simple and intuitive reason is that normally the inversion integral is
not convergent in the regime $J<2$ [13], such that the resulting OPE or
anomalous dimension poses some poles in $J$ within the region $0\leq J<2$.
Fortunately, we can see that our results with finite spin at the heavy-limit
(4.10) and (4.15) is free of any non-negative poles in $J$. In fact, we will
see shortly that the anomalous dimensions with finite spin are well-behaved in
all $J\geq 0$ in any dimensions. However, even though the OPE coefficients and
anomalous dimensions do not have any poles in $J\geq 0$, it is still possible
that the result of $J=0$ is not reliable due to the bad Regge behavior of the
correlator $\mathcal{G}$ in the dDisc: it stays bounded rather than decays
fast enough such that the arcs at infinity cannot be dropped in the derivation
of the Lorentzian inversion formula [12, 13]. In this way, the subtraction is
required such that the Regge behavior is made nicer [51], which would cause
the ambiguity of the correlator associated with spin zero. More precisely, by
perturbing the OPE and anomalous dimensions with $J=0$ appropriately, the
correlator can be enhanced to have additional crossing-symmetric part that
vanishes under dDisc, e.g. see [19], thus the OPE and anomalous dimensions at
$J=0$ obtained from the Lorentzian inversion formula are far from complete.
Nevertheless, the heavy-limit we consider removes all the subtleties. It is
worth noting the correlator $\mathcal{G}$ in our case has an overall
$(z\bar{z})^{(\Delta_{H}+\Delta_{L})/2}$ that highly suppresses the Regge
behavior of the wanted correlator by $z\bar{z}\rightarrow
0,\Delta_{H}\rightarrow\infty$. Another justification is the heavy block that
we would present in the next section (5.6), it is obvious that there is no way
to construct additional crossing-symmetric correlator associated with $J=0$.
Moreover, we can also reproduce the anomalous dimension in (4.15) from
holography by using the Hamiltonian perturbation approach developed in [21],
see Appendix D. Since there is no similar subtlety for $J=0$ has been observed
in holographic theory, we thus verify that our results with $J=0$ should hold
true. Hence, we should trust that they are analytic in all non-negative spin.
This argument is essential, since we are going to sum over conformal blocks
from $J=0$ in the next section.

###  4.2 General dimensions

The obstacle of extracting the double-twist
$[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ with finite spin is that our
knowledge of $\bar{z}\rightarrow 1$ expansion of the single-stress-tensor
conformal block in general dimension is limited. Recently, [52, 53] have
obtained the $\bar{z}\rightarrow 1$ expansion of conformal blocks with any
$(\Delta,J,a,b)$. For our purpose, we need $(\Delta=d,J=2)$ and $a=b=0$.
However, it would be more convenient to have single-stress-tensor block
expanded in term of $(1-\bar{z})/\bar{z}$ rather than $(1-\bar{z})$ as in [52,
53]. In addition, as we mentioned above, we are only interested in the
anomalous dimension in general dimensions, thus the logarithmic part of
$(1-\bar{z})/\bar{z}$ expansion of stress-tensor block is enough for our
purpose. From the geodesic Witten diagram [54], we manage to find a nice
expression for $(1-\bar{z})/\bar{z}$ expansion of stress-tensor conformal
block with parameterizing $y=z/(1-z),\bar{y}=(1-\bar{z})/\bar{z}$

$\displaystyle
G_{T}(y,\bar{y})=\sum_{k=0}\mathcal{N}_{k}{\frac{\bar{y}^{k}\,y^{{\frac{d-2}{2}}}\log\bar{y}}{y+1}}\big{(}(d-2)(3d(y+1)+2(ky+k-2y-1))g_{{\frac{d}{2}},k}(y)$
(4.18)
$\displaystyle-2\big{(}2d^{2}(y+1)+d(k(4y+3)-6y-5)+2(k-1)(ky+k-2y-1)\big{)}g_{{\frac{d-2}{2}},k}(y)\big{)}\,,$
(4.19) (4.20)

where the coefficient $\mathcal{N}_{k}$ and the function $g_{p,k}(y)$ is given
below

$\displaystyle\mathcal{N}_{k}=\frac{2^{d+1}\Gamma\left(\frac{d+3}{2}\right)\Gamma\left(\frac{d}{2}+k+1\right)}{\sqrt{\pi}(d+2k-2)(d+2k)\Gamma\left(\frac{d}{2}+1\right)\Gamma(k+1)^{2}\Gamma\left(\frac{d}{2}-k+1\right)}\,,$
(4.21) (4.22) $\displaystyle
g_{p,k}(y)=\,_{2}F_{1}\big{(}p,-k,{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}(d+2-2k),-y\big{)}\,.$ (4.23)

As far as we know, (4.20) is a new result, and the trick of this guessing job
is listed in Appendix C. We verify that our results (4.20) are consistent with
the general $\bar{z}\rightarrow 1$ expansion of conformal block found in [52,
53] up to very high order of $\bar{y}$. Taking $d=4$, we can easily verify
that (4.20) reproduces the correct logarithmic part. Then we have

$\mathcal{G}_{T}(z,\bar{z})={\frac{(z\bar{z})^{{\frac{\Delta_{H}+\Delta_{L}}{2}}}}{((1-z)(1-\bar{z}))^{\Delta_{L}}}}G_{T}(\bar{y},y)\,.$
(4.24)

We can then apply the formula (A.9) to (4.9) termwise, leading to a final
integral with (4.8) to be integrated over $z$. The remaining integral is not
possible to be done once and for all, we are supposed to expand the integrand
as a series of $z$ and extract the OPE coefficients twist by twist.
Surprisingly, we find that the results show a pattern: there is a series
representation for the anomalous dimension with finite spin!

$\displaystyle\tilde{\gamma}^{(1)}_{n,J}=\sum_{p=0}^{n}\sum_{q=0}^{n-p}{\textstyle{\frac{\scriptstyle(-1)^{p+1}\Gamma(J+1)\Gamma(q+{\frac{d+2}{2}})\Gamma(p+{\frac{d-2}{2}})\Gamma(p+q-n)\Gamma({\frac{d}{2}}-q-\Delta_{L})\Gamma(\Delta_{L}+1)\sin({\frac{\pi}{2}}(d-2\Delta_{L}))}{\scriptstyle
2\pi\Gamma(q+1)\Gamma(p+1)\Gamma({\frac{d+2}{2}}-q)\Gamma({\frac{d-2}{2}}-p)\Gamma({\frac{d}{2}}+J+p)\Gamma(1+p+q)\Gamma(-n)}}}\,.$
(4.25) (4.26)

It can be verified that for $d=4$ (4.26) comes back to the first formula in
(4.15). By the same argument, (4.26) is expected to be valid for $J=0$. The
large $n$ limit looks nice and is given by 222Note this behavior is different
from [41] where it goes $n^{d/2}$. The reason is that the limit $J\gg n\gg 1$
is taken in [41] while we keep $J$ finite.

$\tilde{\gamma}^{(1)}_{n,J}|_{n\rightarrow\infty}={\frac{(d-1)\Gamma(J+1)n^{d-2}}{\Gamma(J+d-2)}}\,.$
(4.27)

Normally, for all light operators, the large $n$ limit OPE and anomalous
dimension is dominated near the bulk-point singularity $z-\bar{z}\rightarrow
0$ (on the second sheet) where the four-point function can be mapped to the
massless four-point amplitude in flat-space [55, 56, 57, 58, 39], for which
the large $n$ limit of anomalous dimension shall predict the coefficients of
partial wave expansion of the massless amplitude with noting $n^{2}\sim s$
[19]. However, the heavy-limit where $\Delta_{H}\gg n$ seems to ruin every
nice set-ups of the standard flat space limit. What is the flat-space
amplitude the large $n$ limit of heavy-light four-point function (4.27)
corresponds to is thus a tantalizing question to ask in the future.

####  4.2.1 Explicit examples in $d=6,8,10$

For latter uses, we would like to present the explicit examples in $d=6,8,10$.

In $d=6$, we find

$\displaystyle\tilde{\gamma}^{(1)}_{n,J}=-{\frac{1}{2(J+1)(J+2)(J+3)}}\big{(}3\left(J(4n-1)+4n^{2}+6n-3\right)\Delta_{L}^{2}+(J+2n+3)\Delta_{L}^{3}$
(4.28)
$\displaystyle+2(3Jn(5n-7)+J+n(n(10n+9)-35)+3)\Delta_{L}+10(n-2)(n-1)n(2J+n+3)\big{)}\,,$
(4.29) (4.30)

In $d=8$ we have

$\displaystyle\tilde{\gamma}^{(1)}_{n,J}=-{\frac{1}{2(J+1)(J+2)(J+3)(J+4)(J+5)}}\big{(}2n^{2}(5J^{2}(9\Delta_{L}^{2}-51\Delta_{L}+77)$
(4.31)
$\displaystyle+15J(\Delta_{L}(\Delta_{L}+6)(2\Delta_{L}-9)+91)+3\Delta_{L}(\Delta_{L}(\Delta_{L}(\Delta_{L}+24)-64)-136)+1036)$
(4.32)
$\displaystyle+2n(5J^{2}(\Delta_{L}-3)(\Delta_{L}(2\Delta_{L}-9)+14)+3J(\Delta_{L}(\Delta_{L}(\Delta_{L}(\Delta_{L}+14)-124)+319)-294)$
(4.33)
$\displaystyle+2\Delta_{L}(3\Delta_{L}(\Delta_{L}(2\Delta_{L}+3)-68)+518)-840)+(J+4)(J+5)(\Delta_{L}-3)(\Delta_{L}-2)(\Delta_{L}-1)\Delta_{L}$
(4.34)
$\displaystyle+84n^{5}(J+\Delta_{L})+10n^{4}(3(7J+4)\Delta_{L}+7(J-3)J+9\Delta_{L}^{2}-42)+20n^{3}(J+\Delta_{L})(7J(\Delta_{L}-3)$
(4.35) $\displaystyle+2\Delta_{L}(\Delta_{L}+6)-42)+28n^{6}\big{)}\,.$ (4.36)

The case $d=10$ is highly complicated and is given by

$\displaystyle\tilde{\gamma}^{(1)}_{n,J}=-{\frac{1}{2(J+1)(J+2)(J+3)(J+4)(J+5)(J+6)(J+7)}}\big{(}28n^{6}(18J^{2}+9J(5\Delta_{L}-4)$
(4.37)
$\displaystyle+5\Delta_{L}(4\Delta_{L}+5)-99)+84n^{5}(J+\Delta_{L})(3J^{2}+3J(5\Delta_{L}-12)+5\Delta_{L}(\Delta_{L}+5)-99)$
(4.38)
$\displaystyle+20n^{3}(J+\Delta_{L})(7J^{2}(\Delta_{L}(4\Delta_{L}-31)+63)+7J(\Delta_{L}(\Delta_{L}(2\Delta_{L}+13)-159)+324)$
(4.39)
$\displaystyle+\Delta_{L}(\Delta_{L}+5)(\Delta_{L}(\Delta_{L}+45)-256)+2709)+2n((6J^{2}+63J+167)\Delta_{L}^{5}+5(3J(J+1)(J+7)$
(4.40)
$\displaystyle-130)\Delta_{L}^{4}-5(3J(J(13J+128)+378)+811)\Delta_{L}^{3}+10(J(J(104J+987)+2989)+2765)\Delta_{L}^{2}$
(4.41)
$\displaystyle-2(J(J(1375J+11976)+33173)+28782)\Delta_{L}+432(J(7J(J+8)+139)+105))$
(4.42)
$\displaystyle+10n^{4}(63J^{3}(\Delta_{L}-4)+42J^{2}(\Delta_{L}(4\Delta_{L}-13)-6)+21J(\Delta_{L}(\Delta_{L}(5\Delta_{L}+9)-119)+108)$
(4.43)
$\displaystyle+\Delta_{L}(\Delta_{L}(15\Delta_{L}(\Delta_{L}+15)-724)-1280)+2709)+2n^{2}(105J^{3}(\Delta_{L}-4)((\Delta_{L}-7)\Delta_{L}+15)$
(4.44)
$\displaystyle+6J^{2}(5\Delta_{L}(\Delta_{L}(3\Delta_{L}(\Delta_{L}+1)-226)+1039)-7392)+3J(5\Delta_{L}(\Delta_{L}(\Delta_{L}(\Delta_{L}(\Delta_{L}+38)-211)$
(4.45)
$\displaystyle-432)+4069)-31332)+5\Delta_{L}(\Delta_{L}(3\Delta_{L}(\Delta_{L}(5\Delta_{L}+47)-470)+1898)+5530)-57564)$
(4.46)
$\displaystyle+(J+5)(J+6)(J+7)(\Delta_{L}-4)(\Delta_{L}-3)(\Delta_{L}-2)(\Delta_{L}-1)\Delta_{L}+360n^{7}(J+\Delta_{L})+90n^{8}\big{)}\,.$
(4.47)

We can also obtain these explicit examples by using the holographic
Hamiltonian perturbation approach [21] in Appendix D.

##  5 On the $\Delta_{L}$ poles

It turns out that in general the multi-stress-tensor OPE coefficients contain
some poles in $\Delta_{L}$ [31, 32, 38, 35]. It is argued that when
$\Delta_{L}$ takes the value of those poles, the multi-stress-tensor sectors
mix with double-trace sectors $[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$ such
that the complete correlator is free of divergence [31]. In addition, the
mixed operators acquire anomalous dimensions, and it turns out that the
product of these anomalous dimensions and the relevant OPE coefficients obey
the so called Residue relation to the corresponding multi-stress-tensor OPE
coefficients [38]: taking the Residue of the relevant multi-stress-tensor OPE
coefficients at the poles correctly give rise to the product of the mixed OPE
coefficient and anomalous dimension. However, the mixed OPE is beyond our
knowledge even in the holographic context [31, 38].

In this section, we aim to discuss the situation when $\Delta_{L}$ approaches
the poles of multi-stress-tensor OPE. Our main focus is $d=4$ where we take
$\Delta_{L}=2$, we extract the mixed OPE coefficient and verify the Residue
relation. For higher dimensions, we take $d=6$ as an example to verify the
Residue relation. With the anomalous dimensions (4.36) and (4.47) in hands, we
can easily verify Residue relation in $d=8,10$, but we will not provide the
details of $d=10$ in the paper, for $d=8$ see Appendix E. We leave the
examples of the OPE coefficients in $d=6,8$ in Appendix E. In the end, we
understand the universality of the Residue relation.

###  5.1 Summation instead of integration

Typically, the $\Delta_{L}$ poles arise from the integration over $J$ in the
large spin limit [35]. It turns out that it boils down to following integral

$\int_{0}^{\infty}dJ{\frac{\bar{z}^{J}}{J^{a}}}=(-1)^{a-1}\Gamma(1-a)(\log\bar{z})^{a-1}\,.$
(5.1)

We thus see the poles $\Gamma(1-a)$ appear and whenever we take, for example,
$a=1$, the integral diverges. This divergence actually comes from the ill-
behaved integrand at $J=0$. To be clear, we can introduce the cut-off
$\epsilon\rightarrow 0$ and then we have

$\int_{\epsilon}dJ{\frac{\bar{z}^{J}}{J}}\sim\log\epsilon\,.$ (5.2)

How do we resolve this thing? Since the large spin limit of the anomalous
dimensions is ill-behaved, the most natural consideration is then to consider
the anomalous dimension with finite spin that have been worked out in the
previous section. It is apparent from the last section that the anomalous
dimensions with finite spin are well-behaved at $J=0$. Since there is no any
large spin limit presumed, we are required to sum over $J$ rather than
integrate over it in the procedure of obtaining the correlators. We would like
to present the specific procedure here.

Like [35], we need the heavy block as the ingredient for us to perform the
summation in order to obtain the required correlators. In $d=4$, we just need
to take the heavy limit of a half of the full conformal block (4.3), resulting
in

$g_{n,J}(z,\bar{z})={\frac{(z\bar{z})^{n+{\frac{\Delta_{H}+\Delta_{L}+\tilde{\gamma}}{2}}}}{\bar{z}-z}}\bar{z}^{J+1}\,,$
(5.3)

where $\tilde{\gamma}$ is the anomalous dimensions for double-twist operators.
In general dimension, we have to solve the Casimir equation for lightcone
expansion of conformal blocks with heavy limit as in [35], i.e. solve
$B^{a,b}_{pq}$ in

$G^{a,b}_{\Delta,J}(z,\bar{z})=\sum_{p}\sum_{q=-p}^{q=p}B^{a,b}_{pq}z^{{\frac{1}{2}}(2(n+p)+\Delta_{L}+\Delta_{H}+\tilde{\gamma})}\bar{z}^{{\textstyle{\frac{\scriptstyle\Delta_{H}+\Delta_{L}+\tilde{\gamma}}{\scriptstyle
2}}}+J+q+n}\,.$ (5.4)

The difference from [35] is that [35] takes the large spin limit in addition
to heavy-limit. Following Appendix A in [35] but keep $J$ finite, we find

$B^{a,b}_{pq}=\frac{\Gamma(J+1)\Gamma\left(\frac{1}{2}(d+2p-2)\right)\Gamma\left(\frac{1}{2}(d+2J-2p-2)\right)}{p!\Gamma\left(\frac{d-2}{2}\right)\Gamma\left(\frac{1}{2}(d+2J-2)\right)\Gamma(J-p+1)}\delta_{p+q,0}\,,$
(5.5)

leading us to

$g_{n,J}(z,\bar{z})=z^{{\frac{\Delta_{H}+\Delta_{L}+\tilde{\gamma}}{2}}+n}\bar{z}^{{\frac{\Delta_{H}+\Delta_{L}+\tilde{\gamma}}{2}}+n+J}\,_{2}F_{1}\big{(}{\frac{d-2}{2}},-J,2-{\frac{d}{2}}-J,{\frac{z}{\bar{z}}}\big{)}\,.$
(5.6)

Then we shall evaluate the part of correlator that remains under dDisc

$\sum_{n=0,J=0}\tilde{c}^{\rm
free}_{n,J}{\frac{\tilde{\gamma}^{(1)2}_{n,J}}{8}}(\log
z)^{2}g_{n,J}(z,\bar{z})\big{|}_{\Delta_{L}\rightarrow{\rm poles}}\,.$ (5.7)

Crossing it to $(z\rightarrow 1-\bar{z},\bar{z}\rightarrow 1-z)$ and keeping
terms up to one that produces the lowest-twist double-stress-tensor, the
resulting correlator should give us the mixed OPE coefficients and anomalous
dimensions. We only consider lowest-twist double-stress-tensor, because this
is the only thing we can trust: only the lowest-twist multi-stress-tensor is
universal [35]. For higher twist cases, there will be non-universal piece
beyond our awareness engaging in the game [36] such that the mixed results
would be unwarranted. The exception is CFT2 where there is no higher twist
double-stress-tensor exists at all! This will be discussed momentarily in
section 6. In general, as we will see in $d=6$ and $d=8$ in Appendix E, the
lowest-twist double-stress-tensor may not be leading-twist double-trace
operator $[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$, in other words, it may mix
with double-trace operators with nonzero $n=n_{0}>0$. At the order
$\mathcal{O}(\mu^{2})$, those double-trace operators are also all universal.

###  5.2 Verify the Residue relation

####  5.2.1 $d=4$

In $d=4$, the only pole is $\Delta_{L}=2$, for which the lowest-twist double-
stress-tensor mixes with lowest-twist of double-trace operator
$[\mathcal{O}_{L}\mathcal{O}_{L}]_{0,J}$. The free OPE and anomalous
dimensions simplify a lot with $\Delta_{L}=2$

$\tilde{c}^{\rm
free}_{n,J}=1+J\,,\qquad\tilde{\gamma}^{(1)}_{n,J}=-{\frac{1+3n+3n^{2}}{1+J}}\,.$
(5.8)

Evaluating from (5.7), we actually find a rather simple crossed correlator

$\mathcal{G}(z,\bar{z})\big{|}_{\log(1-\bar{z})^{2}}=-z^{2}\log({\frac{z}{2}})\,{\frac{216-432\bar{z}+264\bar{z}^{2}-48\bar{z}^{3}+\bar{z}^{4}}{16\bar{z}^{4}}}\,.$
(5.9)

Then we just need to work on the inversion integral, restricting to the
lowest-twist, (A.1) becomes

$\kappa_{\beta}\int
dzd\bar{z}\mu^{0,0}(z,\bar{z})z^{{\frac{2-\tau}{2}}}k_{\beta}^{0,0}{\rm
dDisc}[\mathcal{G}(z,\bar{z})\big{|}_{\log(1-\bar{z})^{2}}]\,.$ (5.10)

Following integral formula would be useful

$\int{\frac{d\bar{z}}{\bar{z}^{2}}}k^{0,0}_{\beta}(\bar{z})\big{(}{\frac{1-\bar{z}}{\bar{z}}}\big{)}^{a}=-{\frac{2\Gamma(a+1)^{2}\Gamma({\frac{\beta}{2}}-a)\Gamma(\beta)}{(2(a+1)-\beta)\Gamma(a+1+{\frac{\beta}{2}})\Gamma({\frac{\beta}{2}})^{2}}}\,.$
(5.11)

We thus find

$\displaystyle c(\beta,J)|_{n\rightarrow 0}=({\frac{1}{2n^{2}}}-{\frac{\log
2}{2n}})\eta(\beta)\,,\qquad\eta(\beta)=\frac{\sqrt{\pi}2^{4-\beta}\left(\beta^{4}-4\beta^{3}+92\beta^{2}-176\beta+384\right)\Gamma\left(\frac{\beta-2}{2}\right)}{(\beta-10)(\beta-6)\beta(\beta+4)(\beta+8)\Gamma\left(\frac{\beta-1}{2}\right)}\,,$
(5.12)

where $\beta=4+2J+2n$. From (E.7), the coefficient of $1/n^{2}$ immediately
gives rise to

$\displaystyle\gamma^{\rm mix}c^{\rm
mix}=-2\eta|_{\beta=4+2J}=-\frac{\sqrt{\pi}2^{-2J-2}J\left(J^{4}+6J^{3}+35J^{2}+78J+72\right)\Gamma(J-1)}{(J-3)(J+2)(J+4)(J+6)\Gamma\left(J+\frac{3}{2}\right)}\,.$
(5.14)

Having the explicit lowest-twist double-stress-tensor OPE $c_{T^{2}}$ [32, 33,
35], we can then verify the Residue relation proposed in [38]

$\gamma^{\rm mix}c^{\rm mix}=-2{\rm Res}_{\Delta_{L}=2}\,c_{T^{2}}\,.$ (5.16)

Furthermore, we can also extract the mixed OPE coefficient from (5.12)

$c^{\rm mix}=(-2\partial_{\beta}+\log 2)\eta(\beta)\big{|}_{\beta=4+2J}\,.$
(5.17)

####  5.2.2 $d=6$

In $d=6$, from the results in [35, 34], there are two poles in $\Delta_{L}$:
$\Delta_{L}=4$ and $\Delta_{L}=3$. For $\Delta_{L}=4$, the lowest-twist
double-stress-tensor mixes with the leading-twist double-trace operator
$[\mathcal{O}_{L}\mathcal{O}_{L}]_{0,J}$; for $\Delta_{L}=3$, the lowest-twist
double-stress-tensor mixes with the sub-leading-twist double-trace operator
$[\mathcal{O}_{L}\mathcal{O}_{L}]_{1,J}$, which will be naturally observed
shortly.

For $\Delta_{L}=4$, the summed correlator $\mathcal{G}$ is cumbersome, and we
are not going to present the complete expression here but just exhibit the
logarithmic part that looks much simpler

$\mathcal{G}(z,\bar{z})|_{\log(1-\bar{z})^{2},\log
z}\sim-\frac{3z^{4}\left(\bar{z}^{6}-96\bar{z}^{5}+1296\bar{z}^{4}-5900\bar{z}^{3}+11700\bar{z}^{2}-10500\bar{z}+3500\right)}{8\bar{z}^{6}}\,.$
(5.18)

Then the inversion integral like (5.10) (replace $2-\tau$ by $4-\tau$) would
give us

$\gamma^{\rm mix}c^{\rm
mix}=-\frac{3\sqrt{\pi}2^{-2J-3}\left(J^{4}+14J^{3}+101J^{2}+364J+540\right)\Gamma(J+4)}{(J-3)(J-1)(J+1)(J+6)(J+8)(J+10)\Gamma\left(J+\frac{7}{2}\right)}\,.$
(5.19)

From lowest-twist double-stress-tensor OPE in $d=6$ obtained in [35, 34], we
can verify the Residue relation

$\gamma^{\rm mix}c^{\rm mix}=-2{\rm Res}_{\Delta_{L}=4}\,c_{T^{2}}\,.$ (5.20)

By including the non-logarithmic part, we can extract the mixed OPE $c^{\rm
mix}$ which is presented in the Appendix E.

For $\Delta_{L}=3$, the summed correlator is still too complicated to be
presented here, but we would like to comment the feature we observe. We find
the correlator behaves as follows

$\mathcal{G}(z,\bar{z})=z^{3}f_{1}(\bar{z})+z^{4}f_{2}(\bar{z})+z^{4}\log
z\,f_{3}(\bar{z})\,,$ (5.21)

where $f_{i}(\bar{z})$ represents something depending on $\bar{z}$. The
logarithmic part appears with higher order in $z$! In fact, the power $z^{3}$
gives rise to the operator with twist given by $\Delta-J=6$. It is not
possible for stress-tensors to have this twist, the only possibility is the
leading-twist double-trace operator $[\mathcal{O}_{L}\mathcal{O}_{L}]_{0,J}$
which is free of mixing, implied by that it contains no $\log z$. The power
$z^{4}$ would lead to the trajectory with $\Delta-J=8$, this is a mixture from
lowest-twist double-stress-tensor and sub-leading-twist double-trace operator
$[\mathcal{O}_{L}\mathcal{O}_{L}]_{1,J}$. $f_{3}(\bar{z})$ is rather simple

$f_{3}(\bar{z})\big{|}_{\log(1-\bar{z})^{2}}=\frac{9\left(\bar{z}^{9}-168\bar{z}^{8}+2688\bar{z}^{7}-13040\bar{z}^{6}+26520\bar{z}^{5}-24000\bar{z}^{4}+8000\bar{z}^{3}\right)}{128\bar{z}^{9}}\,,$
(5.22)

Then we can evaluate the Lorentzian inversion formula to obtain

$\gamma^{\rm mix}c^{\rm
mix}=\frac{9\sqrt{\pi}2^{-2J-7}\left(J^{6}+21J^{5}+283J^{4}+2247J^{3}+10372J^{2}+25956J+27360\right)\Gamma(J+3)}{(J-3)(J-1)(J+1)(J+4)(J+6)(J+8)(J+10)\Gamma\left(J+\frac{7}{2}\right)}\,,$
(5.23)

and then verify

$\gamma^{\rm mix}c^{\rm mix}=-2{\rm Res}_{\Delta_{L}=3}\,c_{T^{2}}\,.$ (5.24)

We leave the leading-twist double-trace OPE coefficient and the mixed OPE
coefficients in Appendix E.

We actually verified the Residue relation for $d=8$ and $d=10$ by using the
obtained double-twist anomalous dimension with finite spin (4.36) and (4.47).
As in $d=6$, the common property is that the maximal pole corresponds to the
mixing with leading-twist double-trace operators, and the next pole
corresponds to sub-leading-twist and so on.

###  5.3 The universality of the Residue relation

In this subsection, we would like to try to understand the universality of the
Residue relation. Since the lowest-twist double-stress-tensor is universally
fixed by large spin region of its cross channel, the Residue relation
immediately implies that for $\Delta_{L}$ located at poles, the logarithmic
part of evaluated correlator is also completely controlled by the large spin
kinematics of cross channel. To see this fact, let us think what might be the
origin of the logarithmic behavior?

As quoted in [35], we can simply take $\Delta_{L}$ approach a certain pole,
then the large spin correlator would create a $\log z$ term in addition to the
divergent part (let us introduce a cut-off to regulate it), with coefficients
given by Residue of double-stress-tensor OPE coefficient. However, naively
doing this bothers us with the divergent part: we should not have divergence
at all with getting the correct logarithmic part. To resolve the divergence,
we shall recall that the divergence comes from $J=0$. However, remember we are
working in the large spin limit, implying $J=0$ is a fake value where we
integrate from. We can simply go from $J=1$ instead, then there will be no any
divergent term but logarithmic term kept same. To be precise, we look at the
integral

$\int_{1}^{\infty}dJ{\frac{\bar{z}^{J}}{J^{a}}}=\int_{\epsilon}^{\infty}dJ{\frac{\bar{z}^{J}}{J^{a}}}-\int_{\epsilon}^{1}dJ{\frac{\bar{z}^{J}}{J^{a}}}\,,$
(5.25)

where $\epsilon\rightarrow 0$ is the cut-off. Sending $a$ to the poles, the
logarithmic term $\log z$ after crossing only appears from the first term in
(5.25) with the divergence canceled identically by the second term. Thus the
Residue relation is guaranteed to be satisfied. Higher $1/J$ terms also create
$\log z$, but they are associated with the mixing of higher twist double-
stress-tensors. In this way, although we can say nothing about the mixed
anomalous dimension associated with higher twist double-stress-tensors because
of the lack of non-universal pieces in higher twist double-stress-tensors, we
can insist that the Residue relation does hold true.

However, the above arguments can not be applied to the mixed OPE coefficients.
It can be easily seen that different values of $J$ we integrate from brings up
different non-logarithmic parts, reflecting that the mixed OPE coefficients
are sensitive to lower spin region and thus the integration approximation
should not be applied any more. We shall start with anomalous dimensions with
finite spin obtained in section 4 and sum them over to produce the correlator
that tells us the mixed OPE coefficients, as we did in this section before.

We actually have another way to think about the Residue relation, similar to
the arguments in [38]. Specifically, we can directly analyze the conformal
blocks. We decompose the correlator into conformal blocks

$\mathcal{G}_{\mu^{2}}=\sum_{J}(c_{T^{2}}G_{2(d-2)+J,J}+\sum_{n}c_{n,J}\,G_{2\Delta_{L}+J+2n,J}+\cdots)\,,$
(5.26)

where $c_{n,J}$ is the OPE coefficient associated with the double-trace
operators $[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$ and dots represents higher
twist double-stress-tensors and other possible operators. We consider the pole
$\Delta^{p}_{L}$ of $c_{T^{2}}$ where $2\Delta^{p}_{L}-J+2n_{p}=2(d-2)$, then
it is expected that $c_{n_{p},J}$ contains the same pole $\Delta^{p}_{L}$ as
discussed in [38, 31]. In general, we may expand $c_{T^{2}}$ and $c_{n_{p},J}$
in terms of $\Delta_{L}-\Delta_{L}^{p}$

$\displaystyle
c_{T^{2}}={\frac{c_{T^{2}}^{-1}}{\Delta_{L}-\Delta_{L}^{p}}}+c_{T^{2}}^{0}+c_{T^{2}}^{1}(\Delta_{L}-\Delta_{L}^{p})+\cdots\,,$
(5.27) $\displaystyle
c_{n_{p},J}={\frac{c_{n_{p},J}^{-1}}{\Delta_{L}-\Delta_{L}^{p}}}+c_{n_{p},J}^{0}+c_{n_{p},J}^{1}(\Delta_{L}-\Delta_{L}^{p})+\cdots\,,$
(5.28)

where $c_{T^{2}}^{-1}={\rm Res}_{\Delta_{L}=\Delta_{L}^{p}}\,c_{T^{2}}$. Then
by taking $\Delta_{L}=\Delta_{L}^{p}$ we have

$\mathcal{G}_{\mu^{2}}=\sum_{J}\big{(}(c_{n_{p},J}^{0}+c_{T^{2}}^{0})G_{2\Delta_{L}+J+2n_{p},J}+{\frac{c_{n_{p},J}^{-1}+c_{T^{2}}^{-1}}{\Delta_{L}-\Delta_{L}^{p}}}G_{2\Delta_{L}+J+2n_{p},J}+c_{n_{p},J}^{-1}G_{2\Delta_{L}+J+2n_{p},J}\log(z\bar{z})\big{)}\,.$
(5.29)

It is then straightforward to see $c_{n_{p},J}^{-1}=c^{\rm mix}\gamma^{\rm
mix}/2$. In addition, the correlator at $\Delta_{L}=\Delta_{L}^{p}$ should be
well-defined, which requires us to cancel the divergence, leading to

$\gamma^{\rm mix}c^{\rm mix}=-2{\rm
Res}_{\Delta_{L}=\Delta_{L}^{p}}\,c_{T^{2}}\,.$ (5.30)

The mixed OPE coefficients are thus given by

$c^{\rm mix}=c_{n_{p},J}^{0}+c_{T^{2}}^{0}\,.$ (5.31)

We can actually clearly extract the contributions from the lowest-twist
double-stress-tensor and double-trace operator separately! In Appendix E, we
extract the full mixed OPE for all pole examples in $d=6,8$. However, it is
still not possible to recover the pure double-trace OPE coefficient before
mixing, since the coefficients that vanish at $\Delta_{L}=\Delta_{L}^{p}$ are
not known, which are, probably non-universal.

##  6 Discussion on CFT2

[35] did not consider the case of $d=2$, but its framework can actually be
applied to CFT2. In the case $d=2$, one can immediately see that all the
lowest-twist multi-stress-tensors have twist zero due to $d-2$ factor, as the
result, there are no poles in $\Delta_{L}$. This is true, in $d=2$, there is
no mixing happen to multi-stress-tensor. In fact, the enhanced conformal
symmetry, i.e. the Virasoro symmetry, allows one to gather all lowest-twist
multi-stress-tensors together to find the vacuum Virasoro conformal block
[21], and moreover, there are no higher twist multi-stress-tensors.

In this section, we would like to discuss CFT2 from the global Lorentzian
inversion formula by following the framework of [35]. By using the extracted
finite spin $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$, we shall then verify, up
to double-stress-tensor, the lowest-twist double-stress-tensor is the only
thing we have, as consistent with Virasoro symmetry. We verify this fact by
using the concept of twist conformal block [9]. These careful double-checks
may allow us to gain some intuitions about how to connect the properties of
OPE coefficients to Virasoro symmetry, which is beneficial for us to study the
asymptotic Virasoro symmetry in high dimensions in the future.

###  6.1 Finite spin double-twist operators

The way to extract the OPE coefficients and anomalous dimensions with finite
spin for double-twist operators $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ in
$d=2$ is exactly the same as in $d=4$ in section 4, even much simpler, since
the conformal block in CFT2 is very simple

$G^{a,b}_{\Delta,J}(z,\bar{z})=k^{a,b}_{\Delta+J}(z)k_{\Delta-J}^{a,b}(\bar{z})+k^{a,b}_{\Delta-J}(z)k_{\Delta+J}^{a,b}(\bar{z})\,.$
(6.1)

We can just use a half of it. The single-stress-tensor correlator is very
simple

$\mathcal{G}_{T}=-{\frac{\Delta_{L}}{2}}{\frac{(z\bar{z})^{{\frac{\Delta_{H}+\Delta_{L}}{2}}}}{((1-z)(1-\bar{z}))^{\Delta_{L}}}}\big{(}1+{\frac{1+z}{2(1-z)}}\log
z\big{)}\,.$ (6.2)

We still decompose this correlator by looking at logarithmic part and non-
logarithmic part separately for simplicity. For non-logarithmic part, we do
not even really do the calculation, since it is nothing but the free
correlator! We can immediately write the answer down

$\tilde{c}_{n,J}^{(1){\rm non-log}}=-{\frac{\Delta_{L}}{2}}\,,$ (6.3)

without any dependence on $J$, since the $J$ dependent part is nothing but
free OPE (2.3). The logarithmic part is not so trivial, but still not hard to
work out. Follow the procedure in $d=4$ case, we find

$\tilde{\gamma}^{(1)}_{n,J}=-{\frac{1}{2}}(\Delta_{L}+2n)\,,\qquad\tilde{c}^{(1)}_{n,J}=-{\frac{\Delta_{L}}{2}}-{\frac{1}{2}}\tilde{\gamma}^{(1)}_{n,J}\,(\psi_{n+J+1}-\psi_{n+J+\Delta_{L}})\,.$
(6.4)

It follows that the anomalous dimension also does not have $J$ dependence at
all! We will see in the next subsection that this is the essential origin of
the uniqueness of lowest-twist double-stress-tensor.

###  6.2 Uniqueness of double-stress-tensor

In order to go to double-stress-tensor, we have to square the anomalous
dimensions. We only keep the term that survives under dDisc

$\sum_{n=0,J=0}\tilde{c}^{\rm
free}_{n,J}{\frac{\tilde{\gamma}^{(1)2}_{n,J}}{8}}(\log
z)^{2}z^{n+{\frac{\Delta_{H}+\Delta_{L}}{2}}}\bar{z}^{n+J+{\frac{\Delta_{H}+\Delta_{L}}{2}}}\,.$
(6.5)

Since $\tilde{\gamma}^{(1)}_{n,J}$ contains no $J$, the above summation over
$J$ is exactly same as in free field theory, which is defined as the twist
conformal block [9]. In our case, we shall call it the heavy twist conformal
block

$H_{n}(z,\bar{z}):=\sum_{J}\tilde{c}^{\rm
free}_{n,J}\,G^{a,b}_{\Delta,J}(z,\bar{z})=\sum_{J}\tilde{c}^{\rm
free}_{n,J}z^{n+{\frac{\Delta_{H}+\Delta_{L}}{2}}}\bar{z}^{n+J+{\frac{\Delta_{H}+\Delta_{L}}{2}}}\,.$
(6.6)

By doing the inversion job for free field theory, we can easily find the heavy
twist conformal block

$H_{n}(z,\bar{z})={\frac{\Gamma(n+\Delta_{L})}{\Gamma(n+1)\Gamma(\Delta_{L})}}{\frac{z^{n+{\frac{\Delta_{H}+\Delta_{L}}{2}}}\bar{z}^{{\frac{\Delta_{H}+\Delta_{L}}{2}}}}{(1-\bar{z})^{\Delta_{L}}}}\,.$
(6.7)

Thus the required correlator can be evaluated by

$\sum_{n=0,J=0}{\frac{\tilde{\gamma}^{(1)2}_{n,J}}{8}}(\log
z)^{2}H_{n}(z,\bar{z})\,,$ (6.8)

and then go to the cross channel by $(z\rightarrow
1-\bar{z},\bar{z}\rightarrow 1-z)$. We end up with

$\mathcal{G}_{T^{2}}(z,\bar{z})\big{|}_{\log(1-\bar{z})^{2}}={\frac{\Delta_{L}\big{(}4(1-\bar{z})+\Delta_{L}(2-\bar{z})^{2}\big{)}}{32\bar{z}^{2}}}\,.$
(6.9)

(6.9) can be immediately verified to be true by comparing with the vacuum
Virasoro conformal block [21]

$\mathcal{V}_{0}=z^{\Delta_{L}}(1-z)^{-{\frac{\Delta_{L}}{2}}(1-\alpha)}\big{(}{\frac{1-(1-z)^{\alpha}}{\alpha}}\big{)}^{-\Delta_{L}}\,.\qquad\alpha=\sqrt{1-\mu}\,.$
(6.10)

Expanding (6.10) in terms of $\mu$ and crossing it $(z\rightarrow
1-\bar{z},\bar{z}\rightarrow 1-z)$ and then extracting the term with $\mu^{2}$
and $\log(1-\bar{z})^{2}$, we can precisely obtain (6.9). It is apparent from
(6.9) that it contains no $z$ dependence, implying it encodes and only encodes
$n=0$ trajectory, i.e. the lowest-twist double-stress-tensor. We shall call
this the uniqueness of double-stress-tensor in CFT2. Using the Lorentzian
inversion formula with putting (6.9) in the dDisc, we can easily find the
double-stress-tensor OPE given by

$c_{T^{2}}={\frac{2^{-1-2J}\sqrt{\pi}\Delta_{L}\big{(}4+(J^{2}-J-2)\Delta_{L}\big{)}\Gamma(J+1)}{(J-3)J(J+2)\Gamma(J-{\frac{1}{2}})}}\,,$
(6.11)

exactly agrees with one found in [33].

In CFT2, it is understood that this uniqueness and universality of double-
stress-tensor is guaranteed by the Virasoro symmetry, but from the global
inversion perspective, it appears that the $J$ independent anomalous
dimensions should be responsible for the uniqueness and universality. In fact,
the origin of this $J$-independence is the $\bar{z}$-independence of the (a
half of) crossed single-stress-tensor conformal block. Looking at the single-
stress-tensor conformal block in $d=2$, we can easily find that $z$-dependence
and $\bar{z}$-dependence is completely factorized. This implies that the
underlying symmetry can be factorized to pure holomorphic and pure anti-
holomorphic part, exactly as Virasoro symmetry indicates. The uniqueness of
double-stress-tensor can be directly understood from this holomorphic
factorization: higher twists $n>0$ are coming with the derivative
$\partial_{z}\partial_{\bar{z}}$ that mixes the holomorphic sector and anti-
holomorphic sector, thus they are enforced to be vanishing, and the
universality comes with this uniqueness.

So the question is then what can we learn about higher dimensional version of
Virasoro-like structure analogous to $d=2$? We should start from the
properties of double-stress-tensor OPE in high dimension. There is no
uniqueness at all, which simply implies that CFT in high dimension does not
have the nice holomorphic factorization. Nevertheless, the lowest-twist
double-stress-tensor shows the universality (ignoring the case that
$\Delta_{L}$ takes values at poles), indicating that the lowest-twist double-
stress-tensor might be controlled by the symmetry similar to Virasoro
symmetry. We can imagine that some operators $(L_{m},\bar{L}_{m})$ can be
constructed from stress-tensor operator by following the construction in
$d=2$, but in general

$[L_{m},\bar{L}_{n}]\neq 0\,.$ (6.12)

However, it is plausible that near the lightcone limit $L_{m}$ and
$\bar{L}_{n}$ is asymptotically commutable

$L_{m}=L^{(0)}_{m}+\cdots\,,\qquad\bar{L}_{m}=\bar{L}^{(0)}_{m}+\cdots\,,\qquad[L^{(0)}_{m},\bar{L}^{(0)}_{n}]=0\,,$
(6.13)

where dots represent terms that are suppressed in the lightcone limit. Then
the commutable subsectors $L^{(0)}_{m}$ form Virasoro-like algebra that
controls lowest-twist multi-stress-tensors. This asymptotic Virasoro structure
was constructed recently in [29, 30].

##  7 Conclusion

In this paper, we continued studying heavy-light four-point function up to
double-stress-tensor, providing the supplements to previous work [35].

We analyzed the leading and next-to-leading Regge behavior (in the large
impact regime) of lowest-twist double-stress-tensor in general even dimensions
by noting the Regge poles in the lowest-twist double-stress-tensor OPE. Then
we used the Lorentzian inversion formula to extend the universality of double-
twist operators $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ at the linear order
$\mathcal{O}(\mu)$ from large spin to finite spin. More specifically, we
extracted the OPE coefficients and anomalous dimension of double-twist
operators with finite spin by using the Lorentzian inversion formula where the
large spin expansion of our results agree with those found in [36]. We also
took advantage of geodesic Witten diagram to find $\bar{z}\rightarrow 1$
expansion for the logarithmic part of single-stress-tensor block in general
dimension, which allowed us to bootstrap the anomalous dimension of double-
twist operators with finite spin in any dimension. The fact that these
anomalous dimensions with finite spin are consistent with holographic
computations were observed.

By using these anomalous dimension with finite spin, we addressed $\Delta_{L}$
poles. Specifying $\Delta_{L}$ to specific poles, we found that we can summed
over spin and twist to obtain $\mathcal{O}(\mu^{2})$ correlators without
divergence in $\Delta_{L}$, from which it is obvious to see the mixing between
double-trace operators $[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$ with certain
twist $n_{0}$ and the lowest-twist double-stress-tensor. The mixed anomalous
dimension arises in the mixing trajectory and we verified that the product of
the mixed anomalous dimension and the mixed OPE coefficient obeys the Residue
relation quoted in [38]. The universality of this Residue relation was also
understood. Besides, the double-trace OPE coefficients with $n\leq n_{0}$ and
the mixed OPE coefficients associated with all poles in $d=4,6$ were computed.
Furthermore, we considered CFT2 within the framework of [35] and this paper.
We extracted the OPE coefficients and anomalous dimensions of double-twist
operators $[\mathcal{O}_{L}\mathcal{O}_{L}]_{n,J}$ with finite spin in $d=2$,
and in particular the anomalous dimensions are actually independent of $J$.
From $J$-independent anomalous dimension, we found we were led to the
uniqueness of double-stress-tensor. In other words, the lowest-twist double-
stress-tensor is only thing exists in the double-stress-tensor sector of
heavy-light four-point function in $d=2$, which is consistent with Virasoro
symmetry [21].

There are other tantalizing questions to investigate in the future. It is
especially interesting to think about how to study heavy-light four-point
function from AdS/CFT in a more formal way, verifying the validity of the
framework proposed in [31]. In other words, how do we take the heavy-limit of
Witten diagram in general higher derivative gravities? This question could
also shed the light on the question of what is the scattering amplitudes
corresponding to the flat-space limit of heavy-light four-point function.

## Acknowledgement

We are grateful to Nikhil Anand, Simon Caron-Huot, Wenliang Li and Hong Lu for
helpful discussions. This work is supported in part by the NSFC (National
Natural Science Foundation of China) Grant No. 11935009 and No. 11875200.

##  Appendix A Lorentzian inversion formula

In this Appendix, we briefly review the Lorentzian inversion formula [12, 13,
14]

$c(\Delta,J)={\frac{1+(-1)^{J}}{4}}\kappa^{a,b}_{\Delta+J}\int
dzd\bar{z}\,\mu^{a,b}(z,\bar{z})G^{a,b}_{J+d-1,\Delta-d+1}(z,\bar{z}){\rm
dDisc}[\mathcal{G}(z,\bar{z})]\,,$ (A.1)

where

$\displaystyle\mu^{a,b}(z,\bar{z})=\Big{|}{\frac{z-\bar{z}}{z\bar{z}}}\Big{|}^{d-2}{\frac{\big{(}(1-z)(1-\bar{z})\big{)}^{a+b}}{(z\bar{z})^{2}}}\,,$
(A.2)
$\displaystyle\kappa^{a,b}_{\beta}={\frac{\Gamma({\frac{\beta}{2}}-a)\Gamma({\frac{\beta}{2}}+a)\Gamma({\frac{\beta}{2}}-b)\Gamma({\frac{\beta}{2}}+b)}{2\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}}\,.$
(A.3)

and ${\rm dDisc}$ is the the double-discontinuity given by the double
commutator
$\langle[\mathcal{O}_{1}\mathcal{O}_{4}][\mathcal{O}_{2}\mathcal{O}_{3}]\rangle$
that can be evaluated by

${\rm
dDisc}[\mathcal{G}(z,\bar{z})]=\cos(\pi(a+b))\mathcal{G}(z,\bar{z})-{\frac{e^{-i(a+b)}}{2}}\mathcal{G}^{\circlearrowleft}(z,\bar{z})-{\frac{e^{i(a+b)}}{2}}\mathcal{G}^{\circlearrowright}(z,\bar{z})\,,$
(A.4)

where $\mathcal{G}^{\circlearrowleft}$ and $\mathcal{G}^{\circlearrowright}$
are two different analytic continuations for $\bar{z}$ around $1$. We can set
$(-1)^{J}=1$ in this paper since the exchanged operators are expected to have
even spin. The physical spectrum and the corresponding OPE coefficients are
encoded in the OPE data $c(\Delta,J)$ [12]

$c(\Delta,J)=-{\frac{c_{\Delta_{\rm phys},J}}{\Delta-\Delta_{\rm phys}}}\,,$
(A.5)

where $\Delta_{\rm phys}$ is the physical spectrum including the anomalous
dimension

$\Delta_{\rm phys}=\Delta_{\rm bare}+\gamma_{\rm phys}\,,$ (A.6)

where $\Delta_{\rm bare}$ is the bare physical spectrum at the degenerate
point or in free field theory.

In the end, there is an integral formula that is used throughout this paper
[12]

$\displaystyle
I^{a,b}_{\hat{\tau}}(\beta)=\int_{0}^{1}{\frac{d\bar{z}}{\bar{z}^{2}}}(1-\bar{z})^{a+b}\kappa^{a,b}_{\beta}k^{a,b}_{\beta}(\bar{z})\,{\rm
dDisc}[\big{(}{\frac{1-\bar{z}}{\bar{z}}}\big{)}^{{\frac{\hat{\tau}}{2}}-b}(\bar{z})^{-b}]$
(A.7) (A.8)
$\displaystyle={\frac{\Gamma({\frac{\beta}{2}}-a)\Gamma({\frac{\beta}{2}}+b)\Gamma({\frac{\beta}{2}}-{\frac{\hat{\tau}}{2}})}{\Gamma(-{\frac{\hat{\tau}}{2}}-a)\Gamma(-{\frac{\hat{\tau}}{2}}+b)\Gamma(\beta-1)\Gamma({\frac{\beta}{2}}+{\frac{\hat{\tau}}{2}}+1)}}\,.$
(A.9)

##  Appendix B Series expansion of conformal blocks

In this Appendix, we review the series expansion of conformal block with using
the Casimir operators [59].

We consider the series expansion of conformal blocks in $u=z\bar{z}\rightarrow
0$ [59]

$G^{a,b}_{\Delta,J}=\sum_{n=0}\sum_{m=-n}^{n}\mathcal{A}^{a,b}_{nm}\mathcal{P}_{\Delta+n,J+m}(z,\bar{z})\,,$
(B.1)

where $\mathcal{P}_{\Delta,J}(z,\bar{z})$ can be simply expressed as the
Gegenbauer polynomial

$\displaystyle\mathcal{P}_{\Delta,J}(z,\bar{z})={\frac{2^{3-d-J}\sqrt{\pi}\Gamma(J+d-2)}{\Gamma({\frac{d-1}{2}})\Gamma({\frac{d}{2}}+J-1)}}(z\bar{z})^{{\frac{\Delta}{2}}}\tilde{C}_{J}({\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}})\,,$
(B.2) (B.3)
$\displaystyle\tilde{C}_{J}(x)={\frac{\Gamma(d+J-2)}{\Gamma(d-2)\Gamma(J+1)}}C_{J}^{{\frac{d}{2}}-1}(x)=\,_{2}F_{1}(-J,d+J-2,\frac{d-1}{2},\frac{1-x}{2})\,.$
(B.4)

Then the task is to find out the coefficients $\mathcal{A}^{a,b}_{nm}$. The
trick is to use the (quadratic) Casimir operator, since the conformal block is
the eigenfunction of the Casimir operator, i.e.

$\displaystyle\mathcal{C}_{2}\,G^{a,b}_{\Delta,J}(z,\bar{z})=C_{\Delta,J}\,G^{a,b}_{\Delta,J}(z,\bar{z})\,,$
(B.5) $\displaystyle C_{\Delta,J}=\Delta(\Delta-d)+J(J+d-2)\,,$ (B.6)

where

$\displaystyle\mathcal{C}_{2}=\mathcal{D}_{z}+\mathcal{D}_{\bar{z}}+2(d-2){\frac{z\bar{z}}{z-\bar{z}}}((1-z)\partial_{z}-(1-\bar{z})\partial_{\bar{z}})\,,$
(B.7)
$\displaystyle\mathcal{D}_{z}=2(z^{2}(1-z)\partial_{z}^{2}-(1+a+b)z^{2}\partial_{z}-abz)\,.$
(B.8)

In fact, it turns out that the Casimir operator $\mathcal{C}_{2}$ can be
separated into two parts $\mathcal{C}_{2}^{1}$ and $\mathcal{C}_{2}^{2}$ where
$\mathcal{C}_{2}^{1}$ is homogeneous in $u$ and $\mathcal{C}_{2}^{2}$ shifts
$u$ by one unit [59]. Specifically, it was found [59]

$\displaystyle\mathcal{C}_{2}^{1}\mathcal{P}^{a,b}_{\Delta,J}=C_{\Delta,J}\mathcal{P}^{a,b}_{\Delta,J}\,,\qquad\mathcal{C}_{2}^{2}\mathcal{P}^{a,b}_{\Delta,J}=\gamma^{-,a,b}_{\Delta,J}\mathcal{P}^{a,b}_{\Delta+1,J-1}+\gamma^{+,a,b}_{\Delta,J}\mathcal{P}^{a,b}_{\Delta+1,J+1}\,,$
(B.9)

where

$\displaystyle\gamma^{+,a,b}_{\Delta,J}=(\Delta+J+2a)(\Delta+J+2b)\,,$ (B.10)
$\displaystyle\gamma^{-,a,b}_{\Delta,J}=\frac{J(d+J-3)(-2a+d-\Delta+J-2)(-2b+d-\Delta+J-2)}{(d+2J-4)(d+2J-2)}\,.$
(B.11)

Then one can have the recursion equation for determining
$\mathcal{A}^{a,b}_{nm}$

$(C_{\Delta+n,J+m}-C_{\Delta,J})\mathcal{P}^{a,b}_{\Delta,J}=\gamma^{+,a,b}_{\Delta+n-1,J+m-1}\mathcal{P}^{a,b}_{\Delta+n-1,J+m-1}+\gamma^{-,a,b}_{\Delta+n-1,J+m+1}\mathcal{P}^{a,b}_{\Delta+n-1,J+m+1}\,.$
(B.12)

##  Appendix C Stress-tensor block from geodesic Witten diagram

In this Appendix, we would briefly review spin-$2$ geodesic Witten diagram
that serves as the integral representation of the stress-tensor conformal
block [54] and then we would describe the trick how we derive the logarithmic
part in $\bar{y}$ expansion of stress-tensor conformal block (4.20), (4.23).

The geodesic Witten diagram is defined holographically in AdSd+1

$\displaystyle\mathcal{W}_{\Delta,J}(x_{i})=\int_{\gamma_{12},\gamma_{34}}G_{b\partial}(y(\lambda),x_{1})G_{b\partial}(y(\lambda),x_{2})G_{bb}(y(\lambda),y(\lambda^{\prime});\Delta,J)G_{b\partial}(y(\lambda^{\prime}),x_{3})G_{b\partial}(y(\lambda^{\prime}),x_{4})\,,$
(C.1) (C.2)

where $G_{b\partial}$ and $G_{bb}$ are bulk-to-boundary and bulk-to-bulk
propagators respectively. Roughly speaking, the geodesic Witten diagram is
just Witten diagram but it integrates along the geodesics that connect the
boundary points $x_{1,3}$ and $x_{2,4}$. It is proved that the geodesic Witten
diagram satisfies the Casimir equation and thus should be viewed as the
conformal block with exchanging $(\Delta,J)$. In our case $(\Delta=d,J=2)$
where (denote $y(\lambda^{\prime})=y^{\prime}$)

$G_{bb}\sim(G_{bb})_{\mu\nu\rho\sigma}{\frac{\partial
y^{\mu}}{\partial\lambda}}{\frac{\partial
y^{\nu}}{\partial\lambda}}{\frac{\partial
y^{\prime\rho}}{\partial\lambda^{\prime}}}{\frac{\partial
y^{\prime\sigma}}{\partial\lambda^{\prime}}}\,.$ (C.3)

We are mainly interested in the case with $a=b=0$, and one may set this from
now on such that the calculation would be more smooth, however, this is not
permitted, since setting $a=b=0$ in (C.2) would cause the divergence of the
integration. The right way is to work out (C.2) at first and then set $a=b=0$,
which is rather complicated. Let us not be so ambitious, we are only
interested in the logarithmic part of the conformal block which is closely
related to the fake divergence, thus the trick is to regulate the divergence
in $a$ or $b$ and its coefficients shall be what we are looking for. For
simplicity, we may set $b=0$ and work on (C.2) to find the regulated
divergence $1/a$. Within the conformal frame $(z,\bar{z})$, we can parametrize
the geodesics as [54]

$\displaystyle ds^{2}={\frac{1}{u^{2}}}(du^{2}+dzd\bar{z}+\cdots)\,,\qquad
u={\frac{\sqrt{z\bar{z}}}{2\cosh\lambda}}\,,$ (C.4) (C.5) $\displaystyle
z(\lambda)={\frac{2-z}{2}}-{\frac{z}{2}}\tanh\lambda\,,\qquad\bar{z}(\lambda)={\frac{2-\bar{z}}{2}}-{\frac{\bar{z}}{2}}\tanh\lambda\,,$
(C.6)

then we find

$\mathcal{W}_{d,2}=\int_{0}^{1}d\sigma{\textstyle{\frac{\scriptstyle\Gamma(d+2)}{\scriptstyle\Gamma(1-a+{\frac{d}{2}})\Gamma(1+a+{\frac{d}{2}})}}}{\textstyle{\frac{\scriptstyle(z\bar{z})^{{\frac{d-2}{2}}}((1-z\sigma)(1-\bar{z}\sigma))^{{\frac{d-2}{2}}}\sigma^{{\frac{d}{2}}+a}(1-\sigma)^{{\frac{d}{2}}-a}}{\scriptstyle
d(1-z(1-\bar{z})\sigma-\bar{z}\sigma)^{2}}}}\mathcal{F}(z,\bar{z},\sigma)\,,$
(C.7)

where

$\displaystyle\mathcal{F}(z,\bar{z},\sigma)=d(\sigma^{2}z^{4}(\bar{z}-1)^{2}(2\sigma\bar{z}(\sigma\bar{z}-1)+1)-2\sigma
z^{3}(\bar{z}-1)(\sigma\bar{z}(\sigma\bar{z}-1)(2\sigma\bar{z}+\bar{z}-3)+\bar{z}-1)$
(C.8)
$\displaystyle+z^{2}(\bar{z}(-6\sigma+\bar{z}(\sigma(14\sigma+\bar{z}(\sigma(-2(5\sigma+4)+2\sigma(\sigma+2)\bar{z}+\bar{z})-2)+4)+2)-2)+1)$
(C.9)
$\displaystyle-2z\bar{z}(\sigma\bar{z}-1)^{2}(\sigma\bar{z}+\bar{z}-1)+\bar{z}^{2}(\sigma\bar{z}-1)^{2})-2z\bar{z}(\sigma
z-1)(\sigma\bar{z}-1)(\sigma^{2}(z(-\bar{z})+z+\bar{z})^{2}$ (C.10)
$\displaystyle+2\sigma z(\bar{z}-1)+z(\bar{z}-1)-2\sigma\bar{z}-\bar{z}+2)\,,$
(C.11)

with $\sigma=e^{2\lambda^{\prime}}/(1+e^{2\lambda^{\prime}})$.

Then our trick of the guessing job to find (4.20) is as follows

  * •

Expand (C.7) as the series of $\bar{y}$ at first, integrate it over $\sigma$
and then send $a\rightarrow 0$ to keep terms with $1/a$. This remaining term
with $1/a$ shall be the logarithmic part, indicating the replacement
$1/a\rightarrow\log\bar{y}$. Expand the resulting expressions as the series of
$y$ for each $\bar{y}^{k}$, it is made possible to find patterns for the
coefficients associated with $y^{m}$ and in the end the pattern for
coefficients $C_{km}$ associated with $y^{m}\bar{y}^{k}$ can be found. With
$C_{km}$ in hand, (4.20) can be deduced by

$\sum_{k}\sum_{m=0}^{k}C_{km}y^{m}\bar{y}^{k}\log\bar{y}\,.$ (C.12)

##  Appendix D $[\mathcal{O}_{H}\mathcal{O}_{L}]_{n,J}$ from holography

In this Appendix, we extract the double-twist anomalous dimensions with finite
spin by using the holographic Hamiltonian perturbation theory [6, 21], and it
is observed that the holographic results precisely agree with the results in
the main text from Lorentzian inversion formula.

From the holographic perspective, the conformal dimension of
$[\mathcal{O}_{1}\mathcal{O}_{2}]_{n,J}$ is interpreted as the energy of the
bound state of two particles $1$ and $2$ circling around each other. In the
heavy-limit we consider, the picture is that the light particle is circling
around the black hole, and the anomalous dimension is the energy-shift of that
particle caused by the interaction between itself and the black hole. We
consider the general higher derivative gravity minimally coupled to massive
scalars in bulk, giving the Lagrangian as

$\displaystyle
L=R-2\Lambda+H(R_{\mu\nu\rho\sigma})-\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi+\Delta_{L}(\Delta_{L}-d)\phi^{2})\,,$
(D.1)

where $H(R_{\mu\nu\rho\sigma})$ is the generic higher derivative terms. We
consider the spherical symmetric black hole constructed from the gravity
sector and the scalar fields are considered as the field that creates
particles moving around such a black hole. Typically, we should consider the
massless higher derivative gravity where the massive modes are decoupled [60]
to guarantee there is no extra operators as input in addition to single-
stress-tensor [38]. In this way, by dimensional analysis, the general metric
ansatz is

$ds^{2}=fdt^{2}+h^{-1}dr^{2}+r^{2}d\Omega_{d-1}^{2}\,,$ (D.2)

where

$\displaystyle
f=1+r^{2}-\frac{\mu}{r^{d-2}}+\sum_{k=2}^{\infty}\frac{\mu^{k}f_{k}}{r^{kd-2}}$
(D.3) $\displaystyle
h=1+r^{2}-\frac{\mu}{r^{d-2}}+\sum_{k=2}^{\infty}\frac{\mu^{k}h_{k}}{r^{kd-2}}\,,$
(D.4)

where $f_{k}$ and $h_{k}$ depend on the detail of higher derivative terms. To
compute the anomalous dimension, we should just do quantum mechanics on this
black hole background: write down the Hamiltonian $H=H_{0}+H_{\rm int}$ where
$H_{0}$ is the free Hamiltonian and $H_{\rm int}$ represents the interaction,
and then compute $\langle n,J|H_{\rm int}|n,J\rangle$. Up to the order
$\mathcal{O}(\mu)$, we find

$\displaystyle H_{\rm int}=\mu\int
d\Omega_{d-1}dr\bigg{(}-\frac{r\dot{\phi}^{2}}{2(1+r^{2})^{2}}-\frac{r}{2}(\partial_{r}\phi)^{2}\bigg{)}\,,$
(D.5)

where all theory dependent terms $f_{k}$ make no contributions, implies that
the anomalous dimensions of double-twist operators are exactly universal at
the order $\mathcal{O}(\mu)$. The quantization can be standardly performed

$\phi=\sum_{n,J,j}(a^{\dagger}_{n,J,j}\psi_{n,J,j}+a_{n,J,j}\psi^{*}_{n,J,j})\,,$
(D.6)

where $a^{\dagger}$ and $a$ are creation operator and annihilation operator
respectively and the mode function $\psi_{n,J,j}$ is given by [61]

$\psi_{n,J,j}(t,r,\Omega)=\frac{1}{N_{\Delta_{L},n,J}}e^{-iE_{n,J}t}Y_{J,j}(\Omega)\frac{r^{J}}{(1+r^{2})^{\frac{\Delta_{L}+J}{2}}}\,_{2}F_{1}\Big{(}-n,\Delta_{L}+J+n,J+\frac{d}{2},\frac{r^{2}}{r^{2}+1}\Big{)}\,,$
(D.7)

where

$\displaystyle E_{n,J}=\Delta_{L}+2n+J\,,$ (D.8) $\displaystyle
N_{\Delta,n,J}=(-1)^{n}\bigg{(}\frac{n!\Gamma(J+\frac{d}{2})\Gamma(\Delta+n-\frac{d-2}{2})}{\Gamma(n+J+\frac{d}{2})\Gamma(\Delta+n+J)}\bigg{)}^{{\frac{1}{2}}}\,.$
(D.9)

It is then straightforward to compute the anomalous dimension by

$\displaystyle\mu\tilde{\gamma}^{(1)}_{n,J}=\langle n,J,j|H_{\rm
int}|n,J,j\rangle\,,$ (D.10) $\displaystyle=\mu\int
drd\Omega\bigg{(}-\frac{r\partial_{t}\psi_{n,J,j}\partial_{t}\psi_{n,J,j}^{*}}{(1+r^{2})^{2}}-r\partial_{r}\psi_{n,J,j}\partial_{r}\psi_{n,J,j}^{*}\bigg{)}\,.$
(D.11)

In the literature, it is normal to just keep
$\partial_{t}\psi_{n,J,j}\partial_{t}\psi_{n,J,j}^{*}$ and drop
$\partial_{r}\psi_{n,J,j}\partial_{r}\psi_{n,J,j}^{*}$, because
$\partial_{r}\psi_{n,J,j}\partial_{r}\psi_{n,J,j}^{*}$ only contributes $1/J$
correction [21, 6, 41]. Substituting (D.7) into (D.11), we can work out the
integral $n$ by $n$ and find the pattern, and the results precisely agree with
the anomalous dimensions obtained in the main text for $d=4,6,8,10$ (4.15),
(4.30), (4.36) and (4.47).

##  Appendix E Double-trace and mixed OPE coeffcieints

The expressions of mixed OPE coefficients and the OPE coefficients associated
with those double-trace operators (without mixing with double-stress-tensor)
are extremely complicated, thus we put what we find for $d=6$ and $d=8$ as
examples in this Appendix. The complexity of these OPE coefficients are
resulted from complicated double-twist anomalous dimensions in section 4: the
resulting correlator with summing $n$ and $J$ over would contain
transcendental functions. In particular, there will be Lerch transcendents
$\Phi\big{(}(1-z)^{2},2,a/2\big{)}$ where $a$ is odd integer. These functions
can actually be decomposed into some PolyLog functions ${\rm Li}_{2}$, for
examples

$\displaystyle\Phi(x,2,{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}})=-\frac{2\left(\text{Li}_{2}\left(-\sqrt{x}\right)-\text{Li}_{2}\left(\sqrt{x}\right)\right)}{\sqrt{x}}\,,$
(E.1) (E.2) $\displaystyle\Phi(x,2,{\textstyle{\frac{\scriptstyle
3}{\scriptstyle
2}}})=-\frac{2\left(\text{Li}_{2}\left(-\sqrt{x}\right)-\text{Li}_{2}\left(\sqrt{x}\right)+2\sqrt{x}\right)}{x^{3/2}}\,,$
(E.3) (E.4) $\displaystyle\Phi(x,2,{\textstyle{\frac{\scriptstyle
5}{\scriptstyle
2}}})=-\frac{2\left(9\text{Li}_{2}\left(-\sqrt{x}\right)-9\text{Li}_{2}\left(\sqrt{x}\right)+2(x+9)\sqrt{x}\right)}{9x^{5/2}}\,.$
(E.5)

Although the expressions of correlators are quite complicated, we still can
expand the correlators in terms of $z$ up to the order where lowest-twist
double-stress-tensor appears and then do the inversion integral to obtain
$c(\Delta,J)$. It is convenient to expand $c(\Delta,J)$ in terms of
$n=1/2(\beta-2J-\tau_{0})\rightarrow 0$ where $\tau_{0}$ is the physical twist
and $\beta=\Delta+J$, to write (A.5) as

$c(\Delta,J)={\frac{S_{-1}(\beta)}{n}}+{\frac{S_{-2}(\beta)}{n^{2}}}+\cdots\,,$
(E.6)

where the coefficients $S_{-1}(\beta)$ and $S_{-2}(\beta)$ are related to OPE
coefficients and anomalous dimensions by

$\displaystyle
c_{\tau_{0},J}=-2(S_{-1}+2\partial_{\beta}S_{-2})\big{|}_{\beta=\tau_{0}+2J}\,,\qquad
c_{\tau_{0},J}\gamma_{\tau_{0},J}=-4S_{-2}\,.$ (E.7)

Below we would just present $S_{-1}$ and $S_{-2}$ for all examples we
consider. Particularly, it would be useful to denote

$\mathcal{I}_{a}={\frac{2^{4-\beta}\Gamma(a+1)^{2}\Gamma({\frac{\beta}{2}}-a-1)\Gamma({\frac{\beta}{2}})}{\pi^{{\frac{3}{2}}}\Gamma({\frac{\beta-1}{2}})\Gamma({\frac{\beta}{2}}+a+1)}}\,.$
(E.8)

###  E.1 $d=6$

In $d=6$, there are two poles $\Delta_{L}=4,3$ [35, 34].

####  E.1.1 $\Delta_{L}=4$

For $\Delta_{L}=4$, the lowest-twist double-stress-tensor mixes with leading-
twist double-trace operator $[\mathcal{O}_{L}\mathcal{O}_{L}]_{0,J}$. The
mixed OPE data is recorded as

$\displaystyle
S_{-2}=\frac{3}{2}\pi^{2}\left(\mathcal{I}_{0}+96\mathcal{I}_{1}+1296\mathcal{I}_{2}+5900\mathcal{I}_{3}+11700\mathcal{I}_{4}+10500\mathcal{I}_{5}+3500\mathcal{I}_{6}\right)\,,$
(E.9)

and

$\displaystyle
S_{-1}=6\pi^{2}(21\mathcal{I}_{1}+2215\mathcal{I}_{2}+46246\mathcal{I}_{3}+383283\mathcal{I}_{4}+1626920\mathcal{I}_{5}+3960290\mathcal{I}_{6}+5772550\mathcal{I}_{7}$
(E.10)
$\displaystyle+4989950\mathcal{I}_{8}+2362500\mathcal{I}_{9}+472500\mathcal{I}_{10})-\frac{3}{2}\pi^{2}\log
2\,(3\mathcal{I}_{0}+744\mathcal{I}_{1}+28024\mathcal{I}_{2}+381740\mathcal{I}_{3}$
(E.11)
$\displaystyle+2557220\mathcal{I}_{4}+9695140\mathcal{I}_{5}+22217580\mathcal{I}_{6}+31393600\mathcal{I}_{7}+26748400\mathcal{I}_{8}+12600000\mathcal{I}_{9}$
(E.12) $\displaystyle+2520000\mathcal{I}_{10})\,.$ (E.13)

It is obvious that (E.9) gives us (5.23) that satisfies the Residue relation.

####  E.1.2 $\Delta_{L}=3$

For $\Delta_{L}=3$, the lowest-twist double-stress-tensor mixes with sub-
leading-twist double-trace, we then would like to extract both the leading-
twist (non-mixing) double-trace OPE and the mixed OPE.

  * 1.

Leading-twist

$\displaystyle
S_{-1}=\frac{9}{16}\pi^{2}\big{(}(31\mathcal{I}_{1}+2490\mathcal{I}_{2}+39276\mathcal{I}_{3}+246200\mathcal{I}_{4}+783384\mathcal{I}_{5}+1393200\mathcal{I}_{6}+1407200\mathcal{I}_{7}$
(E.14) $\displaystyle+756000\mathcal{I}_{8}+168000\mathcal{I}_{9})-\log
2\,(\mathcal{I}_{0}+226\mathcal{I}_{1}+6888\mathcal{I}_{2}+74112\mathcal{I}_{3}+387080\mathcal{I}_{4}+1123632\mathcal{I}_{5}$
(E.15)
$\displaystyle+1909600\mathcal{I}_{6}+1889600\mathcal{I}_{7}+1008000\mathcal{I}_{8}+224000\mathcal{I}_{9})\big{)}\,.$
(E.16)

There is no $S_{-2}$, reflects that the anomalous dimension does not exist and
thus there is no mixing with stress-tensor at all.

  * 2.

Mixed

$\displaystyle
S_{-2}=-\frac{9}{32}\pi^{2}\left(\mathcal{I}_{0}+168\mathcal{I}_{1}+2688\mathcal{I}_{2}+13040\mathcal{I}_{3}+26520\mathcal{I}_{4}+24000\mathcal{I}_{5}+8000\mathcal{I}_{6}\right)\,.$
(E.17) (E.18) $\displaystyle
S_{-1}=\frac{9}{32}\pi^{2}(\mathcal{I}_{0}+288\mathcal{I}_{1}+7016\mathcal{I}_{2}+1316\mathcal{I}_{3}-581016\mathcal{I}_{4}-3953936\mathcal{I}_{5}-12017536\mathcal{I}_{6}$
(E.19)
$\displaystyle-20008800\mathcal{I}_{7}-18929600\mathcal{I}_{8}-9576000\mathcal{I}_{9}-2016000\mathcal{I}_{10})-\frac{27}{32}\pi^{2}(\mathcal{I}_{0}+98\mathcal{I}_{1}-296\mathcal{I}_{2}$
(E.20)
$\displaystyle-40576\mathcal{I}_{3}-443304\mathcal{I}_{4}-2160496\mathcal{I}_{5}-5800576\mathcal{I}_{6}-9156800\mathcal{I}_{7}-8473600\mathcal{I}_{8}-4256000\mathcal{I}_{9}$
(E.21) $\displaystyle-896000\mathcal{I}_{10})\,.$ (E.22)

It can be readily verified that the Residue relation holds true.

###  E.2 $d=8$

In $d=8$, we have poles located at $\Delta_{L}=6,5,4$ [35].

####  E.2.1 $\Delta_{L}=6$

For $\Delta_{L}=6$, the lowest-twist double-stress-tensor mixes with leading-
twist double-trace, and we find

$\displaystyle
S_{-2}=\frac{45}{8}\pi^{2}(3\mathcal{I}_{0}+520\mathcal{I}_{1}+13390\mathcal{I}_{2}+119820\mathcal{I}_{3}+500910\mathcal{I}_{4}+1105440\mathcal{I}_{5}+1328880\mathcal{I}_{6}+823200\mathcal{I}_{7}$
(E.23) $\displaystyle+205800\mathcal{I}_{8})\,,$ (E.24) (E.25) $\displaystyle
S_{-1}=\frac{15}{4}\pi^{2}(9\mathcal{I}_{0}+14121\mathcal{I}_{1}+2084124\mathcal{I}_{2}+91807278\mathcal{I}_{3}+1856575540\mathcal{I}_{4}+21210703202\mathcal{I}_{5}$
(E.26)
$\displaystyle+153524501602\mathcal{I}_{6}+753615785692\mathcal{I}_{7}+2614971373672\mathcal{I}_{8}+6571097962296\mathcal{I}_{9}+12095797182672\mathcal{I}_{10}$
(E.27)
$\displaystyle+16315061347200\mathcal{I}_{11}+15939178794000\mathcal{I}_{12}+10974472821312\mathcal{I}_{13}+5048455828416\mathcal{I}_{14}$
(E.28)
$\displaystyle+1392269598720\mathcal{I}_{15}+174033699840\mathcal{I}_{16})-\frac{45}{8}\pi^{2}\log
2\,(39\mathcal{I}_{0}+26080\mathcal{I}_{1}+2709366\mathcal{I}_{2}+101741580\mathcal{I}_{3}$
(E.29)
$\displaystyle+1907236670\mathcal{I}_{4}+21018198144\mathcal{I}_{5}+149649169616\mathcal{I}_{6}+729638532000\mathcal{I}_{7}+2526439025544\mathcal{I}_{8}$
(E.30)
$\displaystyle+6348500905152\mathcal{I}_{9}+11695090680000\mathcal{I}_{10}+15789391123968\mathcal{I}_{11}+15438335996160\mathcal{I}_{12}$
(E.31)
$\displaystyle+10636095934464\mathcal{I}_{13}+4894641303552\mathcal{I}_{14}+1350079610880\mathcal{I}_{15}+168759951360\mathcal{I}_{16})\,.$
(E.32)

####  E.2.2 $\Delta_{L}=5$

For $\Delta_{L}=5$, what the lowest double-stress-tensor mixes with is sub-
leading twist double-trace.

  * 1.

Leading-twist

$\displaystyle
S_{-1}=\frac{25}{8}\pi^{2}(3\mathcal{I}_{0}+3228\mathcal{I}_{1}+378804\mathcal{I}_{2}+13872662\mathcal{I}_{3}+237168194\mathcal{I}_{4}+2305126614\mathcal{I}_{5}$
(E.33)
$\displaystyle+14206144050\mathcal{I}_{6}+59189235400\mathcal{I}_{7}+173118036924\mathcal{I}_{8}+362574882068\mathcal{I}_{9}+546923130012\mathcal{I}_{10}$
(E.34)
$\displaystyle+589539816936\mathcal{I}_{11}+443012361792\mathcal{I}_{12}+220435235328\mathcal{I}_{13}+65262637440\mathcal{I}_{14}$
(E.35) $\displaystyle+8701684992\mathcal{I}_{15})-\frac{75}{2}\pi^{2}\log
2\,(\mathcal{I}_{0}+622\mathcal{I}_{1}+56850\mathcal{I}_{2}+1849522\mathcal{I}_{3}+29874024\mathcal{I}_{4}$
(E.36)
$\displaystyle+282767886\mathcal{I}_{5}+1722915726\mathcal{I}_{6}+7149558660\mathcal{I}_{7}+20897483544\mathcal{I}_{8}+43797760608\mathcal{I}_{9}$
(E.37)
$\displaystyle+66136501200\mathcal{I}_{10}+71359870176\mathcal{I}_{11}+53663081472\mathcal{I}_{12}+26713978368\mathcal{I}_{13}+7910622720\mathcal{I}_{14}$
(E.38) $\displaystyle+1054749696\mathcal{I}_{15})\,.$ (E.39)

  * 2.

Mixed

$\displaystyle
S_{-2}=-\frac{75}{8}\pi^{2}(\mathcal{I}_{0}+240\mathcal{I}_{1}+7080\mathcal{I}_{2}+67580\mathcal{I}_{3}+292020\mathcal{I}_{4}+655620\mathcal{I}_{5}+794780\mathcal{I}_{6}+493920\mathcal{I}_{7}$
(E.40) $\displaystyle+123480\mathcal{I}_{8})\,,$ (E.41) (E.42) $\displaystyle
S_{-1}=\frac{25}{8}\pi^{2}(12\mathcal{I}_{0}+6999\mathcal{I}_{1}+379278\mathcal{I}_{2}+972310\mathcal{I}_{3}-184455212\mathcal{I}_{4}-3753014472\mathcal{I}_{5}$
(E.43)
$\displaystyle-36152126988\mathcal{I}_{6}-212468552382\mathcal{I}_{7}-837434528352\mathcal{I}_{8}-2318322569528\mathcal{I}_{9}$
(E.44)
$\displaystyle-4609877097728\mathcal{I}_{10}-6626423377980\mathcal{I}_{11}-6831406410864\mathcal{I}_{12}-4926403116096\mathcal{I}_{13}$
(E.45)
$\displaystyle-2359679148288\mathcal{I}_{14}-674380586880\mathcal{I}_{15}-87016849920\mathcal{I}_{16})-\frac{75}{8}\pi^{2}\log
2\,(7\mathcal{I}_{0}+2540\mathcal{I}_{1}$ (E.46)
$\displaystyle+82440\mathcal{I}_{2}-2164092\mathcal{I}_{3}-117428188\mathcal{I}_{4}-1968851772\mathcal{I}_{5}-17936516924\mathcal{I}_{6}-103383558648\mathcal{I}_{7}$
(E.47)
$\displaystyle-404896847592\mathcal{I}_{8}-1119462183840\mathcal{I}_{9}-2227268584128\mathcal{I}_{10}-3204982912320\mathcal{I}_{11}$
(E.48)
$\displaystyle-3307460647680\mathcal{I}_{12}-2386946580480\mathcal{I}_{13}-1143848724480\mathcal{I}_{14}-326972405760\mathcal{I}_{15}$
(E.49) $\displaystyle-42189987840\mathcal{I}_{16})\,.$ (E.50)

It is not hard to verify the Residue relation from $S_{-2}$ above.

####  E.2.3 $\Delta_{L}=4$

In the case of $\Delta_{L}=4$, the mixing happens to sub-sub-leading-twist.

  * 1.

Leading-twist

$\displaystyle
S_{-1}=\frac{1}{2}\pi^{2}(3\mathcal{I}_{0}+2034\mathcal{I}_{1}+184115\mathcal{I}_{2}+5546904\mathcal{I}_{3}+79479952\mathcal{I}_{4}+650656672\mathcal{I}_{5}$
(E.51)
$\displaystyle+3374106104\mathcal{I}_{6}+11765981920\mathcal{I}_{7}+28516757080\mathcal{I}_{8}+48712874000\mathcal{I}_{9}+58497502280\mathcal{I}_{10}$
(E.52)
$\displaystyle+48347712000\mathcal{I}_{11}+26186462400\mathcal{I}_{12}+8367004800\mathcal{I}_{13}+1195286400\mathcal{I}_{14})-3\pi^{2}\log
2\,(\mathcal{I}_{0}$ (E.53)
$\displaystyle+628\mathcal{I}_{1}+50980\mathcal{I}_{2}+1429728\mathcal{I}_{3}+19711944\mathcal{I}_{4}+158456832\mathcal{I}_{5}+815776944\mathcal{I}_{6}+2839435680\mathcal{I}_{7}$
(E.54)
$\displaystyle+6884892120\mathcal{I}_{8}+11774800800\mathcal{I}_{9}+14156731680\mathcal{I}_{10}+11711078400\mathcal{I}_{11}+6346636800\mathcal{I}_{12}$
(E.55)
$\displaystyle+2028364800\mathcal{I}_{13}+289766400\mathcal{I}_{14})\,.$
(E.56)

  * 2.

Sub-leading twist

$\displaystyle
S_{-1}=\frac{1}{2}\pi^{2}(9\mathcal{I}_{0}+3960\mathcal{I}_{1}+179922\mathcal{I}_{2}+814558\mathcal{I}_{3}-49410876\mathcal{I}_{4}-911304904\mathcal{I}_{5}$
(E.57)
$\displaystyle-7596112752\mathcal{I}_{6}-38105022160\mathcal{I}_{7}-126821192680\mathcal{I}_{8}-292642807520\mathcal{I}_{9}-476503711600\mathcal{I}_{10}$
(E.58)
$\displaystyle-546738918320\mathcal{I}_{11}-432930876000\mathcal{I}_{12}-225268209600\mathcal{I}_{13}-69326611200\mathcal{I}_{14}-9562291200\mathcal{I}_{15})$
(E.59) $\displaystyle-6\pi^{2}\log
2\,(\mathcal{I}_{0}+368\mathcal{I}_{1}+13000\mathcal{I}_{2}-84872\mathcal{I}_{3}-7570136\mathcal{I}_{4}-117039648\mathcal{I}_{5}-932926096\mathcal{I}_{6}$
(E.60)
$\displaystyle-4615513760\mathcal{I}_{7}-15307215000\mathcal{I}_{8}-35325058560\mathcal{I}_{9}-57585511200\mathcal{I}_{10}-66154085760\mathcal{I}_{11}$
(E.61)
$\displaystyle-52432665600\mathcal{I}_{12}-27298252800\mathcal{I}_{13}-8403225600\mathcal{I}_{14}-1159065600\mathcal{I}_{15})\,.$
(E.62)

  * 3.

Mixed

$\displaystyle
S_{-2}=\frac{3}{4}\pi^{2}(\mathcal{I}_{0}+440\mathcal{I}_{1}+16280\mathcal{I}_{2}+171680\mathcal{I}_{3}+778840\mathcal{I}_{4}+1792000\mathcal{I}_{5}+2198000\mathcal{I}_{6}$
(E.63) $\displaystyle+1372000\mathcal{I}_{7}+343000\mathcal{I}_{8})\,,$ (E.64)
(E.65) $\displaystyle
S_{-1}=\frac{1}{8}\pi^{2}(63\mathcal{I}_{0}+19296\mathcal{I}_{1}+647940\mathcal{I}_{2}+9031736\mathcal{I}_{3}+199507148\mathcal{I}_{4}+3890478864\mathcal{I}_{5}$
(E.66)
$\displaystyle+42847963888\mathcal{I}_{6}+285325317280\mathcal{I}_{7}+1245992534280\mathcal{I}_{8}+3751666766400\mathcal{I}_{9}$
(E.67)
$\displaystyle+8001253401120\mathcal{I}_{10}+12205757188800\mathcal{I}_{11}+13244672767200\mathcal{I}_{12}+9987643344000\mathcal{I}_{13}$
(E.68)
$\displaystyle+4975891200000\mathcal{I}_{14}+1472592844800\mathcal{I}_{15}+196026969600\mathcal{I}_{16})-\frac{3}{4}\pi^{2}\log
2\,(13\mathcal{I}_{0}+4040\mathcal{I}_{1}$ (E.69)
$\displaystyle+181496\mathcal{I}_{2}+3887328\mathcal{I}_{3}+73926184\mathcal{I}_{4}+1114649088\mathcal{I}_{5}+10983854896\mathcal{I}_{6}+70165706720\mathcal{I}_{7}$
(E.70)
$\displaystyle+302034903480\mathcal{I}_{8}+905878243200\mathcal{I}_{9}+1931862606720\mathcal{I}_{10}+2950250822400\mathcal{I}_{11}$
(E.71)
$\displaystyle+3205199088000\mathcal{I}_{12}+2419240857600\mathcal{I}_{13}+1205970124800\mathcal{I}_{14}+356992204800\mathcal{I}_{15}$
(E.72) $\displaystyle+47521689600\mathcal{I}_{16})\,,$ (E.73)

where $S_{-2}$ can be used to verify the Residue relation easily.

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