# Power Corrections to Energy Flow Correlations from Large Spin Perturbation

Hao Chen<EMAIL_ADDRESS>Zhejiang Institute of Modern Physics,
School of Physics, Zhejiang University, Hangzhou, 310027, China  Xinan Zhou
<EMAIL_ADDRESS>Kavli Institute for Theoretical Sciences, University of
Chinese Academy of Sciences, Beijing 100190, China.  Hua Xing Zhu
<EMAIL_ADDRESS>Zhejiang Institute of Modern Physics, School of Physics,
Zhejiang University, Hangzhou, 310027, China

###### Abstract

Dynamics of high energy scattering in Quantum Chromodynamics (QCD) are
primarily probed through detector energy flow correlations. One important
example is the Energy-Energy Correlator (EEC), whose back-to-back limit probes
correlations of QCD on the lightcone and can be described by a transverse-
momentum dependent factorization formula in the leading power approximation.
In this work, we develop a systematic method to go beyond this approximation.
We identify the origin of logarithmically enhanced contributions in the back-
to-back limit as the exchange of operators with low twists and large spins in
the local operator product expansion. Using techniques from the conformal
bootstrap, the large logarithms beyond leading power can be resummed to all
orders in the perturbative coupling. As an illustration of this method, we
perform an all-order resummation of the leading and next-to-leading logarithms
beyond the leading power in ${\cal N}=4$ Super Yang-Mills theory.

##  I Introduction

Distributions of energy flows in high energy scattering in Quantum
Chromodynamics (QCD) encode unique information of the Lorentizian dynamics of
quantum gauge theory, which is otherwise hard to extract. A famous example is
the formation of jets, which are collimated sprays of hadrons that manifest
the underlying structure of of quark and gluon scattering Hanson  _et al._
(1975); Sterman and Weinberg (1977). From the early days of QCD, global energy
flow distributions have been characterized using infrared and collinear safe
shape functions Bjorken and Brodsky (1970); Ellis  _et al._ (1976); Georgi and
Machacek (1977); Farhi (1977); Parisi (1978); Donoghue  _et al._ (1979); Rakow
and Webber (1981); Berger  _et al._ (2003); Stewart  _et al._ (2010).
Alternatively, it can also be studied using statistical correlation of final-
state energy flows, of which the simplest one is the two-point Energy-Energy
Correlator (EEC) Basham  _et al._ (1978a, b, 1979a, 1979b). In fact, the two
different approaches for analyzing energy flow distributions are closely
related by an integral transformation Belitsky  _et al._ (2001); Sterman
(2004); Komiske  _et al._ (2018); Chen  _et al._ (2020).

In the $e^{+}e^{-}$ scattering, EEC can be written as the Fourier
transformation of the correlation function of energy flow operators

$\text{EEC}(y)=\frac{8\pi^{2}}{q^{2}\sigma_{0}}\\!\int\\!d^{4}x\,e^{iq\cdot
x_{13}}\langle J^{\mu}(x_{1}){\cal E}(n_{2}){\cal
E}(n_{4})J_{\mu}^{\dagger}(x_{3})\rangle\,,$ (1)

where $n_{2}=(1,\vec{n}_{2})$ and $n_{4}=(1,\vec{n}_{4})$ specify the
directions of the detected energy flow in the collider, $y=1-\frac{(n_{2}\cdot
n_{4})q^{2}}{2(n_{2}\cdot q)(n_{4}\cdot q)}=(1+\cos\theta)/2$, $q^{2}>0$ is a
timelike momentum, $J^{\mu}$ is the electromagnetic current, and the energy
flow operator is defined as a detector time integral of the energy-momentum
tensor Sveshnikov and Tkachov (1996); Tkachov (1997); Korchemsky and Sterman
(1999); Bauer  _et al._ (2008); Hofman and Maldacena (2008); Belitsky  _et
al._ (2014a, b); Kravchuk and Simmons-Duffin (2018)

$\mathcal{E}(n_{i})=\int\displaylimits_{-\infty}^{\infty}\frac{d\,n_{i}\\!\cdot\\!x_{i}}{16}\lim_{\bar{n}_{i}\\!\cdot\\!x_{i}\to\infty}(\bar{n}_{i}\cdot
x_{i})^{2}T_{\mu\nu}(x_{i})\bar{n}_{i}^{\mu}\bar{n}_{i}^{\nu}\,.$ (2)

Experimental studies of EEC have a long history Abe  _et al._ (1995); Adrian
_et al._ (1992); Acton  _et al._ (1992); Adachi  _et al._ (1989); Braunschweig
_et al._ (1987); Bartel  _et al._ (1984); Fernandez  _et al._ (1985); Wood
_et al._ (1988); Behrend  _et al._ (1982); Berger  _et al._ (1985). They have
been used to provide precision extraction of strong coupling constant de
Florian and Grazzini (2005); Kardos  _et al._ (2018).

Intuitively, when two narrow jets are produced in $e^{+}e^{-}$, they tend to
be back-to-back due to momentum conservation, reflecting the underlying
$q\bar{q}$ production. This leads to a peak for EEC as $y\to 0$, known as the
Sudakov peak. In perturbation theory, it is manifested as large logarithmic
corrections,

$\text{EEC}(y)\sim\sum_{n=1}^{\infty}\sum_{m=0}^{2n-1}\alpha_{s}^{n}\left(c_{n,m}\frac{\log^{m}y}{y}+d_{n,m}\log^{m}y\right)\,,$
(3)

where we have only shown the Leading Power (LP, $\sim y^{-1}$) and the Next-
to-Leading Power (NLP, $\sim y^{0}$) corrections for simplicity. The leading
power series of EEC resembles the perturbative structure of vector boson
production at small $p_{T}$ and exhibits lightcone divergences. It can be
resummed to all orders in the perturbative coupling by solving a 2d
renormalization group equation in virtuality and rapidity Collins and Soper
(1981); Moult and Zhu (2018). Using the recently available 4-loop rapidity
anomalous dimension Moult  _et al._ (2022); Duhr  _et al._ (2022), its
perturbative resummation has been performed to N4LL accuracy Duhr  _et al._
(2022). However, power corrections to EEC, and in general to event shape
functions and transverse momentum dependent observables, are much less
understood, both perturbatively and non-perturbatively. Recently there have
been significant developments towards a more satisfying picture of power
corrections for various observables, see e.g. Pirjol and Stewart (2003);
Beneke  _et al._ (2002); Bonocore  _et al._ (2015a, b); Moult  _et al._
(2017a); Boughezal  _et al._ (2017); Feige  _et al._ (2017); Del Duca  _et
al._ (2017); Moult  _et al._ (2017b, 2018a); Goerke and Inglis-Whalen (2018);
Beneke  _et al._ (2018a); Moult  _et al._ (2018b); Ebert  _et al._ (2018);
Beneke  _et al._ (2018b); Boughezal  _et al._ (2018); Ebert  _et al._ (2019);
Moult  _et al._ (2019); van Beekveld  _et al._ (2020); Bahjat-Abbas  _et al._
(2019); Bacchetta  _et al._ (2019); Cieri  _et al._ (2019); Buonocore  _et
al._ (2020); Moult  _et al._ (2020a); Beneke  _et al._ (2020a); Moult  _et
al._ (2020b); Liu and Neubert (2020); Ebert  _et al._ (2021); Beneke  _et al._
(2020b); Luisoni  _et al._ (2021); Inglis-Whalen  _et al._ (2021a, b); Liu
_et al._ (2021a, b); van Beekveld  _et al._ (2021a, b); Salam and Slade
(2021); Caola  _et al._ (2022); Ebert  _et al._ (2022); Vladimirov  _et al._
(2022); Buonocore  _et al._ (2022); Camarda  _et al._ (2022); Luke  _et al._
(2022); Liu  _et al._ (2022); Gamberg  _et al._ (2022). Yet the status is
still far from that of leading power terms.

In this work we initiate a study of power corrections to EEC by exploiting
conformal symmetry and techniques from the analytic conformal bootstrap
Rychkov (2016); Simmons-Duffin (2017); Poland  _et al._ (2019); Bissi  _et
al._ (2022).111Applications of conformal symmetry in QCD has a long history
Braun  _et al._ (2003). For recent applications, see e.g. Vladimirov (2017);
Braun  _et al._ (2017, 2019); Chen  _et al._ (2022); Chang and Simmons-Duffin
(2022); Moult  _et al._ (2022); Duhr  _et al._ (2022). Building upon an
important observation by Korchemsky Korchemsky (2020), we connect the
logarithmically enhanced terms in power corrections ($m>0$ in (3)) with the
expansion of correlators around the double lightcone limit, which is
controlled by the twist expansion and the large spin expansion. Using twist
conformal blocks Alday (2017), tails from large spins can be systematically
resummed. Further simplifications come from crossing symmetry, which relates
the twist corrections with large spin corrections. Using this method, we
explicitly carry out a calculation in ${\cal N}=4$ super Yang-Mills (SYM)
theory, and achieve the first Leading and Next-to-Leading Logarithmic
resummation at the subleading power.

##  II Back-to-back v.s. double lightcone

Figure 1: Penrose diagram for the double lightcone limit of the local
correlator.

As is clear from (1), EEC is related to the local Wightman correlator
$\langle\Omega|J(x_{1})T(x_{2})T(x_{4})J(x_{3})|\Omega\rangle$ by detector
time integrals and a Fourier transform Belitsky  _et al._ (2014a, b). It is
useful to understand which region of the local correlator corresponds to the
$y\to 0$ limit of EEC. Let us backtrack the dominant contribution by first
undoing the Fourier transformation. We can apply a Lorentz transformation to
make the two detectors exactly back-to-back:
$n_{4}^{\mu}\to\bar{n}_{2}^{\mu}=(1,-\vec{n}_{2})$. At the same time, the
momentum $q^{\mu}$ gains a small transverse component $\vec{q}^{\perp}$, which
is schematically related to $y$ as $|\vec{q}^{\perp}|^{2}/q^{2}\sim y$. In
position space, this corresponds to the region where $|(n_{2}\cdot
x_{13})(\bar{n}_{2}\cdot x_{13})/x_{13}^{2}|\sim y$. Since
$\mathcal{E}(n_{2})\mathcal{E}(\bar{n}_{2})$ is invariant under the boost
along $\vec{n}_{2}$, we have an extra degree of freedom to choose a frame such
that $n_{2}\cdot x_{13},\bar{n}_{2}\cdot x_{13}\sim\sqrt{y}|x_{13}|$. In such
a frame, the lightcone singularities of $1$ and $3$ are very close on the path
of detector time integrals of $2$ and $4$. Therefore, we expect the dominant
contribution to come from the region where $2$ and $4$ are near the pinch of
lightcone singularities: $x_{12}^{2},x_{23}^{2},x_{34}^{2},x_{14}^{2}\ll
x_{13}^{2},x_{24}^{2}$, which is the null square configuration Alday  _et al._
(2011). Instead, we can also choose the frame that $n_{2}\cdot x_{13}\sim
y|x_{13}|,\bar{n}_{2}\cdot x_{13}\sim|x_{13}|$. Then the leading $y\to 0$
dependence comes from 2 being near the lightcone limit of both $1$ and $3$,
which is called the double lightcone limit. In this work we delineate this
correspondence and systematically go beyond the leading power results in
Korchemsky (2020).

It is well known that the lightcone limit is controlled by twist expansion
Ferrara  _et al._ (1971, 1972); Christ  _et al._ (1972); Muta (1998). The most
singular behavior in the $1,2$-channel as $x_{12}^{2}\to 0$ is produced by
operators with the lowest twist $\tau=\Delta-\ell$. However, any single
operator in the $1,2$-channel cannot generate the lightcone singularity in the
crossed channel with $x_{13}^{2}\to 0$. The $x_{13}^{2}\to 0$ singularity can
only be produced by an infinite sum of operators with different spins at a
given twist. Therefore, the double lightcone limit, or relatedly the back-to-
back limit of EEC, is determined by the large spin asymptotics of low-twist
operators.

To be specific, we now consider EEC in ${\cal N}=4$ SYM. Comments for QCD will
be given at the end. We consider the correlator of scalar operators belonging
to the stress tensor supermultiplet

$\langle\mathcal{O}(x_{1})\mathcal{O}(x_{2})\mathcal{O}(x_{3})\mathcal{O}(x_{4})\rangle_{\rm
dyn}=\frac{1}{(2\pi)^{4}}\frac{x_{13}^{4}x_{24}^{4}}{(x_{12}^{6}x_{34}^{2})^{6}}\mathcal{F}(u,v)\,.$
(4)

Here we have only kept the dynamical part and subtracted the contribution of
protected operators which are not perturbatively corrected, see Supplemental
Material for details. Conformal symmetry ensures that $\mathcal{F}$ is a
function of the conformal cross ratios

$u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=z\bar{z}\;,\quad
v=\frac{x_{23}^{2}x_{14}^{2}}{x_{13}^{2}x_{24}^{2}}=(1-z)(1-\bar{z})\;.$ (5)

Expanded in the small coupling $a=\frac{g^{2}N_{c}}{4\pi^{2}}$,
$\mathcal{F}(u,v)$ reads

$\mathcal{F}(u,v)=\sum_{n=0}^{\infty}a^{n}\mathcal{F}^{(n)}(u,v)=\mathcal{F}^{(0)}(u,v)+\frac{u^{3}}{v}\Phi(u,v)\,,$
(6)

The new function $\Phi(u,v)=\sum_{n\geq 1}a^{n}\Phi^{(n)}(u,v)$ packages all
the coupling dependent information and is crossing symmetric, i.e.,
$\Phi(u,v)=\Phi(v,u)$. It has been calculated to three loops in Eden  _et al._
(2012); Drummond  _et al._ (2013). The correlator admits the following
superconformal block decomposition Dolan and Osborn (2002); Beem  _et al._
(2017):

$\mathcal{F}(u,v)=\sum_{\Delta}\sum_{\text{even
}\ell}a_{\tau,\ell}G_{\Delta+4,\ell}(u,v)\,.$ (7)

Here the sums are over all superconformal primary operators with dimension
$\Delta$, spin $\ell$ and OPE coefficient $a_{\tau,\ell}$. The use of $\tau$
in the label foreshadows the twist expansion later. The 4d bosonic conformal
blocks are given by Dolan and Osborn (2001)

$G_{\Delta,\ell}(u,v)=\frac{z\bar{z}}{\bar{z}-z}\left[k_{\Delta-\ell-2}(z)k_{\Delta+\ell}(\bar{z})-(z\leftrightarrow\bar{z})\right]\,,$
(8)

where
$k_{\beta}(x)=x^{\beta/2}{}_{2}F_{1}(\frac{\beta}{2},\frac{\beta}{2},\beta;x)$.
They are eigenfunctions of the quadratic Casimir operator

$\mathcal{D}_{2}=z^{2}((1-z)\partial_{z}^{2}-\partial_{z})+\frac{(d-2)z\bar{z}}{z-\bar{z}}(1-z)\partial_{z}+(z\leftrightarrow\bar{z})$
(9)

with eigenvalues $\frac{1}{2}\left(\Delta(\Delta-4)+\ell(\ell+2)\right)$ Dolan
and Osborn (2004). Superconformal symmetry causes a shift $\Delta\to\Delta+4$
in the expansion (7), and we denote
$\mathbf{G}_{\tau,\ell}(z,\bar{z})=G_{\Delta+4,\ell}(u,v)$ for convenience.
Perturbative corrections, via conformal block decomposition, are encoded in
the expansions of twists and OPE coefficients

$\tau=\tau_{0}+\sum_{n=1}^{\infty}a^{n}\gamma^{(n)}_{\tau_{0},\ell}\,,\quad
a_{\tau,\ell}=\sum_{n=0}^{\infty}a^{n}a^{(n)}_{\tau_{0},\ell}\,,$ (10)

where $\tau_{0}$ is the classical twist and $\sum_{n\geq
1}a^{n}\gamma^{(n)}_{\tau_{0},\ell}=\gamma_{\tau,\ell}$ is the anomalous
dimension. The anomalous dimensions enter via the expansion for conformal
blocks:
$\mathbf{G}_{\tau,\ell}=\sum_{n=0}^{\infty}\frac{1}{n!}\gamma_{\tau,\ell}^{n}\partial_{\tau_{0}}^{n}\mathbf{G}_{\tau_{0},\ell}$.
Note that $\partial_{\tau_{0}}^{n}\mathbf{G}_{\tau_{0},\ell}$ contains at most
$\log^{n}u$ in the small $u$ limit.

The double lightcone limit corresponds to $u,v\to 0$, or equivalently, $z\to
0,\,\bar{z}\to 1$. Each conformal block in this limit is controlled by the
twist $G_{\Delta,\ell}(z,\bar{z})\sim z^{\tau/2}\log(1-\bar{z})$. The presence
of logarithms partially demonstrates the effect of summing over infinitely
many spinning operators because each conformal block contains infinitely many
descendants in a given conformal family. However, we will encounter more
divergent pieces than a single logarithm, which require the large spin
contribution from infinitely many primary operators. Examples include power
divergences $(1-\bar{z})^{m<0}$ and powers of logarithms
$[\log(1-\bar{z})]^{k\geq 2}$ 222At one loop the perturbative logarithms at
NLP have at most $k=1$, therefore are not fixed by large spins.. The latter
arises from $\log^{k}u$ of $\partial_{\tau_{0}}^{k}\mathbf{G}_{\tau_{0},\ell}$
in small $u$ under crossing. In the following, we refer to these as enhanced
divergences. To lighten the notations, we will denote the logarithms as
$\log(x)=L_{x}$ from now on.

We also point out that crossing symmetry will play an important role in our
computation of the power corrections. As we will see, the $u$, $v$ power
corrections to $\Phi(u,v)$ contribute equally to the $y$ power corrections in
EEC. In particular, EEC at NLP corresponds to order $u^{0}v^{1}$ and
$u^{1}v^{0}$ in $\Phi(u,v)$. The former only requires twist-2 data to
subleading order in the large spin limit ($\mathcal{O}(\ell^{-2})$), while the
latter contains twist-4 contributions. Crossing symmetry equates these two
contributions and therefore avoids the necessity of inputting the twist-4
information.

Table 1: CFT data needed at different orders | Power Corrections | Perturbative Corrections
---|---|---
twist | large spin | LL | NLL
LP | 2 | $\mathcal{O}(\ell^{0})$ | $a_{2,\ell}^{(0)}\,,\gamma_{2,\ell}^{(1)}$ | $a_{2,\ell}^{(1)}\,,\gamma_{2,\ell}^{(2)}$
NLP | 2 | $\mathcal{O}(\ell^{-2})$ | $a_{2,\ell}^{(0)}\,,\gamma_{2,\ell}^{(1)}$ | $a_{2,\ell}^{(1)}\,,\gamma_{2,\ell}^{(2)}$
4 | $\mathcal{O}(\ell^{0})$ | $a_{4,\ell}^{(0)}\,,\gamma_{4,\ell}^{(1)}$ | $a_{4,\ell}^{(1)}\,,\gamma_{4,\ell}^{(2)}$

Before going into the technical details, we provide a brief overview for the
next two sections. We will first use techniques from large spin perturbation
theory to extract the enhanced divergences in $v$ at twist-2 up to NLP. That
is, we will obtain $\Phi(u,v)=p_{0}(L_{u},L_{v})+p_{1}(L_{u},L_{v})v+\cdots$,
in which $p_{0}$, $p_{1}$ are polynomials in $L_{u}$, $L_{v}$ at each
perturbative order. Crossing symmetry then fixes the NLP contribution in $u$
to be
$\Phi(u,v)=p_{0}(L_{u},L_{v})+p_{1}(L_{u},L_{v})v+p_{1}(L_{v},L_{u})u\cdots$.
The relevant data is listed in Table 1, but only the first two rows are needed
thanks to crossing symmetry. Finally, we map the small $u$, $v$ expansion of
$\Phi(u,v)$ to the back-to-back limit of $\mathrm{EEC}(y)$. The rules at LP
and NLP are given in Table 2 and are valid to NLL accuracy. We will explain
these points in more detail in the rest of the paper.

Table 2: Logarithms Map at LP and NLP | $\Phi(u,v)$ | $4y(1-y)^{2}\times\mathrm{EEC}(y)$
---|---|---
LP | $L_{u}^{m}L_{v}^{m}$ | $2(m+n)L^{m+n-1}_{y/(1-y)}$
NLP | $uL_{u}^{m}L_{v}^{n}$ | $\frac{2m(1-m)}{m+n-1}\frac{y}{1-y}L^{m+n-1}_{y/(1-y)}$
$vL_{u}^{m}L_{v}^{n}$ | $\frac{2n(1-n)}{m+n-1}\frac{y}{1-y}L^{m+n-1}_{y/(1-y)}$

##  III Large spin analysis

The enhanced divergences can be systematically handled by using the Large Spin
Perturbation Theory Alday (2017), which culminates an array of earlier works
in the large spin sector Alday and Maldacena (2007); Fitzpatrick  _et al._
(2013); Komargodski and Zhiboedov (2013); Alday and Bissi (2013); Alday  _et
al._ (2015); Alday and Zhiboedov (2016, 2017). The starting point is the free
theory limit where the twists are degenerate. The correlators can be written
as a sum over the twists

$\mathcal{F}^{(0)}(z,\bar{z})=\sum_{\tau_{0}=2,4,\ldots}H_{\tau_{0}}(z,\bar{z})\;,$
(11)

where each $H_{\tau_{0}}(z,\bar{z})$ sums over spins

$H_{\tau_{0}}(z,\bar{z})=\sum_{\ell=0}^{\infty}\langle
a_{\tau_{0},\ell}^{(0)}\rangle\mathbf{G}_{\tau_{0},\ell}(z,\bar{z})\;,$ (12)

and is known as a twist conformal block (TCB). When the interaction is turned
on, the twist degeneracies are lifted and the OPE coefficients are corrected.
In general, we find that the expansion consists of sums of the form

$\sum_{\ell=0}^{\infty}\langle
a_{\tau_{0},\ell}^{(0)}\rangle\kappa_{\tau_{0}}(\ell)\mathbf{G}_{\tau_{0},\ell}(z,\bar{z})\;,$
(13)

where the quantities $\kappa_{\tau_{0}}(\ell)$ admit expansions around large
conformal spins
$J^{2}_{\tau,\ell}=(\ell+\frac{\tau}{2})(\ell+\frac{\tau}{2}-1)$ with the
following schematic form

$\kappa_{\tau_{0}}(\ell)=\sum_{m=0}^{\infty}\sum_{i=0}^{N}\frac{K_{m,i}}{J_{\tilde{\tau}_{0},\ell}^{2m}}\log^{i}J_{\tilde{\tau}_{0},\ell}^{2}\;.$
(14)

Here we introduce shifted twists $\tilde{\tau}_{0}=\tau_{0}+4$ to take into
account the dimension shift in the decomposition (7). An interesting feature,
as we will see in explicit calculations, is that only even negative powers
appear in the expansion. This is closely related to the reciprocity relation
Dokshitzer  _et al._ (2006); Basso and Korchemsky (2007), which has been
explicitly verified in QCD to three loops Chen  _et al._ (2021). Consequently,
the correlator should now be expanded in terms of a more general class of TCBs

$H_{\tau_{0}}^{(m,i)}(z,\bar{z})=\sum_{\ell}a^{(0)}_{\tau_{0},\ell}\frac{\log^{i}J^{2}_{\tilde{\tau}_{0},\ell}}{J_{\tilde{\tau}_{0},\ell}^{2m}}\mathbf{G}_{\tau_{0},\ell}(z,\bar{z})\;.$
(15)

Note that conformal blocks satisfy the Casimir equation

$\mathcal{C}_{\tilde{\tau}_{0}}\mathbf{G}_{\tau_{0},\ell}(z,\bar{z})=J_{\tilde{\tau}_{0},\ell}^{2}\mathbf{G}_{\tau_{0},\ell}(z,\bar{z})\;,$
(16)

where $\mathcal{C}_{\tau}=\mathcal{D}_{2}+\frac{1}{4}\tau(2d-\tau-2)$ is the
shifted conformal Casimir. It follows that TCBs obey the following recursion
relations

$H_{\tau_{0}}^{(m,i)}(z,\bar{z})=\mathcal{C}_{\tilde{\tau}_{0}}H_{\tau_{0}}^{(m+1,i)}(z,\bar{z})\;.$
(17)

The full TCBs are in general difficult to compute. However, we will only need
them in the small $v$ limit where computations become manageable. The TCBs in
4D take a factorized form

$H_{\tau_{0}}^{(m,i)}(z,\bar{z})=\frac{z\,k_{\tilde{\tau}_{0}-2}(z)}{\bar{z}-z}\bar{H}_{\tau_{0}}^{(m,i)}(\bar{z})\;,$
(18)

where we have dropped the regular part when $\bar{z}\to 1$. Moreover, the
recursion relation (17) becomes

$\bar{H}_{\tau_{0}}^{(m,i)}(\bar{z})=\bar{D}\bar{H}_{\tau_{0}}^{(m+1,i)}(\bar{z})\;,$
(19)

where

$\bar{D}=\bar{z}^{2}(1-\bar{z})\frac{d^{2}}{d\bar{z}^{2}}-\bar{z}(2-\bar{z})\frac{d}{d\bar{z}}+2-\bar{z}\;.$
(20)

Using this recursion relation, one can compute the TCBs explicitly in the
small $v$ limit Henriksson and Lukowski (2018). For example, we find

$\displaystyle\bar{H}_{2}^{(0,i)}(\bar{z})$ $\displaystyle=$
$\displaystyle(-1)^{i}\left[\frac{L_{\epsilon}^{i}}{2\epsilon}+\frac{L_{\epsilon}^{i+1}}{6(i+1)}+\frac{\gamma_{E}-3}{3}L_{\epsilon}^{i}\right]+\cdots\,,$
$\displaystyle\bar{H}_{2}^{(1,i)}(\bar{z})$ $\displaystyle=$
$\displaystyle(-1)^{i}\left[\frac{L_{\epsilon}^{i+2}}{2(i+1)(i+2)}+\frac{\gamma_{E}L_{\epsilon}^{i+1}}{i+1}\right]+\cdots\,.$
(21)

where we have defined $\epsilon=1-\bar{z}$.

Let us now focus on the small $u$ limit, i.e., $z\to 0$. Together with
$\bar{z}\to 1$, the correlator takes the form

$\begin{split}\mathcal{F}^{(n)}={}&z^{3}\frac{\log^{n}z}{\epsilon}\big{(}\log^{n}\epsilon(A_{n,1}+A_{n,2}\epsilon+\ldots)\\\
{}&+\log^{n-1}\epsilon(B_{n,1}+B_{n,2}\epsilon+\ldots)+\ldots\big{)}+\mathcal{O}(z^{4})\\\
={}&z^{3}\sum_{m=0}^{\infty}\sum_{i=0}^{n}C_{m,i}\bar{H}_{2}^{(m,i)}(\bar{z})+\mathcal{O}(z^{4})\;.\end{split}$

An important point is that to compute the NqLP in the small $\epsilon$
expansion, i.e., $A_{n,j=1,2,\ldots,q}$, $B_{n,j=1,2,\ldots,q}$ etc, we only
need $\bar{H}_{2}^{(m,i)}$ with $m=0,1,\ldots,q$. To see this, we act on the
two expansions with $\bar{D}$ and compare the power divergences. Note that for
any polynomial $p(\epsilon)$

$\bar{D}(p(\epsilon)\log^{i}\epsilon)=\frac{i(i-1)(1-\epsilon)p(\epsilon)\log^{i-2}\epsilon}{\epsilon}+\mathcal{O}(\epsilon^{0})\;.$

Taking $p(\epsilon)=\epsilon^{q}$ at the $q$-th order, we find the RHS only
becomes a power divergence after acting $q$ times with $\bar{D}$. On the other
hand, it is known that only the TCBs with $m\leq 0$ are power divergent. The
repeated $\bar{D}$ action makes the TCBs with $m\leq q$ power divergent and
therefore responsible for the $q$-th order correction.

We now explicitly compute the power corrections using the TCB decomposition.
From Table 1, to NLL and 2nd order in NLP the needed data is Dolan and Osborn
(2006); Henriksson and Lukowski (2018); Kologlu  _et al._ (2021)

$\displaystyle a_{2,\ell}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\ell+3)^{2}}{\Gamma(2\ell+5)}\,,$ (22)
$\displaystyle\gamma_{2,\ell}^{(1)}$ $\displaystyle=$ $\displaystyle\log
J^{2}_{6,\ell}+2\gamma_{E}+\frac{1}{3J^{2}_{6,\ell}}+\mathcal{O}(J^{-4}_{6,\ell})\,,$
(23) $\displaystyle\frac{a_{2,\ell}^{(1)}}{a_{2,\ell}^{(0)}}$ $\displaystyle=$
$\displaystyle\\!\bigg{[}\frac{1+4J_{6,\ell}}{16J^{2}_{6,\ell}}\\!-\\!\log
2\bigg{]}\\!\gamma^{(1)}_{2,\ell}\\!-\zeta_{2}\\!+\\!\frac{1}{J_{6,\ell}}\\!+\\!\mathcal{O}(J^{-3}_{6,\ell}),$
(24) $\displaystyle\gamma_{2,\ell}^{(2)}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{J_{6,\ell}}-\frac{\zeta_{2}}{2}\right)\gamma_{2,\ell}^{(1)}-\frac{3\zeta_{3}}{2}+\frac{1}{J^{2}_{6,\ell}}+\mathcal{O}(J^{-3}_{6,\ell})\,.$
(25)

From the conformal block decomposition (7), the small $u$ expansion of
$\mathcal{F}^{(n)}$ up to NLL accuracy reads

$\begin{split}\mathcal{F}^{(n)}=z^{3}\sum_{\text{even
}\ell}a^{(0)}_{2,\ell}\Bigg{\\{}\frac{\left(\gamma^{(1)}_{2,\ell}\right)^{n}}{2^{n}n!}L_{z}^{n}+\frac{\left(\gamma_{2,\ell}^{(1)}\right)^{n-1}L_{z}^{n-1}}{2^{n-1}(n-1)!}\\\
\times\left[\frac{a^{(1)}_{2,\ell}}{a^{(0)}_{2,\ell}}+(n-1)\frac{\gamma_{2,\ell}^{(2)}}{\gamma_{2,\ell}^{(1)}}+\frac{\gamma_{2,\ell}^{(1)}\partial_{\ell}}{2}\right]\Bigg{\\}}k_{2\ell+6}(\bar{z})+\cdots\,,\end{split}$
(26)

where all the odd powers in $1/J_{6,\ell}$ cancel out upon using the large
spin expansion of CFT data (23-25). We can therefore rewrite it in terms of
TCBs as

$\displaystyle\mathcal{F}^{(n)}=$ (27)
$\displaystyle\frac{L_{z}^{n}}{n!}\sum_{i}\begin{pmatrix}n\\\
i\end{pmatrix}\left[\frac{\gamma_{E}^{n-i}}{2^{i}}H_{2}^{(0,i)}+\frac{n-i}{3}\frac{\gamma_{E}^{n-1-i}}{2^{i+1}}H_{2}^{(1,i)}\right]+\cdots\,.$

We have only showed the LL part for brevity and left the NLL part to
Supplemental Material. Substituting in the TCBs using (18) and (21) we obtain
$\mathcal{F}^{(n)}$ at LP in $z$ and NLP in $1-\bar{z}$

$\mathcal{F}^{(n)}=\frac{(-1)^{n}z^{3}}{2^{n+1}n!}L_{z}^{n}L_{\epsilon}^{n}\\!\left[\frac{1-\epsilon}{\epsilon}+\frac{2n}{3L_{\epsilon}}+\frac{1}{L_{z}}+\\!\cdots\\!\right]+\mathcal{O}(z^{4})\,,$
(28)

with NLL accuracy. Using
$\mathcal{F}^{(n)}(z,\bar{z})=\frac{v}{u^{3}}\Phi^{(n)}(u,v)$ and crossing
symmetry $\Phi(u,v)=\Phi(v,u)$, we get the NLL prediction for $\Phi^{(n)}$ at
NLP in the double lightcone limit

$\displaystyle\Phi^{(n)}=\frac{(-1)^{n}}{2^{n}n!}L_{u}^{n}L_{v}^{n}\left\\{\frac{1}{2}+(u+v)\right.$
(29)
$\displaystyle\left.+\left[\left(\frac{n+1}{2}u+\frac{n}{3}v\right)\frac{1}{L_{v}}+(u\leftrightarrow
v)\right]+\cdots\right\\}\,,\quad n>1\,.$

As was promised in the last section, only twist-2 CFT data was used in the
whole process. This prediction is checked against the available two- and
three-loop results in the Supplemental Material.

##  IV Power corrections to EEC in $\mathcal{N}=4$ SYM

The final task is to find the explicit relation between the double lightcone
limit series $u^{j_{1}}v^{j_{2}}L_{u}^{m}L_{v}^{n}$ and the back-to-back limit
series $y^{j}L_{y}^{k}$. The answer can be found using the Mellin
representation of EEC Belitsky  _et al._ (2014b, a, c)

$\mathrm{EEC}(y)=\frac{1}{4y(1-y)^{2}}\int\frac{dj_{1}dj_{2}}{(2\pi
i)^{2}}M(j_{1},j_{2})K(j_{1},j_{2};y)\,,$ (30)

where $M(j_{1},j_{2})$ is the Mellin amplitude for $\Phi(u,v)$:
$\Phi(u,v)=\int\frac{dj_{1}dj_{2}}{(2\pi
i)^{2}}M(j_{1},j_{2})u^{j_{1}}v^{j_{2}}$ and the kernel $K(j_{1},j_{2};y)$ is
defined as

$K(j_{1},j_{2};y)=\frac{2\Gamma(1-j_{1}-j_{2})\left(\frac{y}{1-y}\right)^{j_{1}+j_{2}}}{\Gamma(j_{1}+j_{2})\left[\Gamma(1-j_{1})\Gamma(1-j_{2})\right]^{2}}\,.$
(31)

This gives the map
$u^{j_{1}}v^{j_{2}}\to\frac{K(j_{1},j_{2};y)}{4y(1-y)^{2}}$. Taking
derivatives w.r.t. $j_{1}$, $j_{2}$ generates the rules containing logarithms
in $u$, $v$. One subtlety is that, due to the presence of the pole at
$j_{1}+j_{2}=1$ in $K(j_{1},j_{2};y)$, the maps at NLP cannot predict the
$y^{0}$ term without any $\log y$ enhancement. But it can be shown that all
the logarithmic contributions at NLP are preserved (see Supplemental
Material). The rules at LP and NLP up to NLL are summarized in Table 2, and
lead to

$\displaystyle\mathrm{EEC}^{(n>1)}(y)=\frac{(-1)^{n}}{2^{n}(n-1)!}\left[\frac{1}{2y}\left(L_{y}^{2n-1}+{\cal
O}(L_{y}^{2n-3})\right)\right.$
$\displaystyle\left.+\left(\frac{n}{2n-1}L_{y}^{2n-1}+\frac{7n-5}{12}L_{y}^{2n-2}+{\cal
O}(L_{y}^{2n-3})\right)+\cdots\right]\,,$ (32)

which is in full agreement with the full theory calculation up to $n=3$ in
Henn  _et al._ (2019). The $n>3$ terms are new and are one of the main result
of this work. The analytic series in $n$ can be resummed explicitly to all
orders, leading to the following NLL formula at LP and NLP333The one-loop NLL
contribution at NLP has no $L_{y}$ enhancement. Therefore, we need to input
the one-loop EEC to fix the constant $a/12$.

$\displaystyle\mathrm{EEC}(y)=-\frac{aL_{y}e^{-\frac{aL_{y}^{2}}{2}}}{4y}-\frac{1}{4}\left[\sqrt{\frac{\pi}{2}}\sqrt{a}\,\mathrm{erf}\left(\sqrt{\frac{a}{2}}L_{y}\right)\right.$
$\displaystyle\left.+aL_{y}e^{-\frac{aL_{y}^{2}}{2}}\right]+\frac{a}{48}(7aL_{y}^{2}-4)e^{-\frac{aL_{y}^{2}}{2}}+\frac{a}{12}+\cdots\,,\quad$
(33)

where $\mathrm{erf}$ is the error function
$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$ 444We note that
the LL-NLP series has been studied in Moult  _et al._ (2020b). Their results
disagree with ours starting from ${\cal O}(a^{3})$, and seems to be in
conflict with the fixed-order analytic result in Henn  _et al._ (2019).
Further comparison is provided in the Supplemental Material..

In Fig. 2 we plot the ${\cal N}=4$ EEC in the back-to-back limit to illustrate
the importance of NLP resummation. It can be seen that the LL and NLL series
at NLP leads to substantial corrections for not too large $\theta$. For
$\theta>175^{\circ}$ the Sudakov double logs suppressed the NLP contributions.
For comparison we also plot in dashed line the fixed-order NLP results
truncated to NNLO Henn  _et al._ (2019), along with the LP NLL series. In this
case sizable NLP corrections can be found for $\theta>175^{\circ}$, which
however is misleading as they disappear after resumming to all orders in
coupling.

Figure 2: EEC as a function of $\theta$ in the back-to-back limit. We use
$g^{2}/(4\pi)=0.118$ to mimic the QCD strong coupling at Z pole. The dashed
line refers to LP resummed to NLL, with the inclusion of NLP terms up to NNLO
($n\leq 3$).

##  V Discussions

Our results lead to several exciting research avenues. First of all, it is
interesting to apply the results to EEC in QCD, where fixed-order data up to
NLO has become available recently Dixon  _et al._ (2018); Luo  _et al._
(2019); Gao  _et al._ (2021). The local correlator of four electromagnetic
currents in QCD has also been computed at one loop Chicherin  _et al._ (2021).
Secondly, in QCD running coupling corrections will modify NLL series. It would
be important to understand how to incorporate these effects while retaining
the power of conformal symmetry. Thirdly, our results provide concrete data
for quantitative comparison between additive and multiplicative scheme in
resummation matching, see e.g. Bizoń  _et al._ (2018). Fourthly, local
correlators exhibit other interesting limits, such as the Regge limit Costa
_et al._ (2012). It would be interesting to understand what constraints are
imposed on EEC by such limits. Last but not least, it would be worthwhile to
understand the relation between our approach and the conventional approach
based on momentum space renormalization group, in particular the relation
between crossing symmetry for local correlator and the consistency relations
from infrared poles cancellations Moult  _et al._ (2017a).

###### Acknowledgements.

We thank Zhongjie Huang, Kai Yan, and Xiaoyuan Zhang for useful discussions.
H.C. and H.X.Z. are supported by the Natural Science Foundation of China under
contract No. 11975200 and No. 12147103. X.N.Z. is supported by funds from UCAS
and KITS, and by the Fundamental Research Funds for the Central Universities.

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## Supplemental material

##  Appendix A Scalar Local Correlator

To make precise the local correlator in (4), we consider the following
operator made out of the six scalars $\phi^{I=1,\ldots,6}$ of $\mathcal{N}=4$
SYM

$O^{IJ}=\mathrm{tr}\left(\phi^{I}\phi^{J}\right)-\frac{1}{6}\delta^{IJ}\mathrm{tr}\left(\phi^{K}\phi^{K}\right)\,,$
(S.1)

The operator is the superprimary of the stress tensor multiplet and transforms
in the symmetric traceless representation of the $SO(6)$ R-symmetry group. It
is convenient to keep track of the R-symmetry information by contracting the
indices with null polarization vectors $Y_{I}$ or with a traceless symmetric
tensor $S_{IJ}$555The tensor can be built from the vectors, e.g.,
$S_{IJ}=Y^{\prime}_{I}Y^{\prime}_{J}+Y^{\prime\prime}_{I}Y^{\prime\prime}_{J}$.
Therefore, it is sufficient to focus on the former case.

$O(x,Y)\equiv O^{IJ}(x)Y_{I}Y_{J}\,,\quad\text{or}\quad
O(x,S)=O^{IJ}(x)S_{IJ}\,.$ (S.2)

Due to superconformal symmetry, the four-point function has following
“partially non-renormalized” form Eden  _et al._ (2001)

$\langle
O(x_{1},Y_{1})O(x_{2},Y_{2})O(x_{3},Y_{3})O(x_{4},Y_{4})\rangle=G^{(0)}(1,2,3,4)+\frac{2(N_{c}^{2}-1)}{(4\pi^{2})^{4}}\frac{y_{12}^{4}y_{34}^{4}}{x_{12}^{2}x_{34}^{2}x_{14}^{2}x_{23}^{2}}R\;\Phi(u,v)$
(S.3)

where $G^{(0)}(1,2,3,4)$ is the tree-level correlator

$\displaystyle G^{(0)}(1,2,3,4)=$
$\displaystyle\frac{(N_{c}^{2}-1)^{2}}{4(4\pi^{2})^{4}}\left(\left(\frac{y_{12}^{2}y_{34}^{2}}{x_{12}^{2}x_{34}^{2}}\right)^{2}+\left(\frac{y_{13}^{2}y_{24}^{2}}{x_{13}^{2}x_{24}^{2}}\right)^{2}+\left(\frac{y_{41}^{2}y_{23}^{2}}{x_{41}^{2}x_{23}^{2}}\right)^{2}\right)$
(S.4)
$\displaystyle+\frac{N_{c}^{2}-1}{(4\pi^{2})^{4}}\left(\frac{y_{12}^{2}y_{23}^{2}y_{34}^{2}y_{41}^{2}}{x_{12}^{2}x_{34}^{2}x_{23}^{2}x_{41}^{2}}+\frac{y_{12}^{2}y_{24}^{2}y_{34}^{2}y_{13}^{2}}{x_{12}^{2}x_{24}^{2}x_{34}^{2}x_{31}^{2}}+\frac{y_{13}^{2}y_{23}^{2}y_{24}^{2}y_{41}^{2}}{x_{13}^{2}x_{23}^{2}x_{24}^{2}x_{41}^{2}}\right)\,,$

with $y_{ij}^{2}\equiv Y_{i}\cdot Y_{j}$ and the function $\Phi(u,v)$ encodes
all the dynamical information. The factor $R$ is determined by superconformal
symmetry

$R=(1-z\alpha)(1-z\bar{\alpha})(1-\bar{z}\alpha)(1-\bar{z}\bar{\alpha})\;,$
(S.5)

where the conformal cross ratios $z$, $\bar{z}$ have already been introduced
in the main text and $\alpha$, $\bar{\alpha}$ are similarly the R-symmetry
cross ratios defined by

$\frac{y_{13}^{2}y_{24}^{2}}{y_{12}^{2}y_{34}^{2}}=\alpha\bar{\alpha}\;,\quad\frac{y_{14}^{2}y_{23}^{2}}{y_{12}^{2}y_{34}^{2}}=(1-\alpha)(1-\bar{\alpha})\;.$
(S.6)

The weak coupling $g\ll 1$ expansion of $\Phi(u,v)$ reads

$\Phi(u,v)=\sum_{n=1}^{\infty}a^{n}\Phi^{(n)}(u,v)\,,\quad\quad
a=\frac{g^{2}N_{c}}{4\pi^{2}}\;,$ (S.7)

and is known up to three loops Drummond  _et al._ (2013). Since the dynamic
function $\Phi(u,v)$ is R-symmetry independent, we can choose special
polarizations to simplify the correlator (S.3). Following Belitsky  _et al._
(2014a), let us take

$Y_{I}=(1,0,1,0,i,i)\,,\quad S_{IJ}=\mathrm{diag}(1,-1,0,0,0,0)\,,\quad
S^{\prime}_{IJ}=\mathrm{diag}(0,0,1,-1,0,0)\,,$ (S.8)

and define

$\begin{split}&\widetilde{\mathcal{O}}(x_{2})=2O(x_{2},S)\,,\quad\widetilde{\mathcal{O}}^{\prime}(x_{4})=2O(x_{4},S^{\prime})\,,\\\
&\mathcal{O}(x_{3})=\left(\frac{N_{c}^{2}-1}{2\pi^{4}}\right)^{-\frac{1}{2}}O(x_{3},Y)\,,\quad\mathcal{O}^{\dagger}(x_{1})=\left(\frac{N_{c}^{2}-1}{2\pi^{4}}\right)^{-\frac{1}{2}}O(x_{1},Y^{*})\,.\end{split}$
(S.9)

Then the four-point function becomes

$\begin{split}\langle\mathcal{O}^{\dagger}(x_{1})\widetilde{\mathcal{O}}(x_{2})\widetilde{\mathcal{O}}^{\prime}(x_{4})\mathcal{O}(x_{3})\rangle={}&\frac{1}{(2\pi)^{4}}\frac{1}{(x_{12}^{2}x_{34}^{2})^{2}}\left[\frac{N_{c}^{2}-1}{8}\left(1+\frac{u^{2}}{v^{2}}\right)+\frac{u}{v}\left(\frac{1}{2}+\Phi(u,v)\right)\right]\\\
={}&\frac{1}{(2\pi)^{4}}\frac{1}{(x_{12}^{2}x_{34}^{2})^{2}}\left(\frac{N_{c}^{2}-1}{8}G^{\text{short}}(u,v)+\frac{1}{u^{2}}\mathcal{F}(u,v)\right)\;,\end{split}$
(S.10)

where we have further split it into short multiplet contribution
$G^{\text{short}}(u,v)$ and the long multiplet contribution
$\mathcal{F}(u,v)$. The explicit form of $G^{\text{short}}(u,v)$ can be found
in Beem  _et al._ (2017) and is protected from perturbative corrections. As a
result, $\Phi(u,v)$ is essentially the same as all the loop corrections
($n\geq 1$) of $\mathcal{F}(u,v)=\sum_{n\geq 0}a^{n}\mathcal{F}^{(n)}(u,v)$,
i.e.,

$\mathcal{F}(u,v)-\mathcal{F}^{(0)}(u,v)=\frac{u^{3}}{v}\Phi(u,v)\,.$ (S.11)

Another reason for choosing the polarizations (S.8) is the relation to the
scalar detectors

$\mathcal{S}(n_{i})=\frac{1}{4}\int\displaylimits_{-\infty}^{\infty}dn_{i}\cdot
x_{i}\lim_{\bar{n}_{i}\cdot x_{i}\to\infty}(\bar{n}_{i}\cdot
x_{i})^{2}\mathcal{O}(x_{i})\,.$ (S.12)

Thanks to superconformal symmetry, the spinning correlator $\langle
JTTJ\rangle$ is related to the scalar correlator (S.3) by Ward identities.
Moreover, the EEC and the scalar-scalar correlation (SSC) are also
proportional Belitsky  _et al._ (2014a)

$\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle=\frac{4(q^{2})^{2}}{(n_{2}\cdot
n_{4})^{2}}\langle\mathcal{S}(n_{2})\mathcal{S}(n_{4})\rangle\,.$ (S.13)

##  Appendix B Twist Conformal Blocks

The TCBs with logarithms can be computed using the method of Henriksson and
Lukowski (2018). The idea is to apply the recursion relations on
$\bar{H}_{\tau_{0}}^{(0,0)}(\bar{z})$, which is determined by the tree-level
correlator, and compute $\bar{H}_{\tau_{0}}^{(m,0)}(\bar{z})$ at negative
integer $m$. We then analytic continue in $m$ and take derivatives to obtain
the logarithms. For our case, $\tau_{0}=2$ and we have

$\bar{H}_{2}^{(0,0)}(\bar{z})=\frac{\bar{z}^{2}(2-\bar{z})}{2(1-\bar{z})}+\bar{z}\log(1-\bar{z})=\frac{1}{2\epsilon}+\text{regular
terms }\,.$ (S.14)

Repeated $\bar{D}$ action gives $\bar{H}_{2}^{(m,0)}(\bar{z})$ and the first
few terms in small $\epsilon$ for negative integer $m$ are given by (3.37) of
Henriksson and Lukowski (2018)

$\begin{split}\bar{H}^{(m,0)}_{2}(\bar{z})=&\frac{1}{2}\epsilon^{m-1}\Gamma(1-m)^{2}+\frac{1}{6}m\left(2m^{2}-6m+1\right)\epsilon^{m}\Gamma(-m)^{2}\\\
&+\frac{1}{180}(m-1)m(m+1)\left(20m^{3}-54m^{2}-35m+36\right)\epsilon^{m+1}\Gamma(-m-1)^{2}+\cdots\,.\end{split}$
(S.15)

Obtaining the full analytic expression for $\bar{H}^{(m,i)}_{2}(\epsilon)$ is
difficult. But life is much easier if we content ourselves with getting first
few orders in $\epsilon$ and logarithms. Truncated to order $\epsilon^{0}$,
only $\bar{H}^{(0,i)}_{2}(\epsilon)$ and $\bar{H}^{(1,i)}_{2}(\epsilon)$ are
relevant and we find

$\displaystyle\bar{H}^{(0,i)}_{2}(\bar{z})$ $\displaystyle=$
$\displaystyle\frac{(-1)^{i}}{\epsilon}\left[\frac{1}{2}\log^{i}\epsilon+i\gamma_{E}L^{i-1}_{\epsilon}+\frac{i(i-1)}{12}(12\gamma_{E}^{2}+\pi^{2})L^{i-2}_{\epsilon}+\cdots\right]$
$\displaystyle+$
$\displaystyle(-1)^{i}\left[\frac{1}{6(i+1)}L^{i+1}_{\epsilon}+\frac{\gamma_{E}-3}{3}L^{i}_{\epsilon}+\frac{\pi^{2}+12\gamma_{E}^{2}-72\gamma_{E}+12}{36}iL^{i-1}_{\epsilon}+\dots\right]+\cdots\,,$
$\displaystyle\bar{H}^{(1,i)}_{2}(\bar{z})$ $\displaystyle=$
$\displaystyle(-1)^{i}\left[\frac{1}{2(i+1)(i+2)}L^{i+2}_{\epsilon}+\frac{\gamma_{E}}{i+1}L^{i+1}_{\epsilon}+\frac{12\gamma_{E}^{2}+\pi^{2}}{12}L^{i}_{\epsilon}+\cdots\right]+\cdots\,.$
(S.17)

##  Appendix C More Details of (26) and (27)

In this section, we present the details of the large spin perturbation
calculation needed for obtaining EEC in the back-to-back limit to the NLL and
NLP order. As we explained in the main text, only twist-2 contributions are
needed. The expansion of the twist-2 conformal block
$\mathbf{G}_{2+\gamma_{2,\ell},\ell}(z,\bar{z})$ in the $z\to 0$ limit is

$\begin{split}\mathbf{G}_{2+\gamma_{2,\ell},\ell}(z,\bar{z})&=z^{3+\gamma_{2,\ell}/2}k_{6+2\ell+\gamma_{2,\ell}}(\bar{z})+\mathcal{O}(z^{4})\\\
&=z^{3}\sum_{n=0}^{\infty}a^{n}\Bigg{\\{}\left[\frac{1}{2^{n}n!}\left(\gamma^{(1)}_{2,\ell}\right)^{n}\log^{n}z+\frac{1}{2^{n-1}(n-2)!}\gamma^{(2)}_{2,\ell}\left(\gamma^{(1)}_{2,\ell}\right)^{n-2}\log^{n-1}z+\cdots\right]k_{6+2\ell}(\bar{z})\\\
&\qquad\qquad\qquad+\frac{1}{2^{n}(n-1)!}\left(\gamma^{(1)}_{2,\ell}\right)^{n}\log^{n-1}z\;\partial_{\ell}k_{6+2\ell}(\bar{z})+\cdots\Bigg{\\}}+\mathcal{O}(z^{4})\,.\end{split}$
(S.18)

Combined with the expansion of the OPE coefficient $a_{2,\ell}$, we obtain the
leading twist contribution to $\mathcal{F}(z,\bar{z})$

$\begin{split}\mathcal{F}^{(n)}(z,\bar{z})&=z^{3}\sum_{\text{even
}\ell}\Bigg{\\{}\log^{n}z\frac{1}{2^{n}n!}a^{(0)}_{2,\ell}\left(\gamma^{(1)}_{2,\ell}\right)^{n}k_{6+2\ell}(\bar{z})+\log^{n-1}z\Bigg{[}a^{(1)}_{2,\ell}\frac{\left(\gamma^{(1)}_{2,\ell}\right)^{n-1}}{2^{n-1}(n-1)!}+a^{(0)}_{2,\ell}\frac{\gamma^{(2)}_{2,\ell}\left(\gamma^{(1)}_{2,\ell}\right)^{n-2}}{2^{n-1}(n-2)!}\Bigg{]}k_{6+2\ell}(\bar{z})\\\
&\qquad+\log^{n-1}z\;a^{(0)}_{2,\ell}\frac{1}{2^{n}(n-1)!}\left(\gamma^{(1)}_{2,\ell}\right)^{n}\partial_{\ell}k_{6+2\ell}(\bar{z})+\cdots\Bigg{\\}}+\mathcal{O}(z^{4})\,,\end{split}$
(S.19)

which is NLL in $\log z$. Then using the IBP identity

$a^{(0)}_{2,\ell}\partial_{\ell}k_{6+2\ell}(\bar{z})=\partial_{\ell}[a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})]-k_{6+2\ell}(\bar{z})\partial_{\ell}a^{(0)}_{2,\ell}\,,$
(S.20)

and
$a^{(1)}_{2,\ell}=-\zeta_{2}a^{(0)}_{2,\ell}+\frac{1}{2}\partial_{\ell}\left(a^{(0)}_{2,\ell}\gamma^{(1)}_{2,\ell}\right)$,
we rewrite (S.19) as

$\begin{split}\mathcal{F}^{(n)}(z,\bar{z})=&z^{3}\sum_{\text{even
}\ell}\left\\{\frac{1}{2^{n}n!}\log^{n}z\;a^{(0)}_{2,\ell}\left(\gamma^{(1)}_{2,\ell}\right)^{n}k_{6+2\ell}(\bar{z})\right.+\frac{1}{2^{n}(n-1)!}\log^{n-1}z\,\bigg{[}\left(\gamma^{(1)}_{2,\ell}\right)^{n}\partial_{\ell}\left(a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\right)\\\
&\qquad\qquad\left.+a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\left(\gamma^{(1)}_{2,\ell}\right)^{n-2}\left[\left(-2\zeta_{2}+\partial_{\ell}\gamma^{(1)}_{2,\ell}\right)\gamma^{(1)}_{2,\ell}+2(n-1)\gamma^{(2)}_{2,\ell}\right]\bigg{]}+\cdots\right\\}+\mathcal{O}(z^{4})\,.\end{split}$
(S.21)

The use of (S.20) becomes clear when we use the integer-step finite difference
to approximate
$\partial_{\ell}\left(a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\right)$ at large
spin $\ell$. On a general function $f(\ell)$, we approximate
$f^{\prime}(\ell)\approx\frac{f(\ell+2)-f(\ell-2)}{4}$, which is accurate up
to $\mathcal{O}(\ell^{-2})$. Neglecting boundary terms which vanish at large
spins, we can write

$\sum_{\text{even
}\ell}\left(\gamma^{(1)}_{2,\ell}\right)^{n}\partial_{\ell}\left(a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\right)\approx\frac{1}{4}\sum_{\text{even
}\ell}a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\left[\left(\gamma^{(1)}_{2,\ell-2}\right)^{n}-\left(\gamma^{(1)}_{2,\ell+2}\right)^{n}\right]\,,$
(S.22)

Expanding everything other than $a_{2,\ell}^{(0)}$ with respect to the large
conformal spin $J_{6,\ell}^{2}$, we get

$\begin{split}\mathcal{F}^{(n)}(z,\bar{z})=z^{3}\sum_{\text{even
}\ell}&\left\\{\frac{1}{2^{n}n!}\log^{n}z\;a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\left[\left(\log(J_{6,\ell}^{2})+2\gamma_{E}\right)^{n}+\frac{n}{3}\left(\log(J_{6,\ell}^{2})+2\gamma_{E}\right)^{n-1}\frac{1}{J_{6,\ell}^{2}}+\cdots\right]\right.\\\
&+\frac{1}{2^{n}(n-1)!}\log^{n-1}z\;a^{(0)}_{2,\ell}k_{6+2\ell}(\bar{z})\bigg{[}-(n+1)\zeta_{2}\left(\log(J_{6,\ell}^{2})+2\gamma_{E}\right)^{n-1}\\\
&-3(n-1)\zeta_{3}\left(\log(J_{6,\ell}^{2})+2\gamma_{E}\right)^{n-2}+\frac{1}{J^{2}_{6,\ell}}\bigg{(}(n-1)\left(1-\frac{\zeta_{2}}{3}(n+1)\right)\left(\log(J_{6,\ell}^{2})+2\gamma_{E}\right)^{n-2}\\\
&\left.\hskip
113.81102pt-(n-1)(n-2)\zeta_{3}\left(\log(J_{6,\ell}^{2})+2\gamma_{E}\right)^{n-3}\bigg{)}\bigg{]}+\cdots\right\\}+\mathcal{O}(z^{4})\,,\end{split}$
(S.23)

which can be organized into TCBs as

$\begin{split}\mathcal{F}^{(n)}(z,\bar{z})=&\frac{\log^{n}z}{2^{n}n!}\left(\sum_{i=0}^{n}\frac{n!}{i!(n-i)!}(2\gamma_{E})^{i}H_{2}^{(0,n-i)}+\frac{n}{3}\sum_{i=0}^{n-1-i}\frac{(n-1)!}{i!(n-1-i)!}(2\gamma_{E})^{i}H_{2}^{(1,n-1-i)}+\cdots\right)\\\
&+\frac{\log^{n-1}z}{2^{n}(n-1)!}\bigg{\\{}-(n+1)\zeta_{2}\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-1-i)!}(2\gamma_{E})^{i}H_{2}^{(0,n-1-i)}\\\
&\qquad\qquad-3(n-1)\zeta_{3}\sum_{i=0}^{n-2}\frac{(n-2)!}{i!(n-2-i)!}(2\gamma_{E})^{i}H_{2}^{(0,n-2-i)}\\\
&\qquad\qquad+(n-1)\left(1-\frac{\zeta_{2}}{3}(n+1)\right)\sum_{i=0}^{n-2}\frac{(n-2)!}{i!(n-2-i)!}(2\gamma_{E})^{i}H_{2}^{(1,n-2-i)}\\\
&\qquad\qquad-(n-1)(n-2)\zeta_{3}\sum_{i=0}^{n-3}\frac{(n-3)!}{i!(n-3-i)!}(2\gamma_{E})^{i}H_{2}^{(1,n-3-i)}+\cdots\bigg{\\}}+\cdots\,.\end{split}$
(S.24)

Substituting the explicit TCBs (B, S.17), we get

$\displaystyle\mathcal{F}^{(n)}(z,\bar{z})$
$\displaystyle=z^{3}\left\\{\frac{1}{n!}\log^{n}z\left[\frac{1}{\epsilon}\left(\frac{(-1)^{n}}{2^{n+1}}\log^{n}\epsilon+\cdots\right)+\left(\frac{(-1)^{n+1}}{2^{n+1}}\log^{n}\epsilon+\frac{(-1)^{n}n}{3\times
2^{n}}\log^{n-1}\epsilon+\cdots\right)+\cdots\right]\right.$ (S.25)
$\displaystyle\qquad\quad+\frac{\log^{n-1}z}{(n-1)!}\bigg{[}\frac{1}{\epsilon}\left(\frac{(-1)^{n}}{2^{n+1}}(n+1)\zeta_{2}\log^{n-1}\epsilon-\frac{(-1)^{n}}{2^{n+1}}3(n-1)\zeta_{3}\log^{n-2}\epsilon+\cdots\right)$
$\displaystyle\left.\hskip
99.58464pt+\left(\frac{(-1)^{n}}{2^{n+1}n}\log^{n}\epsilon+\frac{(-1)^{n+1}}{2^{n+1}}(n+1)\zeta_{2}\log^{n-1}\epsilon+\cdots\right)\bigg{]}+\cdots\right\\}+\mathcal{O}(z^{4})\,.$

Via $\mathcal{F}^{(n)}(z,\bar{z})=\frac{v}{u^{3}}\Phi^{(n)}(z,\bar{z})$, this
gives the expansion of $\Phi^{(n)}(z,\bar{z})$. Crossing symmetry allows us to
further reconstruct the $\mathcal{O}(u^{1}v^{0})$ contributions, which gives
the results in (29).

##  Appendix D Details of the Map from $u^{j_{1}}v^{j_{2}}L_{u}^{m}L_{v}^{n}$
to $y^{j}L_{y}^{k}$

In this section, we provide more details for establishing the map from the
small $u,v$ expansion of $\Phi(u,v)$ to the small $y$ expansion of
$\mathrm{EEC}(y)$. Instead of electromagnetic current sources $J_{\mu}$, we
consider two scalar operator sources, belonging to the stress tensor
multiplet, in the center of mass frame $q^{\mu}=(Q,0,0,0)$. $\mathrm{EEC}(y)$
relates to $\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle$ by an overall
factor:

$\mathrm{EEC}(y)=\int
d\Omega_{2}d\Omega_{4}\delta(\vec{n}_{2}\cdot\vec{n}_{4}-\cos\theta)\frac{\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle}{Q^{2}}=\frac{8\pi^{2}}{Q^{2}}\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle\,,$
(S.26)

where $\theta$ is the angle between $\vec{n}_{2}$ and $\vec{n}_{4}$ and we
assume the convention that
$\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle$ has already been
normalized to the cross section. The superconformal Ward identities further
reduce the EEC $\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle$ to scalar-
scalar correlation (SSC) $\langle\mathcal{S}(n_{2})\mathcal{S}(n_{4})\rangle$
Belitsky  _et al._ (2014a, c)

$\langle\mathcal{E}(n_{2})\mathcal{E}(n_{4})\rangle=\frac{4(q^{2})^{2}}{(n_{2}\cdot
n_{4})^{2}}\langle\mathcal{S}(n_{2})\mathcal{S}(n_{4})\rangle\,.$ (S.27)

The SSC is related to the local correlator in a simple way in Mellin space
Belitsky  _et al._ (2014a)

$\Phi(u,v)=\int\frac{dj_{1}dj_{2}}{(2\pi
i)^{2}}M(j_{1},j_{2})u^{j_{1}}v^{j_{2}}\;.$ (S.28)

Here $M(j_{1},j_{2})$ is the Mellin amplitude and encodes all the dynamical
information. To compute the SSC, the first step is to obtain the Lorentzian
correlator. This is achieved by using the Wightman prescription
$x_{ij}^{2}\to-x_{ij}^{2}+i\epsilon t_{ij}$, if operator $i$ sits before $j$.
In our case, the operator ordering is $1<2<4<3$. The second step is to perform
the light transform on the Lorentzian correlator

$\prod_{i=2,4}\int_{-\infty}^{\infty}d(n_{i}\cdot x_{i})\lim_{\bar{n}_{i}\cdot
x_{i}\to\infty}\left(\frac{\bar{n}_{i}\cdot x_{i}}{2}\right)^{2}\,,$ (S.29)

which turns local operators into detectors. To obtain SSC in the momentum
space, the last step is the Fourier transformation $\int
d^{4}x_{13}\exp(iq\cdot x_{13})$. After normalized to the total cross section,
the Mellin representation for SSC is

$\langle\mathcal{S}(n_{2})\mathcal{S}(n_{4})\rangle=\frac{1}{(2\pi)^{4}}\frac{\pi^{2}}{(n_{2}\cdot
n_{4})q^{2}}\frac{1-y}{y}\int\frac{dj_{1}dj_{2}}{(2\pi
i)^{2}}M(j_{1},j_{2})K(j_{1},j_{2};y)\,,$ (S.30)

with

$K(j_{1},j_{2};y)=\frac{2\Gamma(1-j_{1}-j_{2})}{\Gamma(j_{1}+j_{2})\left[\Gamma(1-j_{1})\Gamma(1-j_{2})\right]^{2}}\left(\frac{y}{1-y}\right)^{j_{1}+j_{2}}\,.$
(S.31)

Then using $n_{2}\cdot n_{4}=2(1-y)$ in the center of mass frame, we obtain
the Mellin representation for $\mathrm{EEC}(y)$

$\mathrm{EEC}(y)=\frac{8\pi^{2}Q^{2}}{(1-y)^{2}}\langle\mathcal{S}(n_{2})\mathcal{S}(n_{4})\rangle=\frac{1}{4y(1-y)^{2}}\int\frac{dj_{1}dj_{2}}{(2\pi
i)^{2}}M(j_{1},j_{2})K(j_{1},j_{2};y)\,.$ (S.32)

Therefore, we expect the following mapping from the double lightcone limit to
back-to-back limit

$u^{j_{1}}v^{j_{2}}\to\frac{K(j_{1},j_{2};y)}{4y(1-y)^{2}}\,.$ (S.33)

At LP and NLP, we need the following rules containing logarithms in $u,v$

$\displaystyle\log^{m}u\log^{n}v$ $\displaystyle\;\to\;$ $\displaystyle
m!\,n!\oint_{|j_{1}|=\epsilon}\frac{dj_{1}}{2\pi
i}\frac{1}{j_{1}^{m+1}}\oint_{|j_{2}|=\epsilon}\frac{dj_{2}}{2\pi
i}\frac{1}{j_{2}^{n+1}}\frac{K(j_{1},j_{2};y)}{4y(1-y)^{2}}\,,$ (S.34)
$\displaystyle u\log^{m}u\log^{n}v$ $\displaystyle\;\to\;$
$\displaystyle{m!\,n!}\oint_{|j_{1}-1|=\epsilon}\frac{dj_{1}}{2\pi
i}\frac{1}{(j_{1}-1)^{m+1}}\oint_{|j_{2}|=\epsilon}\frac{dj_{2}}{2\pi
i}\frac{1}{j_{2}^{n+1}}\frac{K(j_{1},j_{2};y)}{4y(1-y)^{2}}\,,$ (S.35)
$\displaystyle v\log^{m}u\log^{n}v$ $\displaystyle\;\to\;$
$\displaystyle{m!\,n!}\oint_{|j_{1}|=\epsilon}\frac{dj_{1}}{2\pi
i}\frac{1}{j_{1}^{m+1}}\oint_{|j_{2}-1|=\epsilon}\frac{dj_{2}}{2\pi
i}\frac{1}{(j_{2}-1)^{n+1}}\frac{K(j_{1},j_{2};y)}{4y(1-y)^{2}}\,.$ (S.36)

For the cases we will inspect, the only exception is the constant term at NLP
which is caused by the $j_{1}+j_{2}=1$ pole in $K(j_{1},j_{2};y)$. To see
this, we consider the first derivative of $\frac{1-y}{y}K(j_{1},j_{2};y)$
w.r.t. $\frac{y}{1-y}$. Such an action shifts $\Gamma(1-j_{1}-j_{2})$ to
$\Gamma(2-j_{1}-j_{2})$, which is analytic at $j_{1}+j_{2}=1$. The constant
term is killed after taking the derivative, while the logarithmic information
remains. The explicit expressions for general $m,n$ up to NLL accuracy are
shown in Table 2.

##  Appendix E Comparison with Existing Results

In this section we provide a comparison of our predictions with the existing
results in the literature. We begin with the local correlator in the double
lightcone limit to NLL accuracy. The full three-loop correlator can be found
in Drummond  _et al._ (2013)666There is an overall normalization difference
$\left(-\frac{1}{4\pi}\right)^{n}$ at the $n$-th loop.. Upon expanding their
results in the double lightcone limit we find

$\displaystyle\Phi^{(1)}(u,v)$ $\displaystyle=$
$\displaystyle\left[-\frac{1}{4}\log u\log
v+0\cdot\log(uv)+\cdots\right]-{\color[rgb]{1,0,0}\left[\frac{1}{4}(u+v)\log
u\log v+\frac{1}{2}(u\log u+v\log v)+\cdots\right]}+\cdots\,,$
$\displaystyle\Phi^{(2)}(u,v)$ $\displaystyle=$
$\displaystyle\left[\frac{1}{16}\log^{2}u\log^{2}v+0\cdot\log u\log
v\log(uv)+\cdots\right]+\left[\frac{1}{8}(u+v)\log^{2}u\log^{2}v\right.$
$\displaystyle\qquad\left.+\frac{3}{16}\log u\log v(u\log u+v\log
v)+{\color[rgb]{1,0,0}\frac{1}{8}\log u\log v(v\log u+u\log
v)}+\cdots\right]+\cdots\,,$ $\displaystyle\Phi^{(3)}(u,v)$ $\displaystyle=$
$\displaystyle\left[-\frac{1}{96}\log^{3}u\log^{3}v+0\cdot\log^{2}u\log^{2}v\log(uv)+\cdots\right]-\left[\frac{1}{48}(u+v)\log^{3}u\log^{3}v\right.$
(S.37) $\displaystyle\left.\qquad+\frac{1}{24}\log^{2}u\log^{2}v(u\log u+v\log
v)+\frac{1}{48}\log^{2}u\log^{2}v(v\log u+u\log v)+\cdots\right]+\cdots\,.$

Compared with our NLL prediction (29), we find perfect agreement except for
the NLP terms at one loop and a single NLL-NLP term at two loops (shown in red
explicitly). Such terms do not correspond to enhanced divergence in $v$ and
hence cannot be captured by the large spin analysis. Moreover, the two-loop
NLL-NLP mismatched term ${\color[rgb]{1,0,0}\frac{1}{8}\log u\log v(v\log
u+u\log v)}$ does not map to NLL-NLP term in EEC, as can be checked from Table
2.

For EEC in ${\cal N}=4$ SYM results up to three loops with full $y$ dependence
have been computed in Henn  _et al._ (2019), using the local correlator from
Drummond  _et al._ (2013) as input. The back-to-back expansion of Henn  _et
al._ (2019) up to NLL at NLP is

$\displaystyle\mathrm{EEC}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4y}\log y{\color[rgb]{1,0,0}-\frac{1}{2}\log
y}{\color[rgb]{1,0,0}+0\cdot y^{0}}+\mathcal{O}(y)\,,$
$\displaystyle\mathrm{EEC}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{y}\left(\frac{\log^{3}y}{8}+0\cdot\log^{2}y+\cdots\right)+\left(\frac{\log^{3}y}{6}+\frac{3}{16}\log^{2}y+\cdots\right)+\mathcal{O}(y)$
$\displaystyle\mathrm{EEC}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{y}\left(-\frac{\log^{5}y}{32}+0\cdot\log^{4}y+\cdots\right)+\left(-\frac{3\log^{5}y}{80}-\frac{\log^{4}y}{12}+\cdots\right)+\mathcal{O}(y)\,.$
(S.38)

To obtain (S.38), it is necessary to expand the following integral to NLP,

$\mathrm{E}=\int_{0}^{1}d\bar{z}\int_{0}^{\bar{z}}dt\frac{-1}{t(1-y-\bar{z})+y\bar{z}}\left[\frac{z\bar{z}}{1-z-\bar{z}}P_{1}+\frac{z^{2}\bar{z}}{(1-z)^{2}(1-z\bar{z})}P_{2}\right]\,,$
(S.39)

where $z=(1-y)t(t-\bar{z})/(t(1-y-\bar{z})+y\bar{z})$ and $P_{1}$ and $P_{2}$
are lengthy combinations of weight-3 harmonic polylogarithms and can be found
in Henn  _et al._ (2019). The expansion of this integral begins from NLP and
reads

$\mathrm{E}=\frac{\log^{5}y}{60}+0\cdot\log^{4}y+\frac{1}{12}\zeta_{2}\log^{3}y+\cdots$
(S.40)

where we have neglected terms of ${\cal O}(\log^{2}y)$ and beyond. The
expansion of this integral was also performed in Moult  _et al._ (2020b), but
the result reported in Eq. (5.17) of Moult  _et al._ (2020b) is larger than
(S.40) by a factor of two. Using (S.40) and expanding the remaining results in
Henn  _et al._ (2019) with the package HPL, we obtain $\mathrm{EEC}^{(3)}$
presented in (S.38).

Our results in (32) for $n>1$ are in full agreement with $\mathrm{EEC}^{(2)}$
and $\mathrm{EEC}^{(3)}$ in (S.38). This provides a strong check for our
results. Our large spin analysis does not expect to capture the NLP terms at
one loop (shown in red in (S.38)), for the same reason as explained for local
correlator.

We note that a LL study at NLP has also been performed in Moult  _et al._
(2020b), where an RG equation at NLP has been derived. After changing to our
normalization, their predicted LL coefficient at NLP and three loop, given in
Eq. (5.21) of Moult  _et al._ (2020b), is $-1/30\log^{5}y$, which disagrees
with our results in (33), as well as with the full results from Henn  _et al._
(2019).


# Re-crowning The Queen: Membership, Age and Rotation Periods for
the Open Cluster Coma Berenices

Kyle Singh  Department of Astronomy, Columbia University, 550 West 120th
Street, New York, NY 10027, USA  Peter Rothstein  Department of Astronomy,
Columbia University, 550 West 120th Street, New York, NY 10027, USA  Jason L.
Curtis  Department of Astronomy, Columbia University, 550 West 120th Street,
New York, NY 10027, USA  American Museum of Natural History, 200 Central Park
West, New York, NY 10024, USA  Alejandro Núñez  Department of Astronomy,
Columbia University, 550 West 120th Street, New York, NY 10027, USA  Marcel A.
Agüeros  Department of Astronomy, Columbia University, 550 West 120th Street,
New York, NY 10027, USA

###### Abstract

Coma Berenices (Coma Ber), an open cluster about the same age as Praesepe and
the Hyades (700-800 Myr) is, despite being only 85 pc away, less well studied
than its famous cousins. This is due principally to its sparseness and low
proper motion, which together made Coma Ber’s membership challenging to
establish pre-Gaia. We have curated a new list of its members based on Gaia
DR2 astrometry, derived its metallicity and interstellar reddening using
LAMOST data, and inferred the cluster’s age by fitting PARSEC isochrones to
its color–magnitude diagram. We then measured rotation periods for Coma Ber’s
low-mass members using TESS and ZTF photometry. Our isochrone fitting and the
TESS- and ZTF-derived rotation periods confirm that Coma Ber is coeval with
the Hyades and Praesepe. This work is the first step toward re-establishing
Coma Ber as another valuable benchmark cluster for age–rotation–activity
studies.

##  1 Introduction

A low-mass ($\buildrel<\over{\sim}$1 M⊙) star’s angular-momentum content
depends on its internal structure, core–envelope coupling, magnetic field
geometry, and tidal effects due to any companions. Determining the
distribution of rotation periods ($P_{\mathrm{rot}}$) for these stars in open
clusters, whose ages are well known, provides empirical constraints on how the
angular-momentum content changes over time and as a function of mass,
informing theoretical models. However, there are only a limited number of
nearby open clusters whose low-mass members can be examined in detail. Studies
of the age–rotation relation have generally relied on observations of rich,
close-by clusters such as the Hyades, Pleiades, and Praesepe. With new
observing capabilities, however, this work has been extended in the last few
years to more distant and challenging clusters.

We report on our work to add Coma Berenices (Coma Ber) to the small number of
benchmark open clusters for age–rotation–activity studies. Coma Ber, which is
about the same age as Praesepe ($\approx$700 Myr) and, at 87 pc, not much
farther away than the Hyades, has been far less well studied than its famous
cousins, mainly because its sparseness and low proper motion made its
membership challenging to establish. With Gaia data, however, it is now
possible to assemble a high-confidence membership catalog for this cluster.
Below we describe how we constructed such a catalog, used it to find an
isochrone age for the cluster, and then determined the color–period
distribution for low-mass stars in Coma Ber based on light curves from the
Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015) and the
Zwicky Transient Factory (ZTF; Masci et al., 2018).

Our isochrone fitting and the TESS- and ZTF-derived $P_{\mathrm{rot}}$ confirm
that Coma Ber is coeval with the Hyades and Praesepe, making Coma Ber another
valuable benchmark for studies of the age–rotation relation.

Figure 1: Left—Gaia EDR3 CMD for Coma Ber members (291 stars) with best-fit
PARSEC isochrone overlaid. The subset of cluster members that have Gaia DR2
RVs and RV errors and EDR3 RUWEs and proper motions consistent with being
single stars are highlighted in light green (208 stars). Right—Distribution of
$P_{\mathrm{rot}}$ for stars consistent with being single members of Coma Ber
(62 stars; teal points) together with data for single-star members of Praesepe
(orange points; Douglas et al., 2019); the orange line is a polynomial fit to
the slowly rotating sequence in Praesepe. Dark blue points show Coma Ber
astrometric binary candidates that are photometrically and spectroscopically
single (i.e., there no evidence that they have short orbital periods where
tides operate, nor is the secondary bright enough to impact the time-series
analysis; 19 stars). The slowly rotating sequences of FGK stars for the two
clusters are remarkably similar, despite the 0.2 dex difference in metallicity
between them.

##  2 A new membership catalog

We combined the pre-Gaia list from Kraus & Hillenbrand (2007) with Gaia-
DR2-based lists from Gaia Collaboration et al. (2018), Fürnkranz et al.
(2019), Tang et al. (2019), and results of our own query of the DR2 database.
We later updated the astrometric and photometric values for our candidate
members to their Early Data Release 3 (EDR3; Gaia Collaboration et al., 2020)
values.

By plotting the EDR3 parallaxes for these stars, we found that $\approx$10% of
our candidate members have very low parallaxes, indicating that they are
likely background giants. We also used Gaia data to identify binary candidates
following criteria based on those used in Curtis et al. (2020). Our new
catalog will be described fully in a forthcoming paper (Agüeros et al. in
prep.).

##  3 Coma Ber’s Fundamental Properties

We collected spectroscopic properties for Coma Ber and Praesepe members from
Xiang et al. (2019), which is based on the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope (LAMOST) DR5. Using Praesepe as a reference, we fit
and removed a systematic trend in [Fe/H] vs. effective temperature, thus
placing the values for Coma Ber on a differential scale, and found that the
metallicity of Coma Ber is 0.21 dex lower than Praesepe. Applying [Fe/H] =
+0.15 dex for Praesepe, as determined from Keck/HIRES spectra, we find [Fe/H]
= $-$0.06 dex for Coma Ber.

We computed the interstellar reddening for Coma Ber by comparing spectroscopic
temperatures from LAMOST with photometric $(G_{\rm BP}-G_{\rm RP})$ colors
from Gaia. Here again, we obtain a relative value by calculating the reddening
relative to that for Praesepe. Applying the visual extinction value of
$A_{V}=0.04$ mag for Praesepe (Douglas et al., 2019), we find zero reddening
in the Coma Ber foreground.

To determine the age of Coma Ber, we constructed EDR3 color-magnitude diagrams
(CMDs) for it and for Praesepe. We then used PAdova and TRieste Stellar
Evolution Code (PARSEC) isochrones (Bressan et al., 2012) to determine the
ages for both clusters. We visually fit the main sequence turnoff and subgiant
members. The best fit for Coma Ber is a 700 Myr isochrone (see left panel,
Figure 1). For Praesepe, the best fit is 730 Myr, an age that is slightly
older than the median literature age (670 Myr) for this cluster. We conclude
that the clusters are coeval.

##  4 Rotation In Coma Ber

TESS surveyed members of Coma Ber in Sectors 15 or 16. Given the relatively
short sector duration ($\approx$27 days) and large data gaps midway through
each sector during the data downlink, our analysis was restricted to searching
for $P_{\mathrm{rot}}$ $\leq$ 10 days.

ZTF has observed Coma Ber regularly since 2018 March 25. Following Curtis et
al. (2020), we extracted differential photometry from the archival ZTF
imaging. Although the resulting light curves are much sparser and more
irregularly sampled than those from TESS, they can provide confirmation of the
TESS-derived periods in cases where a star was observed from the ground and
from space, and can be used to measure periods for slower rotators in Coma
Ber.

We merged our TESS (58 stars) and ZTF measurements (26 stars), with data from
Collier Cameron et al. (2009, 15 stars) and Terrien et al. (2014, 4 stars),
yielding a total of 81 Coma Ber stars with measured $P_{\mathrm{rot}}$. We
then filtered out stars that are spectroscopic and/or photometric
binaries111For example, we removed stars with Gaia renormalized unit weight
error (RUWE) values $>$1.2, which are likely to be in wide binaries. and stars
with bright members blended in the large TESS pixels. The resulting
color–period distribution for Coma Ber is shown in the right panel of Figure
1, where we also include the distribution of single-star rotators in Praesepe.
The slowly rotating sequences of FGK stars for the two clusters are remarkably
similar, despite the 0.2 dex difference in metallicity between them.

###### Acknowledgements.

We thank John Brewer for sharing spectroscopic results for Hyades, Praesepe,
and Coma Ber, and Soichiro Hattori for assistance with tess_cpm.py.

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