symmetry applied to the four-point function $\langle TTTO_{i}\rangle$. How is
it possible then that the identity (149) fails?

The resolution of the puzzle is that the commutator $[{\cal
E}_{\omega_{1}}(n_{1}),{\cal E}_{\omega_{2}}(n_{2})]$ in $d>2$ CFTs requires a
regularization. In this section we defined the commutator by first separating
the light-ray operators in the ‘$+$’ direction by distance $\epsilon$, and
then taking $\epsilon\to 0$. Let us denote the commutator at finite $\epsilon$
as $[{\cal E}_{\omega_{1}}(n_{1}),{\cal E}_{\omega_{2}}(n_{2})]_{\epsilon}$.
We then expect that the following sum rule holds

$\displaystyle\lim_{\epsilon\to 0}\Big{(}[[{\cal E}_{\omega_{1}}(n_{1}),{\cal
E}_{\omega_{2}}(n_{2})]_{\epsilon},{\cal
E}_{\omega_{3}}(n_{3})]_{\epsilon}+{\rm cyclic}\Big{)}=0.$ (150)

There is an important difference between (149) and (150). According to (130),
the contribution of the operator $O_{\Delta,S}$ with twist $\tau=\Delta-S$ to
the commutator $[{\cal E}_{\omega_{1}}(n_{1}),{\cal
E}_{\omega_{2}}(n_{2})]_{\epsilon}$ vanishes as $\epsilon^{{\tau-(d-2)\over
2}}{\cal J}_{\omega_{1}+\omega_{2},S}$ for $\tau>d-2$. Performing the light-
ray transform of the three-point function $\langle TTO_{\Delta,S}\rangle$, one
can show that $[{\cal J}_{\omega_{1}+\omega_{2}}(n_{1}),{\cal
E}_{\omega_{3}}(n_{2})]_{\epsilon}$ receives a singular contribution
$\epsilon^{{(d-2)-\tau\over 2}}{\cal E}_{\omega_{1}+\omega_{2}+\omega_{3}}$
which precisely compensates the factor $\epsilon^{{\tau-(d-2)\over 2}}$ coming
from the commutator of the energy flow operators. Therefore, we conclude that
the operators with twist $\tau>d-2$ do not contribute to (149), but they do
contribute to (150).

The conclusion is that in a generic CFT we expect that the Jacobi identity
(149) receives contributions from infinitely many operators when inserted
inside the correlation function, e.g.

$\lim_{\epsilon\to 0}\langle\Big{(}[[{\cal E}_{\omega_{1}}(n_{1}),{\cal
E}_{\omega_{2}}(n_{2})]_{\epsilon},{\cal
E}_{\omega_{3}}(n_{3})]_{\epsilon}+{\rm cyclic}\Big{)}T\rangle=0\ .$

One might wonder what kind of sum rules can be obtained in this way, but this
goes beyond the scope of the present paper.

Instead, as we discuss in the concluding remarks, we can use (148) to write
simpler sum rules which generalize the superconvergence sum rules
Kologlu:2019bco  and connect to the dispersive sum rules Caron-Huot:2020adz ;
Caron-Huot:2021enk .

##  4 Contact terms in Mellin space

In this section, we apply the obtained results for the algebra of light-ray
operators to understand the properties of (generalized) energy-energy
correlation in ${\cal N}=4$ SYM

$\displaystyle\text{EEC}(z,\omega_{1},\omega_{2})=\langle{0|\phi^{\dagger}\mathcal{E}_{\omega_{1}}(n_{1})\mathcal{E}_{\omega_{2}}(n_{2})\phi(q)|0}\rangle\,,$
(151)

where $\phi$ and $\phi^{\dagger}$ are the half-BPS scalar primary operators in
the representation $\mathbf{20^{\prime}}$ of the $R-$symmetry group (see Ref.
Korchemsky:2021okt  for details).

The initial state $\phi(q)|0\rangle$ carries the total momentum $q^{\mu}$ and
the energy flow operators $\mathcal{E}_{\omega_{1}}(n_{1})$ and
$\mathcal{E}_{\omega_{2}}(n_{2})$ mimic two calorimeters detecting the
particles propagating in the directions specified by null vectors $n_{1}$ and
$n_{2}$. The function (151) depends on dimensionless scaling variable

$\displaystyle z={q^{2}(n_{1}n_{2})\over 2(qn_{1})(qn_{2})}\,,$ (152)

satisfying $0\leq z\leq 1$, and two dimensionless frequencies
$\hat{\omega}_{i}=2\omega_{i}(qn_{i})/q^{2}$. To simplify notations, we will
not display the additional factor in the expression for $\hat{\omega}_{i}$ and
shall treat $\omega_{i}$ as dimensionless variables.

For $\omega_{1}=\omega_{2}=0$, the function (151) coincides with the
conventional energy-energy correlation. A powerful approach to computing this
function was developed in Refs. Belitsky:2013xxa ; Belitsky:2013bja . It was
recently applied to the generalized energy-energy correlation (151) with
$\omega_{i}\neq 0$ in Korchemsky:2021okt . The approach relies on the Mellin
representation of the correlation functions and yields the following
expression for the generalized energy-energy correlation

$\displaystyle\text{EEC}(z,\omega_{1},\omega_{2})=\text{EEC}^{\text{tree}}(z,\omega_{1},\omega_{2})+\text{EEC}^{\text{Mellin}}(z,\omega_{1},\omega_{2})\
,$ (153)

where $\text{EEC}^{\text{tree}}$ describes the tree-level (zero coupling)
contribution and $\text{EEC}^{\text{Mellin}}$ is given by a convolution of the
Mellin amplitude of the correlator and a kinematical kernel depending on
dimensionless variables $z$ and $\omega_{i}$

$\displaystyle\text{EEC}^{\text{Mellin}}(z,\omega_{1},\omega_{2})=\int{dj_{1}dj_{2}\over(2\pi
i)^{2}}K_{\text{EEC}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})M(j_{1},j_{2})\,.$
(154)

The explicit form of $K_{\text{EEC}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})$ can
be found in the ancillary file contained in the submission of Ref.
Korchemsky:2021okt .

The results obtained in Refs. Belitsky:2013xxa ; Belitsky:2013bja ;
Korchemsky:2021okt  apply when the detectors are located at separated points
on the celestial sphere $n_{1}\neq n_{2}$. In this case, the energy flow
operators commute and the function $\text{EEC}(z,\omega_{1},\omega_{2})$ is
symmetric under the exchange of the detectors. The commutation relations (1)
imply that this property does not hold if the detectors are oriented along the
same direction $n_{1}=n_{2}$, or equivalently $z=0$. It follows from (1) that
$\text{EEC}(z,\omega_{1},\omega_{2})-\text{EEC}(z,\omega_{2},\omega_{1})\sim\delta(z)$.171717The
locus $z=0$ corresponds to $n_{1}=n_{2}$, as such $\delta^{d-2}(n_{1},n_{2})$
that we used in the previous sections becomes $\delta(z)$ up to a trivial
constant pre-factor. It is natural to ask to what extent the analysis in
Mellin space can be generalized to also cover such contact terms. This is the
subject of the present section.

Let us start with an important subtlety which we already discussed at the
beginning of Section 3. In doing the computations in the papers mentioned
above, the detector operators were first placed to future null infinity, and
only then the integral over working time of the detector, or the light
transform, was performed. It is possible to check that if we were to do this
in the previous sections of the present paper we would miss important contact
terms that produce nontrivial contributions to the commutator. As a result, if
we want to correctly capture the commutator the analysis of Belitsky:2013xxa ;
Korchemsky:2021okt  is not obviously applicable.

At the technical level, the subtlety arises from the treatment of the
separation between detectors $(x_{12}^{2})^{-\delta_{12}}$ in the definition
of the Mellin amplitude. If we do the light transform first, we can
potentially pick the poles from $x_{12}^{2}=0$ which are sensitive to the
light-ray commutators. If we, on the other hand, take the detectors to null
infinity first, the $x_{12}^{2}=0$ poles never contribute to the light
transform.

There is another possibility for poles $x_{12}^{2}=0$ not to contribute even
if we want to discuss the locus $n_{1}=n_{2}$. If the light-ray operators are
ordered as ${\cal E}_{\omega_{1}\geq 0}(n_{1}){\cal E}_{\omega_{2}\leq
0}(n_{2})$ then by evaluating the light transform we will not get the
contribution from the $x_{12}^{2}=0$ poles due to the $i\epsilon$
prescription, which takes the form $t_{1}-t_{2}-i\epsilon$, where we used the
detector time from the definition (1). For $\omega_{1}\geq 0$ we can deform
the $t_{1}$ contour to the lower half-plane, and given that $\omega_{2}\leq
0$, the $t_{2}$ contour to the upper half-plane, so that the pole at ${\rm
Im}[t_{1}-t_{2}]=\epsilon$ never contributes.

In principle, it should be possible to generalize the analysis of
Belitsky:2013xxa ; Korchemsky:2021okt  to include such contributions, but here
for simplicity we only restrict our consideration to the particular ordering
${\cal E}_{\omega_{1}\geq 0}(n_{1}){\cal E}_{\omega_{2}\leq 0}(n_{1})$ for
which the contact terms in the event shapes can be simply computed using the
formulas of Belitsky:2013xxa ; Korchemsky:2021okt  as we explain below.

###  4.1 Zero frequencies

Let us first consider the conventional energy-energy correlation corresponding
to $\omega_{1}=\omega_{2}=0$. In this case, we have (see Refs.
Belitsky:2013xxa ; Korchemsky:2021okt )

$\displaystyle\text{EEC}^{\text{tree}}(z,0,0)={1\over
4}\Big{(}\delta(z)+\delta(1-z)\Big{)}\,,$ $\displaystyle
K_{\text{EEC}}(j_{1},j_{2}|0,0,z)={\pi\over
2\sin\pi(j_{1}+j_{2})}z^{-2-j_{1}-j_{2}}\left(1-z\right)^{j_{1}+j_{2}-1}\,.$
(155)

In order to reveal the origin of contact term $\delta(z)$ in the Mellin
integral (154), we change integration variable to $j=j_{1}+j_{2}$ and rewrite
(154) as

$\displaystyle\text{EEC}^{\text{Mellin}}(z,0,0)=\int_{-\epsilon-i\infty}^{-\epsilon+i\infty}{dj\over
2\pi i}{1\over z^{1+(1+j)}}f(j,z)\,,$ (156)

where the notation was introduced for

$\displaystyle f(j,z)={2\pi\over\sin\pi
j}\left(1-z\right)^{j-1}\int{dj_{1}\over 2\pi i}M(j_{1},j-j_{1})\,.$ (157)

To find the leading behaviour of (156) at small $z$, we deform the integration
contour to the left and pick up a residue at the rightmost pole of $f(j,z)$.
Imagine that it is located at $j=-1$ and the function $f(j,z)$ has the
following expansion around it

$\displaystyle f(j,z)={f_{-1}(z)\over 1+j}+f_{0}(z)+\dots\ .$ (158)

Then, we expand ${1/z^{1+(1+j)}}$ around $j=-1$ using the identity

$\displaystyle{1\over z^{1+(1+j)}}=-{1\over 1+j}\delta(z)+\Big{[}{1\over
z}\Big{]}_{+}+\dots\ ,$ (159)

where the plus distribution is defined as
$\int_{0}^{1}dz\,\phi(z)\big{[}{1\over
z}\big{]}_{+}=\int_{0}^{1}dz(\phi(z)-\phi(0))/z$ for an arbitrary test
function. In this way, closing the contour in (156) to the left we get

$\displaystyle\text{EEC}^{\text{Mellin}}(z,0,0)=-f_{0}(0)\delta(z)+f_{-1}(0)\Big{[}{1\over
z}\Big{]}_{+}+\dots\,.$ (160)

Let us apply the relations (157) – (160) to the leading expressions for the
Mellin amplitude $M(j_{1},j_{2})$ at weak and strong coupling

$\displaystyle M^{\text{weak}}(j_{1},j_{2})=-{a\over
4}{(j_{1}+j_{2})^{2}\over(j_{1}j_{2})^{2}}\,,$ $\displaystyle
M^{\text{strong}}(j_{1},j_{2})=-{1\over
2}{(j_{1}+j_{2})^{2}(1+j_{1}+j_{2})\over j_{1}j_{2}}\,,$ (161)

where $a=g_{\text{YM}}^{2}N/(4\pi^{2})$ is the ’t Hooft coupling constant.
Substituting these expressions into (157) and (158) we get

$\displaystyle f_{-1}^{\text{weak}}(0)$ $\displaystyle={a\over 4},$
$\displaystyle f_{0}^{\text{weak}}(0)={a\over 4}\,,$ $\displaystyle
f_{-1}^{\text{strong}}(0)$ $\displaystyle=0,$ $\displaystyle
f_{0}^{\text{strong}}(0)={1\over 4}\,.$ (162)

Combining these relations with (160) we reproduce the known expressions for
the energy-energy correlation at small $z$.

For example, at strong coupling we get from (153) and (160)

$\displaystyle\text{EEC}^{\text{strong}}(z,0,0)={1\over
4}\Big{(}\delta(z)+\delta(1-z)\Big{)}-{1\over
4}\Big{(}\delta(z)+\delta(1-z)-2\Big{)}={1\over 2}\,,$ (163)

where the first term comes from $\text{EEC}^{\text{tree}}$ and the second one
from $\text{EEC}^{\text{Mellin}}$. Notice that the contact terms cancel in the
sum of two terms as expected (see Ref. Hofman:2008ar ). The origin of
$-{1\over 4}\delta(z)$ term in $\text{EEC}^{\text{Mellin}}$ was explained
above. The presence of $-{1\over 4}(\delta(1-z)-2)$ can be equally detected by
computing the moments $\int_{0}^{1}dzz^{N}\text{EEC}^{\text{Mellin}}(z,0,0)$.

###  4.2 Finite frequencies

Let us now consider generalized energy-energy correlation (153) and (154) for
$\omega_{i}\neq 0$. For the sake of simplicity we restrict our consideration
to $\omega_{1}\geq 0$ and $\omega_{2}\leq 0$ and denote the corresponding
function (153) as $\text{EEC}_{+-}(z,\omega_{1},\omega_{2})$.

In this case, we have in $\mathcal{N}=4$ SYM theory 181818Here we ignore the
contribution of the disconnected correlator which is localized at $z=0$ and in
addition is proportional to $\delta(\omega_{1}+\omega_{2})$. We also write
down the tree-level expression that is valid for any signs of $\omega_{i}$.

$\displaystyle\text{EEC}^{\text{tree}}(z,\omega_{1},\omega_{2})$
$\displaystyle=\frac{(\omega_{2}^{2}+6\omega_{2}+6)(\omega_{1}^{2}+6(\omega_{2}+1)\omega_{1}+6(\omega_{2}+1)^{2})}{144(\omega_{2}+1)}\theta(1+\omega_{2})\delta(z)$
$\displaystyle-\frac{(\omega_{1}^{2}+6\omega_{1}+6)(\omega_{2}^{2}+6(\omega_{1}+1)\omega_{2}+6(\omega_{1}+1)^{2})}{144(\omega_{1}+1)}\theta(-1-\omega_{1})\delta(z)$
$\displaystyle+\frac{(\omega_{1}^{2}+6\omega_{1}+6)(\omega_{2}^{2}+6\omega_{2}+6)}{144}\theta(1+\omega_{1})\theta(1+\omega_{2})\delta(1-z)\,.$
(164)

It is easy to check that for $\omega_{1}=\omega_{2}=0$ this relation coincides
with the first relation in (4.1).

As in the previous section, the contact term $\delta(z)$ in
$\text{EEC}_{+-}^{\text{Mellin}}(z,\omega_{1},\omega_{2})$ arises from small
$z$ behaviour of the kernel
$K_{\text{EEC}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})$ in (154). Using the
explicit form of the kernel we find

$\displaystyle K_{\text{EEC}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})\sim{1\over
z^{1+(1+j_{1}+j_{2})}}\,.$ (165)

Going through the same steps as in the previous section, we obtain the
following result for $\text{EEC}_{+-}^{\text{Mellin}}$ at strong coupling for
$\omega_{1}\geq 0$ and $-1<\omega_{2}\leq 0$

$\displaystyle\text{EEC}_{+-}^{\text{Mellin}}(z,\omega_{1},\omega_{2})$
$\displaystyle=\frac{(\omega_{2}^{2}+6\omega_{2}+6)(6(\omega_{2}+1)^{2}+6\omega_{1}(\omega_{2}+1)(\omega_{2}z+1)+\omega_{1}^{2}(\omega_{2}z+1)^{2})}{72(\omega_{2}z+1){}^{3}}$
$\displaystyle-\frac{(\omega_{2}^{2}+6\omega_{2}+6)(\omega_{1}^{2}+6(\omega_{2}+1)\omega_{1}+6(\omega_{2}+1)^{2})}{144(\omega_{2}+1)}\delta(z)$
$\displaystyle-\frac{(\omega_{1}^{2}+6\omega_{1}+6)(\omega_{2}^{2}+6\omega_{2}+6)}{144}\delta(1-z)\,.$
(166)

Substituting (4.2) and (4.2) into (153) we observe that the contact terms
cancel in the sum of two expressions leading to

$\displaystyle\text{EEC}^{\text{strong}}_{+-}(z,\omega_{1},\omega_{2})$
$\displaystyle=\frac{(\omega_{2}^{2}+6\omega_{2}+6)(6(\omega_{2}+1)^{2}+6\omega_{1}(\omega_{2}+1)(\omega_{2}z+1)+\omega_{1}^{2}(\omega_{2}z+1)^{2})}{72(\omega_{2}z+1)^{3}}\,.$
(167)

Thus, the generalized energy-energy correlation does not contain any
distributional terms at strong coupling. This is indeed consistent with the
following simple observation. Taking into account that the energy flow
operator $\mathcal{E}_{\omega}(n)$ annihilates the left (or right) vacuum for
$\omega>0$ (or $\omega<0$) we can write for $\omega_{1}\geq 0$ and
$\omega_{2}\leq 0$

$\displaystyle\text{EEC}_{+-}(z,\omega_{1},\omega_{2})=\langle
0|[\phi^{\dagger},{\cal E}_{\omega_{1}}(n_{1})][{\cal
E}_{\omega_{2}}(n_{2}),\phi]|0\rangle\,.$ (168)

Following Kologlu:2019bco , we can insert a complete state of states between
the two commutators to get the OPE representation of
$\text{EEC}_{+-}(z,\omega_{1},\omega_{2})$. At strong coupling only $\phi(x)$
itself contributes at leading order in ${1/N_{c}^{2}}$, due to the familiar
suppression of the double trace operators in the double commutator. We do not
expect to get any distributional terms from the exchange of $\phi(x)$, hence
we get a regular result in agreement with (4.2). This is a very nontrivial
check of consistency of our approach.

If we, on the other hand, exchange the flow operators in (151) the same
argument does not apply because for $\omega_{1}\geq 0$ and $\omega_{2}\leq 0$
the function $\text{EEC}_{-+}(z,\omega_{2},\omega_{1})$ cannot be writen as a
double commutator. In particular, the double trace operators are not
suppressed in this case and they eventually generate the nontrivial contact
term $\delta(z)$ in $\text{EEC}_{-+}(z,\omega_{2},\omega_{1})$.

###  4.3 Asymmetry

As was mentioned above, the commutator of the energy flow operator controls
the asymmetry of the energy-energy correlation,
$\text{EEC}(z,\omega_{1},\omega_{2})-\text{EEC}(z,\omega_{2},\omega_{1})\sim\delta(z)$.

Let us first consider the limit of zero coupling in $\mathcal{N}=4$ SYM when
the Mellin contribution to (153) is absent. It follows from the algebra (9)
that

$\displaystyle\text{EEC}^{\text{tree}}(z,\omega_{1},\omega_{2})-\text{EEC}^{\text{tree}}(z,\omega_{2},\omega_{1})={1\over
4}\delta(z)\langle(\omega_{2}-\omega_{1}){\cal
E}_{\omega_{1}+\omega_{2}}(n_{1})+\omega_{1}^{2}\omega_{2}^{2}{\cal
O}^{-}_{\omega_{1},\omega_{2}}(n_{1})\rangle_{\text{tree}}\,,$ (169)

where an average on the right-hand side is taken with respect to the state
$|{\phi(q)}\rangle$ containing in the free theory two on-shell scalar
particles with the total momentum $q$, in the representation
$\mathbf{20^{\prime}}$ of the $R-$symmetry group (see Ref. Korchemsky:2021okt
for details). The relevant one-point functions take the form

$\displaystyle\langle{\cal E}_{\omega}(n)\rangle_{\text{tree}}$
$\displaystyle=1+\omega+{\omega^{2}\over 6}\,,$ $\displaystyle\langle{\cal
O}^{-}_{\omega_{1},\omega_{2}}(n)\rangle_{\text{tree}}$
$\displaystyle={\omega_{1}-\omega_{2}\over 36(1+\omega_{1})(1+\omega_{2})}\,.$
(170)

We can use (4.2) to check that the light-ray algebra identity (169) is indeed
satisfied. Notice that the first relation in (4.3) is protected from
corrections in $\mathcal{N}=4$ SYM.

The situation becomes more interesting at finite coupling. We show in Appendix
F that the operator ${\cal O}^{(\phi)}_{\omega_{1},\omega_{2}}$ is given by
the sum of light-ray operators (1) of an arbitrary spin $S$. In
$\mathcal{N}=4$ SYM the operators with $S\neq 2$ acquire anomalous dimensions.
Their twist satisfies $\tau>2$ and, as a consequence, they do not contribute
to the light-ray algebra. The only operator that contributes is the stress-
energy tensor. This leads to the algebra (15). As a consequence, the energy-
energy correlation satisfies

$\displaystyle\text{EEC}(z,\omega_{1},\omega_{2})-\text{EEC}(z,\omega_{2},\omega_{1})={1\over
4}\delta(z)(\omega_{2}-\omega_{1}+{1\over
10}f_{\phi}(\omega_{1},\omega_{2}))\langle{\cal
E}_{\omega_{1}+\omega_{2}}(n_{1})\rangle\,.$ (171)

Combining this relation with (153) and (169) we expect that the following
identity holds in $\mathcal{N}=4$ SYM at finite coupling

$\displaystyle\text{EEC}^{\text{Mellin}}(z,\omega_{1},\omega_{2})-\text{EEC}^{\text{Mellin}}(z,\omega_{2},\omega_{1})$
$\displaystyle={1\over 4}\delta(z)\left[{1\over
10}f_{\phi}(\omega_{1},\omega_{2})\langle{\cal
E}_{\omega_{1}+\omega_{2}}(n_{1})\rangle-\omega_{1}^{2}\omega_{2}^{2}\langle{\cal
O}^{-}_{\omega_{1},\omega_{2}}(n_{1})\rangle\right],$ (172)

where the one-point functions on the right-hand side are given by (4.3). It
would be interesting to derive this relation by an explicit computation.

##  5 Concluding remarks

In this paper we explored commutation relations of the light-ray operators (1)
built out of local operators in a $d-$dimensional CFT. The commutator of these
operators
$[\mathcal{J}_{\omega,S}(n),\mathcal{J}_{\omega^{\prime},S^{\prime}}(n^{\prime})]$
is localized at $n=n^{\prime}$, when both operators are pointed in the same
direction on the celestial sphere. We argued that it has different form in
free and interacting CFTs.

We demonstrated that in a free theory in $d$ dimensions, the flow operators
form a closed algebra and derived the explicit expressions for the
corresponding structure constants and the central charges. In $d=2$ the light-
ray operators are closely related to generators of infinite-dimensional
symmetries of CFT2. More precisely, the light-ray operators of spin $S=2$, or
equivalently the energy flow operators, are generating functions of the
generators of Virasoro algebra. In the similar manner, for $d=2$, the light-
ray operators of higher spin are related to the generators of the $W_{\infty}$
algebra. We found that in the free theory of complex scalar and fermion in
$d=4$ dimensions (and, more generally, for any even $d$), the structure
constants of the algebra of the light-ray operators are related to those of
the Virasoro and $W_{\infty}$ algebras by a linear finite-difference relation.

Defining the light-ray operators (1) in free CFTs, we have chosen the
conformal operators to be built from two fundamental fields of the same type
(scalars, fermions and gauge fields). In a similar manner, we can consider a
free CFT describing one fermion, one complex scalar and gauge field, define
“mixed” operators (built out of fields of different type) and construct the
corresponding flow operators. In $d=2$ dimensions, this theory respects
$\mathcal{N}=2$ supersymmetry and, as a consequence, such flow operators form
a $\mathcal{N}=2$ supermultiplet. Their algebra coincides with
super$-W_{\infty}$ algebra Bergshoeff:1988uc ; Bergshoeff:1990cz ;
Bergshoeff:1991dz . In $d=4$ dimensions, all leading twist operators belong to
the same $\mathcal{N}=4$ supermultiplet Belitsky:2003sh . This suggests that
the flow operators generate an $\mathcal{N}=4$ superalgebra whose bosonic
sector is given by (8) and (9). It would be interesting to compute the
structure constants of this superalgebra and to establish their relation to
those of super$-W_{\infty}$ algebra.

We found that in an interacting CFT in $d>2$ dimensions, the commutator of the
energy flow operators only receives contribution from the stress-energy tensor
itself and light scalar primary operators (which appear in the OPE of two
stress-energy tensors) with dimension $\Delta\leq d-2$. The latter
contribution is divergent for $\Delta<d-2$.

As opposed to the free theories considered in Section 2, the light-ray
operators do not form an algebra in interacting CFTs. As explained in Section
3.4, this is related to the careful, regularized definitions of the
commutators. It would be interesting to understand if there is a sense in
which the light-ray algebra that we found in the free CFTs is weakly broken at
weak coupling, see e.g. Refs. Maldacena:2012sf ; Gerasimenko:2021sxj .

We believe that the significance of the commutation relations considered in
the present paper is two-fold. First, the algebra of the light-ray operators
is connected to the symmetries of the theory, be it global conformal
invariance in interacting CFTs or higher spin symmetry in the free CFTs. For
example, the only spinning operators that contribute to the commutator of the
energy flow operators are conserved currents out of which the symmetry
generators are constructed. The result (1) contains a universal term
$[\mathcal{E}_{\omega_{1}}(n_{1}),\mathcal{E}_{\omega_{2}}(n_{2})]=\delta^{(d-2)}(n_{1},n_{2})(\omega_{2}-\omega_{1}){\cal
E}_{\omega_{1}+\omega_{2}}(n_{1})+\dots$ which comes from the OPE $T\times
T\sim T+\dots$ and is closely related to the conformal Ward identities. In a
free theory, in a close analogy with the $W$ symmetry in CFT2, the algebra
gets extended to include the light-ray operators built out of higher spin
currents.

Second, the commutator of light-ray operators naturally leads to a family of
sum rules

$\displaystyle\int\prod_{i=1}^{2}d\omega_{i}d\vec{n}_{i}\,\delta(\vec{n}_{i}^{2}-1)g(n_{i},\omega_{i})\langle\phi_{i}|(\text{light-
ray commutator})|\phi_{j}\rangle=0,$ (173)

where $g(n_{i},\omega_{i})$ is a priori an arbitrary function. The sum rules
are obtained by inserting a complete set of states between the light-ray
operators in the commutator. For example, let us consider a generic
interacting CFT, where the light-ray algebra takes the form (1). The presence
of the divergent contributions in the algebra due to light scalar primary
operators, $\Delta<d-2$, signify the necessity of subtractions to get well-
defined CFT sum rules. For the identity operator subtraction boils down to
considering the connected correlator on the left-hand side of (173). For other
light scalar primary operators of dimension ${d-2\over 2}\leq\Delta<d-2$, that
appear in the OPE of stress-energy tensors, we can for example subtract the
corresponding scalar exchange Witten diagram which effectively removes light
scalars from the $[{\cal E}_{\omega_{1}}(n_{1}),{\cal
E}_{\omega_{2}}(n_{2})]$. The sum rules (173) generalize the superconvergence
relations of Ref. Kologlu:2019bco  and we can refer to the sum rules that
follow from the light-ray algebra as generalized superconvergence relations.
We do not attempt studying the consequences of these sum rules for the OPE
data of CFTs in the present paper.

Let us note in this context that the generalized superconvergence relations,
analogous to (173), were shown to be equivalent to the dispersive sum rules in
some interesting cases Caron-Huot:2020adz ; Caron-Huot:2021enk . It would be
very interesting to explore this connection in a greater depth. For example,
by making $g(n_{i},\omega_{i})$ vanishing in the limit
$\omega_{1}-\omega_{2}\to 0$ with $\omega_{1}+\omega_{2}$ fixed, one damps the
contribution of the Regge limit to the sum rule. Functionals with this
property are related to subtractions in the CFT dispersion relations
Carmi:2019cub ; Mazac:2019shk ; Penedones:2019tng .

The light-ray algebra has important application to the generalized event
shapes introduced in Ref. Korchemsky:2021okt . It allows one to control the
contribution to these observables of distributions localized at the coinciding
points on the celestial sphere. Already for the conventional event shapes, for
$\omega_{i}=0$, understanding of such distributional terms was important in
order to verify the conformal Ward identities Kologlu:2019mfz ;
Korchemsky:2019nzm ; Dixon:2019uzg , which serve as a nontrivial check of the
computations. The same applies to the generalized event shapes for
$\omega_{i}\neq 0$. In this context, we have explained how the distributional
terms naturally emerge in Mellin space in ${\cal N}=4$ SYM and presented
expected expression for the commutator (4.3). It would be interesting to
derive this relation from the first principles.

Relatedly, generalized event shapes satisfy the following positivity condition

$\displaystyle\int\prod_{i=1}^{2}d\omega_{i}d\vec{n}_{i}\,\delta(\vec{n}_{i}^{2}-1)\,g^{*}(n_{1},\omega_{1})g(n_{2},\omega_{2})\langle\Psi|\mathcal{E}_{-\omega_{1}}(n_{1})\mathcal{E}_{\omega_{2}}(n_{2})|\Psi\rangle\geq
0\,.$ (174)

It follows from the fact that
$\mathcal{E}_{\omega_{1}}^{\dagger}(n_{1})=\mathcal{E}_{-\omega_{1}}(n_{1})$
and therefore (174) can be interpreted as a norm, or the total cross-section
of the scattering $|{\Psi}\rangle$ off the state created by the integrated
energy flow operator. Computing (174) necessarily involves evaluation of the
generalized event shapes at coincident point $n_{1}=n_{2}$, which we
considered in Mellin space in Section 4.

## Acknowledgments

We are grateful to Alex Belin, Andrei Belitsky, Diego Hofman, Denis Karateev,
Marco Meineri, Emery Sokatchev, Eric Perlmutter, Slava Rychkov, and Matthew
Walters for useful discussions. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement number 949077). The work of
GK was supported by the French National Agency for Research grant
ANR-17-CE31-0001-01.

##  Appendix A Conventions

For an arbitrary $d-$dimensional vector $x^{\mu}=(x^{0},\dots,x^{d-1})$ the
light-like coordinates are defined as

$\displaystyle x^{+}={1\over\sqrt{2}}(x^{0}+x^{d-1})\,,\qquad\quad
x^{-}={1\over\sqrt{2}}(x^{0}-x^{d-1})\,,\qquad\quad\mbox{\boldmath$x$}=(x^{1},\dots,x^{d-2})\,.$
(175)

The metric in these coordinates is

$\displaystyle
dx^{2}=2dx^{+}dx^{-}-d\mbox{\boldmath$x$}d\mbox{\boldmath$x$}\,,$ (176)

and the scalar product looks as

$\displaystyle(xy)=x^{+}y^{-}+x^{-}y^{+}-\mbox{\boldmath$x$}\cdot\mbox{\boldmath$y$}\,.$
(177)

The definition of the flow operator (1) involves an auxiliary light-like
vector $\bar{n}^{\mu}$ that we choose to be along the ‘$-$’ direction,
$\bar{n}^{-}=1$ and $\bar{n}^{+}=\bar{\mbox{\boldmath$n$}}=0$. In $d=2$
dimensions, the light-like vectors satisfying $(n\bar{n})\neq 0$ take the form
$n^{\mu}\sim(1,1)$, or equivalently $n^{-}=0$.

To obtain representation of the light-ray operator (16), we apply a conformal
transformation Cornalba:2007fs ; Hofman:2008ar

$\displaystyle z^{+}=-{1\over 2x^{+}}\,,\qquad\qquad
z^{-}=x^{-}-{\mbox{\boldmath$x$}^{2}\over
2x^{+}}\,,\qquad\qquad\mbox{\boldmath$z$}={\mbox{\boldmath$x$}\over\sqrt{2}x^{+}}\,,$
(178)

where $dz^{2}=dx^{2}/(2(x^{+})^{2})$. It maps a light ray at null infinity
$x^{\mu}=nr+\bar{n}t$ with $r\to\infty$ and $-\infty<t<\infty$ to a line
$-\infty<z^{-}<\infty$ with $z^{+}=0$,
$n^{\mu}=(1,\mbox{\boldmath$z$}^{2},\sqrt{2}\mbox{\boldmath$z$})$ and
$\bar{n}^{\mu}=(0,1,\mbox{\boldmath$0$})$ in the light-cone coordinates

The commutation relations of the flow operators (2.3) involve the delta-
function on the sphere $S^{d-2}$. For two light-like vectors
$n^{\mu}=(1,\vec{n})$ and $n^{\prime}{}^{\mu}=(1,\vec{n}^{\prime})$ it is
defined as

$\displaystyle\delta^{(d-2)}(n,n^{\prime})=\lim_{\epsilon\to 0}{1\over(2\pi
i\epsilon)^{d/2-1}}\mathop{\rm e}\nolimits^{i(nn^{\prime})/\epsilon}\,.$ (179)

It is straightforward to verify that it satisfies

$\displaystyle\int
d\vec{n}\,\delta(\vec{n}^{2}-1)f(n)\delta^{(d-2)}(n,n^{\prime})=f(n^{\prime})\,,$
(180)

where $\int d\vec{n}=\int dn_{1}\dots dn_{d-1}$ and $f(n)$ is a test function.
For $d=2$, it follows from $(\bar{n}n)\neq 0$ and $(\bar{n}n^{\prime})\neq 0$
that $n\sim n^{\prime}$ and, therefore, $\delta^{(0)}(n,n^{\prime})=1$.

Taking into account (179) we obtain

$\displaystyle\lim_{\epsilon\to 0}{1\over(2\pi
i\epsilon)^{d/2-1}}{\Gamma(2j)(-i)^{2j}\over(-(nn^{\prime})/\epsilon+t-i0)^{2j}}$
$\displaystyle=\int_{0}^{\infty}d\alpha\,\alpha^{2j-d/2}\mathop{\rm
e}\nolimits^{-i\alpha t}\lim_{\epsilon\to 0}{\mathop{\rm
e}\nolimits^{is(nn^{\prime})/\epsilon}\over(2\pi i\epsilon/s)^{d/2-1}}$
$\displaystyle=\delta^{(d-2)}(n,n^{\prime})\int_{0}^{\infty}d\alpha\,\alpha^{2j-d/2}\mathop{\rm
e}\nolimits^{-i\alpha t}\,,$ (181)

where in the first relation we applied (188).

##  Appendix B Gegenbauer polynomials

The leading twist operators (29) and (31) are expressed in terms of Gegenbauer
polynomials $C_{n}^{(\lambda)}(z)$. Their index $\lambda=2j-1/2$ is related to
the conformal spin of the field (25). In particular, $j=0$ for a scalar field
in $d=2$ dimensions. In this case, we have

$\displaystyle C_{n}^{(-1/2)}(x)={(1-x^{2})\over n(n-1)}C_{n-2}^{(3/2)}(x)$
(182)

for $n\geq 2$, or equivalently in terms of polynomials (31)

$\displaystyle P_{n}^{(0)}(p_{1},p_{2})={4p_{1}p_{2}\over
n(n-1)}P_{n-2}^{(1)}(p_{1},p_{2})\,,$ (183)

where the superscript refers to the conformal spin $j$.

The Gegenbauer polynomials satisfy a recurrence relation

$\displaystyle z\,C_{n}^{(\lambda)}(z)={(2\lambda+n-1)\over
2(\lambda+n)}C_{n-1}^{(\lambda)}(z)+{(n+1)\over
2(\lambda+n)}C_{n+1}^{(\lambda)}(z)\,.$ (184)

One can use it to verify the following identity

$\displaystyle(t-\varepsilon)$ $\displaystyle
P_{\ell}(t,-t+\varepsilon)P_{\ell^{\prime\prime}}(t,1-t)$
$\displaystyle=\frac{\varepsilon}{2(1-\varepsilon)}\Bigg{[}\frac{(4j+{\ell^{\prime\prime}}-2)}{(4j+2{\ell^{\prime\prime}}-1)}\mathop{\rm
e}\nolimits^{-\partial_{\ell^{\prime\prime}}}+\frac{({\ell^{\prime\prime}}+1)}{(4j+2{\ell^{\prime\prime}}-1)}\mathop{\rm
e}\nolimits^{\partial_{\ell^{\prime\prime}}}$
$\displaystyle+\frac{\varepsilon(1-2\varepsilon)(4j+\ell-2)}{(4j+2\ell-1)}\mathop{\rm
e}\nolimits^{-\partial_{\ell}}+\frac{(1-2\varepsilon)(\ell+1)}{\varepsilon(4j+2\ell-1)}\mathop{\rm
e}\nolimits^{\partial_{\ell}}\Bigg{]}P_{\ell}(t,-t+\varepsilon)P_{\ell^{\prime\prime}}(t,1-t)\,,$
(185)

where $P_{\ell}$ is given by (31).

The Gegenbauer polynomials are orthogonal with respect to inner product

$\displaystyle\int_{-1}^{1}dx\,(1-x^{2})^{\lambda-\frac{1}{2}}C_{n}^{(\lambda)}(x)C_{m}^{(\lambda)}(x)=\delta_{nm}{\pi
2^{1-2\lambda}\Gamma(n+2\lambda)\over n!(n+\lambda)\Gamma^{2}(\lambda)}\,.$
(186)

This relation ensures that the two-point correlation function of conformal
operators (33) is diagonal in spins. To show this, we consider the correlation
function $\langle{O_{S_{1}}(x_{1})O_{S_{2}}(x_{2})}\rangle$ for
$x_{i}=(x_{i}^{+},x_{i}^{-},\mbox{\boldmath$0$})$. Using (29) one obtains

$\displaystyle\langle{O_{S_{1}}(x_{1})O_{S_{2}}(x_{2})}\rangle=\langle{\Phi(x_{1})\bar{\Phi}(x_{2})}\rangle
P_{\ell_{1}}(\stackrel{{\scriptstyle\leftarrow}}{{i\partial_{1-}}},\stackrel{{\scriptstyle\rightarrow}}{{i\partial_{1-}}})P_{\ell_{2}}(\stackrel{{\scriptstyle\rightarrow}}{{i\partial_{2-}}},\stackrel{{\scriptstyle\leftarrow}}{{i\partial_{2-}}})\langle{\bar{\Phi}(x_{1})\Phi(x_{2})}\rangle\,,$
(187)

where $\partial_{i-}\equiv\partial_{x_{i}^{-}}$ and the propagator is given by
(2.1). Introducing Schwinger parametrization for $x^{-}-$dependent part of the
propagator (2.1)

$\displaystyle{\Gamma(2j)\over(x_{12}^{-}-i0)^{2j}}=i^{2j}\int_{0}^{\infty}dp\,p^{2j-1}\mathop{\rm
e}\nolimits^{-ipx_{12}^{-}}\,,$ (188)

one finds from (187)

$\displaystyle\langle{O_{S_{1}}(x_{1})O_{S_{2}}(x_{2})}\rangle\sim\int_{0}^{\infty}dp_{1}dp_{2}(p_{1}p_{2})^{2j-1}\mathop{\rm
e}\nolimits^{-i(p_{1}+p_{2})x_{12}^{-}}P_{\ell_{1}}(p_{1},p_{2})P_{\ell_{2}}(p_{1},p_{2})\,.$
(189)

After change of the integration variables, $p_{1}=tp$ and $p_{2}=(1-t)p$, with
$0\leq x\leq 1$ and $0\leq p<\infty$, the integral over $t$ takes the form
(34) and leads to (33).

##  Appendix C Commutation relation for scalar operators

In this appendix, we examine the correlation function
$\langle{\phi(x_{1})\mathcal{J}_{\omega,S}(n)\mathcal{J}_{\omega^{\prime},S^{\prime}}(n^{\prime})\bar{\phi}(x_{2})}\rangle_{c}$
involving scalar flow operators and derive their commutation relations. For
fermions and gauge fields, the corresponding correlation function is given by
(55) and (2.2). Substituting $s=0$ into there relations we find that the
integral in (2.2) diverges logarithmically.

To elucidate its origin, we repeat the calculation of diagrams shown in Figure
2. Introducing notation for
$\epsilon=(n\bar{n})/r^{\prime}-(n^{\prime}\bar{n})/r$ we find for
$r,r^{\prime}\to\infty$

$\displaystyle\langle{\Phi(x_{1})\mathcal{J}_{\omega,S}(n)\mathcal{J}_{\omega^{\prime},S^{\prime}}(n^{\prime})\bar{\Phi}(x_{2})}\rangle_{c}={1\over(2\pi)^{d-1}}\lim_{\epsilon\to
0}{1\over(2\pi
i\epsilon)^{d/2-1}}\,\int_{0}^{\infty}d\alpha_{1}\,(\alpha_{1}\alpha_{2})^{2j-1}\theta(\alpha_{2})$
$\displaystyle\times\Big{[}\mathop{\rm
e}\nolimits^{i(\alpha_{1}-\omega)(nn^{\prime})/\epsilon}\mathop{\rm
e}\nolimits^{-i\alpha_{1}(n_{1}x_{1})+i\alpha_{2}(n_{2}x_{2})}\theta(\alpha_{1}-\omega)(\alpha_{1}-\omega)^{2j-1}P_{\ell}(\alpha_{1},-\alpha_{1}+\omega)P_{\ell^{\prime}}(\alpha_{1}-\omega,-\alpha_{2})$
$\displaystyle{}+(-1)^{2s}\mathop{\rm
e}\nolimits^{i(\omega^{\prime}-\alpha_{1})(nn^{\prime})/\epsilon}\mathop{\rm
e}\nolimits^{-i\alpha_{1}(n_{2}x_{1})+i\alpha_{2}(n_{1}x_{2})}\theta(\omega^{\prime}-\alpha_{1})(\omega^{\prime}-\alpha_{1})^{2j-1}P_{\ell^{\prime}}(\alpha_{1},-\alpha_{1}+\omega^{\prime})P_{\ell}(\alpha_{1}-\omega^{\prime},-\alpha_{2})\Big{]}\,,$
(190)

where integration goes over the energy of the outgoing particle and
$\alpha_{2}=\alpha_{1}-\omega^{\prime\prime}$ (with
$\omega^{\prime\prime}=\omega+\omega^{\prime}$) is the energy of the incoming
particle. The two terms inside the brackets correspond to the diagrams shown
in Figure 2(a) and (b), respectively.

Applying the identity (179), one is tempting to replace the first exponential
factor on the second line of (C) with
$\delta^{(d-2)}(n,n^{\prime})/(\alpha_{1}-\omega)$ and similar on the last
line. This would lead to relations (55) and (2.2). Such transformations are
only legitimate for $\alpha_{1}-\omega\neq 0$ and
$\omega^{\prime}-\alpha_{1}\neq 0$. The boundary values $\alpha_{1}=\omega$
and $\alpha_{1}=\omega^{\prime}$ correspond to the vanishing energy of the
particle exchanged between the detectors, see Figure 2. Assuming that
$(nn^{\prime})\neq 0$ and integrating in the vicinity of the boundary values,
we get from (C)

$\displaystyle\langle{\Phi(x_{1})\mathcal{J}_{\omega,S}(n)\mathcal{J}_{\omega^{\prime},S^{\prime}}(n^{\prime})\bar{\Phi}(x_{2})}\rangle_{c}={\Gamma(2j)\over(2\pi)^{3d/2-2}}\,(-\omega\omega^{\prime})^{2j-1}$
$\displaystyle\times{(i\epsilon)^{2s}\over(nn^{\prime})^{2j}}\Big{[}P_{\ell}(\omega,0)P_{\ell^{\prime}}(0,\omega^{\prime})\theta(\omega)\theta(-\omega^{\prime})+(-1)^{2s}P_{\ell^{\prime}}(\omega^{\prime},0)P_{\ell}(0,\omega)\theta(-\omega)\theta(\omega^{\prime})\Big{]}+\dots\,,$

where dots denote terms localized at $n=n^{\prime}$ and we took into account
that $j=s+(d-2)/4$. We observe that for $s>0$ this expression vanishes whereas
for $s=0$ it leads to (60).

Thus, for scalar fields the four-point function (C) receives an additional
contribution that does not vanish for $n\neq n^{\prime}$ and scales as
$1/(nn^{\prime})^{(d-2)/2}$. It is symmetric under the exchange of the
detectors and does not contribute to the difference of the correlation
functions (37). Taking into account (C) we obtain from (37)

$\displaystyle\langle{\phi(x_{1})[\mathcal{J}_{\omega,S}(n),\mathcal{J}_{\omega^{\prime},S^{\prime}}(n^{\prime})]\bar{\phi}(x_{2})}\rangle_{c}={1\over(2\pi)^{d-1}}\lim_{\epsilon\to
0}{1\over(2\pi
i\epsilon)^{d/2-1}}\,\int_{0}^{\infty}d\alpha_{1}\,(\alpha_{1}\alpha_{2})^{d/2-2}\theta(\alpha_{2})$
$\displaystyle\times\Big{[}\mathop{\rm
e}\nolimits^{i(\alpha_{1}-\omega)(nn^{\prime})/\epsilon}\mathop{\rm
e}\nolimits^{-i\alpha_{1}(n_{1}x_{1})+i\alpha_{2}(n_{2}x_{2})}(\alpha_{1}-\omega)^{d/2-2}P_{\ell}(\alpha_{1},-\alpha_{1}+\omega)P_{\ell^{\prime}}(\alpha_{1}-\omega,-\alpha_{2})$
$\displaystyle{}+\mathop{\rm
e}\nolimits^{i(\omega^{\prime}-\alpha_{1})(nn^{\prime})/\epsilon}\mathop{\rm
e}\nolimits^{-i\alpha_{1}(n_{2}x_{1})+i\alpha_{2}(n_{1}x_{2})}(\omega^{\prime}-\alpha_{1})^{d/2-2}P_{\ell^{\prime}}(\alpha_{1},-\alpha_{1}+\omega^{\prime})P_{\ell}(\alpha_{1}-\omega^{\prime},-\alpha_{2})\Big{]},$
(191)

where $\alpha_{2}=\alpha_{1}-\omega^{\prime\prime}$ and we took into account
that $\epsilon$ changes the sign under the exchange of the detectors. As
follows from the above discussion, the contribution from $\alpha_{1}=\omega$
and $\alpha_{1}=\omega^{\prime}$ cancels on the right-hand side of (C). This
allows us to deform the integration contour in the vicinity of the two points.
Then, applying the identity (179) we arrive at

$\displaystyle\langle{\phi(x_{1})[\mathcal{J}_{\omega,S}(n),\mathcal{J}_{\omega^{\prime},S^{\prime}}(n^{\prime})]\bar{\phi}(x_{2})}\rangle_{c}={\delta^{(d-2)}(n,n^{\prime})\over(2\pi)^{d-1}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{\infty}d\alpha_{1}\,(\alpha_{1}\alpha_{2})^{d/2-2}\theta(\alpha_{2})\mathop{\rm
e}\nolimits^{-i\alpha_{1}(nx_{1})+i\alpha_{2}(nx_{2})}$
$\displaystyle\times\Big{[}{P_{\ell}(\alpha_{1},-\alpha_{1}+\omega)P_{\ell^{\prime}}(\alpha_{1}-\omega,-\alpha_{2})\over\alpha_{1}-\omega}-{P_{\ell^{\prime}}(\alpha_{1},-\alpha_{1}+\omega^{\prime})P_{\ell}(\alpha_{1}-\omega^{\prime},-\alpha_{2})\over\alpha_{1}-\omega^{\prime}}\Big{]},$
(192)

where dash denotes the principal value prescription. The expression in the
second line of (C) coincides with (75).

##  Appendix D Relation to the Virasoro and $W$ algebras

In this appendix we show that the algebra of the flow operators (2.3) in $d=2$
dimensions is equivalent to the $W-$algebra in two-dimensional Euclidean CFT
Bouwknegt:1992wg .

The commutation relations in the $W$ algebra are Pope:1989ew ; Pope:1989sr ;
Pope:1991ig ; Korybut:2020ncv

$\displaystyle[V_{m}^{i},V_{n}^{j}]=\sum_{p\geq
0}g_{2p}^{ij}(m,n)V_{m+n}^{i+j-2p}+c_{i}(m)\delta^{ij}\delta_{m+n,0}\,,$ (193)

where the operators $V_{m}^{i}$ are modes in the Laurent expansion of the
currents $O_{S}(z)$ of spin $S=i+2$ on two-dimensional Euclidean plane

$\displaystyle O_{S}(z)=\sum_{m=-\infty}^{\infty}V_{m}^{i}\,z^{-m-i-2}\,,$
(194)

where $z=x^{1}+ix^{2}$. For the $W_{\infty}$ and $W_{1+\infty}$ algebras the
spins satisfy condition $S\geq 2$ and $S\geq 1$, respectively.

For $S=2$ the current $O_{S=2}(z)$ coincides with the stress-energy tensor and
the corresponding modes $L_{n}=V_{n}^{0}$ satisfy the Virasoro algebra

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+{c\over
12}m(m-1)(m+1)\delta_{m+n,0}\,.$ (195)

The structure constants and the central charges in (193) can be determined by
requiring the relations (193) to be consistent with the Jacobi identities. In
particular, the central charges take the form

$\displaystyle c_{i}(m)=m(m^{2}-1)(m^{2}-4)\dots(m^{2}-(i+1)^{2})c_{i}\,,$
(196)

where $c_{i}$ is proportional to the central charge $c$.

To compare (193) with the algebra (2.3) one has to extend the above relations
to Lorentzian signature. In two-dimensional Euclidean CFT, the modes
$V_{n}^{j}$ are given by

$\displaystyle V_{n}^{j}=\oint{dz\over 2\pi i}z^{n+j+1}O_{S=j+2}(z)\,,$ (197)

where the integration contour encircles the origin. After Wick rotation
$x^{2}\to ix^{0}$, the complex variable $z=x^{1}+ix^{2}$ turns into the light-
cone coordinate $z=-\sqrt{2}x^{-}$. Then, one may try to generalize the
relation (197) as $V_{n}^{j}\sim\int dx^{-}\,(x^{-})^{n+j+1}O_{S=j+2}(x^{-})$.
However, as was noticed in Refs. Belin:2020lsr ; Besken:2020snx , such
operator is not well-defined because inserted inside a correlation function it
leads to integrals that are divergent as $x^{-}\to\infty$. 191919These papers
deal with spin-two operators, the same arguments apply to operators of high
spin.

Following Ref. Besken:2020snx  we can generalize (197) to Lorentzian signature
by applying a (complexified) conformal transformation $z\to w=i(1+iz)/(1-iz)$
that maps a circle of unit radius $|z|=1$ to a line $-\infty<w<\infty$. Taking
into account transformation properties of the currents $O_{S}(z)$, we obtain
from (197)

$\displaystyle V_{n}^{j}={1\over
2^{S-1}}\int_{-\infty}^{\infty}dw\,(1-iw)^{S-1-n}(1+iw)^{S-1+n}O_{S}(w)\Big{|}_{S=j+2}\,,$
(198)

where the additional factor comes from $(\partial w/\partial z)^{S-1}$. These
operators satisfy the same commutation relations as the operators (197) but
they do not suffer from divergences since the large $w$ asymptotics of the
integrand in (198) does not depend on $n$.

To express (198) in terms of the flow operators (16), it is convenient to
introduce the weight function

$\displaystyle\mu_{n,S}(\omega)$ $\displaystyle={1\over
2^{S-1}}\int_{-\infty}^{\infty}{dx\over 2\pi}\mathop{\rm e}\nolimits^{i\omega
x}(1-ix)^{S-1-n}(1+ix)^{S-1+n}\,.$ (199)

Then, the operators (198) can be written as weighted integrals of the flow
operators

$\displaystyle
V_{n}^{j}=\int_{-\infty}^{\infty}d\omega\,\mu_{n,S}(\omega)\,\mathcal{J}_{\omega,S}\Big{|}_{S=j+2}\,.$
(200)

This relation establishes the correspondence between the $W-$algebra (193) and
the algebra of the flow operators (2.3) in $d=2$ dimensions.

The weight function (199) verifies relations

$\displaystyle\mu_{n,S}(\omega)=\mu_{-n,S}(-\omega)={\left({\mu_{n,S}(\omega)}\right)}^{*}\,.$
(201)

It has different properties depending on the value of integer $n$.

For $-S<n<S$, the expression on the right-hand side of (199) is given by a
linear combination of $\delta(w)$ and its derivatives

$\displaystyle\mu_{n,S}(\omega)$ $\displaystyle={1\over
2^{S-1}}(1-\partial_{\omega})^{S-1-n}(1+\partial_{\omega})^{S-1+n}\delta(\omega)\,.$
(202)

Its substitution to (200) yields expansion of $\mathcal{J}_{\omega,S}$ at
small $\omega$

$\displaystyle V_{n}^{j}={1\over
2^{S-1}}(1+\partial_{\omega})^{S-1-n}(1-\partial_{\omega})^{S-1+n}\mathcal{J}_{\omega,S}\Big{|}_{\omega=0}\,,$
(203)

where $j=S-2$. For $S=2$ this relation looks as

$\displaystyle L_{-1}={1\over
2}(1+\partial_{\omega})^{2}\mathcal{E}_{\omega=0}\,,\qquad\quad L_{0}={1\over
2}(1-\partial_{\omega}^{2})\mathcal{E}_{\omega=0}\,,\qquad\quad L_{1}={1\over
2}(1-\partial_{\omega})^{2}\mathcal{E}_{\omega=0}\,,$ (204)

where we replaced $\mathcal{J}_{\omega,S=2}=\mathcal{E}_{\omega}$ and
$V_{n}^{0}=L_{n}$. It is straightforward to verify that the algebra (96) of
the energy flow operators translates to the commutation relations (195) for
the modes $L_{-1}$, $L_{0}$ and $L_{1}$. Notice that the expression on the
right-hand side of (203) is given by a linear combination of the first
$(2S-1)$ terms of the small $\omega$ expansion of the flow operator
$\mathcal{J}_{\omega,S}$. This is in agreement with the observation made in
Section 2.2 that the flow operators have a well-defined expansion at small
$\omega$ up to order $O(\omega^{2S-1})$.

For $n\geq S$ the function (199) can be expressed in terms of (generalized)
Laguerre polynomial

$\displaystyle\mu_{n,S}(\omega)$ $\displaystyle=(-1)^{n-S}2^{S}\mathop{\rm
e}\nolimits^{\omega}L_{n-S}^{(2S-1)}(-2\omega)\theta(-\omega)\,.$ (205)

For $n\leq-S$ the function $\mu_{n,S}(\omega)$ can be found using the first
relation in (201). Taking into account the properties of the Laguerre
polynomials, we find that $\mu_{n,S}(\omega)$ satisfies the orthogonality
condition

$\displaystyle\int_{-\infty}^{\infty}d\omega\,\omega^{2S-1}\mu_{m,S}(\omega)\mu_{-n,S}(\omega)=-{\Gamma(m+S)\over\Gamma(m-S+1)}\delta_{n+m,0}\,.$
(206)

Applying (200) we obtain

$\displaystyle[V_{m}^{i},V_{n}^{j}]=\int_{-\infty}^{\infty}d\omega
d\omega^{\prime}\,\mu_{m,S}(\omega)\mu_{n,S^{\prime}}(\omega^{\prime})\,[\mathcal{J}_{\omega,S},\mathcal{J}_{\omega^{\prime},S^{\prime}}]\,,$
(207)

where $S=i+2$ and $S^{\prime}=j+2$. Replacing the commutator of the flow
operators with (2.3) we can reproduce the $W-$algebra (193).

The contribution of the second term in (2.3) to (207) yields the central
charge

$\displaystyle-\delta_{SS^{\prime}}\Omega_{S}\int_{-\infty}^{\infty}d\omega\,\mu_{m,S}(\omega)\mu_{n,S}(-\omega)\omega^{2S-1}=\delta_{ij}\delta_{n+m,0}\Omega_{i+2}m(m^{2}-1)\dots(m^{2}-(i+1)^{2}),$
(208)

where we applied (201) and (206) and replaced $S=i+2$ and $S^{\prime}=j+2$.
This relation coincides with the expression for the central charge (196) of
the $W-$algebra (193). The central charge of Virasoro algebra (195) is given
by $c=12\Omega_{2}$, see Eq.(2.2). For the central charge of the $W-$algebra
we get $c_{i}=\Omega_{i+2}$. Notice that the central charge (208) vanishes for
$-(i+1)\leq m\leq i+1$. We recall that the corresponding operators $V_{m}^{i}$
form a finite-dimensional representation of the $SL(2;\mathbb R)$ subgroup the
conformal group in $d=2$ dimensions and their commutation relations do not
contain central extension.

The contribution of the first term in (2.3) to (207) can be written as (for
$\ell_{i}=S_{i}$)

$\displaystyle\int d\omega
d\omega^{\prime}\mu_{m,S}(\omega)\mu_{n,S^{\prime}}(\omega^{\prime})C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega,\omega^{\prime})\mathcal{J}_{\omega+\omega^{\prime},S^{\prime\prime}}=\int
dx\,O_{S^{\prime\prime}}(x)f(x)\,,$ (209)

where we replaced $\mathcal{J}_{\omega,S}=\int dx\mathop{\rm
e}\nolimits^{-i\omega x}O_{S}(x)$ and introduced notation for

$\displaystyle f(x)=\int d\omega
d\omega^{\prime}\mu_{m,S}(\omega)\mu_{n,S^{\prime}}(\omega^{\prime})C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega,\omega^{\prime})\mathop{\rm
e}\nolimits^{-ix(\omega+\omega^{\prime})}\,.$ (210)

As we will see in a moment, this function is given by

$\displaystyle f(x)=g_{2p}^{ij}(m,n)\int
d\omega^{\prime\prime}\mu_{n+m,S^{\prime\prime}}(\omega^{\prime\prime})\mathop{\rm
e}\nolimits^{-ix\omega^{\prime\prime}}\,,$ (211)

where $S=2+i$, $S^{\prime}=2+j$ and $S^{\prime\prime}=S+S^{\prime}-2(p+1)$.
Then, replacing $f(x)$ in (209) with this expression and applying (200), we
can express the right-hand side of (209) as
$g_{2p}^{ij}(m,n)V_{m+n}^{i+j-2p}$, thus reproducing the first term in (193).

To find the relation between the structure constants $g_{2p}^{ij}(m,n)$ and
$C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega,\omega^{\prime})$, we apply (199)
and match the expression on the right-hand side of (210) and (211)

$\displaystyle{(1+ix)^{m+S-1}\over(1-ix)^{m-S+1}}C_{SS^{\prime}}{}^{S^{\prime\prime}}(i\\!\\!\stackrel{{\scriptstyle\leftarrow}}{{\partial_{x}}},i\\!\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial_{x}}}){(1+ix)^{n+S^{\prime}-1}\over(1-ix)^{n-S^{\prime}+1}}=2^{2p+1}g_{2p}^{ij}(m,n){(1+ix)^{m+n+S^{\prime\prime}-1}\over(1-ix)^{m+n-S^{\prime\prime}+1}}\,,$
(212)

where $i=S-2$, $j=S^{\prime}-2$ and $2p=S+S^{\prime}-S^{\prime\prime}-2$. This
relation holds for arbitrary $x$. Choosing $x=i(1+\epsilon)$ and taking the
limit $\epsilon\to 0$ we get

$\displaystyle
g_{2p}^{ij}(m,n)=-{1\over\epsilon^{m+n+S^{\prime\prime}-1}}{\left({\epsilon^{m+S-1}C_{SS^{\prime}}{}^{S^{\prime\prime}}(\stackrel{{\scriptstyle\leftarrow}}{{\partial_{\epsilon}}},\stackrel{{\scriptstyle\rightarrow}}{{\partial_{\epsilon}}})\,\epsilon^{n+S^{\prime}-1}}\right)}+O(\epsilon)\,.$
(213)

We recall that $C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega,\omega^{\prime})$
is a homogenous polynomial of degree $S+S^{\prime}-S^{\prime\prime}-1$. It is
easy to check that the first term in (213) approaches a finite value as
$\epsilon\to 0$. For instance, for $p=0$, or equivalently
$S^{\prime\prime}=S+S^{\prime}-2$, we apply the relation (98) to find from
(213)

$\displaystyle g_{0}^{ij}(m,n)$
$\displaystyle=c^{(s)}_{SS^{\prime}}{\left({m(S^{\prime}-1)-n(S-1)}\right)}\big{|}_{S=i+2,\,S^{\prime}=j+2}\,,$
(214)

in agreement with the known result of Ref. Pope:1991ig .

The structure constants (213) should be independent on $\epsilon$. 202020One
can show that this condition fixes
$C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega,\omega^{\prime})$ up to a
normalization constant. This imposes nontrivial restrictions on
$C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega,\omega^{\prime})$. One can show
that vanishing of the $O(\epsilon)$ term in (213) follows from invariance of
the commutation relations (2.4) under the $SL(2)$ conformal transformations.
We can invert the relation (213) to get

$\displaystyle
C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega_{1},\omega_{2})=-\sum_{m,n\geq
0\atop m+n=2p+1}{\omega^{m}\over m!}{(\omega^{\prime})^{n}\,\over
n!}g_{2p}^{ij}(m-S+1,n-S^{\prime}+1)\,,$ (215)

where $S=i+2$, $S^{\prime}=2+j$ and $S^{\prime\prime}=S+S^{\prime}-2(p+1)$.
This provides the relation between the structure constants of the algebra of
flow operators in $d=2$ dimensions and the $W-$algebra.

The structure constants in (193) are sensitive to the normalization of the
operators $V_{m}^{i}$. Replacing $V_{m}^{i}\to f_{i}V_{m}^{i}$ in (193) one
finds that the structure constants transform as

$\displaystyle g_{2p}^{ij}(m,n)\to{f_{i}f_{j}\over
f_{i+j-2p}}g_{2p}^{ij}(m,n)\,.$ (216)

Applying this transformation to (215) with
$f_{i}=-2^{1-2i}(2i+2)!/((i+1)!(i+2)!)$ and replacing
$C_{SS^{\prime}}{}^{S^{\prime\prime}}(\omega_{1},\omega_{2})$ with its
expression (2.4), we find that $g_{2p}^{ij}(m,n)$ coincide with the known
expressions for the structure constants of the $W_{\infty}$ algebra, see Eqs.
(3.4) – (3.6) in Ref. Pope:1991ig .

##  Appendix E Scalar integral

In this appendix, we compute the integral (124) that appeared in the
discussion of the light-ray algebra in interacting CFTs. It takes the
following general form

$\displaystyle
I_{\alpha\beta}(\omega_{1},\omega_{2})=\int_{-\infty}^{\infty}dx_{1}^{-}dx_{2}^{-}e^{-i\omega_{1}x_{1}^{-}-i\omega_{2}x_{2}^{-}}{1\over(x_{12}^{2})^{\alpha}(x_{13}^{2}x_{23}^{2})^{\beta}}\,,$
(217)

where $x_{i}=(x_{i}^{+},x_{i}^{-},\mbox{\boldmath$x$}_{i})$ in the light-cone
coordinates and singularities of the denominator at $x_{12}^{2}=0$ are
resolved as

$\displaystyle{1\over(x_{12}^{2})^{\alpha}}\equiv{1\over(2x_{12}^{+}(x_{12}^{-}-i\epsilon)-\mbox{\boldmath$x$}_{12}^{2})^{\alpha}}={1\over(2x_{12}^{+})^{\alpha}}{i^{\alpha}\over\Gamma(\alpha)}\int_{0}^{\infty}dss^{\alpha-1}e^{-is\left(x_{12}^{-}-{\boldsymbol{x}_{12}^{2}\over
2x_{12}^{+}}\right)},$ (218)

and similar for the two remaining factors in (217). Here the ‘$-i\epsilon$’
prescription reflects the ordering of the operators in the correlation
function (122).

Applying the last relation and performing integration in (217), we encounter
integrals of the form

$\displaystyle\int_{-\infty}^{\infty}{dx_{1}^{-}\,e^{-i\omega
x_{1}^{-}}\over(2x_{13}^{+}(x_{13}^{-}-i\epsilon)-\mbox{\boldmath$x$}_{13}^{2})^{\beta}}={2\pi\theta(-\omega)\over(2x_{13}^{+})^{\beta}}{i^{\beta}\over\Gamma(\beta)}(-\omega)^{\beta-1}e^{i\omega\left(x_{3}^{-}+{\boldsymbol{x}_{13}^{2}\over
2x_{13}^{+}}\right)}.$ (219)

Combining various factors together we obtain in the limit $x_{12}^{+}\to 0$

$\displaystyle I_{\alpha\beta}(\omega_{1},\omega_{2})=$
$\displaystyle(2\pi)^{2}\theta(-\omega_{1})\theta(-\omega_{1}-\omega_{2}){i^{\alpha+2\beta}\over\Gamma(\alpha)\Gamma^{2}(\beta)}$
$\displaystyle\times$
$\displaystyle(2x_{12}^{+})^{-\alpha}(4x_{13}^{+}x_{23}^{+})^{-\beta}\int_{\omega_{2}}^{-\omega_{1}}ds\,s^{\alpha-1}((-s-\omega_{1})(s-\omega_{2}))^{\beta-1}e^{is{\boldsymbol{x}_{12}^{2}\over
2x_{12}^{+}}}\,.$ (220)

We deduce from this relation that for $\boldsymbol{x}_{12}^{2}\neq 0$ the
integral decreases exponentially fast as $x_{12}^{+}\to 0$. At the same time,
for $\boldsymbol{x}_{12}^{2}=0$ it scales as $(x_{12}^{+})^{-\alpha}$. This
suggests that $I_{\alpha\beta}(\omega_{1},\omega_{2})$ is proportional to a
distribution localized at $\boldsymbol{x}_{12}=0$. To identify its form we
integrate both sides of the last relation over
$\mbox{\boldmath$x$}=\boldsymbol{x}_{12}$

$\displaystyle\int
d^{d-2}\mbox{\boldmath$x$}\,I_{\alpha\beta}(\omega_{1},\omega_{2})\sim(x_{12}^{+})^{-\alpha+(d-2)/2}\,,$
(221)

leading to
$I_{\alpha\beta}(\omega_{1},\omega_{2})\sim(x_{12}^{+})^{-\alpha+(d-2)/2}\delta^{(d-2)}(\mbox{\boldmath$x$}_{12})$.

##  Appendix F Extra light-ray operator in the free scalar theory

In this Appendix, we examine the properties of nonlocal operators
$\mathcal{O}^{\pm}_{\omega_{1},\omega_{2}}(n)$ that enters the commutation
relations (9) and (11) of the light-ray operators in a free scalar theory.
Using (1) it is convenient to write
$\mathcal{O}^{\pm}_{\omega_{1},\omega_{2}}(n)$ as

$\displaystyle{\cal O}^{\pm}_{\omega_{1},\omega_{2}}(n)\sim{\cal
O}_{\omega_{1},\omega_{2}}(n)\pm{\cal O}_{\omega_{2},\omega_{1}}(n)\,,$ (222)

where the notation was introduced for

$\displaystyle{\cal
O}_{\omega_{1},\omega_{2}}(n)=i(n\bar{n})^{2}\lim_{r\to\infty}r^{d-2}\int_{-\infty}^{\infty}dt_{1}dt_{2}\,\mathop{\rm
e}\nolimits^{-i(t_{1}\omega_{1}+t_{2}\omega_{2})(n\bar{n})}{\rm
sign}(t_{1}-t_{2})\phi(rn+t_{1}\bar{n})\bar{\phi}(rn+t_{2}\bar{n}).$ (223)

In the embedding space, the light-ray operator (223) takes the form ${\cal
O}_{\omega_{1},\omega_{2}}(n)\equiv{1\over 4}\mathbb{O}_{{\omega_{1}\over
2},{\omega_{2}\over 2}}(X_{\infty},Z_{\infty})$, see Korchemsky:2021okt  for
the details,

$\displaystyle\mathbb{O}_{\omega_{1},\omega_{2}}(X,Z)=i\int_{-\infty}^{\infty}d\alpha_{1}\int_{-\infty}^{\infty}d\alpha_{2}e^{-i\omega_{1}\alpha_{1}}e^{-i\omega_{2}\alpha_{2}}{\rm
sign}(\alpha_{1}-\alpha_{2})\phi(Z-\alpha_{1}X)\bar{\phi}(Z-\alpha_{2}X).$
(224)

The product of scalar fields on the right-hand side of (223) can be expanded
over local conformal operators $O_{S}(x)\equiv
O_{\mu_{1}\dots\mu_{S}}(x)\bar{n}^{\mu_{1}}\dots\bar{n}^{\mu_{S}}$ and their
descendants (see e.g. Braun:2003rp )

$\displaystyle\phi(rn+t_{1}\bar{n})\bar{\phi}(rn+t_{2}\bar{n})=\sum_{S\geq
0}c_{S}{(it_{12})^{S}\over
S!}\int_{0}^{1}du\,u^{S+\nu}(1-u)^{S+\nu}O_{S}(rn+(t_{1}-ut_{12})\bar{n})\,,$
(225)

where $t_{12}=t_{1}-t_{2}$ and $\nu=2j_{\phi}-1=(d-4)/2$ is related to the
conformal spin of the scalar field.212121In writing (225) we implicitly
subtracted the contribution of the identity operator
$\langle\phi(rn+t_{1}\bar{n})\bar{\phi}(rn+t_{2}\bar{n})\rangle$ on the right-
hand side. The integral over $u$ takes into account the contribution of
descendants. The OPE coefficient $c_{S}$ depends on the normalization of the
local operator $O_{S}$. At $S=2$, for the stress-energy tensor, we have
$c_{2}={\Gamma(6+2\nu)/\Gamma^{2}(3+\nu)}$.

Combining the relations (223) and (225) we get for $(n\bar{n})=1$

$\displaystyle{\cal O}_{\omega_{1},\omega_{2}}(n)$ $\displaystyle=\sum_{S\geq
0}{c_{S}\over
S!}\lim_{r\to\infty}r^{d-2}\int_{-\infty}^{\infty}dt\,O_{S}(rn+t\bar{n})\mathop{\rm
e}\nolimits^{-it(\omega_{1}+\omega_{2})}\left[I_{S}(\omega_{1},\omega_{2})-(-1)^{S}I_{S}(\omega_{2},\omega_{1})\right]$
(226)

where the notation was introduced for

$\displaystyle
I_{S}(\omega_{1},\omega_{2})=i^{S+1}\int_{0}^{1}du\,(u(1-u))^{S+\nu}\int_{-\infty}^{\infty}dt_{1}dt_{2}\,\mathop{\rm
e}\nolimits^{-i((t_{1}-t)\omega_{1}+(t_{2}-t)\omega_{2})}$
$\displaystyle\times\theta(t_{1}-t_{2})(t_{1}-t_{2})^{S}\delta(t_{1}-t-u(t_{1}-t_{2}))$
(227)

Shifting the integration variables, $t_{i}\to t_{i}+t$, it is easy to see that
$I_{S}(\omega_{1},\omega_{2})$ does not depend on $t$. Going through
calculation we obtain

$\displaystyle I_{S}(\omega_{1},\omega_{2})$
$\displaystyle=\int_{0}^{1}{dx\,(x(1-x))^{S+\nu}\over[x\omega_{1}-(1-x)\omega_{2}-i0]^{S+1}}={\Gamma^{2}(S+\nu+1)\over\Gamma(2S+2\nu+2)}f_{S}(\omega_{1},\omega_{2})\,,$
$\displaystyle f_{S}(\omega_{1},\omega_{2})$
$\displaystyle={(-\omega_{2})^{-S-1}\,_{2}F_{1}\left({S+1,S+\nu+1\atop
2(S+\nu+1)}\Big{|}1+\frac{\omega_{1}-i0}{\omega_{2}}\right)}\,,$ (228)

where $\nu=(d-4)/2$. Notice that ${\rm
Re}[I_{S}(\omega_{1},\omega_{2})]=-(-1)^{S}{\rm
Re}[I_{S}(\omega_{2},\omega_{1})]$, and ${\rm
Im}[I_{S}(\omega_{1},\omega_{2})]=(-1)^{S}{\rm
Im}[I_{S}(\omega_{2},\omega_{1})]$.

Taking into account (1), we can express (226) as the sum over light-ray
operators of arbitrary spin

$\displaystyle{\cal O}_{\omega_{1},\omega_{2}}(n)$ $\displaystyle=\sum_{S\geq
0}{c_{S}\over
S!}\left[I_{S}(\omega_{1},\omega_{2})-(-1)^{S}I_{S}(\omega_{2},\omega_{1})\right]\mathcal{J}_{\omega_{1}+\omega_{2},S}(n)\,$
$\displaystyle=2\sum_{S\geq 0}{c_{S}\over S!}{\rm
Re}[I_{S}(\omega_{1},\omega_{2})]\mathcal{J}_{\omega_{1}+\omega_{2},S}(n)\ .$
(229)

Substituting this relation into (222), we find that ${\cal
O}^{\pm}_{\omega_{1},\omega_{2}}(n)$ is given by the sum over odd and even
spins, respectively.

For $S=2$ the contribution of the energy flow operator
$\mathcal{E}_{\omega_{1}+\omega_{2}}(n)=\mathcal{J}_{\omega_{1}+\omega_{2},S=2}(n)$
to the right-hand side of (F) takes the form

$\displaystyle{\cal O}_{\omega_{1},\omega_{2}}(n)$
$\displaystyle=f_{\phi}(\omega_{1},\omega_{2})\mathcal{E}_{\omega_{1}+\omega_{2}}(n)+\dots\,,$
$\displaystyle f_{\phi}(\omega_{1},\omega_{2})$ $\displaystyle=-{\rm
Re}\Big{[}\omega_{2}^{-3}{\,{}_{2}F_{1}\left({3,\frac{d}{2}+1\atop
d+2}\Big{|}1+\frac{\omega_{1}-i0}{\omega_{2}}\right)}\Big{]},$ (230)

where $f_{\phi}=f_{S=2}(\omega_{1},\omega_{2})$ and dots denote the
contribution of the spins $S\neq 2$. Its explicit expression for different
values of $d$ looks as

$\displaystyle
f_{\phi}\big{|}_{d=3}=\frac{16\left(4(\omega_{2}-\omega_{1})+\left(\omega_{1}^{2}-6\omega_{1}\omega_{2}+\omega_{2}^{2}\right)(-\omega_{1}\omega_{2})^{-1/2}[\theta(\omega_{2})-\theta(\omega_{1})]\right)}{\left(\omega_{1}+\omega_{2}\right){}^{4}}\
,$ $\displaystyle
f_{\phi}\big{|}_{d=4}=\frac{30\left(3(\omega_{2}^{2}-\omega_{1}^{2})+\left(\omega_{1}^{2}-4\omega_{2}\omega_{1}+\omega_{2}^{2}\right)\log|\frac{\omega_{1}}{\omega_{2}}|\right)}{\left(\omega_{1}+\omega_{2}\right){}^{5}}.$
(231)

We recall that for $d=2$ the operator (1) vanishes and does not contribute to
the algebra of the light-ray operators.

## References

  * (1) N. A. Sveshnikov and F. V. Tkachov, _Jets and quantum field theory_ , _Phys. Lett. B_ 382 (1996) 403 [hep-ph/9512370].
  * (2) G. P. Korchemsky, G. Oderda and G. F. Sterman, _Power corrections and nonlocal operators_ , _AIP Conf. Proc._ 407 (1997) 988 [hep-ph/9708346].
  * (3) G. P. Korchemsky and G. F. Sterman, _Power corrections to event shapes and factorization_ , _Nucl. Phys. B_ 555 (1999) 335 [hep-ph/9902341].
  * (4) D. M. Hofman and J. Maldacena, _Conformal collider physics: Energy and charge correlations_ , _JHEP_ 05 (2008) 012 [0803.1467].
  * (5) H. Chen, I. Moult and H. X. Zhu, _Spinning Gluons from the QCD Light-Ray OPE_ , 2104.00009.
  * (6) G. Korchemsky, E. Sokatchev and A. Zhiboedov, _Generalizing event shapes: In search of lost collider time_ , 2106.14899.
  * (7) P. Kravchuk and D. Simmons-Duffin, _Light-ray operators in conformal field theory_ , _JHEP_ 11 (2018) 102 [1805.00098].
  * (8) T. Hartman, S. Kundu and A. Tajdini, _Averaged Null Energy Condition from Causality_ , _JHEP_ 07 (2017) 066 [1610.05308].
  * (9) M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, _Shocks, Superconvergence, and a Stringy Equivalence Principle_ , _JHEP_ 11 (2020) 096 [1904.05905].
  * (10) C. Córdova and S.-H. Shao, _Light-ray Operators and the BMS Algebra_ , _Phys. Rev. D_ 98 (2018) 125015 [1810.05706].
  * (11) A. Belin, D. M. Hofman, G. Mathys and M. T. Walters, _On the Stress Tensor Light-ray Operator Algebra_ , 2011.13862.
  * (12) M. Beşken, J. De Boer and G. Mathys, _On Local and Integrated Stress-Tensor Commutators_ , 2012.15724.
  * (13) A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, _Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory_ , _Nucl. Phys. B_ 241 (1984) 333.
  * (14) A. B. Zamolodchikov, _Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory_ , _Theor. Math. Phys._ 65 (1985) 1205.
  * (15) P. Bouwknegt and K. Schoutens, _W symmetry in conformal field theory_ , _Phys. Rept._ 223 (1993) 183 [hep-th/9210010].
  * (16) A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, _Holographic GB gravity in arbitrary dimensions_ , _JHEP_ 03 (2010) 111 [0911.4257].
  * (17) S. El-Showk, Y. Nakayama and S. Rychkov, _What Maxwell Theory in D $<$$>$4 teaches us about scale and conformal invariance_, _Nucl. Phys. B_ 848 (2011) 578 [1101.5385].
  * (18) A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev and A. Zhiboedov, _Event shapes in $\mathcal{N}=4$ super-Yang-Mills theory_, _Nucl. Phys._ B884 (2014) 206 [1309.1424].
  * (19) M. S. Costa, T. Hansen, J. a. Penedones and E. Trevisani, _Projectors and seed conformal blocks for traceless mixed-symmetry tensors_ , _JHEP_ 07 (2016) 018 [1603.05551].
  * (20) A. Zhiboedov, _A note on three-point functions of conserved currents_ , 1206.6370.
  * (21) A. Zhiboedov, _On Conformal Field Theories With Extremal a/c Values_ , _JHEP_ 04 (2014) 038 [1304.6075].
  * (22) D. Meltzer and E. Perlmutter, _Beyond $a=c$: gravitational couplings to matter and the stress tensor OPE_, _JHEP_ 07 (2018) 157 [1712.04861].
  * (23) H. Casini, E. Teste and G. Torroba, _Modular Hamiltonians on the null plane and the Markov property of the vacuum state_ , _J. Phys. A_ 50 (2017) 364001 [1703.10656].
  * (24) K.-W. Huang, _Lightcone Commutator and Stress-Tensor Exchange in $d>2$ CFTs_, _Phys. Rev. D_ 102 (2020) 021701 [2002.00110].
  * (25) A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev and A. Zhiboedov, _From correlation functions to event shapes_ , _Nucl. Phys. B_ 884 (2014) 305 [1309.0769].
  * (26) V. M. Braun, G. P. Korchemsky and D. Müller, _The Uses of conformal symmetry in QCD_ , _Prog. Part. Nucl. Phys._ 51 (2003) 311 [hep-ph/0306057].
  * (27) Y. M. Makeenko, _Conformal Operators in Quantum Chromodynamics_ , _Sov. J. Nucl. Phys._ 33 (1981) 440.
  * (28) T. Ohrndorf, _Constraints From Conformal Covariance on the Mixing of Operators of Lowest Twist_ , _Nucl. Phys. B_ 198 (1982) 26.
  * (29) H. Osborn and A. C. Petkou, _Implications of conformal invariance in field theories for general dimensions_ , _Annals Phys._ 231 (1994) 311 [hep-th/9307010].
  * (30) A. C. Wall, _A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices_ , _Phys. Rev. D_ 85 (2012) 104049 [1105.3445].
  * (31) C. N. Pope, L. J. Romans and X. Shen, _$W$(infinity) and the Racah-wigner Algebra_, _Nucl. Phys. B_ 339 (1990) 191.
  * (32) J. Maldacena and A. Zhiboedov, _Constraining Conformal Field Theories with A Higher Spin Symmetry_ , _J. Phys. A_ 46 (2013) 214011 [1112.1016].
  * (33) J. Maldacena and A. Zhiboedov, _Constraining conformal field theories with a slightly broken higher spin symmetry_ , _Class. Quant. Grav._ 30 (2013) 104003 [1204.3882].
  * (34) S. Giombi, S. Prakash and X. Yin, _A Note on CFT Correlators in Three Dimensions_ , _JHEP_ 07 (2013) 105 [1104.4317].
  * (35) M. S. Costa, J. Penedones, D. Poland and S. Rychkov, _Spinning Conformal Correlators_ , _JHEP_ 11 (2011) 071 [1107.3554].
  * (36) P. Kravchuk and D. Simmons-Duffin, _Counting Conformal Correlators_ , _JHEP_ 02 (2018) 096 [1612.08987].
  * (37) C.-H. Chang, M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, _Transverse spin in the light-ray OPE_ , 2010.04726.
  * (38) E. Elkhidir, D. Karateev and M. Serone, _General Three-Point Functions in 4D CFT_ , _JHEP_ 01 (2015) 133 [1412.1796].
  * (39) G. F. Cuomo, D. Karateev and P. Kravchuk, _General Bootstrap Equations in 4D CFTs_ , _JHEP_ 01 (2018) 130 [1705.05401].
  * (40) D. Karateev, P. Kravchuk, M. Serone and A. Vichi, _Fermion Conformal Bootstrap in 4d_ , _JHEP_ 06 (2019) 088 [1902.05969].
  * (41) C. Cordova and K. Diab, _Universal Bounds on Operator Dimensions from the Average Null Energy Condition_ , _JHEP_ 02 (2018) 131 [1712.01089].
  * (42) S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, _Dispersive CFT Sum Rules_ , _JHEP_ 05 (2021) 243 [2008.04931].
  * (43) S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, _AdS Bulk Locality from Sharp CFT Bounds_ , 2106.10274.
  * (44) E. Bergshoeff, M. J. Duff, C. N. Pope and E. Sezgin, _Compactifications of the Eleven-Dimensional Supermembrane_ , _Phys. Lett. B_ 224 (1989) 71.
  * (45) E. Bergshoeff, M. A. Vasiliev and B. de Wit, _The SuperW(infinity) (lambda) algebra_ , _Phys. Lett. B_ 256 (1991) 199.
  * (46) E. Bergshoeff, B. de Wit and M. A. Vasiliev, _The Structure of the superW(infinity) (lambda) algebra_ , _Nucl. Phys. B_ 366 (1991) 315.
  * (47) A. V. Belitsky, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, _Superconformal operators in N=4 superYang-Mills theory_ , _Phys. Rev._ D70 (2004) 045021 [hep-th/0311104].
  * (48) P. Gerasimenko, A. Sharapov and E. Skvortsov, _Slightly Broken Higher Spin Symmetry: General Structure of Correlators_ , 2108.05441.
  * (49) D. Carmi and S. Caron-Huot, _A Conformal Dispersion Relation: Correlations from Absorption_ , _JHEP_ 09 (2020) 009 [1910.12123].
  * (50) D. Mazáč, L. Rastelli and X. Zhou, _A basis of analytic functionals for CFTs in general dimension_ , _JHEP_ 08 (2021) 140 [1910.12855].
  * (51) J. Penedones, J. A. Silva and A. Zhiboedov, _Nonperturbative Mellin Amplitudes: Existence, Properties, Applications_ , _JHEP_ 08 (2020) 031 [1912.11100].
  * (52) M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, _The light-ray OPE and conformal colliders_ , _JHEP_ 01 (2021) 128 [1905.01311].
  * (53) G. P. Korchemsky, _Energy correlations in the end-point region_ , _JHEP_ 01 (2020) 008 [1905.01444].
  * (54) L. J. Dixon, I. Moult and H. X. Zhu, _Collinear limit of the energy-energy correlator_ , _Phys. Rev. D_ 100 (2019) 014009 [1905.01310].
  * (55) L. Cornalba, _Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange_ , 0710.5480.
  * (56) C. N. Pope, L. J. Romans and X. Shen, _The Complete Structure of W(Infinity)_ , _Phys. Lett. B_ 236 (1990) 173.
  * (57) C. N. Pope, _Lectures on W algebras and W gravity_ , in _Summer School in High-energy Physics and Cosmology_ , 12, 1991, hep-th/9112076.
  * (58) A. V. Korybut, _Star product for deformed oscillator algebra $\mathsf{Aq}(2,\nu)$_, 2006.01622.