For $\omega_{1}>0$, $\omega_{2}<0$ and $j_{1}+j_{2}$ fixed the kernel decays
if we deform the $j_{1}$ contour to the left. The fact that the kernel is
analytic means that we only pick the contribution from the poles of
$M(j_{1},j_{2})$. As can be seen from (4), both at weak and at strong coupling
only a single pole at $j_{2}=0$ contributes. This dramatically simplifies the
calculation.

Thirdly, note that $\text{SSC}_{+-}$ should go into itself under the
combination of the crossing symmetry transformation (2.6) and the permutation
of the detectors. This is indeed reflected in the following property of the
kernel,

$\displaystyle
K^{+-}_{\text{SS}}(j_{2},j_{1}|\omega_{2},\omega_{1},z)={1\over(1+\omega_{1}z)(1+\omega_{2}z)}K^{+-}_{\text{SS}}(j_{2},j_{1}|\omega_{1}^{\prime},\omega_{2}^{\prime},z^{\prime})\,.$
(155)

To check this property note that the argument of the $s$-integral in (4.1) is
invariant under this transformation, namely

$\displaystyle-{\omega_{1}\omega_{2}(1-z)\over
1+\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z}=-{\omega_{1}^{\prime}\omega_{2}^{\prime}(1-z^{\prime})\over
1+\omega_{1}^{\prime}+\omega_{2}^{\prime}+\omega_{1}^{\prime}\omega_{2}^{\prime}z^{\prime}}\
,$ (156)

which makes checking (155) trivial.

Using the properties of the kernel described above and the explicit
expressions (4) for the Mellin amplitudes it is easy to compute

$\displaystyle\text{SSC}_{+-}^{\text{weak}}$ $\displaystyle={a\over
4}\bigg{[}\frac{\log\left(\frac{1+\omega_{2}z}{1-z}\right)}{1-z}+{(1+\omega_{2})z\over
2(1-z)(1+z\omega_{2})}\log\left({\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z+1\over(\omega_{2}+1)^{2}}\right)\bigg{]},$
(157) $\displaystyle\text{SSC}_{+-}^{\text{strong}}$
$\displaystyle={z^{2}\over
2}\frac{\left(\omega_{2}+1\right)^{2}}{\left(\omega_{2}z+1\right)^{3}}\,\,.$
(158)

Let us reiterate that only the $j_{2}=0$ pole of the Mellin amplitude
contributes to the above computations. The expressions for $\text{SSC}_{-+}$
can be obtained from $\text{SSC}_{+-}$ by permuting
$\omega_{1}\leftrightarrow\omega_{2}$. We have also checked our calculation by
performing the Mellin integrals numerically and found perfect agreement with
the formulas above.

###  4.2 SSC kernel: same sign

Consider next the case when the frequencies of the detectors have the same
sign, $\omega_{1}\omega_{2}>0$. This covers both cases $++$ and $--$. The
convenient representation of the kernel (4) takes the form (see appendix E for
the derivation)

$\displaystyle{K^{\text{++}}_{\text{SS}}(j_{1},j_{2}|\omega_{1},\omega_{2},z)\over
K_{\text{S}\text{S}}(j_{1},j_{2}|0,0,z)}=(1+\omega_{1})^{-j_{2}}(1+\omega_{2})^{-j_{1}}$
$\displaystyle\times\int_{{\cal C}_{K}}{ds\over 2\pi
i}\Gamma(-s){\Gamma(1-j_{1}-j_{2}-s)\Gamma(j_{2}+s)\Gamma(j_{1}+s)\over\Gamma(1-j_{1}-j_{2})\Gamma(j_{2})\Gamma(j_{1})}\Big{(}{\omega_{1}\omega_{2}(1-z)\over(1+\omega_{1})(1+\omega_{2})}\Big{)}^{s},$
(159)

where the contour of integration ${\cal C}_{K}$ is defined in the same way as
in (4.1).

Let us comment on the relevant properties of (4.2). Firstly, as expected, the
kernel is invariant under the permutation of the detectors
$\omega_{1}\leftrightarrow\omega_{2}$ combined with $j_{1}\leftrightarrow
j_{2}$. The latter exchange does not affect the event shape since
$M(j_{1},j_{2})=M(j_{2},j_{1})$. Due to the relation (which one can explicitly
check using (4.2))

$\displaystyle
K^{\text{++}}_{\text{SS}}(j_{1},j_{2}|\omega_{1},\omega_{2},z)={1\over(1+\omega_{1}z)(1+\omega_{2}z)}K^{\text{++}}_{\text{SS}}(j_{2},j_{1}|\omega_{1}^{\prime},\omega_{2}^{\prime},z^{\prime})\,,$
(160)

where $\omega_{i}^{\prime}<0$, and the crossing symmetry relations (2.6), the
right-hand side of (4.2) defines the kernel for the $--$ case as well.

Secondly, as before it is trivial to extract the $\omega_{i}\to 0$ limit from
(4.2) by closing the Mellin contour in (4.2) to the right and picking the
contributions form the poles at $s=n$ and $s=1-j_{1}-j_{2}+n$,
$n\in\mathbb{Z}_{+}$. For example, the case $\omega_{i}=0$ is simply given by
the residue at $s=0$, which produces $1$ for the ratio (4.2). These two series
of poles can be resummed into a pair of hypergeometric functions (E).

Thirdly, there is an important difference in the analytic properties of the
kernel (4.2) compared to the $+-$ case. Namely, for fixed $j_{1}+j_{2}$ the
kernel has an infinite number of singularities both to the left and to the
right of the $j_{1}$ contour. This means that in evaluating the Mellin
integrals we have to deal with infinitely many poles in both Mellin variables,
even though the Mellin amplitudes of interest (4) are relatively simple. We
will see that the result can still be computed analytically.

Because of the complicated properties of the Mellin kernel, the computation of
$\text{SSC}_{++}$ is more difficult than in the $+-$ case. The result agrees
with the amplitude computation and takes the following form (the details of
the computation can be found in Appendix F)

$\displaystyle\text{SSC}_{++}^{\text{weak}}$ $\displaystyle={a\over
4}\bigg{[}\frac{1}{1-z}\log\left(\frac{1}{1-z}\right)$
$\displaystyle+\frac{(\omega_{1}+1)z}{2(1-z)(\omega_{1}z+1)}\log{\left({\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z+1\over\omega_{1}+1}\right)}$
$\displaystyle+\frac{(\omega_{2}+1)z}{2(1-z)(\omega_{2}z+1)}\log{\left({\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z+1\over\omega_{2}+1}\right)}\bigg{]},$
(161) $\displaystyle\text{SSC}_{++}^{\text{strong}}$
$\displaystyle={z^{2}\over
2}\bigg{[}\frac{6\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}+\omega_{2}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{5}}+\frac{3\omega_{1}\left(2\omega_{2}\omega_{1}^{2}+2\omega_{2}^{2}\omega_{1}+\omega_{1}^{2}+\omega_{2}^{2}\right)\omega_{2}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{4}}$
$\displaystyle+\frac{\omega_{2}^{2}\omega_{1}^{3}+\omega_{2}^{3}\omega_{1}^{2}+2\omega_{2}\omega_{1}^{3}+2\omega_{2}^{3}\omega_{1}+\omega_{1}^{3}+\omega_{2}^{3}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}\bigg{]}\,.$
(162)

As a consistency check, we verify that the relations (157) and (4.2) coincide
for $\omega_{2}=0$. The singularities at
$\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z=0$ are related to the bulk point
singularities of the correlator. They only appear at strong coupling and we
discuss them in more details in Section 4.5.

For the $--$ case, we use the transformation (2.6) to get

$\displaystyle\text{SSC}_{--}^{\text{weak}}$ $\displaystyle={a\over
4}\bigg{[}\frac{1}{1-z}\log\left(\frac{(\omega_{1}z+1)(\omega_{2}z+1)}{1-z}\right)$
$\displaystyle+\frac{(\omega_{1}+1)z}{2(1-z)(\omega_{1}z+1)}\log{\left({1\over\omega_{1}+1}\right)}+\frac{(\omega_{2}+1)z}{2(1-z)(\omega_{2}z+1)}\log{\left({1\over\omega_{2}+1}\right)}\bigg{]},$
(163) $\displaystyle\text{SSC}_{--}^{\text{strong}}$
$\displaystyle={z^{2}\over
2}\bigg{[}\frac{\left(\omega_{1}+1\right)^{2}}{\left(\omega_{1}z+1\right)^{3}}+\frac{\left(\omega_{2}+1\right)^{2}}{\left(\omega_{2}z+1\right)^{3}}$
$\displaystyle-\frac{6\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}+\omega_{2}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{5}}-\frac{3\omega_{1}\left(2\omega_{2}\omega_{1}^{2}+2\omega_{2}^{2}\omega_{1}+\omega_{1}^{2}+\omega_{2}^{2}\right)\omega_{2}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{4}}$
$\displaystyle-\frac{\omega_{2}^{2}\omega_{1}^{3}+\omega_{2}^{3}\omega_{1}^{2}+2\omega_{2}\omega_{1}^{3}+2\omega_{2}^{3}\omega_{1}+\omega_{1}^{3}+\omega_{2}^{3}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}\bigg{]}\,.$
(164)

We recall that the above relations are valid for $0<z<1$ and $\omega_{i}>-1$.

###  4.3 Closing the contour

The reader might have noted something puzzling about our result above: if the
kernel does not depend on the choice $++$ versus $--$, as explained around
(160), how is it possible that the expressions for $\text{SSC}_{++}$ and
$\text{SSC}_{--}$ are different?

To understand the reason, it is instructive to consider the following simple
integral

$\displaystyle\int_{-\epsilon-i\infty}^{-\epsilon+i\infty}{dj\over 2\pi
i}{(1+\omega)^{-j}\over j}=-\theta(\omega)\,.$ (165)

For negative $\omega$ we evaluate the integral by closing the contour to the
left and get zero. For positive $\omega$ we are allowed to close the contour
to the right. Picking up the residue of the pole at $j=0$ we arrive at (165).

The same mechanism is at work for our observables. Consider first the simpler
case $\omega_{1}=0$ with the kernel given by (152). Switching to the
integration variables $j_{1}$ and $j=j_{1}+j_{2}$ in the Mellin integral, we
see that the dependence on $j_{1}$ enters only through the factor
$(1+\omega_{2})^{-j_{1}}$. Therefore, as in the simple example (165) above,
the allowed direction of closing the $j_{1}$ contour depends on the sign of
$\omega_{2}$. For $\omega_{2}\geq 0$ we can close the contour to the right,
whereas for $\omega_{2}\leq 0$ we can close the contour to the left. This can
lead to different results and it explains the difference between
$\text{SSC}(0,\omega_{2},z)$ for positive and negative $\omega_{2}$, even
though the Mellin kernel does not depend on the sign of $\omega_{2}$.

For $\omega_{1},\omega_{2}\neq 0$, the kernels become much more complicated.
Nevertheless, one can identify by inspection the allowed directions of closing
the $j_{1},j_{2}$ contours, depending on the parameters of the problem. We
found that the representations of the kernel (E) and (E) are particularly
useful for this purpose. The result is, as expected, that the generalized
event shapes are not analytic around $\omega_{1},\omega_{2}=0$.

###  4.4 Detector identity

We observe that both at weak and strong coupling the scalar-scalar
correlations satisfy the identity (cf. (137))

$\displaystyle\text{SSC}_{--}+\text{SSC}_{++}-\text{SSC}_{+-}-\text{SSC}_{-+}=0\,.$
(166)

It is understood as an analytic continuation from the region where each
function has been originally computed to some common values of $\omega_{i}$.
In Section 6 we check that this relation also holds for the other detectors
except for the energy one. In the latter case, the identity is violated at
weak coupling due to the contribution from the cross-talk between the
detectors. Here we derive a general formula for the left-hand side of (166)
which is valid at finite coupling and from which its vanishing at weak and
strong coupling follows immediately.

In order to proceed we have to specify the analytic continuation in $\omega$
of the event shapes in (166). Consider for example $\text{SSC}_{++}$ written
for the detector operators defined in the null plane conformal frame (2.2). We
have schematically

$\displaystyle\text{SSC}_{++}=\int_{-\infty}^{\infty}du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle
O_{4}O_{1}(u_{1})O_{2}(u_{2})O_{3}\rangle\,,$ (167)

where we suppressed the dependence on the positions of the source and the
sink, as well as the transverse coordinates of the detector operators. Such
details will not be relevant for what follows.

Since the light-ray operators with $\hat{\omega}_{i}\geq 0$ annihilate the
left vacuum (see Eq. (2.3)), we can rewrite (167) in the following equivalent
way

$\displaystyle\text{SSC}_{++}=\int_{-\infty}^{\infty}du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle[[O_{4},O_{1}(u_{1})],O_{2}(u_{2})]O_{3}\rangle\,.$
(168)

Now we can ‘analytically continue’ $\text{SSC}_{++}$ to any sign of
$\hat{\omega}_{i}$ by using Eq. (168) instead of (167) for any signs of
$\hat{\omega}_{i}$. Let us emphasize again that (168) is equivalent to (167)
only for $\hat{\omega}_{i}>0$ but they differ otherwise. Similarly, we get the
following relations for other choices of the signs of $\omega_{i}$

$\displaystyle\text{SSC}_{--}$
$\displaystyle=\int_{-\infty}^{\infty}du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle
O_{4}[O_{1}(u_{1}),[O_{2}(u_{2}),O_{3}]\rangle,$
$\displaystyle\text{SSC}_{+-}$
$\displaystyle=\int_{-\infty}^{\infty}du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle[O_{4},O_{1}(u_{1})][O_{2}(u_{2}),O_{3}]\rangle,$
$\displaystyle\text{SSC}_{-+}$
$\displaystyle=\int_{-\infty}^{\infty}du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle[O_{4},O_{2}(u_{2})][O_{1}(u_{1}),O_{3}]\rangle,$
(169)

where we used again Eq. (2.3). Switching to commutators does not produce any
effect in the region where each of the event shapes was originally defined.
However, as we go away from this region (by flipping some of the signs of
$\omega_{i}$), formulas (168) and (4.4) define the analytic continuation
needed to check the identity (166).

Now we can evaluate the formulas above for any sign of $\hat{\omega}_{i}$. We
get the following result (where we also use the fact that the detectors
commute with each other for $z\neq 0$)

$\displaystyle\text{SSC}_{--}+\text{SSC}_{++}-\text{SSC}_{+-}-\text{SSC}_{-+}=\int_{-\infty}^{\infty}du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\Big{[}\langle
O_{1}(u_{1})O_{2}(u_{2})O_{4}O_{3}\rangle$ $\displaystyle+\langle
O_{4}O_{3}O_{1}(u_{1})O_{2}(u_{2})\rangle-\langle
O_{1}(u_{1})O_{4}O_{3}O_{2}(u_{2})\rangle-\langle
O_{2}(u_{2})O_{4}O_{3}O_{1}(u_{1})\rangle\Big{]}\,.$ (170)

Let us now specify the signs of $\hat{\omega}_{i}$ and choose, for example,
$\hat{\omega}_{i}>0$. Since the positive frequency detectors annihilate the
left vacuum (see Eq. (2.3)), we find that only one term survives,

$\displaystyle{\text{SSC}_{--}+\text{SSC}_{++}-\text{SSC}_{+-}-\text{SSC}_{-+}\stackrel{{\scriptstyle\hat{\omega}_{i}>0}}{{=}}\int
du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle
O_{4}O_{3}O(u_{1})O(u_{2})\rangle\,.}$ (171)

This formula is completely general and it should hold for any choice of the
detectors.

Compared to (167), the Wightman function on the right-hand side of (171) has
different operator ordering. In a close analogy with (148), the integral over
$u_{i}$ in (171) can be expressed as the convolution of the Mellin amplitude
$M(j_{1},j_{2})$ and the kernel
$\tilde{K}_{SS}(j_{1},j_{2}|\omega_{1},\omega_{2},z)$. Going through the same
steps as in Appendix E, we find that this kernel takes the form

$\displaystyle\tilde{K}_{SS}(j_{1},j_{2}|$
$\displaystyle\omega_{1},\omega_{2},z)={(j_{1}+j_{2}-1)(qn_{1})(qn_{2})\over
2^{j_{1}+j_{2}-1}(n_{1}n_{2})^{j_{1}+j_{2}}}\int_{0}^{\hat{\omega}_{1}}ds_{1}\int_{0}^{\hat{\omega}_{2}}ds_{2}$
$\displaystyle\times\big{[}(\hat{\omega}_{1}-s_{1})s_{2}\big{]}^{-j_{1}}\big{[}s_{1}(\hat{\omega}_{2}-s_{2})\big{]}^{-j_{2}}\big{[}(q+n_{1}s_{1}+n_{2}s_{2})^{2}\big{]}^{j_{1}+j_{2}-2}\,.$
(172)

When integrated against the Mellin amplitudes (4) it gives zero both at weak
and strong coupling. Indeed, from the integral representation (4.4) it follows
that $\tilde{K}_{SS}$ is an analytic function of $j_{2}$ for ${\rm
Re}\,j_{2}<0$. The same is true for the Mellin amplitudes (4). Closing the
integration contour for ${\rm Re}\,j_{2}<0$ we get that the integral on the
right-hand side of (171) is zero. For more complicated Mellin amplitudes with
extra poles at negative $j_{2}$ the integral in (171) will not vanish. This
happens in particular for the correlations involving the energy detector. We
checked that (171) correctly reproduces our results in these cases as well.

###  4.5 Bulk point singularity

The reader might have noticed something peculiar about the expressions for SSC
at strong coupling, Eqs. (158) and (4.2). We see that there is a new type of
singularity that appears in $\text{SSC}_{++}^{\text{strong}}$ at

$\displaystyle\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z=0\,,$ (173)

but is absent in $\text{SSC}_{+-}^{\text{strong}}$. This locus cannot be
reached for $\omega_{1,2}>0$ and $0<z<1$ but a singularity appears if we
analytically continue $\omega_{2}$ to negative values. Using (168) and (4.4)
we get

$\displaystyle\text{SSC}_{++}\stackrel{{\scriptstyle\hat{\omega}_{2}<0}}{{=}}\text{SSC}_{+-}-\int
du_{1}du_{2}\,e^{-i\sum_{k=1}^{2}\hat{\omega}_{k}u_{k}}\langle
O_{2}(u_{2})O_{4}O_{1}(u_{1})O_{3}\rangle\,.$ (174)

Given that $\text{SSC}_{+-}^{\text{strong}}$ is regular at (173), we deduce
that the singular contribution is generated by the second term in (174).

Coming back to the four-point correlation function (138), it is a well-known
fact that there are certain Lorentzian singularities which are present at
strong coupling but are absent at weak coupling Gary:2009ae , namely the bulk
point limit singularities Maldacena:2015iua . It is therefore natural to
conjecture that the singularity at (173) is nothing but an avatar of the bulk
point singularity in the $\omega$-deformed event shape. We will now
demonstrate that this is indeed the case.

Recall that the emergence of the bulk point singularity in Mellin space is
related to the behavior of the Mellin amplitude (143) in the limit
$j_{i}\to\infty$ with ${j_{1}/j_{2}}$ fixed, see Penedones:2010ue . We see
from (4) that

$\displaystyle M^{\text{weak}}(\lambda j_{1},\lambda
j_{2})=O(\lambda^{-2})\,,\qquad\qquad M^{\text{strong}}(\lambda j_{1},\lambda
j_{2})=O(\lambda)\,,$ (175)

and indeed at strong coupling the Mellin amplitude grows as
$\lambda\to\infty$.

To make the connection between the bulk point singularity and the structure of
the $\omega$-deformed event shapes more direct, let us consider the following
toy model

$\displaystyle
M_{p}(j_{1},j_{2})=(j_{1}+j_{2})^{2}(1+j_{1}+j_{2})^{2}\big{[}j_{1}j_{2}(1+j_{1}+j_{2})\big{]}^{p-2}\,,$
(176)

where $p$ is positive integer. This Mellin amplitude is polynomial for $p\geq
2$ and thus corresponds to a contact quartic interaction in the AdS dual
theory Penedones:2010ue . In the large $j$ limit we have $M_{p}(\lambda
j_{1},\lambda j_{2})=O(\lambda^{3p-2})$.

We now want to compute the $\omega$-deformed event shapes (148) for the toy
Mellin amplitude (176). First, one can readily see that

$\displaystyle\text{SSC}_{p,+-}=0\,.$ (177)

This immediately follows from the analytic properties of the kernel described
in the previous sections. A more interesting case is $\text{SSC}_{p,++}$ for
which we get

$\displaystyle\text{SSC}_{p,++}=z^{2}{f_{p}(\omega_{1},\omega_{2})\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{6p-1}}+\dots\
\,.$ (178)

The details of the computation are given in Appendix G. For the first few
values of $p$ we find

$\displaystyle f_{1}(\omega_{1},\omega_{2})$
$\displaystyle=-12(\omega_{1}\omega_{2})^{2}(\omega_{1}+\omega_{2})\,,$
$\displaystyle f_{2}(\omega_{1},\omega_{2})$ $\displaystyle=6\times
7!\times(\omega_{1}\omega_{2})^{3}(\omega_{1}+\omega_{2})^{2}\,.$ (179)

As expected, the maximal power of the singularity in (178) is linked to the
growth of the amplitude in the large $\lambda$ limit.

Let us also write down the complete answer for $p=2$

$\displaystyle\text{SSC}_{p=2,++}={12\omega_{1}^{3}\omega_{2}^{3}\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{11}}\sum_{k=0}^{7}c_{k}(\omega_{1},\omega_{2})(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{k}\,,$
(180)

where the explicit form of the coefficients $c_{k}(\omega_{1},\omega_{2})$ can
be found in appendix G.

The function $\text{SSC}_{p=2,++}$ has interesting properties in the zero
frequency limit $\omega_{i}\to 0$. If we consider $\omega_{1}\to 0$ with
$\omega_{2}$ fixed the result vanishes as $O(\omega_{1}^{3})$. If on the other
hand we take both $\omega_{i}\to 0$ at the same rate, the result diverges as
$\text{SSC}_{p=2,++}(z,\lambda\omega_{1},\lambda\omega_{2})\sim{1/\lambda}$.

Another interesting subtlety related to polynomial Mellin amplitudes of the
type (176) is that they lead to nontrivial distributional terms in the
generalized event shapes which are supported at $\omega_{i}=0$. Let us
consider for simplicity $\omega_{1}=0$ and $p=2$ (this example will also be
relevant in section 7 when we discuss one-loop supergravity and stringy
corrections). In this case we apply (148), (149) and (152) to get the
following expression for the scalar-scalar correlation

$\displaystyle\text{SSC}_{p=2}(0,\omega_{2},z)$
$\displaystyle=\int{djdj_{1}\over(2\pi i)^{2}}{\pi\over 2\sin\pi
j}z^{-j}\left(1-z\right)^{j-1}j^{2}(1+j)^{2}(1+\omega_{2})^{-j_{1}}$
$\displaystyle=2z^{2}(1-6z+6z^{2})\delta(\omega_{2})\,.$ (181)

Here the $j_{1}$ integral was obtained by differentiating (165) with respect
to $\omega$.

##  5 ${\cal N}=4$ supersymmetry Ward identities for the kernels

In the previous sections we used the known results for the four-point
correlation functions of scalar half-BPS operators in $\mathcal{N}=4$ SYM to
compute the correlation between the scalar detectors SSC. In this section, we
extend the consideration to more complicated correlations that involve charge
and energy flow operators. According to their definition (31), they are light-
ray operators built out of the $R-$current $J^{R}_{\mu}$ and the stress-energy
tensor $T_{\mu\nu}$. As a consequence, their correlations can be expressed in
terms of the four-point correlation functions of these operators.

We start in Section 5.1 with a brief review of the ${\cal N}=4$ supersymmetry
Ward identities for the four-point correlation functions and for the
undeformed flow correlations, i.e. for $\omega_{i}=0$. Then in Section 5.2 we
give a summary of the new relationships between the Mellin kernels of the
various flow correlations with the frequencies $\omega_{i}$ turned on. The
detailed derivation is presented in Appendix H.

###  5.1 Stress-energy tensor multiplet in ${\cal N}=4$ SYM

In what follows, we assume that the source and sink are given by scalar
operators $\phi(3)\equiv O_{\bf 20^{\prime}}(x_{3},Y)$ and
$\phi^{\dagger}(4)\equiv O_{\bf 20^{\prime}}(x_{4},\bar{Y})$ (see (99)) and
consider the following correlation functions202020For definiteness we always
assume that $J_{1}\geq J_{2}$.

$\displaystyle
G_{J_{1}J_{2}}=\langle{\phi^{\dagger}(4)O_{J_{1}}(1)O_{J_{2}}(2)\phi(3)}\rangle\,,$
(182)

where $O_{J}(i)\equiv O_{J}(x_{i})$ denotes an operator with spin $J$ whose
Lorentz indices are projected on the light cone,

$\displaystyle O_{J=1}(i)=\bar{n}_{i}^{\mu}J_{\mu}^{R}(x_{i})\,,\qquad\qquad
O_{J=2}(i)=\bar{n}_{i}^{\mu}\bar{n}_{i}^{\nu}T_{\mu\nu}(x_{i})\,,$ (183)

and $O_{J=0}$ was defined in (100). In a generic CFT, the correlation
functions (182) are very complicated and are not related to each other.

A unique feature of $\mathcal{N}=4$ SYM is that the operators (183) belong to
the same stress-tensor supermultiplet. As a result, their correlation
functions are related to each other by $\mathcal{N}=4$ superconformal Ward
identities. Solving them one can express (182) in terms of the simplest four-
point function involving scalar operators only Belitsky:2014zha ;
Korchemsky:2015ssa . The correlation functions (182) have a rather complicated
R-symmetry structure. It corresponds to the overlap of the tensor product of
two real $SO(6)\sim SU(4)$ representations
$\mathbf{20^{\prime}}\times\mathbf{20^{\prime}}=\mathbf{1}+\mathbf{15}+\mathbf{20^{\prime}}+\mathbf{84}+\mathbf{105}+\mathbf{175}$
for the scalar source ($\phi$) and sink ($\phi^{\dagger}$), and that for the
detectors $O_{J}$, for example for two R-symmetry currents
$\mathbf{15}\times\mathbf{15}=\mathbf{1}+\mathbf{15}_{s}+\mathbf{15}_{a}+\mathbf{20^{\prime}}+\mathbf{45}+\mathbf{\overline{45}}+\mathbf{84}$.
We are not going to study this rich structure in full generality. Instead, we
will concentrate on a single R-symmetry channel in each correlation function,
namely the one corresponding to the top value of the $SU(4)$ quadratic
Casimir. In the example above this is the channel $\mathbf{84}$. The reason
for this choice is the remarkably simple relation between the various flow
correlations in this channel for vanishing $\omega_{i}$ discovered in
Belitsky:2013bja ; Belitsky:2014zha  (see (193) below).

We wish to find out to what extent these relations are preserved in the case
of interest $\omega_{i}\neq 0$. So, we concentrate on the preferred channel
and show only the space-time dependence of the correlation functions, dropping
the isotopic variables $Y$ and $S$ of Section 3.4. For instance, the all-
scalar correlation function in the top Casimir channel $\mathbf{105}$ (denoted
by $G^{\mathbf{(105)}}$ in (138)) is

$\displaystyle G_{00}=(x_{12}^{2}x_{34}^{2})^{2}F(x)\,.$ (184)

Here the dependence on the coupling resides in the function $F(x)$ given by

$\displaystyle F(x)={1\over(2\pi)^{4}}{\Phi(u,v)\over
uv(x_{12}^{2}x_{34}^{2})^{4}}\,,\qquad u={x_{12}^{2}x_{34}^{2}\over
x_{13}^{2}x_{24}^{2}}\,,\qquad v={x_{23}^{2}x_{14}^{2}\over
x_{13}^{2}x_{24}^{2}}\,,$ (185)

where $\Phi(u,v)$ is a function of the conformal cross-ratios $u,v$ that
admits the Mellin representation (143).212121Note that in this section we do
not consider the explicit rational (Born-level) terms in (140).

Following Korchemsky:2015ssa , in order to present explicit expressions for
the higher-spin correlator components $G_{J_{1}J_{2}}$, it is convenient to
introduce a pair of auxiliary commuting Weyl spinors, $\lambda_{i,\alpha}$ and
$\bar{\lambda}_{i,\dot{\alpha}}$, for each spinning operator in (182). They
form the null vectors $\bar{n}_{i}$ in (183),

$\displaystyle\bar{n}_{i,\mu}\sigma^{\mu}_{\alpha\dot{\alpha}}=\lambda_{i,\alpha}\bar{\lambda}_{i\dot{\alpha}}\,,\qquad
i=1,2\,,$ (186)

where $\sigma^{\mu}=(1,\vec{\sigma})$ is the four-vector of Pauli matrices.
The general form of the correlation function (182) is (up to normalization and
only for the top Casimir channel), see Ref. Korchemsky:2015ssa ,

$\displaystyle G_{J_{1}J_{2}}={\cal D}_{1}^{J_{1}}{\cal
D}_{2}^{J_{2}}\Big{[}{\cal M}_{J_{1},J_{2}}F(x)\Big{]}\ ,$ (187)

where ${\cal M}_{J_{1},J_{2}}$ is a homogenous polynomial in
$\lambda_{1}^{\alpha}$ and $\lambda_{2}^{\alpha}$ of degree $2J_{1}$ and
$2J_{2}$, respectively. ${\cal D}_{i}$ is a differential operator acting both
on the coordinates of the operator $x_{i}$ and the spinors $\lambda_{i}$,

$\displaystyle{\cal
D}_{i}=\bar{\lambda}_{i\dot{\alpha}}{\left({\sigma^{\mu}}\right)}^{\dot{\alpha}\alpha}{\partial\over\partial\lambda_{i}^{\alpha}}{\partial\over\partial
x_{i}^{\mu}}\,.$ (188)

The functions ${\cal M}_{J_{1},J_{2}}$ are expressed in terms of three
building blocks,

$\displaystyle{\cal
X}_{1[234]}=\lambda_{1}^{\alpha}(x_{12})_{\alpha\dot{\alpha}}(\tilde{x}_{23})^{\dot{\alpha}\beta}(x_{34})_{\beta\dot{\gamma}}(\tilde{x}_{41})^{\dot{\gamma}\delta}\lambda_{1,\delta}\equiv\langle{1|x_{12}\tilde{x}_{23}x_{34}\tilde{x}_{41}|1}\rangle\,,\qquad\qquad$
$\displaystyle{\cal
X}_{[12]i}=\lambda_{1}^{\alpha}(x_{1i})_{\alpha\dot{\alpha}}(\tilde{x}_{i2})^{\dot{\alpha}\beta}\lambda_{2,\beta}\equiv\langle{1|x_{1i}\tilde{x}_{i2}|2}\rangle\,,\qquad(i=3,4)\,,$
(189)

as follows:

$\displaystyle{\cal M}_{0,0}=1\,,$ $\displaystyle{\cal
M}_{1,0}=x_{12}^{2}x_{34}^{2}{\cal X}_{1[234]}\,,$ $\displaystyle{\cal
M}_{2,0}=({\cal X}_{1[234]})^{2}\,,$ $\displaystyle{\cal
M}_{1,1}=x_{12}^{2}x_{34}^{2}{\cal X}_{[12]3}{\cal X}_{[12]4}\,,$
$\displaystyle{\cal M}_{2,1}={\cal X}_{1[234]}{\cal X}_{[12]3}{\cal
X}_{[12]4}\,,$ $\displaystyle{\cal M}_{2,2}=({\cal X}_{[12]3}{\cal
X}_{[12]4})^{2}\,.$ (190)

As explained in Section 4, we can use the correlation functions
$G_{J_{1}J_{2}}$ to obtain the various flow correlations. To this end, we have
to apply the transformations specified in (31) to the operators at points 1
and 2. This amounts to sending both operators to the celestial sphere in the
direction of the null vectors $n_{i}$, that is
$x_{i}=r_{i}n_{i}+\alpha_{i}\bar{n}_{i}$ with $r_{i}\to\infty$, and
integrating along the light-rays $\bar{n}_{i}$ with the exponential weight
$\mathop{\rm e}\nolimits^{-i(n_{i}\bar{n}_{i})\hat{\omega}_{i}\alpha_{i}}$. As
a result the operators at points 1 and 2 in the four-point function (182) are
converted into flow operators and the correlator becomes a function of the two
sets of detector variables $(n_{i},\hat{\omega}_{i})$ and of the separation
between the source and sink $x_{34}$. This is followed by a Fourier transform
to momentum space, $x_{34}\to q$. In the simplest case of SSC , i.e.
$J_{1}=J_{2}=0$, the result is the convolution of the Mellin amplitude
$M(j_{1},j_{2})$ and the detector kernel
$K_{\text{SS}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})$ (see Eq. (148)). It is
important to remember that the dynamical information (i.e. the coupling
dependence) resides in the Mellin amplitude while the kernel is universal, so
it applies to both weak and strong coupling.

Because the various correlation functions (187) are determined by the same
function $\Phi(u,v)$, we expect that the same Mellin representation (148)
should hold for the two-point correlations of flow operators with arbitrary
spin, schematically

$\displaystyle\text{XYC}(\omega_{1},\omega_{2},z)=\int{dj_{1}dj_{2}\over(2\pi
i)^{2}}\,K_{\text{XY}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})\,M(j_{1},j_{2})\,\,.$
(191)

Here $X$ and $Y$ stand for the scalar $(S)$, charge $(Q)$ or energy $(E)$ flow
operators. The kernels $K_{\text{XY}}$ are coupling-independent and have a
fixed form for each pair of detectors. They are functions of the scaling
variable $z$ (the angle between the detectors) and the energies $\omega_{i}$
transferred by the detectors.

The special form of the correlation functions (187) is dictated by
$\mathcal{N}=4$ superconformal symmetry. It is natural to ask whether
analogous relations exist between the correlations (191). Indeed, in Ref.
Belitsky:2014zha  it was shown that for $\omega_{i}=0$ the correlation of two
flow operators of spins $J_{1},J_{2}$ in the top $SU(4)$ Casimir channel,
before the final Fourier transform $x\to q$,222222Without loss of generality
we can set $x_{3}\equiv x$ and $x_{4}=0$. is a total space-time derivative of
the fundamental scalar-scalar one (for $J_{1}\geq J_{2}$):

$\displaystyle{(\Box_{x})^{J_{2}}(n_{2}\partial_{x})^{J_{1}-J_{2}}\over
2^{J_{1}-1}(n_{1}n_{2})^{J_{1}+1}}\left(\frac{{\cal
G}(\gamma)}{x^{2}}\right)={2^{J_{2}+1}(xn_{2})^{J_{1}-J_{2}}\over(n_{1}n_{2})^{J_{1}+1}(x^{2})^{J_{1}+1}}{d^{J_{1}}\over
d\gamma^{J_{1}}}\left[(1-\gamma)^{J_{1}}\gamma^{J_{2}}{\cal
G}^{(J_{2})}(\gamma)\right]\,.$ (192)

Here $\gamma={2(xn_{1})(xn_{2})/(x^{2}(n_{1}n_{2}))}$ is the Lorentz-invariant
and dimensionless angle variable. The function ${\cal G}(\gamma)$ contains the
coupling constant dependence. After the Fourier transform $x\to q$ the left-
hand side of (192) gives rise to a very simple relation between the
corresponding kernels $K_{\text{XY}}$ in (191). Stripping off the kinematic
prefactor $\sim(q^{2})^{J_{2}}(qn_{2})^{J_{1}-J_{2}}/(n_{1}n_{2})^{J_{1}+1}$
(i.e., the Fourier transform of the derivatives on the left-hand side of
(192)), and replacing it with the factor
$\sim{(q^{2})^{J_{1}+J_{2}}/((qn_{1})^{J_{1}+1}(qn_{2})^{J_{2}+1})}$ from the
definition (61), one finds

$\displaystyle
K_{\text{XY}}(j_{1},j_{2}|z,0,0)=z^{-J_{X}}K_{\text{SS}}(j_{1},j_{2}|z,0,0)\,,$
(193)

where $J_{X}\geq J_{Y}$. In application to (191) this implies that for
$\omega_{i}=0$ all the flow correlations are given by the same universal
function of $z$, up to an overall power of $z$.

###  5.2 Summary of the relationships between the kernels for $\omega_{i}\neq
0$

Below we show that for $\omega_{i}\neq 0$ the relation (193) is generalized in
the form of differential operators expressing each $K_{\text{XY}}$ in terms of
the basic scalar one $K_{\text{SS}}$:

$\displaystyle
K_{\text{XY}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})=z^{-J_{X}}\sum_{p,q=0}^{2}C_{\text{XY}}^{p,q}(j_{1},j_{2}|z,\omega_{1},\omega_{2})\partial^{p}_{\omega_{1}}\partial^{q}_{\omega_{2}}\,K_{\text{SS}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})\,,$
(194)

where $J_{X}\geq J_{Y}$ and the coefficients $C_{\text{XY}}$ are polynomials
in $z$ and $\omega_{i}$ and rational functions of the Mellin parameters
$j_{1}$ and $j_{2}$. The order of the differential operator is at most 2 in
each variable, depending on the choice of the detectors.

Here we list some of these non-vanishing coefficients:

(i) Charge-scalar correlation:

$\displaystyle
C_{\text{QS}}^{0,0}=1+\frac{j_{2}\omega_{1}z}{j_{1}+j_{2}}\,,\qquad
C_{\text{QS}}^{1,0}=\frac{\omega_{1}\left(\omega_{1}z+1\right)}{2\left(j_{1}+j_{2}\right)}\,,\qquad
C_{\text{QS}}^{0,1}=-\frac{\omega_{1}\left(\omega_{2}z+1\right)}{2\left(j_{1}+j_{2}\right)}\,;$
(195)

(ii) Charge-charge correlation:

$\displaystyle
C_{\text{QQ}}^{0,0}=1+\frac{\left(j_{1}+j_{2}\right)^{2}(\omega_{1}+\omega_{2}+z\omega_{1}\omega_{2})+j_{1}j_{2}(\omega_{1}+\omega_{2})}{2\left(j_{1}+j_{2}\right){}^{2}}\,,$
$\displaystyle
C_{\text{QQ}}^{1,0}=\frac{\omega_{1}\left(2j_{1}\left(\omega_{1}+1\right)+\left(j_{1}+j_{2}\right)\omega_{2}\left(\omega_{1}z+1\right)\right)}{4\left(j_{1}+j_{2}\right){}^{2}}\,,$
$\displaystyle
C_{\text{QQ}}^{0,1}=\frac{\omega_{2}\left(2j_{2}\left(\omega_{2}+1\right)+\left(j_{1}+j_{2}\right)\omega_{1}\left(\omega_{2}z+1\right)\right)}{4\left(j_{1}+j_{2}\right){}^{2}}\,;$
(196)

(i) Energy-scalar correlation:

$\displaystyle
C_{\text{ES}}^{0,0}=1+\frac{z\omega_{1}\left[j_{2}\left((j_{1}+j_{2})^{2}+j_{1}-1\right)\omega_{1}z+\left((j_{1}+j_{2})^{2}-1\right)\left(j_{1}+2j_{2}\right)\right]}{\left(j_{1}+j_{2}-1\right)\left(j_{1}+j_{2}\right)\left(j_{1}+j_{2}+1\right)}\,,$
$\displaystyle
C_{\text{ES}}^{1,0}=\frac{\omega_{1}\left(\omega_{1}z+1\right)\left[\left(\omega_{1}z+1\right)((j_{1}+j_{2})^{2}-1)+2j_{1}j_{2}\omega_{1}z\right]}{\left(j_{1}+j_{2}-1\right)\left(j_{1}+j_{2}\right)\left(j_{1}+j_{2}+1\right)}\,,$
$\displaystyle
C_{\text{ES}}^{0,1}=\frac{\omega_{1}\left[j_{2}\omega_{1}z\left(\omega_{2}z+1\right)+((j_{1}+j_{2})^{2}-1)(z-1)\right]}{\left(j_{1}+j_{2}-1\right)\left(j_{1}+j_{2}\right)\left(j_{1}+j_{2}+1\right)}\,,$
$\displaystyle
C_{\text{ES}}^{1,1}=\frac{\omega_{1}^{2}\Big{[}\left(j_{1}+j_{2}+2\right)(1+z\omega_{1}+z\omega_{2}+z^{2}\omega_{1}\omega_{2})+(j_{1}+j_{2})(2z-1)+z+1\Big{]}}{3\left(j_{1}+j_{2}-1\right)\left(j_{1}+j_{2}\right)\left(j_{1}+j_{2}+1\right)}\,,$
$\displaystyle
C_{\text{ES}}^{2,0}=\frac{\omega_{1}^{2}\left(z\omega_{1}+1\right){}^{2}}{6\left(j_{1}+j_{2}\right)\left(j_{1}+j_{2}+1\right)}\,,\qquad
C_{\text{ES}}^{0,2}=\frac{\omega_{1}^{2}\left(z\omega_{2}+1\right){}^{2}}{6\left(j_{1}+j_{2}\right)\left(j_{1}+j_{2}+1\right)}\,.$
(197)

We see that setting $\omega_{i}=0$ reproduces the relations (193). The
differential operators (194) for EQC and EEC are presented in the ancillary
file. The detailed derivation of the various differential operators from the
${\cal N}=4$ supersymmetry Ward identity (187) is explained in Appendix H.

Another interesting result of the above analysis is the observation that the
basic kernel $K_{\text{SS}}$ satisfies a simple homogeneous partial
differential equation:

$\displaystyle\Big{[}j_{1}j_{2}\left(j_{1}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)+j_{2}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)-\omega_{1}-\omega_{2}\right)$
$\displaystyle+j_{1}\omega_{1}\left(\left(j_{1}+j_{2}-1\right)\left(\omega_{1}+1\right)+\left(j_{1}+j_{2}\right)\omega_{2}\left(\omega_{1}z+1\right)\right)\,\partial_{\omega_{1}}$
$\displaystyle+j_{2}\omega_{2}\left(\left(j_{1}+j_{2}-1\right)\left(\omega_{2}+1\right)+\left(j_{1}+j_{2}\right)\omega_{1}\left(\omega_{2}z+1\right)\right)\,\partial_{\omega_{2}}$
$\displaystyle+\left(j_{1}+j_{2}\right)\omega_{1}\omega_{2}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)\,\partial_{\omega_{1}}\partial_{\omega_{2}}\Big{]}K_{\text{SS}}(j_{1},j_{2}|z,\omega_{1},\omega_{2})=0\,.$
(198)

Its solutions are discussed in Appendix H.

We remark that the differential operators that relate the various kernels
depend on the Mellin parameters $j_{1},j_{2}$, so they do not commute with the
Mellin integral (191). Consequently, the event shapes (191) are not related to
each other in a simple way, except for two special cases discussed in the next
subsection.

###  5.3 Relationship between generalized event shapes in special cases

We might ask the question what happens if we switch off one of the
frequencies? It turns out that the relations between the kernels described
above simplify dramatically. The differential operators (194) are reduced to
simple factors independent of the Mellin parameters $j_{1},j_{2}$.
Consequently, we can establish very simple relations between the Mellin
integrals (191), i.e. between the flow correlations, and not only between the
kernels.

#### Vanishing frequency $\omega_{1}=0$

Let us start with the asymmetric correlations QSC, ESC and EQC. From (195) and
(197) it is obvious that setting $\omega_{1}=0$ trivializes the differential
operators, reducing them to 1. The situation is similar for EQC. Denoting the
ratio

$\displaystyle
r_{\text{XY}}(z,\omega_{2})={K_{\text{XY}}(j_{1},j_{2}|z,0,\omega_{2})\over
K_{\text{SS}}(j_{1},j_{2}|z,0,\omega_{2})}\,,$ (199)

we find

$\displaystyle r_{\text{QS}}=\frac{1}{z}\,,\qquad
r_{\text{ES}}=\frac{1}{z^{2}}\,,\qquad
r_{\text{EQ}}=\frac{1}{z^{2}}\left(1+\frac{\omega_{2}}{2}\right)\,.$ (200)

At the same time, setting $\omega_{2}=0$ in (195) leaves the derivative
$\left.\partial_{\omega_{2}}K_{\text{SS}}\right|_{\omega_{2}=0}$ in place, and
we do not observe any substantial simplification of the differential operator.
The same applies to the other two asymmetric correlations ESC and EQC.

The two symmetric correlations QQC and EEC also undergo drastic
simplifications if we set $\omega_{1}=0$ (or similarly $\omega_{2}=0$, due to
the symmetry). Take for example QQC: putting $\omega_{1}=0$ in (5.2) we get

$\displaystyle
C_{{QQ}}^{0,0}=1+\frac{\left(j_{1}^{2}+3j_{2}j_{1}+j_{2}^{2}\right){\omega_{2}}}{2\left(j_{1}+j_{2}\right)^{2}}\,,\qquad
C_{{QQ}}^{0,1}=\frac{j_{2}{\omega_{2}}({\omega_{2}}+1)}{2\left(j_{1}+j_{2}\right)^{2}}\,,\qquad
C_{{QQ}}^{1,0}=0\,.$ (201)

This is not all, we still have to explore the consequences of the differential
equation (5.2), which now reads simply

$\displaystyle\big{[}(\omega_{2}+1)\partial_{\omega_{2}}+j_{1}\big{]}K_{\text{SS}}(z,\omega_{1}=0,\omega_{2})=0\,.$
(202)

This equation allows us to eliminate the term with $\partial_{\omega_{2}}$ in
(194),

$\displaystyle
C_{{QQ}}^{0,0}-\frac{j_{1}}{\omega_{2}+1}C_{{QQ}}^{0,1}=1+\frac{\omega_{2}}{2}\,,$
(203)

so that the differential operator (194) collapses to (compare with the third
relation in (200))

$\displaystyle
r_{\text{QQ}}=\frac{1}{z}\left(1+\frac{\omega_{2}}{2}\right)\,.$ (204)

The analysis of the differential operator for EEC yields an equally simple
relation,

$\displaystyle
r_{\text{EE}}=\frac{1}{z^{2}}\left(1+\omega_{2}+\frac{\omega_{2}^{2}}{6}\right)\,.$
(205)

As mentioned earlier, the remarkable feature of the relations (200), (204) and
(205) is that they do not depend on the Mellin parameters $j_{i}$. Inserting
them in (191) yields equivalent relations between the various event shapes,

$\displaystyle{\text{XYC}(0,\omega_{2},z)\over\text{SSC}(0,\omega_{2},z)}=r_{\text{XY}}\,,\qquad
J_{X}\geq J_{Y}\,.$ (206)

Most importantly, these relations hold both perturbatively (at weak and at
strong coupling) and nonperturbatively (at finite coupling).

#### Event shapes ${\rm XYC}_{+-}$ at strong coupling

As explained in Section 4.1, the computation of the event shape ${\rm
SSC}_{+-}^{\rm strong}$ is drastically simplified by using the integral
representation (4.1) of the kernel $K^{+-}_{\rm SS}$. The expression in (4.1)
is analytic in $j_{1},j_{2}$, while the Mellin amplitude $M^{\text{strong}}$
from (4) has simple poles. Consequently, the Mellin integral over $j_{2}$ with
$j_{1}+j_{2}$ fixed amounts to evaluating (4.1) at $j_{2}=0$. To this end we
have to analyze the behavior of the factors
$\Gamma(-s)\Gamma(j_{2}+s)/\Gamma(j_{2})$ under the $s-$integral, when
$s,j_{2}\to 0$. The pole in $\Gamma(j_{2}+s)$ at $s=-j_{2}\to 0$ lies outside
the contour whereas the pole in $\Gamma(-s)$ at $s=0$ has the trivial residue
$-1$. After setting $j_{2}=0$, the only remaining $\omega$ dependence is in
the factor $(1+\omega_{2})^{-j_{1}}$ in front of the integral. The Mellin
integral over $j_{1}$ is easily done by residues resulting in the simple
expression (158).

The same scenario applies when we act with the differential operators (194).
We observe another drastic simplification, namely the Mellin integral over
$j_{2}$ turns the differential operators into simple factors independent of
$j_{1}$. They go through the $j_{1}-$integral and the resulting event shapes
${\rm XYC}_{+-}^{\rm strong}$ are obtained by multiplying ${\rm SSC}_{+-}^{\rm
strong}$ by these factors.

Let us illustrate the mechanism on the simplest example of ${\rm
QSC}_{+-}^{\rm strong}$. When the differential operator (194) with
coefficients from (195) acts under the integral in (4.1), it produces terms
$\sim s\to 0$. Similarly, its action on the factor $(1+\omega_{1})^{-j_{2}}$
in front of the integral produces terms $\sim j_{2}\to 0$. So, we just need to
evaluate

$\displaystyle\frac{1}{(1+\omega_{2})^{-j_{1}}}\left[\frac{1}{z}\sum_{q=0}^{2}C^{0,q}_{\rm
QS}(j_{1},0|\omega_{1},\omega_{2})\partial^{q}_{\omega_{2}}\,(1+\omega_{2})^{-j_{1}}\right]$
$\displaystyle=\frac{1}{(1+\omega_{2})^{-j_{1}}}\left[\frac{1}{z}\left(1-\frac{1}{2j_{1}}\omega_{1}(\omega_{2}z+1)\partial_{\omega_{2}}\right)\,(1+\omega_{2})^{-j_{1}}\right]=\frac{\omega_{1}+2\omega_{2}+\omega_{2}\omega_{1}z+2}{2z\left(\omega_{2}+1\right)}\,.$
(207)

As explained above, this result is independent of $j_{1}$, so it immediately
gives the ratio of the event shapes ${\rm QSC}_{+-}^{\rm strong}/{\rm
SSC}_{+-}^{\rm strong}$ listed in (229) and (227), respectively. The
calculation with all the other differential operators (194) is equally simple
and we obtain all the event shapes ${\rm XYC}_{+-}^{\rm strong}$ from Section
6.2. The non-trivial feature of the operators (194) is that their action on
$(1+\omega_{2})^{-j_{1}}$ gives a result independent of $j_{1}$.

Note that the situation is different if we employ the Mellin amplitude
$M^{\text{weak}}$ from (4). In order to evaluate the residue at the double
pole $j_{2}=0$, we need to consider the complete sequence of poles of the
integrand in (4.1) at $s=0,1,2,\ldots$. This results in an infinite sum which,
together with that originating from the poles at $j_{1}=-1,-2,-3,\ldots$ gives
rise to the logs in (157). Further, the action of the differential operators
(194) on the kernel (4.1) modifies the residue at the double pole $j_{2}=0$,
so the result is not as simple as in (5.3).

##  6 Summary of the generalized event shapes in ${\cal N}=4$ SYM

In this section, we present the explicit expressions for the two-point
correlations (50) in $\mathcal{N}=4$ SYM at weak and strong coupling.
232323For the reader’s convenience, we presented the expressions for all two-
point correlations in a Mathematica readable format in the ancillary file
included with an arXiv submission. Following (2.6) we distinguish six
different functions $\rm XYC$ with $X,Y=\\{S,Q,E\\}$.

###  6.1 Weak coupling

At weak coupling, the correlations take the form

$\displaystyle{\rm XYC}(\omega_{1},\omega_{2},z)={\rm XYC}^{(0)}+{a\over
4}\,{\rm XYC}^{(1)}+O(a^{2})\,,$ (208)

where $a=g^{2}N/(4\pi^{2})$. In the Born approximation, the correlations are
given by (107) and (3.4). The functions ${\rm XYC}^{(0)}$ vanish for $0<z<1$
and $\omega_{i}>-1$. To one-loop order, depending on the signs of $\omega_{1}$
and $\omega_{2}$ we find the following expressions:

### Scalar-scalar correlation

$\displaystyle\text{SSC}^{(1)}_{+-}$
$\displaystyle=\frac{\log\left(\frac{\omega_{2}z+1}{1-z}\right)}{1-z}+\frac{\left(\omega_{2}+1\right)z\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{2}+1\right)^{2}}\right)}{2(1-z)\left(\omega_{2}z+1\right)}$
(209) $\displaystyle\text{SSC}^{(1)}_{++}$
$\displaystyle=\frac{\log\left(\frac{1}{1-z}\right)}{1-z}+\frac{(\omega_{1}+1)z\log{\left({\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z+1\over\omega_{1}+1}\right)}}{2(1-z)(\omega_{1}z+1)}+\frac{(\omega_{2}+1)z\log{\left({\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z+1\over\omega_{2}+1}\right)}}{2(1-z)(\omega_{2}z+1)}$
(210)

### Charge-scalar correlation

$\displaystyle\text{QSC}^{(1)}_{+-}$
$\displaystyle=\frac{\left(\frac{\omega_{1}z}{2}+1\right)\log\left(\frac{\omega_{2}z+1}{1-z}\right)}{(1-z)z}+\frac{\left(\omega_{1}+2\omega_{2}+\omega_{2}\omega_{1}z+2\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{2}+1\right)^{2}}\right)}{4(1-z)\left(\omega_{2}z+1\right)}$
(211) $\displaystyle\text{QSC}^{(1)}_{-+}$
$\displaystyle=\frac{\left(\frac{\omega_{1}z}{2}+1\right)\log\left(\frac{\omega_{1}z+1}{1-z}\right)}{(1-z)z}+\frac{\left(\omega_{1}+2\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{1}+1\right)^{2}}\right)}{4(1-z)}$
(212) $\displaystyle\text{QSC}^{(1)}_{++}$
$\displaystyle=\frac{\left(1+\frac{\omega_{1}z}{2}\right)\log\left(\frac{1}{1-z}\right)}{(1-z)z}+\frac{\left(\omega_{1}+2\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{1}+1}\right)}{4(1-z)}$
$\displaystyle+\frac{\left(\omega_{1}(1+\omega_{2}z)+2(\omega_{2}+1)\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{2}+1}\right)}{4(1-z)\left(\omega_{2}z+1\right)}$
(213)

### Charge-charge correlation

$\displaystyle\text{QQC}^{(1)}_{+-}$
$\displaystyle=\frac{(\omega_{1}\omega_{2}z+2(\omega_{1}+\omega_{2}+2))\log\left(\frac{\omega_{2}z+1}{1-z}\right)}{4(1-z)z}$
$\displaystyle+\frac{(\omega_{2}+2)(\omega_{2}(\omega_{1}z+2)+\omega_{1}+2)\log\left(\frac{\omega_{1}\omega_{2}z+\omega_{1}+\omega_{2}+1}{(\omega_{2}+1)^{2}}\right)}{8(1-z)(\omega_{2}z+1)}$
(214) $\displaystyle\text{QQC}^{(1)}_{++}$
$\displaystyle=\frac{\left(2\omega_{1}+2\omega_{2}+\omega_{2}\omega_{1}z+4\right)\log\left(\frac{1}{1-z}\right)}{4(1-z)z}$
$\displaystyle+\frac{\left(\omega_{1}+2\right)\left(\omega_{2}(1+\omega_{1}z)+2(\omega_{1}+1)\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{1}+1}\right)}{8(1-z)\left(\omega_{1}z+1\right)}$
$\displaystyle+\frac{\left(\omega_{2}+2\right)\left(\omega_{1}(1+\omega_{2}z)+2(\omega_{2}+1)\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{2}+1}\right)}{8(1-z)\left(\omega_{2}z+1\right)}$
(215)

### Energy-scalar correlation

$\displaystyle\text{ESC}^{(1)}_{+-}$
$\displaystyle=\frac{\left(\omega_{1}^{2}z^{2}+6\omega_{1}z+6\right)\log\left(\frac{\omega_{2}z+1}{1-z}\right)}{6(1-z)z^{2}}-\frac{\omega_{1}^{2}\log\left(\frac{\left(\omega_{2}+1\right)z}{\omega_{2}z+1}\right)}{6(\omega_{2}+1)z}$
(216)
$\displaystyle+\frac{\left(6\left(\omega_{2}+1\right)^{2}+6\omega_{1}\left(\omega_{2}+1\right)\left(\omega_{2}z+1\right)+\left(\omega_{1}+\omega_{2}\omega_{1}z\right)^{2}\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{2}+1\right)^{2}}\right)}{12\left(\omega_{2}+1\right)(1-z)z\left(\omega_{2}z+1\right)}$
$\displaystyle\text{ESC}^{(1)}_{-+}$
$\displaystyle=\frac{\left(\omega_{1}^{2}z^{2}+6\omega_{1}z+6\right)\log\left(\frac{\omega_{1}z+1}{1-z}\right)}{6(1-z)z^{2}}-\frac{\omega_{1}^{2}\log\left(\frac{\left(\omega_{1}+1\right)z}{\omega_{1}z+1}\right)}{6\left(\omega_{1}+1\right)z}$
$\displaystyle+\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(\omega_{1}z+1\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{1}+1\right)^{2}}\right)}{12\left(\omega_{1}+1\right)(1-z)z}$
(217) $\displaystyle\text{ESC}^{(1)}_{++}$
$\displaystyle=\frac{\left(\omega_{1}^{2}z^{2}+6\omega_{1}z+6\right)\log{\left({1\over
1-z}\right)}}{6(1-z)z^{2}}$
$\displaystyle+\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(\omega_{1}z+1\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{1}+1}\right)}{12\left(\omega_{1}+1\right)(1-z)z}$
$\displaystyle+\frac{\left(6\left(\omega_{2}+1\right)^{2}+6\omega_{1}\left(\omega_{2}+1\right)\left(\omega_{2}z+1\right)+\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{2}+1}\right)}{12\left(\omega_{2}+1\right)(1-z)z\left(\omega_{2}z+1\right)}$
$\displaystyle-\omega_{1}^{2}\left(\frac{\log\left(\frac{\left(\omega_{1}+1\right)^{2}\omega_{2}z}{\omega_{1}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{12\left(\omega_{1}+1\right)z}+\frac{\log\left(\frac{\omega_{1}\left(\omega_{2}+1\right)^{2}z}{\omega_{2}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{12\left(\omega_{2}+1\right)z}\right)$
(218)

### Energy-charge correlation

$\displaystyle\text{EQC}^{(1)}_{+-}$
$\displaystyle=\frac{\left(6\left(\omega_{2}+2\right)+\omega_{1}\left(z\left(2\omega_{1}+6\omega_{2}+\omega_{2}\omega_{1}z+6\right)+6\right)\right)\log\left(\frac{\omega_{2}z+1}{1-z}\right)}{12(1-z)z^{2}}$
$\displaystyle+\frac{\left(\omega_{2}+2\right)\left(\omega_{2}+1\right)\left(6+6{\omega_{1}\left(\omega_{2}z+1\right)\over\left(\omega_{2}+1\right)}+{\omega_{1}^{2}\left(1+\omega_{2}z\right)^{2}\over\left(\omega_{2}+1\right)^{2}}\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{2}+1\right)^{2}}\right)}{24(1-z)z\left(\omega_{2}z+1\right)}$
$\displaystyle+\frac{\omega_{2}\omega_{1}^{2}\log\left(\frac{z\left(\omega_{2}+1\right)}{\omega_{2}z+1}\right)}{12z(\omega_{2}+1)}$
(219) $\displaystyle\text{EQC}^{(1)}_{-+}$
$\displaystyle=\frac{\left(6\left(\omega_{2}+2\right)+\omega_{1}\left(z\left(2\omega_{1}+6\omega_{2}+\omega_{2}\omega_{1}z+6\right)+6\right)\right)\log\left(\frac{\omega_{1}z+1}{1-z}\right)}{12(1-z)z^{2}}$
$\displaystyle+\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(\omega_{2}+\omega_{1}\left(\omega_{2}z+2\right)+2\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\left(\omega_{1}+1\right)^{2}}\right)}{24\left(\omega_{1}+1\right)(1-z)z}$
$\displaystyle-\frac{\omega_{2}\omega_{1}^{2}\log\left(\frac{z\left(\omega_{1}+1\right)}{\omega_{1}z+1}\right)}{12z\left(\omega_{1}+1\right)}$
(220) $\displaystyle\text{EQC}^{(1)}_{++}$
$\displaystyle=\frac{\left(6\left(\omega_{2}+2\right)+\omega_{1}\left(z\left(2\omega_{1}+6\omega_{2}+\omega_{2}\omega_{1}z+6\right)+6\right)\right)\log{\left({1\over
1-z}\right)}}{12(1-z)z^{2}}$
$\displaystyle+\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(\omega_{2}+\omega_{1}\left(\omega_{2}z+2\right)+2\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{1}+1}\right)}{24\left(\omega_{1}+1\right)(1-z)z}$
$\displaystyle+\frac{\left(\omega_{2}+2\right)\left(6\left(\omega_{2}+1\right)^{2}+6\omega_{1}\left(\omega_{2}+1\right)\left(\omega_{2}z+1\right)+\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{2}+1}\right)}{24\left(\omega_{2}+1\right)(1-z)z\left(\omega_{2}z+1\right)}$
$\displaystyle-\omega_{2}\omega_{1}^{2}\left(\frac{\log\left(\frac{\left(\omega_{1}+1\right)^{2}\omega_{2}z}{\omega_{1}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{24\left(\omega_{1}+1\right)z}-\frac{\log\left(\frac{\omega_{1}\left(\omega_{2}+1\right)^{2}z}{\omega_{2}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{24\left(\omega_{2}+1\right)z}\right)$
(221)

### Energy-energy correlation

$\displaystyle\text{EEC}^{(1)}_{+-}$
$\displaystyle=\frac{\left(\omega_{1}^{2}(\omega_{2}z(\omega_{2}z+6)+6)+6\omega_{1}(\omega_{2}((\omega_{2}+3)z+3)+6)+6(\omega_{2}(\omega_{2}+6)+6)\right)\log{\left({\omega_{2}z+1\over
1-z}\right)}}{36(1-z)z^{2}}$
$\displaystyle+\frac{(\omega_{2}+1)(\omega_{2}^{2}+6\omega_{2}+6)\left({\omega_{1}^{2}(\omega_{2}z+1)^{2}\over(\omega_{2}+1)^{2}}+{6\omega_{1}(\omega_{2}z+1)\over(\omega_{2}+1)}+6\right)\log\left(\frac{\omega_{1}\omega_{2}z+\omega_{1}+\omega_{2}+1}{(\omega_{2}+1)^{2}}\right)}{72(1-z)z(\omega_{2}z+1)}$
$\displaystyle-\frac{\omega_{1}^{2}\omega_{2}^{2}}{36(\omega_{2}+1)z}\log{\left({z(\omega_{2}+1)\over\omega_{2}z+1}\right)}$
(222) $\displaystyle\text{EEC}^{(1)}_{++}$
$\displaystyle=\frac{\left(6\left(\omega_{1}^{2}+6\omega_{1}+6\right)+\omega_{2}^{2}\left(\omega_{1}^{2}z^{2}+6\omega_{1}z+6\right)+6\omega_{2}\left(\omega_{1}^{2}z+3\omega_{1}(z+1)+6\right)\right)\log{\left({1\over
1-z}\right)}}{36(1-z)z^{2}}$
$\displaystyle+\frac{\left(\omega_{1}+1\right)\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(\frac{\omega_{2}^{2}\left(\omega_{1}z+1\right)^{2}}{\left(\omega_{1}+1\right)^{2}}+\frac{6\omega_{2}\left(\omega_{1}z+1\right)}{\omega_{1}+1}+6\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{1}+1}\right)}{72(1-z)z\left(\omega_{1}z+1\right)}$
$\displaystyle+\frac{\left(\omega_{2}+1\right)\left(\omega_{2}^{2}+6\omega_{2}+6\right)\left(\frac{\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}}{\left(\omega_{2}+1\right)^{2}}+\frac{6\omega_{1}\left(\omega_{2}z+1\right)}{\omega_{2}+1}+6\right)\log\left(\frac{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1}{\omega_{2}+1}\right)}{72(1-z)z\left(\omega_{2}z+1\right)}$
$\displaystyle-\omega_{1}^{2}\omega_{2}^{2}\left(\frac{\log\left(\frac{\left(\omega_{1}+1\right)^{2}\omega_{2}z}{\omega_{1}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{72\left(\omega_{1}+1\right)z}+\frac{\log\left(\frac{\left(\omega_{2}+1\right)^{2}\omega_{1}z}{\omega_{2}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{72\left(\omega_{2}+1\right)z}\right)$
$\displaystyle-\omega_{1}^{2}\omega_{2}^{2}\frac{\log\left(\frac{z\left(\omega_{1}\omega_{2}z+\frac{1}{2}\left(\omega_{1}+\omega_{2}+\sqrt{\left(\omega_{1}+\omega_{2}\right)^{2}-4\omega_{1}\omega_{2}z}\right)\right)^{2}}{\omega_{1}\omega_{2}\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+1\right)}\right)}{12z\sqrt{\left(\omega_{1}+\omega_{2}\right)^{2}-4\omega_{1}\omega_{2}z}}$
(223)

The following comments are in order.

The above relations are valid for $0<z<1$ and $\omega_{i}>-1$. We verify that
for $\omega_{1}=\omega_{2}=0$ they simplify to

$\displaystyle\text{XYC}^{(1)}(0,0,z)={1\over(1-z)z^{J_{X}}}\log{\left({1\over
1-z}\right)}\,,$ (224)

where $J_{X}\geq J_{Y}$. This relation is in agreement with the findings of
Ref. Belitsky:2013bja .

For $\omega_{1}=0$ and nonzero $\omega_{2}$ it is possible to show that the
above expressions verify the relation (206). For nonzero $\omega_{i}$, the
first term in the above expressions for the correlations describes the elastic
contribution discussed in Section 3.4. For the correlations SSC, QSC and QQC,
the remaining terms describe the annihilation and production of particles by
the detectors, depending on the sign of $\omega_{i}$ (see Section 3.4). In
addition to such terms, the correlations ESC, EQC and EEC also contain terms
that scale at small $z$ as $(\log z)/z$. They are accompanied by powers of
$\omega_{1}$ and $\omega_{2}$ and take the form

$\displaystyle\text{EXC}^{(1)}(\omega_{1},\omega_{2},z)\stackrel{{\scriptstyle
z\to 0}}{{\sim}}\omega_{1}^{2}\omega_{2}^{J_{X}}\times{\log z\over z}\,,$
(225)

where $J_{X}$ is the spin of the detector $X=\\{S,Q,E\\}$. These additional
terms come from the cross-talk between the detectors. For the energy-energy
correlation (6), such terms are accompanied by a square-root
$\sqrt{\left(\omega_{1}+\omega_{2}\right)^{2}-4\omega_{1}\omega_{2}z}$. Its
origin and physical interpretation are discussed in Appendix I.

We did not display the expressions for $\text{XYC}_{--}$ and
$\text{XXC}_{-+}$. They can be obtained by applying the crossing relation
(2.6) to $\text{XYC}_{++}$ and $\text{XXC}_{+-}$, respectively. Doing so, we
examined the relation

$\displaystyle\text{XYC}_{--}+\text{XYC}_{++}\stackrel{{\scriptstyle?}}{{=}}\text{XXC}_{+-}+\text{XXC}_{-+}\,,$
(226)

which was previously observed to hold for the scalar-scalar correlation (137).
We found that it is satisfied for the correlations SSC, QSC and QQC, which do
not receive contributions from the cross-talk between the detectors. At the
same time, it is violated for the correlations involving the energy detector.
As explained in Section 4.4, this can be understood using the general formula
(171) and the properties of the corresponding kernels.

###  6.2 Strong coupling

At strong coupling, in the leading supergravity approximation, we find the
correlations:

### Scalar-scalar correlation

$\displaystyle\text{SSC}_{+-}=$
$\displaystyle\frac{z^{2}\left(\omega_{2}+1\right)^{2}}{2\left(\omega_{2}z+1\right)^{3}}$
(227) $\displaystyle\text{SSC}_{++}=$ $\displaystyle{z^{2}\over
2}\bigg{[}\frac{\left(\omega_{1}+1\right)^{2}}{\left(\omega_{1}z+1\right)^{3}}+\frac{\left(\omega_{2}+1\right)^{2}}{\left(\omega_{2}z+1\right)^{3}}$
$\displaystyle-\frac{6\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}+\omega_{2}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{5}}-\frac{3\omega_{1}\left(2\omega_{2}\omega_{1}^{2}+2\omega_{2}^{2}\omega_{1}+\omega_{1}^{2}+\omega_{2}^{2}\right)\omega_{2}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{4}}$
$\displaystyle-\frac{\omega_{2}^{2}\omega_{1}^{3}+\omega_{2}^{3}\omega_{1}^{2}+2\omega_{2}\omega_{1}^{3}+2\omega_{2}^{3}\omega_{1}+\omega_{1}^{3}+\omega_{2}^{3}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}\bigg{]}$
(228)

### Charge-scalar correlation

$\displaystyle\text{QSC}_{+-}$
$\displaystyle=\frac{z\left(\omega_{2}+1\right)\left(\omega_{1}+2\omega_{2}+\omega_{2}\omega_{1}z+2\right)}{4\left(\omega_{2}z+1\right)^{3}}$
(229) $\displaystyle\text{QSC}_{-+}$
$\displaystyle=\frac{z\left(\omega_{1}+1\right)\left(\omega_{1}+2\right)}{4\left(\omega_{1}z+1\right)^{2}}$
(230) $\displaystyle\text{QSC}_{++}$ $\displaystyle={z\over
4}\bigg{[}\frac{\left(\omega_{1}+1\right)\left(\omega_{1}+2\right)}{\left(\omega_{1}z+1\right)^{2}}+\frac{\left(\omega_{2}+1\right)\left(\omega_{1}+2\omega_{2}+\omega_{2}\omega_{1}z+2\right)}{\left(\omega_{2}z+1\right)^{3}}$
$\displaystyle-\frac{6\omega_{1}^{2}\omega_{2}\left(\omega_{1}+\omega_{2}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{4}}-\frac{2\left(3\omega_{2}\omega_{1}^{2}+3\omega_{2}^{2}\omega_{1}+\omega_{1}^{2}+2\omega_{2}^{2}\right)\omega_{1}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}$
$\displaystyle-\frac{\omega_{2}\omega_{1}^{3}+\omega_{2}^{2}\omega_{1}^{2}+3\omega_{2}^{2}\omega_{1}+\omega_{1}^{3}+2\omega_{2}^{2}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{2}}\bigg{]}$
(231)

### Charge-charge correlation

$\displaystyle\text{QQC}_{+-}=$
$\displaystyle\frac{z(\omega_{2}+1)(\omega_{2}+2)(\omega_{1}\omega_{2}z+\omega_{1}+2\omega_{2}+2)}{8(\omega_{2}z+1)^{3}}$
(232) $\displaystyle\text{QQC}_{++}=$ $\displaystyle{z\over
16}\bigg{[}\frac{2\left(\omega_{1}+1\right)\left(\omega_{1}+2\right)\left(2\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z+2\right)}{\left(\omega_{1}z+1\right)^{3}}$
$\displaystyle+\frac{2\left(\omega_{2}+1\right)\left(\omega_{2}+2\right)\left(\omega_{1}+2\omega_{2}+\omega_{2}\omega_{1}z+2\right)}{\left(\omega_{2}z+1\right)^{3}}$
$\displaystyle-\frac{2\left(\omega_{2}\omega_{1}+\omega_{1}+\omega_{2}\right)\left(\omega_{2}\omega_{1}^{2}+\omega_{2}^{2}\omega_{1}-2\omega_{2}\omega_{1}+2\omega_{1}^{2}+2\omega_{2}^{2}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{2}}$
$\displaystyle-\frac{4\left(\omega_{2}^{3}\omega_{1}^{3}+7\omega_{2}^{2}\omega_{1}^{3}+7\omega_{2}\omega_{1}^{3}+7\omega_{2}^{3}\omega_{1}^{2}+7\omega_{2}^{3}\omega_{1}+2\omega_{1}^{3}+2\omega_{2}^{3}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}$
$\displaystyle-\frac{24\omega_{1}\omega_{2}\left(\omega_{2}^{2}\omega_{1}^{2}+3\omega_{2}\omega_{1}^{2}+3\omega_{2}^{2}\omega_{1}+\omega_{1}^{2}+\omega_{2}^{2}\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{4}}$
$\displaystyle-\frac{24\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}\omega_{2}+2(\omega_{1}+\omega_{2})\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{5}}\bigg{]}$
(233)

### Energy-scalar correlation

$\displaystyle\text{ESC}_{+-}$
$\displaystyle=\frac{\left(\omega_{2}+1\right)^{2}\left(\frac{\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}}{\left(\omega_{2}+1\right)^{2}}+\frac{6\omega_{1}\left(\omega_{2}z+1\right)}{\omega_{2}+1}+6\right)}{12\left(\omega_{2}z+1\right)^{3}}$
(234) $\displaystyle\text{ESC}_{-+}$
$\displaystyle=\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)}{12(\omega_{1}z+1)}$
(235) $\displaystyle\text{ESC}_{++}$ $\displaystyle={1\over
12}\bigg{[}\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)}{(\omega_{1}z+1)}+\frac{\left(\omega_{2}+1\right)^{2}}{\left(\omega_{2}z+1\right)^{3}}\left(\frac{\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}}{\left(\omega_{2}+1\right)^{2}}+\frac{6\omega_{1}\left(\omega_{2}z+1\right)}{\omega_{2}+1}+6\right)$
$\displaystyle-\frac{\omega_{1}^{2}}{z\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)}-\frac{\left(\omega_{1}^{3}+(\omega_{1}^{2}+6\omega_{1}+6)\omega_{2}\right)}{(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z)}$
$\displaystyle-\frac{\left(6\omega_{1}^{2}+(\omega_{1}^{2}+6\omega_{1}+6)\omega_{2}\right)\omega_{1}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{2}}-\frac{2\left(\omega_{2}\omega_{1}+3(\omega_{1}+\omega_{2})\right)\omega_{1}^{2}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}\bigg{]}$
(236)

### Energy-charge correlation

$\displaystyle\text{EQC}_{+-}$
$\displaystyle=\frac{\left(\omega_{2}+2\right)\left(\omega_{2}+1\right)^{2}\left({\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}\over(\omega_{2}+1)^{2}}+{6\omega_{1}\left(\omega_{2}z+1\right)\over\left(\omega_{2}+1\right)}+6\right)}{24\left(\omega_{2}z+1\right)^{3}}$
(237) $\displaystyle\text{EQC}_{-+}$
$\displaystyle=\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(2+2\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z\right)}{24\left(\omega_{1}z+1\right)^{2}}$
(238) $\displaystyle\text{EQC}_{++}$ $\displaystyle={1\over
24}\bigg{[}\frac{4\left(\omega_{2}+2\right)\left(\omega_{2}+1\right)^{2}}{\left(\omega_{2}z+1\right)^{3}}\left({\omega_{1}^{2}\left(\omega_{2}z+1\right)^{2}\over(\omega_{2}+1)^{2}}+{6\omega_{1}\left(\omega_{2}z+1\right)\over\left(\omega_{2}+1\right)}+6\right)$
$\displaystyle+\frac{\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(2+2\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z\right)}{\left(\omega_{1}z+1\right)^{2}}$
$\displaystyle-\frac{\omega_{1}^{2}\omega_{2}(\omega_{1}-\omega_{2})(2+\omega_{1}+\omega_{2})}{z(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{3}}-\frac{12\omega_{1}^{2}\omega_{2}\left(2\omega_{1}\omega_{2}+3(\omega_{1}+\omega_{2})\right)}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{4}}$
$\displaystyle+\frac{\left(\omega_{2}^{2}\omega_{1}^{3}-\left(\omega_{2}\left(\omega_{2}\left(\omega_{2}+24\right)+54\right)+12\right)\omega_{1}^{2}-54\omega_{2}^{2}\omega_{1}-24\omega_{2}^{2}\right)\omega_{1}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{3}}$
$\displaystyle-\frac{20\omega_{2}\omega_{1}^{3}+12\omega_{1}^{3}+2\left(\omega_{1}+2\right)\left(2\omega_{1}\left(\omega_{1}+3\right)+3\right)\omega_{2}^{2}}{\left(\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z\right)^{2}}$
$\displaystyle-\frac{(\omega_{2}+2)\omega_{1}^{3}+\left(\omega_{1}^{2}+6\omega_{1}+6\right)\omega_{2}^{2}}{\omega_{1}+\omega_{2}+\omega_{2}\omega_{1}z}\bigg{]}$
(239)

### Energy-energy correlation

$\displaystyle\text{EEC}_{+-}$
$\displaystyle=\frac{(\omega_{2}+1)^{2}\left(\omega_{2}^{2}+6\omega_{2}+6\right)\left(\frac{\omega_{1}^{2}(\omega_{2}z+1)^{2}}{(\omega_{2}+1)^{2}}+\frac{6\omega_{1}(\omega_{2}z+1)}{\omega_{2}+1}+6\right)}{72(\omega_{2}z+1)^{3}}$
(240) $\displaystyle\text{EEC}_{++}$ $\displaystyle={1\over
72}\bigg{[}\frac{(\omega_{2}+1)^{2}}{(\omega_{2}z+1)^{3}}\left(\omega_{2}^{2}+6\omega_{2}+6\right)\left(\frac{\omega_{1}^{2}(\omega_{2}z+1)^{2}}{(\omega_{2}+1)^{2}}+\frac{6\omega_{1}(\omega_{2}z+1)}{\omega_{2}+1}+6\right)$
$\displaystyle+\frac{(\omega_{1}+1)^{2}}{(\omega_{1}z+1)^{3}}\left(\omega_{1}^{2}+6\omega_{1}+6\right)\left(\frac{\omega_{2}^{2}(\omega_{1}z+1)^{2}}{(\omega_{1}+1)^{2}}+\frac{6\omega_{2}(\omega_{1}z+1)}{\omega_{1}+1}+6\right)$
$\displaystyle-\frac{\omega_{1}^{2}\omega_{2}^{2}(\omega_{1}^{2}-4\omega_{1}\omega_{2}+\omega_{2}^{2})}{z(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{5}}((\omega_{1}+\omega_{2})^{2}+6(\omega_{1}+\omega_{2})+6)$
$\displaystyle-{\omega_{1}^{3}(\omega_{2}^{2}+6\omega_{2}+6)+\omega_{2}^{3}(\omega_{1}^{2}+6\omega_{1}+6)\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)}$
$\displaystyle-{2(4\omega_{1}^{3}\omega_{2}^{3}+24\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}+\omega_{2}\right)+39\omega_{1}\omega_{2}\left(\omega_{1}^{2}+\omega_{2}^{2}\right)+18\left(\omega_{1}+\omega_{2}\right)\left(\omega_{1}^{2}-\omega_{2}\omega_{1}+\omega_{2}^{2}\right))\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{2}}$
$\displaystyle-{116\omega_{1}^{3}\omega_{2}^{3}+\omega_{1}^{3}\omega_{2}^{3}\left(\omega_{1}+\omega_{2}\right)+264\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}+\omega_{2}\right)+180\omega_{1}\omega_{2}\left(\omega_{1}^{2}+\omega_{2}^{2}\right)\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{3}}$
$\displaystyle-{36\left(\omega_{1}+\omega_{2}\right)\left(\omega_{1}^{2}-\omega_{2}\omega_{1}+\omega_{2}^{2}\right)\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{3}}-{\omega_{1}^{3}\omega_{2}^{3}(312+6\left(\omega_{1}+\omega_{2}\right))\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{4}}$
$\displaystyle-{\omega_{1}\omega_{2}(\omega_{1}^{2}\omega_{2}^{2}\left(\omega_{1}^{2}-4\omega_{2}\omega_{1}+\omega_{2}^{2}\right)+432\omega_{1}\omega_{2}\left(\omega_{1}+\omega_{2}\right)+108\left(\omega_{1}^{2}+\omega_{2}^{2}\right))\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{4}}$
$\displaystyle-{\omega_{1}^{2}\omega_{2}^{2}(6\omega_{1}\omega_{2}\left(\omega_{1}+\omega_{2}\right)+\omega_{1}\omega_{2}\left(6+\omega_{1}+\omega_{2}\right)\left(\omega_{1}^{2}-4\omega_{2}\omega_{1}+\omega_{2}^{2}\right)+216(\omega_{1}\omega_{2}+\omega_{1}+\omega_{2}))\over(\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z)^{5}}\bigg{]}$
(241)

Let us summarize the properties of the above results.

We verify that for $\omega_{1}=\omega_{2}=0$ they are in agreement with the
findings of Ref. Belitsky:2013bja ; Belitsky:2014zha ,

$\displaystyle\text{XYC}(0,0,z)=\frac{1}{2}z^{2-J_{X}}\,,$ (242)

where $J_{X}\geq J_{Y}$. For $\omega_{1}=0$ and $\omega_{2}\neq 0$ they
satisfy the relation (206).

We observe that the expressions for the correlations $\text{XYC}_{++}$ are
more complicated compared to $\text{XYC}_{+-}$ and $\text{XYC}_{-+}$. They
contain additional terms involving inverse powers of
$\tilde{q}^{2}/q^{2}-1=\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z$. As we
explained in Section 4.5, such terms are generated by a bulk point
singularity.

As at weak coupling, the remaining correlations $\text{XYC}_{--}$ can be found
using the crossing relation (2.6). They have a form similar to
$\text{XYC}_{++}$. Then, examining the sum $\text{XYC}_{--}+\text{XYC}_{++}$
we observe that all the terms with the bulk singularity induced scale
$\omega_{1}+\omega_{2}+\omega_{1}\omega_{2}z$ cancel against each other and
the resulting expression exactly matches the sum
$\text{XYC}_{-+}+\text{XYC}_{+-}$. We therefore conclude that the relation
(226) holds at strong coupling. We recall that at weak coupling this relation
is broken for the correlations that receive a contribution from the detector
cross-talk. In terms of the OPE, the cross-talk is associated with the
contribution of the leading, twist-two operators in the product of the
detector operators. At weak coupling, these operators develop anomalous
dimensions that manifest themselves through $\log z$ terms in the small $z$
limit. At strong coupling, these operators become infinitely heavy and do not
contribute. This explains why the cross-talk contribution is not present at
strong coupling.

##  7 Quantum gravity corrections to event shapes

In this section we analyze ${1/c_{T}}\sim 1/N_{c}^{2}$ corrections to the
generalized energy-energy correlations at strong coupling. Such corrections
come from gravitational loops in the bulk theory. Here we only consider the
case $\omega_{1}=0$ which simplifies the analysis dramatically. We use the
Mellin amplitude recently computed in Alday:2018kkw  to study the structure of
the generalized event shapes at one loop in supergravity. As expected on
general grounds Kologlu:2019bco , we find that the result is divergent when
$\omega_{i}\to 0$. To understand the nature of this divergence we derive a
dispersive representation of the generalized event shapes using CFT dispersion
relations in Mellin space Penedones:2019tng . We then discuss the expected
structure of the undeformed ($\omega_{i}=0$) event shapes at finite $\lambda$
and $c_{T}$.

###  7.1 One-loop supergravity

Conventional event shapes, like the energy-energy correlation, cannot be
computed perturbatively in ${1/c_{T}}$ at strong coupling. The reason is that
the correlation function becomes more and more singular in the Regge limit at
higher loops which renders the event shape of interest divergent. This
divergence is an artifact of perturbation theory and, based on general
grounds, it is absent at finite $c_{T}$, see Kologlu:2019bco . Since the
correlation functions at finite $c_{T}$ are not accessible, we can
alternatively deform the observable in a way that makes it finite order by
order in ${1/c_{T}}$. We can then try to understand what happens as we take
the deformation parameter to zero. Introducing non-zero $\omega_{i}$ is one
natural way to do it.

The divergence of the undeformed event shape manifests itself in the
appearance of divergences in the $\omega_{i}\to 0$ limit. In other words, the
$c_{T}\to\infty$ and $\omega_{i}\to 0$ limits do not commute. Here we compute
the $\omega$-deformed event shapes using the one-loop Mellin amplitude in
supergravity. The relevant one-loop supergravity correlation function was
recently found in Alday:2017xua ; Aprile:2017bgs ; Alday:2017vkk .

The starting point of our analysis is the one-loop Mellin amplitude
Alday:2018kkw

$\displaystyle M^{\text{1-loop sugra}}(j_{1},j_{2})$ $\displaystyle={1\over
4c_{T}}(j_{1}+j_{2})^{2}(j_{1}+j_{2}+1)^{2}\sum_{m,n=0}^{\infty}c_{m,n}\Big{[}{1\over(j_{1}-m-1)(j_{2}-n-1)}$
$\displaystyle-{1\over(j_{1}-m-1)(j_{1}+j_{2}+2+n)}-{1\over(j_{2}-m-1)(j_{1}+j_{2}+2+n)}\Big{]}\,.$
(243)

The expression inside the brackets is fully crossing symmetric. The expansion
coefficients are given by

$\displaystyle c_{m,n}$ $\displaystyle=c_{n,m}={1\over
5(m+n-1)_{5}}\Big{(}15m^{4}n^{2}+25m^{4}n+12m^{4}+30m^{3}n^{3}+120m^{3}n^{2}+114m^{3}n$
$\displaystyle+36m^{3}+15m^{2}n^{4}+120m^{2}n^{3}+216m^{2}n^{2}+77m^{2}n-8m^{2}+25mn^{4}+114mn^{3}$
$\displaystyle+77mn^{2}-76mn-40m+12n^{4}+36n^{3}-8n^{2}-40n\Big{)}\,.$ (244)

The double sum in (7.1) is divergent. This is a reflection of the UV
divergence in the bulk theory at one loop which needs to be renormalized. The
divergence does not depend on $j_{i}$ and leads to an ambiguity in the
renormalized one-loop supergravity amplitude which we denote by $c_{\infty}$.

We introduce a crossing-symmetric regulator in the sum,
$\sum_{m,n}c_{m,n}\to\sum_{m,n}c_{m,n}{x^{m}+x^{n}\over 2}$. To recover the
original sum we take $x\to 1$. The divergent part of the sum is given by
$\propto{1/(1-x)}$ and is indeed $j_{i}$ independent. We then arrive at

$\displaystyle M^{\text{1-loop sugra}}(j_{1},j_{2})$
$\displaystyle=(j_{1}+j_{2})^{2}(j_{1}+j_{2}+1)^{2}\left(c_{\infty}+\hat{M}^{\text{reg}}(j_{1},j_{2})\right),$
(245)

where $\hat{M}^{\text{reg}}(j_{1},j_{2})$ can be easily computed explicitly
using the regulator above and dropping the divergent ${1/(1-x)}$ part. It
takes a relatively simple form of a sum of various polygamma functions with
rational coefficients, and we do not present it here to save space.242424It
can be found in the ancillary file that we provided together with the
submission. In fact, the explicit expression for the one-loop Mellin amplitude
was recently found in Aprile:2020luw . We checked that our results agree. We
thank Francesco Aprile for pointing this out to us. After this step, the
Mellin integral representation for the the generalized event shapes becomes
well defined and it can be studied both analytically and numerically. The
result takes the following form

$\displaystyle\text{SSC}_{++}^{\text{1-loop
sugra}}(\omega_{1},\omega_{2},z)=c_{\infty}\,\text{SSC}_{++}^{p=2}(\omega_{1},\omega_{2},z)+\text{SSC}_{++}^{\text{reg}}(\omega_{1},\omega_{2},z)\,,$
(246)

where $\text{SSC}_{++}^{p=2}$ is given in (180) and $c_{\infty}$ is fixed in
the nonperturbative theory. It is known explicitly using the power of
supersymmetric locailzation Chester:2019pvm

$\displaystyle c_{\infty}={1\over c_{T}}\Big{(}{5\sqrt{\lambda}\over
16}+18\zeta(3)-{3\pi^{2}\over 2}-{2141\over 192}+O(\lambda^{-{3\over
2}})\Big{)}\,.$ (247)

Instead of keeping $\omega_{i}$ general, here we focus our attention on
understanding the small $\omega_{i}$ limit. Setting $\omega_{1}=0$ does not
spoil the finiteness of the result but dramatically simplifies the kernel, see
(152). According to (4.5),
$\text{SSC}_{++}^{p=2}(0,\omega_{2},z)\sim\delta(\omega_{2})$ and, therefore,
the $\text{SSC}_{++}^{\text{1-loop sugra}}(0,\omega_{2},z)$ does not depend on
the coefficient $c_{\infty}$ for $\omega_{2}\neq 0$. In fact, any four-point
contact interaction in AdS does not contribute to the generalized event shape
for $\omega_{1}=0$ and $\omega_{2}\neq 0$. 252525Microscopically this is
related to the extra suppression of the double trace operators familiar from
studies of undeformed event shapes.

The result for the generalized energy-energy correlation at one loop in
supergravity takes the form

$\displaystyle\text{EEC}^{\text{1-loop sugra}}(0,\omega_{2},z)={5\over
4c_{T}}\bigg{[}{1-6z+6z^{2}\over\omega_{2}^{2}}+{3\over
2}{2-15z+24z^{2}-10z^{3}\over\omega_{2}}$ $\displaystyle+{12\over
25}(1-36z+216z^{2}-400z^{3}+225z^{4})\log|\omega_{2}|\log(z^{2}|\omega_{2}|)$
$\displaystyle+\Big{(}{716-20496z+111396z^{2}\over
125}-1560z^{3}+846z^{4}\Big{)}\log|\omega_{2}|+O(\omega_{2}^{0})\bigg{]}\ ,$
(248)

where we only presented the terms that are singular in the $\omega_{2}\to 0$
limit and did not include the contact term $\delta(\omega_{2})$.262626The
leading contact term contribution can be read off from (4.5) and (247) and it
is given by ${5\sqrt{\lambda}\over 8c_{T}}\delta(\omega_{2})(1-6z+6z^{2})$.
One can also check that the expression above correctly implements the
transformation property (2.6) which relates the cases $\omega_{2}>0$ and
$\omega_{2}<0$.

As an immediate consistency check, we verified that the stress-tensor Ward
identities (63) continue to hold when we include the one-loop correction,
because (7.1) satisfies

$\displaystyle\int_{0}^{1}dz\,\text{EEC}^{\text{1-loop
sugra}}(z,0,\omega_{2})=\int_{0}^{1}dz\,z\,\text{EEC}^{\text{1-loop
sugra}}(z,0,\omega_{2})=0\,.$ (249)

###  7.2 Dispersive representation of the energy-energy correlator

To elucidate the origin of the ${1/\omega_{2}}$ divergences in (7.1) and to
understand what happens to them at finite coupling, it is useful to consider
the dispersive representation for the Mellin amplitude Penedones:2019tng
.272727It would be very interesting to generalize this analysis to the
coordinate space as well using the coordinate space CFT dispersion relations
Carmi:2019cub ; Mazac:2019shk ; Caron-Huot:2020adz .

To make as many symmetries as possible manifest, we find it convenient to
write down dispersion relations for the Mellin amplitude
$\tilde{M}(j_{1},j_{2})\equiv{M(j_{1},j_{2})/((j_{1}+j_{2})^{2}(j_{1}+j_{2}+1)^{2})}$
introduced in (145). The advantage of using $\tilde{M}(j_{1},j_{2})$ is that
it is fully crossing-symmetric function of $j_{1}$ and $j_{2}$

$\displaystyle\tilde{M}(j_{1},j_{2})=\tilde{M}(j_{2},j_{1})=\tilde{M}(j_{1},-1-j_{1}-j_{2})\
\,.$ (250)

Consistency with the OPE implies that $M(j_{1},j_{2})$ has simple poles at the
positions dictated by the quantum numbers of the exchanged operators
Mack:2009mi ; Mack:2009gy . Let us start by recalling the structure of these
poles in variable $j$ defined as

$\displaystyle j=j_{1}+j_{2}\,.$ (251)

Poles in $j$ correspond to the $\text{detector}\times\text{detector}$ OPE
channel which due to our choice of detectors is the same as the ${\bf 105}$
R-symmetry channel. The residues at the poles are fixed in terms of the three-
point functions of the exchanged operator, see e.g. Costa:2012cb  for the
detailed derivation,

$\displaystyle M(j-j_{2},j_{2})=-{1\over 2}{C_{\tau,J}^{2}{\cal
Q}_{J,m}^{\tau}(2(1-j_{2}))\over j+{\tau\over 2}+m-2}+O\left(j+{\tau\over
2}+m-2\right).$ (252)

Here $\tau$ and $J\in 2\mathbb{Z}_{+}$ are the twist and spin of the exchanged
operator correspondingly. At finite $c_{T}$ and $\lambda$ these include all
operators present in the theory. Below, when analyzing the one-loop
supergravity, only double trace operators will contribute. Descendants are
labeled by integer $m\geq 0$, and ${\cal Q}_{J,m}^{\tau}(2(1-j_{2}))$ are the
so-called Mack polynomials known explicitly, see e.g. Gopakumar:2021dvg  for
an explicit formula. $C_{\tau,J}^{2}$ is the square of the three-point
function of the external operators and the operator that appears in the OPE.
Only even spin operators appear in the formula above due to the crossing
symmetry $M(j_{1},j_{2})=M(j_{2},j_{1})$.

Let us now translate the relation (252) to the statement about the
singularities of the crossing-symmetric modified Mellin amplitude
$\tilde{M}(j_{1},j_{2})$. The relation (252) implies that it has simple poles
in $j=j_{1}+j_{2}$ with the following residues

$\displaystyle\tilde{M}(j-j_{2},j_{2})=-{1\over 2}{C_{\tau,J}^{2}{\cal
Q}_{J,m}^{\tau}(2(1-j_{2}))\over j+{\tau\over 2}+m-2}{1\over({\tau\over
2}+m-2)^{2}({\tau\over 2}+m-3)^{2}}+O\left(j+{\tau\over 2}+m-2\right),$ (253)

where $J\in 2\mathbb{Z}_{+}$. Of course, in addition it could have
singularities at $j=0$ and $j=-1$ since
$\tilde{M}(j-j_{2},j_{2})\equiv{M(j-j_{2},j_{2})/(j^{2}(j+1)^{2})}$.

Since $\tilde{M}(j_{1},j_{2})$ is fully crossing-symmetric, see (250), by
replacing $j_{2}\to-(1+j)$ and $j\to-(1+j_{2})$ in (253) we find that
$\tilde{M}(j-j_{2},j_{2})$ has additional poles in $j_{2}$. They correspond to
the OPE channel detector $\times$ source. The residues at these poles can be
found by applying the crossing to (253)

$\displaystyle\tilde{M}(j-j_{2},j_{2})={1\over 2}{C_{\tau,J}^{2}{\cal
Q}_{J,m}^{\tau}(-2(2+j))\over j_{2}-{\tau\over 2}-m+3}{1\over({\tau\over
2}+m-2)^{2}({\tau\over 2}+m-3)^{2}}+O(j_{2}-{\tau\over 2}-m+3)\,.$ (254)

The additional singularities at $j=-1,0$ are mapped to $j_{2}=0,-1$ under the
crossing.

We now write down the dispersive representation for $\hat{M}(j_{1},j_{2})$ for
fixed $j=j_{1}+j_{2}$

$\displaystyle\tilde{M}(j-j_{2},j_{2})$
$\displaystyle=\oint{dj_{2}^{\prime}\over 2\pi
i}{\tilde{M}(j-j_{2}^{\prime},j_{2}^{\prime})\over
j_{2}^{\prime}-j_{2}}=-\sum_{\alpha=\\{0,-1,j,j-1\\}}{\rm
Res}_{j_{2}^{\prime}=\alpha}{\tilde{M}(j-j_{2}^{\prime},j_{2}^{\prime})\over
j_{2}^{\prime}-j_{2}}$ $\displaystyle-{1\over
2}\sum_{\tau,J-\text{even},m}{C_{\tau,J}^{2}{\cal
Q}_{J,m}^{\tau}(-2(2+j))\over({\tau\over 2}+m-2)^{2}({\tau\over
2}+m-3)^{2}}\Big{(}{1\over{\tau\over 2}+m-3-j_{2}}+{1\over{\tau\over
2}+m-3-(j-j_{2})}\Big{)}\,.$ (255)

In deriving this relation we used the non-perturbative bound on the Mellin
amplitude $|M(j_{1},j_{2})|\leq|j_{2}|$ as $|j_{2}|\to\infty$, see
Penedones:2019tng , which translates to
$|\tilde{M}(j_{1},j_{2})|\leq|j_{2}|^{-3}$ as $|j_{2}|\to\infty$. Using the
crossing symmetry this also implies that
$|\tilde{M}(j-j_{2},j_{2})|\leq|j_{2}|^{-3}$ for fixed $j$, therefore the arc
at infinity does not contribute to (7.2). The sum on the first line of (7.2)
comes from the contribution of protected twist-two operators and it is fixed
by the OPE.282828Here we assume that the coupling is finite and there are no
higher spin conserved currents in the spectrum.

As a result we get the following representation for the nonperturbative Mellin
amplitude

$\displaystyle M(j_{1},j_{2})$ $\displaystyle=-{1\over
2}{(j_{1}+j_{2})^{2}(1+j_{1}+j_{2})\over
j_{1}j_{2}}-{(j_{1}+j_{2})^{2}(j_{1}+j_{2}+1)^{2}\over 2}\ $
$\displaystyle\times\sum_{\tau,J-\text{even},m}{C_{\tau,J}^{2}{\cal
Q}_{J,m}^{\tau}(-2(2+j_{1}+j_{2}))\over({\tau\over 2}+m-2)^{2}({\tau\over
2}+m-3)^{2}}\Big{(}{1\over{\tau\over 2}+m-3-j_{2}}+{1\over{\tau\over
2}+m-3-j_{1}}\Big{)}\,.$ (256)

Let us emphasize that this formula is exact. The first term comes from the
protected twist two operators and it coincides with the leading result at
strong coupling (4). The infinite sum encodes corrections to this result.

For the one loop supergravity Mellin amplitude, the only operators that appear
in the sum in (7.2) are the double trace operators which have twist
$\tau_{n,J}=8+2n+\gamma(n,J)$ with $\gamma(n,J)\sim{1/c_{T}}$.292929The shift
of the twist $\tau$ by $(+4)$ is a consequence of superconformal symmetry, see
e.g. Eqs. (2.5) and (2.6) in Alday:2017vkk . In the language of Aprile:2017bgs
, we have $\tau=4+2t$, where $t\geq 2$. The Mack polynomials ${\cal
Q}_{J,m}^{\tau}$ that enter (7.2) are importantly proportional to

$\displaystyle{\cal Q}_{J,m}^{\tau}\sim\sin^{2}{\left({\pi\tau\over
2}\right)}\,.$ (257)

For the double trace operators this becomes

$\displaystyle\sin^{2}{\left({\pi\tau_{n,J}\over
2}\right)}=\sin^{2}{\left({\pi\gamma_{n,J}\over 2}\right)}\sim{1\over
c_{T}^{2}}$ (258)

Double trace operators have three-point functions that scale as
$C_{\tau,J}^{2}\sim c_{T}$ which can be seen from form of the disconnected
part of the correlator in (140). Combining this fact with (258) we correctly
recover the ${1/c_{T}}$ scaling of the leading correction to (7.2) in
supergravity.

Next we can use the dispersive representation (7.2) to compute the
$\omega$-deformed event shapes at finite coupling in terms of the OPE data.
For simplicity we consider $\omega_{1}=0$ and $\omega_{2}>0$. Using relations
(205), (206), and (152), we get the following representation for the event
shape

$\displaystyle\text{EEC}_{+}(0,\omega_{2},z)$
$\displaystyle=\left(1+\omega_{2}+\frac{\omega_{2}^{2}}{6}\right)\Big{[}{1\over
2}+\sum_{\tau,J-\text{even}}2\sin^{2}{\left({\pi\tau\over
2}\right)}C_{\tau,J}^{2}(1+\omega_{2})^{-({\tau\over
2}-1)}\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})\Big{]},$ (259)

where we have defined a function

$\displaystyle\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})$
$\displaystyle\equiv-{1\over 2}\int{dj\over 2\pi i}{\pi\over 2\sin\pi
j}z^{-j-2}\left(1-z\right)^{j-1}j^{2}(1+j)^{2}$
$\displaystyle\times\sum_{m=0}^{\infty}(1+\omega_{2})^{-m}{{\cal
Q}_{J,m}^{\tau}(-2(j+2))\over 2\sin^{2}{\left({\pi\tau\over
2}\right)}}{1\over({\tau\over 2}+m-2)^{2}({\tau\over 2}+m-3)^{2}}\,.$ (260)

It encodes the contribution to the energy-energy correlation of a set of poles
in (7.2) associated to a given primary operator. 303030For correlation
functions these are known as Polyakov blocks, see e.g. Carmi:2019cub ;
Mazac:2019shk ; Caron-Huot:2020adz  for details. Therefore
$\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})$ encodes the contribution of
a given Polyakov block to the energy-energy correlation. In writing the
formula above we used the fact that for $\omega_{2}>0$ we can close the
$j_{1}$ contour to the right half-plane since this is the direction in which
$(1+\omega_{2})^{-j_{1}}$ decays.

It is possible to compute $\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})$
explicitly for the first few values of $J$. For $J=0$ we get

$\displaystyle\widetilde{\text{EEC}}_{\tau,J=0}(z,0,\omega_{2})$
$\displaystyle=-(1-6z+6z^{2})\frac{\Gamma\left(\frac{\tau-6}{2}\right)^{2}\Gamma(\tau)}{\pi^{2}\Gamma\left(\frac{\tau}{2}\right)^{4}}\
_{2}F_{1}\left({\tau-6\over 2},{\tau-6\over 2},\tau-1,{1\over
1+\omega_{2}}\right)\,.$ (261)

For $J>0$ we similarly get polynomials in $z$,
$\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})=\sum_{i=0}^{J+2}c_{i}(\tau,\omega_{2})z^{i}$.
Curiously, we also observed the relation

$\displaystyle\lim_{\tau\to\infty}\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})\sim(1-6z+6z^{2})\Big{(}1+O(\tau^{-1})\Big{)},$
(262)

which will be useful for our discussion below. Note that the dependence of
(262) on $z$ matches the leading ${1/\omega_{2}^{2}}$ term in (7.1).

It is now easy to understand how the terms singular for $\omega_{2}\to 0$ are
generated in (7.1). The underlying mechanism has already been discussed in the
literature, see for example Maldacena:2015iua , and it is related to the
contribution of double trace operators to the sum (259). In this case the
factor $\sin^{2}{\left({\pi\tau/2}\right)}$ becomes
$\sin^{2}{\left({\pi\gamma_{n,J,i}/2}\right)}$ where $n$ is related to the
twist, $J$ is spin, and $i$ accounts for the degeneracies of the double trace
operators in question. We now see that

$\displaystyle\sin^{2}{\left({\pi\gamma_{n,J,i}\over 2}\right)}\leq 1\,,\qquad
c_{T}-\text{finite}.$ (263)

If on the other hand we first expand at large $c_{T}$ using the fact that
$\gamma_{n,J,i}\sim{1/c_{T}}\ll 1$ we get

$\displaystyle\sin^{2}{\left({\pi\gamma_{n,J,i}\over 2}\right)}={\pi^{2}\over
4}\gamma_{n,J,i}^{2}+O(1/c_{T}^{4})\,.$ (264)

To compute the contribution of double trace operators to (259) we need to sum
both over spin $J$ and twists of the operators labeled by $n$. Because the
anomalous dimensions $\gamma_{n,J,i}$ are growing functions of $n$, it is the
sum over twists of the operators in (259) that produces terms that diverge in
the limit $\omega_{2}\to 0$. Non-perturbatively we know that an infinite
series of ${1/c_{T}}$ corrections turn $\gamma_{n,J,i}^{2}$ into (263) in this
way regulating the sum. This mechanism of regulating the divergences in
perturbation theory is sometimes called “eikonalization” Fitzpatrick:2015qma .
To summarize, we see that the large twist operators can generate divergences
in ${1/c_{T}}$ perturbation theory but they are absent at finite $c_{T}$.

Let us next work out the form of the ${1/\omega_{2}^{2}}$ term in (7.1) in
more detail. In order to do it we replace $C_{\tau,J}^{2}$ in (259) with its
leading asymptotic behaviour in mean field theory at tree level

$\displaystyle\lim_{\tau\to\infty}\langle
C_{\tau,J}^{2}\rangle=c_{T}2^{5}\pi(J+1)4^{-J}4^{-\tau}\tau^{2}\,.$ (265)

For the anomalous dimensions we use the results of Aprile:2017bgs  to get

$\displaystyle\lim_{c_{T}\to\infty}{\langle 2\sin^{2}{\left({\pi\tau\over
2}\right)}C_{\tau,J}^{2}\rangle\over\langle
C_{\tau,J}^{2}\rangle}\sim{\pi^{2}\over
2}{\langle\gamma_{\tau,J}^{2}C_{\tau,J}^{2}\rangle\over\langle
C_{\tau,J}^{2}\rangle}\sim{\tau^{11}\over c_{T}^{2}}\,.$ (266)

Substituting these relations into (259) and taking into account (261) we find
that the contribution of the operators with large twist and $J=0$ involves the
following integral

$\displaystyle\int_{\tau_{0}}^{\infty}d\tau\ \langle
C_{\tau,J}^{2}\rangle{\langle 2\sin^{2}{\left({\pi\tau\over
2}\right)}C_{\tau,J}^{2}\rangle\over\langle
C_{\tau,J}^{2}\rangle}\widetilde{\text{EEC}}_{\tau,J=0}(z,0,\omega_{2})(1+\omega_{2})^{-{\tau\over
2}}$ $\displaystyle\sim{1\over
c_{T}}\int_{\tau_{0}}^{\infty}d\tau\,2^{-\tau}\tau^{{15\over 2}}\
_{2}F_{1}\left({\tau-6\over 2},{\tau-6\over 2},\tau-1,{1\over
1+\omega_{2}}\right)(1+\omega_{2})^{-{\tau\over 2}}\,,$ (267)

where we replaced the sum over twist by an integral over $\tau>\tau_{0}$. To
understand its behaviour for $\omega_{2}\to 0$ it is convenient to use the
integral representation of the hypergeometric function. The result then takes
the following form at large $\tau$

$\displaystyle{1\over
c_{T}}\int_{0}^{1}dt\int_{\tau_{0}}^{\infty}d\tau\tau^{8}{(1-t)(1-t+\omega_{2})^{3}\over
t^{4}}\Big{(}{1-t+\omega_{2}\over t(1-t)}\Big{)}^{-{\tau\over 2}}\sim{1\over
c_{T}\omega_{2}^{2}}\,.$ (268)

The integral over $\tau$ is trivial to do. The integral over $t$ can be easily
analyzed numerically or by the saddle point method, with the expected result
that it behaves as ${1/\omega_{2}^{2}}$. In this way we have also correctly
reproduced the dependence on $\omega_{2}$ of the first term in (7.1). We have
not tried to reproduce the ${1/\omega_{2}}$ term using the formulas above. It
would be interesting to do it by analyzing the ${1/\tau}$ corrections as well
as to generalize the above analysis to higher loops.

Let us also comment on how contact terms $\delta(\omega_{2})$ originate from
the finite coupling formulas above in the perturbative expansion. Consider an
exchange by a heavy operator ($\tau\gg 1$), which could be for example a
stringy mode, in the dispersion representation (7.2) of the Mellin amplitude
$\tilde{M}(j-j_{2},j_{2})$. Expanding the right-hand side of (7.2) in
${1/\tau}$, we get polynomial in $j$ and $j_{2}$ corrections to the Mellin
amplitude. As we explained, these yield contact terms in the generalized event
shape $\text{EEC}(0,\omega_{2},z)$ localized at $\omega_{2}=0$. Let us now see
how this comes about directly from (259). In this case we can write the
following distributional identity

$\displaystyle\lim_{\tau\to\infty}(1+\omega_{2})^{-{\tau\over
2}}={2\over\tau}\delta(\omega_{2})+{4\over\tau^{2}}\Big{(}\delta(\omega_{2})+\delta^{\prime}(\omega_{2})\Big{)}+O(1/\tau^{3})\,.$
(269)

These are terms that will appear in the generalized event shapes in string
perturbation theory, with the ’t Hooft coupling $\lambda$ controlling the
twist of the stringy modes. On the other hand, by going back to the finite
coupling we get

$\displaystyle\lim_{\tau\to\infty}\lim_{\omega_{2}\to
0}(1+\omega_{2})^{-{\tau\over 2}}=1,$ (270)

so that the limits $\tau\to\infty$ and $\omega_{2}\to 0$ do not commute. To
summarize, the contact terms in the perturbative expansion of generalized
event shapes signal the presence of enhanced terms in the undeformed event
shapes. In the example above, $2\delta(\omega_{2})/\tau$ gets replaced with
$1$ after resummation of corrections in $1/\tau$.

###  7.3 Small $\omega$ and the Regge limit

Let us show that the leading behavior of the EEC as $\omega_{2}\to 0$ is
related to the behavior of the Mellin amplitude in the Regge limit. We use
(149), (152) and (205) to write down the expression for
$\text{EEC}(0,\omega_{2},z)$ in the Mellin space

$\displaystyle\text{EEC}(0,\omega_{2},z)=\left(1+\omega_{2}+\frac{\omega_{2}^{2}}{6}\right)\int{dj\over
2\pi i}{1\over 2}{\pi\over\sin\pi j}{z^{-j-2}\left(1-z\right)^{j-1}\over
j^{2}(j+1)^{2}}$ $\displaystyle\times\int{dj_{2}\over 2\pi
i}(1+\omega_{2})^{-j_{2}}\tilde{M}(j-j_{2},j_{2})\,\,.$ (271)

In order to understand the small $\omega_{2}$ expansion, the relevant integral
is thus

$\displaystyle\int_{-\delta-i\infty}^{-\delta+i\infty}dj_{2}\,(1+\omega_{2})^{-j_{2}}\tilde{M}(j-j_{2},j_{2})=\int_{0}^{\infty}dj_{2}(1+\omega_{2})^{-j_{2}}{\rm
Disc}_{j_{2}}\tilde{M}(j-j_{2},j_{2})\,,$ (272)

where we evaluated the integral by closing the contour to the right for
$\omega_{2}>0$ and picking the contribution from the poles in $j_{2}$, or,
equivalently, the discontinuity of the Mellin amplitude. Imagine that the
discontinuity of the Mellin amplitude has the following Regge behavior as
$j_{2}\to\infty$ 313131Since $\tilde{M}(j-j_{2},j_{2})$ is a meromorphic
function of $j_{2}$, this formula should be understood as a statement about
averaged discontinuity.

$\displaystyle{\rm Disc}_{j_{2}}\tilde{M}(j-j_{2},j_{2})\sim
j_{2}^{\alpha}\,.$ (273)

Via the integral (272) it will produce terms as singular as
${1/\omega_{2}^{1+\alpha}}$ in the Regge limit. At one loop we get ${\rm
Disc}_{j_{2}}\tilde{M}(j-j_{1},j_{2})\sim j_{2}$ as $j_{2}\to\infty$, which
via (272) produces ${1/\omega_{2}^{2}}$. It would be interesting to explore
the limit $\omega_{2}\to 0$ and correspondingly the Regge limit of the Mellin
amplitude at two and higher loops using the results of Bissi:2020woe .

We however did check the relevant scalings by looking at the discontinuity
${\rm Disc}_{s}M(s,t)$ of the corresponding flat space supergravity amplitude
$M(s,t)$, where $s$ and $t$ are the standard Mandelstam invariants in flat
space. At one loop the supergravity amplitude takes the following form in the
Regge limit, see for example formula (39) in Alday:2018kkw ,

$\displaystyle{\rm Disc}_{s}M^{1-\text{loop}}(s,t)\sim s\,.$ (274)

The leading term being analytic in $t$ corresponds to scattering at zero
impact parameters, which is consistent from the analysis above where it
emerged from intermediate operators of fixed spin and large twist. A more
familiar eikonal phase in ten dimensions behaves instead as
${t^{2}\log(-t)/s}$. It controls the leading Regge behavior of scattering at
non-zero impact parameters which manifests itself in the fact that it is not
analytic in $t$. We observe the same phenomenon for the expansion of ${\rm
Disc}_{j_{2}}\tilde{M}(j-j_{2},j_{2})$ in one loop supergravity. The term
${1/j_{2}}$ comes with the $j$-dependent pre-factor which is non-analytic. Via
(272) it will contribute to $\log\omega_{2}$ term in (7.1). As opposed to
${1/\omega_{2}^{2}}$ and ${1/\omega_{2}}$ terms, it is non-analytic in $z$
(due to the presence of $\log z$). This can be traced to non-analyticity in
$j$ of the ${1/j_{2}}$ term in the large $j_{2}$ expansion of ${\rm
Disc}_{j_{2}}\tilde{M}(j-j_{2},j_{2})$. This in turn can be traced back to the
universal high energy limit of the eikonal phase in gravity at non-zero impact
parameters, see e.g. Amati:1990xe . It would be interesting to make this
connection more explicit using the eikonal phase in the AdS analysis of
Cornalba:2007zb .

##  8 Ordinary event shapes at finite $\lambda$ and finite $c_{T}$

At weak coupling, given the perturbative expansion of the correlation function
one can systematically compute the perturbative corrections to the event
shapes. The situation is very different at strong coupling. In this case there
is no simple way of translating the perturbative expansion of the correlation
function to the perturbative expansion of the event shape. At the technical
level, this is related to the fact that both the stringy ${1/\lambda}$ and the
${1/c_{T}}$ gravitational loop corrections yield expressions for the
correlation functions that grow too fast in the Regge limit for the event
shape of interest to be well defined. In the collider physics setup studied in
this paper, the Regge limit divergence is translated to a large detector time
effect. It arises because the correlation function describes the energy
correlations with the property that the energy flux does not decay with time
and generates a divergence when the working time of the detector goes to
infinity.

The divergences are absent at finite $\lambda$ and finite $c_{T}$ where, based
on general arguments, we know that the Regge limit is such that, say, the
energy-energy correlation is well defined. Computing the corrections to the
event shapes thus requires resummation of the naive perturbative expansion.
Let us illustrate this phenomenon with a few examples.

###  8.1 Stringy corrections

The simplest example of this phenomenon arises when we consider stringy
corrections. The leading order correction to the Mellin amplitude takes the
form Goncalves:2014ffa ; Binder:2019jwn

$\displaystyle\tilde{M}(j_{1},j_{2})=\tilde{M}^{\text{strong}}(j_{1},j_{2})+{60\zeta(3)\over\lambda^{3/2}}+O(\lambda^{-5/2})\
\,.$ (275)

Substituting this relation into (7.3) and taking into account (4.5), we can
immediately write down the result for the generalized energy-energy
correlations with non-zero $\omega_{2}$,

$\displaystyle\text{EEC}(0,\omega_{2},z)$
$\displaystyle=\left(1+\omega_{2}+{\omega_{2}^{2}\over 6}\right)$
$\displaystyle\left[{1\over
2}\left(\theta(\omega_{2})+\theta(-\omega_{2}){(1+\omega_{2})^{2}\over(1+\omega_{2}z)^{3}}\right)+{120\zeta(3)\over\lambda^{3/2}}\delta(\omega_{2})(1-6z+6z^{2})+\dots\right]\,.$
(276)

On the other hand, the direct computation of the $\omega_{2}=0$ event shape in
string theory gives Hofman:2008ar

$\displaystyle\text{EEC}(z)\equiv\text{EEC}(0,0,z)={1\over
2}\left(1+{4\pi^{2}\over\lambda}(1-6z+6z^{2})+\dots\right)\,,$ (277)

where dots in this and the previous expression denote corrections suppressed
by powers of $\lambda^{-1/2}$.

Let us discuss how the three formulas above are related to each other. The
relation (277) can be understood by taking into account that the leading
correction to the Mellin amplitude (275) scales as
$\tilde{M}(j_{1},j_{2})|_{\lambda^{-3/2}}\sim 1$ in the Regge limit
$j_{2}\to\infty$. For this reason the integral $\int
dj_{2}\tilde{M}(j-j_{2},j_{2})$ that appears in the computation of the energy-
energy correlation (7.3) is divergent at $\omega_{2}=0$. The stringy effects
effectively cut the Mellin integral off at $j_{2}\sim\sqrt{\lambda}$ leading
to (277). The evaluation of the precise coefficient in (277) is a nontrivial
task, requiring the resummation of the string perturbative corrections. Yet it
can be done explicitly, see Goncalves:2014ffa  for details. In terms of the
dispersion relations the mechanism is precisely the same as described above in
Section 7.2, which also explains the dependence on $z$.

Comparing the relations (8.1) and (277), we observe that the leading
correction to both expressions is proportional to the same function of $z$ but
has different dependence on $\lambda$. The presence of the
$\delta(\omega_{2})$ term in (8.1) signifies that the corresponding detector
measures a time-independent energy flux. If we imagine a detector with a
finite working time $T$ this would produce an effect of order
${T/\lambda^{3/2}}$. Letting the working time $T\to\infty$ produces then a
divergent result. What happens at finite $\lambda$ however is that at times
$T_{\text{str}}\sim\lambda^{1/2}$ string perturbation theory breaks down and
the effect of the resummation is that the energy flux effectively becomes zero
for times $T>T_{\text{str}}$.

###  8.2 Gravitational loops

Let us now try to generalize this discussion to include gravitational loops.
It is natural to expect that the same mechanism is at play: the parts of the
one-loop result (7.1) that diverge in the $\omega_{2}\to 0$ limit get enhanced
by $\lambda$ or $c_{T}$ dependent factors coming from the long time effects in
the detector. The presence of such terms, which are enhanced compared to the
naive expectation coming from the perturbative expansion of the correlation
function, is a characteristic feature of strongly coupled conformal field
theories with semi-classical gravity duals.

Based on the OPE data of double trace operators and the mechanism discussed in
section 7.2, we associate the terms ${1/\omega_{2}^{2}}$ and ${1/\omega_{2}}$
with high energy scattering at zero impact parameters. In the flat space limit
this manifests itself through the fact that the corresponding terms in the
Regge limit of ${\rm Disc}_{s}M(s,t)$ are analytic in $t$. In string theory, a
fixed-angle scattering is very soft in the high energy limit Gross:1987ar . We
therefore expect that the divergences for $\omega_{2}\to 0$ get regularized in
the same way as in the example with string corrections above: namely the
$j_{2}$ integral is cut off at $j_{2}\sim\sqrt{\lambda}$. Equivalently, we
expect that the divergences to leading order get substituted by
${1/\omega_{2}}\to\sqrt{\lambda}$. From this argument we expect the leading
stringy correction to be323232The same term originates from the correction
$c_{\infty}\delta(\omega_{2})$ discussed in the previous section.

$\displaystyle\text{EEC}^{\text{QG}}(z)\propto{\lambda\over
c_{T}}(1-6z+6z^{2})+\dots\ \,.$ (278)

Computing the precise coefficient requires a more detailed analysis. The
correction (278) originates from two different sources:
$T{\sqrt{\lambda}/c_{T}}$ and $T^{2}/c_{T}$ which grow with time and both get
saturated at $T\sim T_{\text{str}}\sim\sqrt{\lambda}$.

We expect a different mechanism to be at play for the terms that diverge in
the $\omega_{i}\to 0$ limit which originate from scattering at non-zero impact
parameters. Such divergences will be present in the ${1/c_{T}}$ expansion of
the event shape even at finite string tension, or, equivalently, $\lambda$.
These terms get unitarized at finite $c_{T}$ via the eikonalization of the
gravitational loops. Because of this we also expect to have $c_{T}$-enhanced
terms in the expansion of the event shapes. Effects coming from non-zero
impact parameters originates from operators of arbitrary large spin, see e.g.
Cornalba:2006xk ; Cornalba:2006xm . Given that
$\widetilde{\text{EEC}}_{\tau,J}(z,0,\omega_{2})$ are polynomials in $z$ of
the maximal power $z^{J}$, any non-analyticity in $z$ can only come from the
large spin operators. As an example at one loop, the divergent and non-
analytic in $z$ piece $\sim{1\over c_{T}}\log|\omega_{2}|\log z$ should
originate from the expansion of the eikonal phase in AdS Cornalba:2007zb .
Indeed, the flat space limit of the part of the one-loop supergravity Mellin
amplitude that generates ${1\over c_{T}}\log|\omega_{2}|\log z$ term matches
the expected expansion of the eikonal phase DiVecchia:2019myk ;
DiVecchia:2019kta .

The $\log|\omega_{2}|$ divergence in (7.1) corresponds to the long-time tail
which grows as $\log T$ as a function of the detector working time $T$. We
thus see that at finite but large working times the leading contribution to
the energy correlation takes the form

$\displaystyle\text{EEC}^{\text{QG}}(z,T)\propto{\log T\over
c_{T}}\left(1-36z+216z^{2}-400z^{3}+225z^{4}\right)\log z\,.$ (279)

This correction is universal (it does not depend on the details of the
spectrum of the bulk theory) and originates from the fact that the dual theory
is gravitational, with gravity being the leading interaction at high energies
and large impact parameters. As a nontrivial consistency check, we verified
that (280) satisfies the Ward identities (249).

To recover the energy-energy correlation $\text{EEC}(z)$ we have to take the
limit $T\to\infty$. Doing it rigorously requires the resummation of the
$1/c_{T}$ perturbation theory. This might be possible using the eikonalization
of the gravitational amplitudes in AdS but we do not attempt it in the present
paper. Still, let us discuss the simplest possible scenario in which the
energy flux that grows as $\log T$ gets saturated at some time
$T_{\text{QG}}$. Here $T_{\text{QG}}$ is some characteristic time that goes to
infinity as $c_{T}\to\infty$. From the eikonalization of the gravitational
amplitudes we expect that $\log T_{\text{QG}}\sim\log c_{T}$. In this simple
scenario, we get that the leading quantum gravity correction to the energy-
energy correlation takes the form

$\displaystyle\text{EEC}^{\text{QG}}(z)$
$\displaystyle\stackrel{{\scriptstyle?}}{{\propto}}{\log c_{T}\over
c_{T}}\left(1-36z+216z^{2}-400z^{3}+225z^{4}\right)\log z\,.$ (280)

It would be very interesting to derive the leading quantum gravity correction
to the energy-energy correlation (280) rigorously.

##  9 Conclusions and future directions

In this paper we have introduced a new class of observables in a collider
physics setting which we dubbed generalized event shapes. They are defined as
the matrix elements of light-ray operators (2) with non-zero momentum, (7).
These light-ray operators have a finite resolution in time, given by
${1/\hat{\omega}}$, and thus allow us to probe the longitudinal structure of
the state. By setting $\hat{\omega}=0$ they smoothly transit over to the
familiar event shapes.

The generalized event shapes are not the standard differential cross sections,
since they are sensitive to the relative phases of the various scattering
amplitudes. This makes them harder to measure, but from the theoretical point
of view they are very natural observables. They are IR safe and can be
evaluated both in terms of scattering amplitudes and correlation functions. If
defined in terms of integrated Wightman functions, for $\hat{\omega}\neq 0$
the new observables retain all the information contained in the former. We
therefore expect that they form a convenient basis of collider-type
observables (at least in a CFT). Indeed, many popular jet observables can be
restated in terms of energy correlations, see e.g. Chen:2020vvp .

The generalized event shapes exhibit new features compared to the usual event
shapes. One interesting example is that they develop a bulk point singularity
at strong coupling. Approaching the singularity requires analytic continuation
of the physical parameters and in this way it is reminiscent of the flat space
singularity in the de Sitter correlators, see e.g. Arkani-Hamed:2015bza .
Another characteristic feature of the generalized event shapes is that they
are non-analytic around $\hat{\omega}=0$. This is related to the fact that,
depending on the sign of $\hat{\omega}$, the detector operators can either
create or annihilate particles. Nonperturbatively this fact manifests itself
in Eq. (1).

We studied the generalized event shapes perturbatively in planar ${\cal N}=4$
SYM both at weak and strong coupling. The results are summarized in Section 6.
First, we considered the case where the source, the sink and the detectors are
built out of local scalar operators. In this case, at weak coupling we
obtained the leading-order result using both scattering amplitudes and
correlation functions. At strong coupling we have only the correlation
functions approach at our disposal. Second, we used the ${\cal N}=4$
supersymmetry Ward identities to compute generalized event shapes with
detectors defined by conserved currents, namely charge and energy
correlations. In the special case where only one of the detectors has
$\hat{\omega}\neq 0$, in Section 5.3 we found remarkably simple relations
between the various observables. When both detectors carry non-zero momentum
we found that relatively simple relations exist not among the observables
themselves but among their Mellin kernels, see Eq. (194).

Unlike the ordinary event shapes, the generalized event shapes can be computed
perturbatively order by order in the ${1/c_{T}}$ and ${1/\lambda}$ at strong
coupling. A characteristic feature of the stringy and quantum-gravitational
corrections to the generalized event shapes is that they are divergent in the
limit $\hat{\omega}\to 0$. These divergences signify a longitudinal broadening
of the state and the presence of long-time tails in the measured radiation.
For ordinary event shapes, which perform measurements at arbitrarily late
times, this leads to corrections which are enhanced compared to those coming
from the perturbative expansion of the correlation function. We studied the
one-loop supergravity corrections to the generalized event shapes in Section
7. We also derived a dispersive representation of the generalized event shape
and used it to analyze the result. The structure of the ordinary event shapes
at finite $\lambda$ and finite $c_{T}$ is discussed in Section 8.

Let us list some interesting open questions that we did not address in the
present paper:

  * •

Probably the most important question is to understand whether the generalized
event shapes, or some of their cousins, can be measured in an experiment. We
commented on the special case of low energy elastic scattering, where this
should be possible indirectly. It would be interesting to find out if the same
or a similar method generalizes to higher energies.

  * •

It would be very interesting to study generalized event shapes in QCD. The
computation is analogous to the scattering amplitude one in Section 3. We
expect that the final result will be qualitatively similar to that in ${\cal
N}=4$ SYM, though the details will differ. This is what happens to the
ordinary event shapes and we expect this relation between ${\cal N}=4$ SYM and
QCD to hold for the generalized event shapes as well.

  * •

We motivated our study by thinking about energy fluxes in real time, but for
technical reasons it was more convenient to work with detectors that carry a
definite momentum, rather than detectors that work for a given period of time.
We demonstrated the transition between the two pictures on the simple example
of the one-point generalized event shapes in Appendix J. We observed the
effect of the longitudinal broadening at strong coupling which manifests
itself through the divergences in the $\hat{\omega}\to 0$ limit, or,
equivalently, the long-time tails of the radiation measured by the detectors.
It would be interesting to connect our results to the broadening picture
explored in Hatta:2012kn .

  * •

We emphasized that for $\hat{\omega}\neq 0$ we can render a broader class of
correlations well defined. A prominent example is the charge-charge
correlation which is not IR safe for $\hat{\omega}=0$ beyond one-loop order.
It would be interesting to understand the physics of the $\hat{\omega}\to 0$
limit of such observables and see if there is a way of constructing an
observable that stays finite in that limit.

  * •

We did not discuss the OPE of light-ray operators with non-zero momentum. It
would be useful to generalize the analysis of Kologlu:2019bco ;
Kologlu:2019mfz ; Chang:2020qpj  to this case. As we have seen, one obvious
difference is that the spin selection rule $J=J_{1}+J_{2}-1$ (here $J$ is the
spin of a light-ray operator that appears in the OPE of light-ray operators of
spin $J_{1}$ and $J_{2}$) does not hold anymore since $\hat{\omega}$ carries
non-zero spin. The light-ray OPE can also shed interesting light on the
$\hat{\omega}\to 0$ limit of the observables discussed in this paper.

  * •

We found that thinking about generalized event shapes is helpful for
understanding the stringy and quantum gravity corrections to the ordinary
event shapes. The basic phenomenon is that the ${1/\hat{\omega}}$ divergences
present in the generalized event shapes turn into $\lambda$ or $c_{T}$
enhanced terms in the ordinary event shapes. To establish this fact it is
convenient to use a dispersive representation of the event shapes (we did it
for the leading stringy corrections). For gravitational loop corrections that
unitarize the scattering amplitudes at large impact parameters, a more
thorough analysis should be possible.

  * •

The way how we introduced time dependence in the measurement at infinity is
not unique. Our choice of generalized event shapes was dictated by the
symmetries of the problem: given a source that carries a definite momentum, it
is natural to have detectors that carry definite momenta as well. Another
advantage of these detectors is that they naturally suppress the potential
divergences at large working times $u$, see Eq. (2.2). This makes the
corresponding matrix elements well defined, even if the original event shape
at $\hat{\omega}=0$ was not. Other bases of time-dependent light-ray operators
have been discussed in the literature, see for instance Besken:2020snx .

  * •

It is known that the light transform of the stress-energy tensor, also called
ANEC operator, is a non-negative operator in any unitary QFT Faulkner:2016mzt
; Hartman:2016lgu . If instead we restrict the time integration to some finite
interval we get what is known as the quantum null energy condition (QNEC)
Bousso:2015mna ; Bousso:2015wca ; Balakrishnan:2017bjg . It bounds the energy
flux measured over a time interval $T$ from below by the shape variations of
the entanglement entropy. It would be interesting to explore the implications
of QNEC to the generalized event shapes.

  * •

In this paper, we only studied one- and two-point correlations. It would be
interesting to extend the consideration to multi-point correlations, see e.g.
Chen:2019bpb ; Chen:2021gdk . The computation of the corrections to the lowest
order in the coupling within the amplitude approach would require
incorporating tree-level form factors with an arbitrary number of on-shell
particles. Results for such form factors have been obtained in $\mathcal{N}=4$
SYM Bork:2014eqa ; Bianchi:2018peu . In the correlation function approach one
would need expressions for the mutli-point correlation functions involving
conserved currents. Such correlation functions can be computed following the
approach developed in Refs. Chicherin:2015bza ; Chicherin:2014uca ;
Fleury:2019ydf .

  * •

By placing the detector operators with non-zero null momentum both at past and
future null infinity we can naturally define transition amplitudes in CFT. It
would be interesting to explore the properties of such objects using CFT
methods. It would be also very interesting to understand the connection
between such objects and the transitions studied in Korchemsky:2018hnb ;
Gillioz:2020mdd .

  * •

Generalized event shapes can be defined and studied in gravity (see
Gonzo:2020xza  for a recent discussion of the ordinary event shapes). Indeed,
they are IR safe observables and can be computed using the standard scattering
amplitudes techniques. Moreover, in the case of gravitational waves for
obvious reasons we cannot place the detectors around the collision point and
essentially all the information acquired in the observation comes from the
longitudinal, or time-dependent, details of the signal.

## Acknowledgments

We are grateful to Alex Belin, Andrei Belitsky, Agnese Bissi, Cyuan-Han Chang,
Dima Chicherin, Miguel Correia, Murat Kologlu, Petr Kravchuk, David Simmons-
Duffin, Gabriele Veneziano, Matthew Walters for useful discussions. AZ is
grateful to the participants of the SwissMap workshop “Advances in Quantum
Gravity” for stimulating discussions on related topics. 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 Conformal properties of the deformed light-ray operators

Here we recall some key points about the embedding formalism and index-free
notation used in Section 2. A more detailed introduction to it can be found,
e.g., in Costa:2011mg . We then apply the formalism to give an alternative
derivation of the conformal (21) and dilatation (2.1) transformations of the
deformed light-ray operators.

### Embedding formalism and index-free notation

The embedding formalism provides a redundant description of $d-$dimensional
conformal tensors by raising them to $(d+2)-$dimensional ones. For example, a
vector $V_{A}(X)$ with index $A=(+,-,\mu)$ has $d+2$ components instead of the
expected $d$ for a vector $V_{\mu}(x)$. The $(d+2)-$dimensional null vector
$X^{A}$ (with $X\cdot X\equiv X^{+}X^{-}+X^{\mu}X_{\mu}=0$) is also redundant,
having $d+1$ independent coordinates instead of $d$.

This redundancy is removed in three steps:

1) Require scale invariance,

$\displaystyle\Phi(\lambda X)=\lambda^{-\Delta}\Phi(X)$ (281)

with some weight which will later on be identified with the conformal weight.
Assuming that $X^{+}\neq 0$ this allows us to define the scale invariant field

$\displaystyle\phi(x)=(X^{+})^{\Delta}\Phi(X)\,,$ (282)

where the $d-$dimensional vector $x^{\mu}$ is identified with

$\displaystyle{X^{A}\over X^{+}}=\left(1,\ {X^{-}\over X^{+}}\equiv-x^{2},\
{X^{\mu}\over X^{+}}\equiv x^{\mu}\right)\ \Rightarrow\ X\cdot X=0\,.$ (283)

2) Impose transversality of the tensor fields, e.g. for the vector

$\displaystyle X\cdot V\equiv
X^{A}V_{A}(X)=\frac{1}{2}(X^{+}V^{-}+X^{-}V^{+})+X^{\mu}V_{\mu}=0\,.$ (284)

This condition removes one degree of freedom.

3) Consider the tensor field modulo gauge transformations, e.g. for the vector

$\displaystyle V_{A}(X)\ \to\ V_{A}(X)+X_{A}\,s(X)\,,$ (285)

where $s(X)$ is an arbitrary scalar field that scales as $s(\lambda
X)=\lambda^{-\Delta-1}s(X)$. This is the second redundant degree of freedom of
the vector field that needs to be removed. Clearly, the transversality
condition (284) is invariant under such gauge transformations.

The index-free notation consists in projecting the index $A$ of the vector
field with an auxiliary null vector $Z^{A}$ (with $Z\cdot Z=0$),

$\displaystyle V(X,Z)=Z^{A}V_{A}(X)\,.$ (286)

To respect the gauge invariance (285) the vectors $X$ and $Z$ are assumed
orthogonal,

$\displaystyle Z\cdot X=0\,.$ (287)

Then the transversality (284) takes the form of a gauge invariance condition
(recall (14)),

$\displaystyle V(X,Z+\beta X)=V(X,Z)\,.$ (288)

The auxiliary vector $Z^{A}$ undergoes independent scale transformations
reflecting the homogeneous polynomial dependence on $Z$ of the projected
tensor. In the example of the vector field (286) we obviously have $V(X,\rho
Z)=\rho V(X,Z)$. This scaling, together with (281), is generalized to the case
of arbitrary positive integer spin $J$, see (13), or even to non-integer and
negative spins by analytic continuation.

One can reduce the $(d+2)-$dimensional null vector $Z^{A}$, satisfying the
orthogonality condition (287), to a $d-$dimensional null vector $z^{\mu}$ by
exploiting the gauge invariance (288). Indeed, consider the linear combination
$Z^{A}+\beta X^{A}$ and choose $\beta=-Z^{+}/X^{+}$:

$\displaystyle Z^{A}+\beta X^{A}\ \stackrel{{\scriptstyle\beta=-{Z^{+}\over
X^{+}}}}{{=}}\ \left(0,\ Z^{-}-{Z^{+}\over X^{+}}X^{-}\equiv-2z\cdot x,\
Z^{\mu}-{Z^{+}\over X^{+}}X^{\mu}\equiv z^{\mu}\right)\Rightarrow\ z^{2}=0\,.$
(289)

The vector $z^{\mu}$ is traditionally used for describing symmetric traceless
tensors, see e.g. Dobrev:1977qv .

### Conformal transformations

The main point of the embedding formalism is that the $d-$dimensional
conformal group acts linearly on the $(d+2)-$dimensional vectors, e.g.
$X^{A}\to\Lambda^{A}{}_{B}X^{B}$ with $\Lambda\in SO(2,d)$. The benefit is
that one can easily build invariants of such linear transformations, see e.g.
Costa:2011mg . The price to pay is the redundancy discussed above and the
necessity to fix various gauges, if we want to recover the familiar Lorentz
tensors, e.g. the vector field $V(X,Z)\to V_{\mu}(x)$.

According to this concept, a $Z-$projected tensor field is invariant under
$SO(2,d)$,

$\displaystyle O(\Lambda X,\Lambda Z)=O(X,Z)\,,$ (290)

and in addition has the scaling properties (13).

The conformal group $SO(2,d)$ is extended to $O(2,d)$ by adding the discrete
operation of inversion, $X^{d+2}\to-X^{d+2}$ or equivalently,
$X^{+}\leftrightarrow-X^{-}$. In the fixed frame (283) we find

$\displaystyle x^{\mu}={X^{\mu}\over X^{+}}\ \stackrel{{\scriptstyle
I}}{{\rightarrow}}\ -{X^{\mu}\over X^{-}}=-{X^{+}\over X^{-}}{X^{\mu}\over
X^{+}}={x^{\mu}\over x^{2}}\,.$ (291)

The inversion of the vector $Z^{A}$ in the fixed frame (289) is accompanied by
a compensating gauge transformation (288) with parameter
$\beta=Z^{-}/X^{-}-Z^{+}/X^{+}$. This implies Dobrev:1977qv

$\displaystyle z^{\mu}\ \stackrel{{\scriptstyle I}}{{\rightarrow}}\
z^{\mu}-\frac{2z\cdot x}{x^{2}}x^{\mu}\,,\qquad(I[z^{\mu}])^{2}=0\,,\qquad
I^{2}[z^{\mu}]=z^{\mu}\,.$ (292)

#### Infinitesimal translations and conformal boosts

The conformal algebra is generated by $d-$dimensional translations $P^{+\mu}$
and conformal boosts $K^{-\mu}$. In the embedding space they are realized by
$SO(2,d)$ matrices of the block form

$\displaystyle\Lambda^{A}{}_{B}=\left(\begin{array}[]{c|c}\frac{1}{2}\delta^{\pm}{}_{\pm}&2b^{\pm}{}_{\nu}\\\
\hline\cr-b^{\mu}{}_{\pm}&\delta^{\mu}{}_{\nu}\end{array}\right)$ (295)

with infinitesimal parameters $b^{\pm}\equiv b_{\mp}$. We find

$\displaystyle\delta X^{\pm}=2b^{\pm\mu}X_{\mu}\,,\qquad\delta
X^{\mu}=-b^{-\mu}X^{+}-b^{+\mu}X^{-}\quad\Rightarrow\quad\delta(X\cdot
X)=0\,,$ (296)

so that

$\displaystyle\delta x^{\mu}$ $\displaystyle=\delta\left({X^{\mu}\over
X^{+}}\right)=-b^{-\mu}+b^{+\mu}\,x^{2}-2(b^{+}x)\,x^{\mu}\,,$ (297)

where $b^{-\mu}$ is the parameter of infinitesimal translations and $b^{+\mu}$
of conformal boosts.

The auxiliary null vector $Z^{A}$ transforms in the same way,

$\displaystyle\delta Z^{\pm}=2b^{\pm\mu}Z_{\mu}\,,\qquad\delta
Z^{\mu}=-b^{-\mu}Z^{+}-b^{+\mu}Z^{-}\quad\Rightarrow\quad\delta(Z\cdot
Z)=0\,.$ (298)

Applied to the fixed frame variable $z^{\mu}$ in (289) this transformation
becomes333333The transformation (299) can also be obtained from the inversion
(292) and the relation $K=IPI$.

$\displaystyle\delta z^{\mu}=\delta\left(Z^{\mu}-{Z^{+}\over
X^{+}}X^{\mu}\right)=2b^{+\mu}(xz)-2x^{\mu}(zb^{+})\quad\Rightarrow\
\delta(z^{2})=0\,.$ (299)

We see that $z^{\mu}$ transforms only under conformal boosts and $\delta
z^{\mu}|_{x=0}=0$.

#### Transformations of tensor fields

As we explained, the $Z-$projected fields $O(X,Z)$ are inert under conformal
transformations, see (290). To recover the familiar tensor field
transformations we have to ‘undress’ $O(X,Z)$ to an ordinary field with
Lorentz indices. Let us see how this works in the example of the vector field
(286). First, we use the gauge freedom (285) and (288) to fix the gauges

$\displaystyle V^{-}(X)=Z^{+}=0\,.$ (300)

A conformal boost with parameter $b^{+}$ should be accompanied by compensating
gauge transformations to maintain the gauges (300). Then we use the fact that
the projected vector field $V(X,Z)$ is invariant under the conformal
transformations to write down

$\displaystyle 0$ $\displaystyle=\delta V(X,Z)$
$\displaystyle=z^{\mu}\left(\delta V_{\mu}(x)+\delta
x^{\nu}\partial_{\nu}V_{\mu}(x)-2\Delta(b^{+}x)\,V_{\mu}(x)+2b^{+}_{\mu}x^{\nu}V_{\nu}(x)-2x_{\mu}b^{+\nu}V_{\nu}(x)\right)\,.$
(301)

Here $\delta x^{\nu}$ is the coordinate transformation (297) and the weight
factor comes from the definition (282) of the ordinary field and from (296).
We have also used the transformation (299) of $z^{\mu}$. Stripping off
$z^{\mu}$, we find the familiar conformal transformation of a vector field.
The generalization to tensors of arbitrary integer spin is straightforward.

### Transformations of the deformed light-ray operators

Comparing (18) and (14), we observe that the light-ray operators acquire a
phase under the gauge transformation. When we fix the frame (289), the gauge
parameter $\beta=-Z^{+}/X^{+}$ appears in the exponential factor

$\displaystyle{\bf L}_{\hat{\omega}}[O]\left(X,Z-{Z^{+}\over
X^{+}}X\right)=e^{i\hat{\omega}Z^{+}/X^{+}}{\bf L}_{\hat{\omega}}[O](X,Z)\,.$
(302)

According to (297) and (299) both arguments of
$O\left(x-{z\over\alpha},z\right)$ in (20) are inert under conformal boosts at
the origin $x=0$. However, the exponential factor in (302) transforms:

$\displaystyle\delta_{b^{+}}\left({Z^{+}\over X^{+}}\right)={2b^{+}\cdot
Z\over X^{+}}-{2b^{+}\cdot X\,Z^{+}\over(X^{+})^{2}}\ \ \underset{X^{+}=1,\
Z^{\mu}=z^{\mu}}{\overset{x^{\mu}=X^{\mu}/X^{+}=0}{\xrightarrow{\hskip
56.9055pt}}}\ \ 2(b^{+}z)\,.$ (303)

This leads to the relation (21) but its derivation is different from that in
(22). There we implicitly ‘undressed’ the tensors from the projection
variables $z^{\mu}$ and then applied the standard tensor transformation rules
like (A). Here we treated both $x$ and $z$ on the same footing and we saw that
the effect (21) originates from the compensating $\beta-$gauge transformations
needed to restore the special frame (289).

The dilatation transformation can be obtained from the commutator
$[P^{+\mu},K^{-\nu}]=\eta^{\mu\nu}D+L^{\mu\nu}$. Commuting two variations with
parameters $b^{+}$ and $b^{-}$, we find

$\displaystyle\delta_{D}x^{\mu}=\tau\,x^{\mu}\,,\quad\delta_{D}z^{\mu}=0\,,\quad\delta_{D}X^{+}=-\tau
X^{+}\,,\quad\delta_{D}Z^{+}=-\tau Z^{+}\,,$ (304)

where $\tau=-2b^{+}\cdot b^{-}$ is the infinitesimal dilatation parameter. The
variation of $x$ in the combination $x-z/\alpha$ on the right-hand side of
(20) is accompanied by a compensating transformation of the integration
parameter $\alpha$, $\delta_{D}\alpha=-\tau\,\alpha$. The change of variable
in the integral in (20) produces the weight $\lambda^{1-J}$ (with
$\lambda=1+\tau$) in (2.1). The extra weight $\lambda^{-\Delta}$ is absorbed
by the defining factor $(X^{+})^{-\Delta}$ in (12) (see also (282)), which is
implicit in (20). This reproduces (2.1), without setting $x=0$ in the
dilatation transformations.

##  Appendix B Reality property of generalized event shapes

In this appendix we derive the reality condition for the one-point generalized
event shapes (47). We have

$\displaystyle\langle{\cal O}(\hat{\omega},n)\rangle_{q}$
$\displaystyle={(-1)^{J}\over\eta_{{\cal
O}}}\langle\phi^{\dagger}(\tilde{q})\Big{(}(\mathsf{CRT}){\cal
O}(\hat{\omega},n)(\mathsf{CRT})^{-1}\Big{)}^{\dagger}\phi(q)\rangle$
$\displaystyle={(-1)^{J}\over\eta_{{\cal O}}}\langle(\mathsf{CRT}){\cal
O}(\hat{\omega},n)(\mathsf{CRT})^{-1}\phi(\tilde{q})|(\mathsf{CRT})(\mathsf{CRT})^{-1}\phi(q)\rangle$
$\displaystyle={(-1)^{J}\over\eta_{{\cal
O}}}\langle(\mathsf{CRT})^{-1}\phi(q)|{\cal
O}(\hat{\omega},n)(\mathsf{CRT})^{-1}\phi(\tilde{q})\rangle$
$\displaystyle={(-1)^{J}\over\eta_{{\cal
O}}}\langle\phi^{\dagger}(q)(\mathsf{CRT}){\cal
O}(\hat{\omega},n)(\mathsf{CRT})^{-1}\phi(\tilde{q})\rangle$
$\displaystyle=\langle\phi(q)^{\dagger}{\cal
O}(-\hat{\omega},n)\phi(\tilde{q})\rangle=\langle{\cal
O}(-\hat{\omega},n)\rangle_{q+\hat{\omega}_{1}n_{1}+\hat{\omega}_{2}n_{2}},$
(305)

Let us set, for example, $\Delta_{\phi}=2$, $d=4$. We then get for the
detectors of interest

$\displaystyle f_{\text{S}}(\omega)$
$\displaystyle=f_{\text{S}}\left(-{\omega\over\omega+1}\right)\ ,$
$\displaystyle f_{\text{Q}}(\omega)$
$\displaystyle=(1+\omega)f_{\text{Q}}\left(-{\omega\over\omega+1}\right)\ ,$
$\displaystyle f_{\text{E}}(\omega)$
$\displaystyle=(1+\omega)^{2}f_{\text{E}}\left(-{\omega\over\omega+1}\right)\
\,.$ (306)

The explicit expressions (58) and (59) satisfy (B) indeed.

For the one-point generalized event shapes (2.5) we have

$\displaystyle\langle{\cal O}_{1}(\hat{\omega}_{1},n){\cal
O}_{2}(\hat{\omega}_{2},n_{2})\rangle_{q}$
$\displaystyle={(-1)^{J_{1}+J_{2}}\over\eta_{{\cal O}_{1}}\eta_{{\cal
O}_{2}}}\langle\phi^{\dagger}(\tilde{q})\Big{(}(\mathsf{CRT}){\cal
O}_{1}(\hat{\omega}_{1},n_{1}){\cal
O}_{2}(\hat{\omega}_{2},n_{2})(\mathsf{CRT})^{-1}\Big{)}^{\dagger}\phi(q)\rangle$
$\displaystyle={(-1)^{J_{1}+J_{2}}\over\eta_{{\cal O}_{1}}\eta_{{\cal
O}_{2}}}\langle(\mathsf{CRT}){\cal O}_{1}(\hat{\omega}_{1},n_{1}){\cal
O}_{2}(\hat{\omega}_{2},n_{2})(\mathsf{CRT})^{-1}\phi(\tilde{q})|(\mathsf{CRT})(\mathsf{CRT})^{-1}\phi(q)\rangle$
$\displaystyle={(-1)^{J_{1}+J_{2}}\over\eta_{{\cal O}_{1}}\eta_{{\cal
O}_{2}}}\langle(\mathsf{CRT})^{-1}\phi(q)|{\cal
O}_{1}(\hat{\omega}_{1},n_{1}){\cal
O}_{2}(\hat{\omega}_{2},n_{2})(\mathsf{CRT})^{-1}\phi(\tilde{q})\rangle$
$\displaystyle={(-1)^{J_{1}+J_{2}}\over\eta_{{\cal O}_{1}}\eta_{{\cal
O}_{2}}}\langle\phi^{\dagger}(q)(\mathsf{CRT}){\cal
O}_{1}(\hat{\omega}_{1},n_{1}){\cal
O}_{2}(\hat{\omega}_{2},n_{2})(\mathsf{CRT})^{-1}\phi(\tilde{q})\rangle$
$\displaystyle=\langle\phi^{\dagger}(q){\cal
O}_{1}(-\hat{\omega}_{1},n_{1}){\cal
O}_{2}(-\hat{\omega}_{2},n_{2})\phi(\tilde{q})\rangle$
$\displaystyle=\langle{\cal O}_{1}(-\hat{\omega}_{1},n_{1}){\cal
O}_{2}(-\hat{\omega}_{2},n_{2})\rangle_{q+\hat{\omega}_{1}n_{1}+\hat{\omega}_{2}n_{2}}\
\,.$ (307)

Going from the second to the third line we used the anti-unitary property of
$\mathsf{CRT}$. Together with (50) this results in the relation (54).

##  Appendix C One-point function

In this appendix we consider the simplest correlation function involving a
single flow operator

$\displaystyle\langle{\phi(\tilde{q})\mathcal{O}_{J}(\hat{\omega},n)\phi(q)}\rangle=\int
d^{d}x_{1}e^{-i\tilde{q}x_{1}}\int
d^{d}x_{2}e^{-iqx_{2}}\langle{\phi(x_{1})\mathcal{O}_{J}(\hat{\omega},n)\phi(x_{2})}\rangle\,,$
(308)

where the scalar operators $\phi(x_{2})$ and $\phi(x_{1})$ define the source
and sink, respectively, and the flow operator is given by (31). Its
calculation will serve as an illustration of the general formalism for
computing more complicated observables from correlation functions.

For simplicity we work in $d=4$ dimensions, for arbitrary $d$ the formulae
become more complicated. For the lowest values of the spin $J=0,1,2$, the flow
operator in (308) is identified as the scalar, charge or energy detector,
respectively. After inserting the definition (31) of
$\mathcal{O}_{J}(\hat{\omega},n)$ in (308), we encounter a three-point
Wightman correlation function involving two scalar operators $\phi$ with
scaling dimension $\Delta$ and the spinning twist-two operator