$x_{1}$$x_{4}$$x_{3}$$x_{2}$$\mathcal{P}_{4}$$\mathcal{P}_{1}$$\mathcal{P}_{3}$$\mathcal{P}_{2}$
Figure 5: The external operators at coordinates $x_{i}$ in their respective
Poincaré patches $\mathcal{P}_{i}$. The black dotted lines are identified when
the Poincaré patches are wrapped on the boundary of the global AdS cylinder.

In the new $x_{i}$ coordinates, the Regge limit corresponds to placing the
four external points at the origin of their respective Poincaré patches,

$x_{1},x_{2},x_{3},x_{4}\to 0\,.$ (108)

However, $x_{5}$ to $x_{8}$ are integrated over in the CFT optical theorem.

Conformal correlators transform covariantly under the transformation (107). In
the scalar case we have

$A\left(y_{i}\right)=(-y_{1}^{+})^{-\Delta_{1}}(y_{2}^{+})^{-\Delta_{2}}(-y_{3}^{-})^{-\Delta_{3}}(y_{4}^{-})^{-\Delta_{4}}A\left(x_{i}\right)\,.$
(109)

In the spinning case, one must additionally account for the Jacobian matrix
$\partial y^{a}/\partial x^{m}$.555For external spinning operators, the
conformal transformations have a non-trivial rotation matrix $\partial
y^{a}/\partial x^{m}$. Conformal covariance of the correlators gives, in the
representative example of two vectors and two scalars Cornalba:2009ax ,
$A^{ab}\left(y_{i}\right)=\left(-y_{1}^{+}y_{2}^{+}\right)^{-1-\Delta_{V}}\left(-y_{3}^{-}y_{4}^{-}\right)^{-\Delta_{S}}\frac{\partial
y_{1}^{a}}{\partial x_{1}^{m}}\frac{\partial y_{2}^{b}}{\partial
x_{2}^{n}}A^{mn}\left(x_{i}\right)$. These matrices ensure that the inversion
tensors are correctly mapped from $y_{i}$ to $x_{i}$ variables, preserving
their form.

Next we use conformal symmetry to express the correlator in terms of two
vectors. This is similar to expressing the correlator in terms of two scalar
cross-ratios, with the difference that here we fix two, instead of the
customary three, positions using translations and special conformal
transformations to express the correlator in terms of the remaining two
position vectors. We can follow Cornalba:2009ax  and use a translation to send
$x_{1}$ to 0 which, due to the different transformations in (107) (see also
Kulaxizi_2018 ), will act as a special conformal transformation on the
Poincaré patches for $x_{3}$ and $x_{4}$,

$x_{1}\to 0\,,\quad x_{2}\to x_{2}-x_{1}\,,\quad
x_{3,4}\to\frac{x_{3,4}-x_{3,4}^{2}x_{1}}{1-2x_{3,4}\cdot
x_{1}+x_{3,4}^{2}x_{1}^{2}}\,.$ (110)

Next we implement a translation on the $x_{3}$ and $x_{4}$ patches (acting as
special conformal transformation on $x_{1,2}$) to also map $x_{4}$ to 0 in its
own patch and find that the correlator as a function of the Poincaré patch
coordinates $A(x_{i})$, as defined in (109), can always be expressed as

$A(x_{1},x_{2},x_{3},x_{4})\approx
A(0,-x,\overline{x}/\overline{x}^{2},0)\equiv A(x,\overline{x})\,,$ (111)

with

$x\approx x_{1}-x_{2}\,,\qquad\overline{x}\approx x_{3}-x_{4}\,,$ (112)

in the Regge limit (108).

It is further convenient to implement the coordinate change using embedding
space coordinates $P^{M}\in\mathbb{R}^{2,d}$

$P^{M}=\big{(}P^{+},P^{-},P^{m}\big{)}\,,\qquad P\cdot
P=-P^{+}P^{-}+\eta_{mn}P^{m}P^{n}\,.$ (113)

These are related to the coordinates $y^{m}\in\mathbb{R}^{1,d-1}$ of physical
Minkowski space by Cornalba:2009ax

$P^{M}=\big{(}y^{+},y^{-},1,y^{2},y_{\perp}\big{)}\quad\Rightarrow\quad
P_{ij}\equiv-2P_{i}\cdot P_{j}=(y_{i}-y_{j})^{2}\,,$ (114)

and to the coordinates $x_{i}$ by

$\displaystyle P_{1}^{M}$
$\displaystyle=-y_{1}^{+}\left(-1,-x_{1}^{2},x_{1}^{m}\right)\,,$
$\displaystyle P_{2}^{M}=y_{2}^{+}\left(-1,-x_{2}^{2},x_{2}^{m}\right)\,,$
(115) $\displaystyle P_{3}^{M}$
$\displaystyle=-y_{3}^{-}\left(-x_{3}^{2},-1,x_{3}^{m}\right),$ $\displaystyle
P_{4}^{M}=y_{4}^{-}\left(-x_{4}^{2},-1,x_{4}^{m}\right)\,.$

One can easily show that the cross-ratios (26) are given in terms of $x$ and
$\overline{x}$ as

$z\overline{z}=x^{2}\overline{x}^{2}\,,\qquad(1-z)(1-\overline{z})=1+x^{2}\overline{x}^{2}+2x\cdot\overline{x}\,,$
(116)

and the kinematic prefactor (25) becomes

$T^{1234}=\frac{(-y_{1}^{+})^{-\Delta_{1}}(y_{2}^{+})^{-\Delta_{2}}(-y_{3}^{-})^{-\Delta_{3}}(y_{4}^{-})^{-\Delta_{4}}}{x^{\Delta_{1}+\Delta_{2}}\overline{x}^{\Delta_{3}+\Delta_{4}}}\,.$
(117)

When combining (24) and (109), the numerator of the last expression cancels
the Jacobian prefactor in (109) to give,

$A\left(x_{i}\right)=\frac{A^{1234}(z,\overline{z})}{x^{\Delta_{1}+\Delta_{2}}\overline{x}^{\Delta_{3}+\Delta_{4}}}\,.$
(118)

If we now study the correlator $A^{3652}(x_{i})$, a priori we have to take
into account that only $x_{2}$ and $x_{3}$ are affected by the Regge limit.
However, we will assume that the integration will be dominated by the region
where the integration points are also boosted. Using the embedding space
coordinates

$\displaystyle
P_{5}^{M}=-y_{5}^{+}\left(-1,-x_{5}^{2},x_{5}^{m}\right)\,,\qquad P_{6}^{M}$
$\displaystyle=y_{6}^{-}\left(-x_{6}^{2},-1,x_{6}^{m}\right)\,,$ (119)

we find

$x^{\prime}\approx x_{5}-x_{2}\,,\qquad\overline{x}^{\prime}\approx
x_{3}-x_{6}\,.$ (120)

Where the primed variables are meant to emphasize that the points $1,4$ are
replaced by $5,6$ in this correlator as compared to (111). Performing these
steps for all the correlators in (103) we find,

$\displaystyle\text{dDisc}_{t}A_{\text{1-loop}}(x_{12},x_{34})=-\frac{1}{2}\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\int
dx_{5}dx_{6}\,\operatorname{Disc}_{23}A^{3652}_{\text{tree}}(x_{36},x_{52})$
(121)
$\displaystyle\mathbf{S}_{5}\mathbf{S}_{6}\operatorname{Disc}_{14}A^{1564}_{\text{tree}}(x_{15},x_{64})\Big{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,,$

where we stop explicitly mentioning that we are dealing with only the
contribution of double-trace operators as single-trace contributions are
subleading in the limit considered. Let us stress that in order to write the
correlators on the right hand side in terms of two differences, we assumed
that each of the individual tree-level correlators are in the Regge limit
themselves. The easiest way to justify this is in Fourier space using the
impact parameter transform defined below. Each tree-level position space
correlator is dominated by a power $\sigma^{1-j(\nu)}$ in the Regge limit,
which maps to a power of the AdS center of mass energy $S^{j(\nu)-1}$ in
impact parameter space. Since the optical theorem is multiplicative in impact
parameter space, subleading Regge trajectories or kinematical corrections from
the conformal block at finite boost get mapped to smaller powers of $S$, and
therefore do not contribute to the leading behavior. The eikonal approximation
in AdS Cornalba:2007zb  gives additional intuition for this, since it means
that even in AdS, the particles remain essentially undeflected, scattering
forward each time they exchange a Pomeron. Furthermore, we will show in
section 5 that this configuration reproduces the behavior at one-loop derived
in Meltzer:2019pyl .

###  4.2 Impact parameter space

Let us now consider the two-point functions
$\langle\widetilde{\mathcal{O}}_{5}\widetilde{\mathcal{O}}_{7}\rangle$ and
$\langle\widetilde{\mathcal{O}}_{8}\widetilde{\mathcal{O}}_{6}\rangle$ for the
shadow transforms in (121). In the Regge configuration, $x_{5}$ is the patch
of $x_{1}$, $x_{6}$ is the patch of $x_{4}$, $x_{7}$ is the patch of $x_{2}$
and $x_{8}$ is the patch of $x_{3}$. As explained in Cornalba:2007zb ;
Kravchuk:2018htv , the two-point function between two coordinates on adjacent
Poincaré patches has an additional phase factor $e^{i\pi\Delta}$ and this
phase can be accounted for by switching from the $i\epsilon$ prescription of a
Feynman propagator to that of a Wightman propagator (see Cornalba:2007zb ).
The normalization of the shadow transform in (10) and (11) is obtained from
the Fourier transform of a two-point function as the shadow transform acts
multiplicatively in Fourier space (see section 3.2 of Karateev:2018oml ). The
normalization in (11) is obtained from the Fourier transform of a Euclidean
two-point function, which matches the one of a Lorentzian two-point function
with Feynman $i\epsilon$ prescription. The Wightman propagator in momentum
space however has support only on the future lightcone and the coefficient of
the Fourier transform is different (see Appendix B of Cornalba:2007zb  and
section 2.1 of Gillioz:2018mto )

$\displaystyle\int dx\frac{e^{-2iq\cdot
x}}{\left[-(x^{0}-i\epsilon)^{2}+\vec{x}^{2}\right]^{\Delta}}=\mathcal{M}_{\mathcal{O}}\,\Theta(q^{0})\,\Theta(-q^{2})\left(-q^{2}\right)^{\Delta-\frac{d}{2}}\,,$
(122)

with

$\mathcal{M}_{\mathcal{O}}=\frac{2\pi^{\frac{d}{2}+1}}{\Gamma(\Delta)\Gamma\big{(}\Delta-\frac{d}{2}+1\big{)}}\,.$
(123)

Consequently we change the normalization $N_{\mathcal{O}}$ of the shadow
transform in (10) to (for scalar operators)

$\displaystyle\mathcal{N}_{\mathcal{O}}=\mathcal{M}_{\mathcal{O}}\mathcal{M}_{\widetilde{\mathcal{O}}}\,.$
(124)

Next we define, following Cornalba:2007fs , the impact parameter
representation as the Fourier transform of the discontinuity of the correlator
in the two remaining vectors666The AdS impact parameter representation was
previously defined in Cornalba:2007fs ; Cornalba:2006xm ; Cornalba:2008qf  as
the Fourier transform of the correlator on the second sheet
$A(z,\overline{z}^{\circlearrowleft})$. Since this contribution dominates in
the Regge limit, it is indistinguishable from the discontinuity of the
correlator in this limit. However, in the $t$-channel the two notions are
clearly different and it was necessary to take the discontinuity to derive
(5). In the next section we will see that the discontinuity is the better
choice also in the $s$-channel.

$\operatorname{Disc}_{14}A^{1jk4}(x_{1j},x_{k4})=\int
dp\,d\overline{p}\,e^{-2ip\cdot x_{1j}-2i\overline{p}\cdot
x_{k4}}B^{1jk4}(p,\overline{p})\,,$ (125)

where the function $B(p,\overline{p})$ has support only on the future Milne
wedge of $p$ and $\overline{p}$. Using (105) on (125) we get the Fourier
transform of $\operatorname{Disc}_{23}$.

$\displaystyle\operatorname{Disc}_{23}A^{3kj2}(x_{3k},x_{j2})=-\int
dp\,d\overline{p}\,e^{-2ip\cdot x_{3k}-2i\overline{p}\cdot
x_{j2}}B^{3kj2}(-p,-\overline{p})^{*}\,.$ (126)

The causal relations and thus the $i\epsilon$ prescription in (126) are
opposite to those in (125) and the complex conjugation prescribed in (105)
compensates for that. Inserting (125) into (121) and using that
$A^{1\tilde{5}\tilde{6}4}_{\text{tree}}$ is a double shadow transform of
$A^{1784}_{\text{tree}}$ we obtain, upon using (105) for the
$\operatorname{Disc}_{23}$,

$\displaystyle\text{dDisc}_{t}A_{\text{1-loop}}(x_{12},x_{34})=\frac{1}{2}\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\int
dx_{5}\,dx_{6}\,dx_{7}\,dx_{8}\;\int
dp\,d{\overline{p}}\,dp^{\prime}\,d\overline{p}^{\prime}$ (127)
$\displaystyle\qquad\times e^{-2i(p^{\prime}\cdot
x_{36}+\overline{p}^{\prime}\cdot x_{52}+p\cdot x_{17}+\overline{p}\cdot
x_{84})}B^{3652}_{\text{tree}}(-p^{\prime},-{\overline{p}}^{\prime})^{*}B^{1564}_{\text{tree}}(p,{\overline{p}})$
$\displaystyle\qquad\times\frac{T^{(\rho_{5})}(x_{75})}{\mathcal{N}_{\mathcal{O}_{5}}\left[-(x^{0}_{75}-i\epsilon)^{2}+\vec{x}_{75}^{2}\right]^{d-\Delta_{5}}}\frac{T^{(\rho_{6})}(x_{68})}{\mathcal{N}_{\mathcal{O}_{6}}\left[-(x^{0}_{68}-i\epsilon)^{2}+\vec{x}_{68}^{2}\right]^{d-\Delta_{6}}}\Bigg{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,.$

$T^{(\rho)}(x_{ij})$ is the tensor structure for the two-point function of an
operator with $SO(d)$ quantum number $\rho$. For example it is the familiar
inversion tensor $\eta^{\mu\nu}-2\frac{x^{\mu}x^{\nu}}{x^{2}}$ for spin 1
operators. Note that we have $B^{1564}_{\text{tree}}$ instead of
$B^{1784}_{\text{tree}}$, as the superscripts now only indicate the dimensions
of the corresponding operators and $\Delta_{5}=\Delta_{7}$,
$\Delta_{6}=\Delta_{8}$.

We can now express the two-point functions from the shadow transforms in
Fourier space by inverting (122)

$\displaystyle\frac{T^{(\rho)}(x)}{\left[-(x^{0}-i\epsilon)^{2}+\vec{x}^{2}\right]^{d-\Delta}}={}\frac{\mathcal{M}_{\widetilde{\mathcal{O}}}}{\pi^{d}}\int\limits_{M}dq\,e^{-2iq\cdot
x}\,\widehat{T}^{(\rho)}(q)\,(-q^{2})^{\frac{d}{2}-\Delta}\,.$ (128)

The Fourier space integral is over the future Milne wedge $M$ as the Fourier
transform has support only on this domain. $\widehat{T}^{(\rho)}(q)$ is the
tensor structure of the two-point function in Fourier space, which has been
discussed for example in Gillioz:2018mto . It is a tensor composed of
$q^{\mu}$ and $\eta^{\mu\nu}$ that can be factorized into a product of new
tensors $t^{(\rho)}(q)$ as follows

$\widehat{T}^{(\rho)}(q)^{\mu_{1}\ldots\mu_{|\rho|}}_{\nu_{1}\ldots\nu_{|\rho|}}=t^{(\rho)}(q)^{\mu_{1}\ldots\mu_{|\rho|}}_{\sigma_{1}\ldots\sigma_{|\rho|}}t^{(\rho)}(q)^{\sigma_{1}\ldots\sigma_{|\rho|}}_{\nu_{1}\ldots\nu_{|\rho|}}\,.$
(129)

Using (128) in (127) we end up with four position integrals over
$x_{5},x_{6},x_{7},x_{8}$ and six integrals over $q,\overline{q}$ (from the
four-point functions) and $p,\overline{p},p^{\prime},\overline{p}^{\prime}$.
The four position integrals give four Dirac delta functions with which we can
eliminate the $q,\overline{q},p^{\prime},\overline{p}^{\prime}$ integrals to
obtain

$\displaystyle\text{dDisc}_{t}A_{\text{1-loop}}(x_{12},x_{34})=$
$\displaystyle\frac{\pi^{2d}}{2}\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\frac{1}{\mathcal{M}_{\mathcal{O}_{5}}\mathcal{M}_{\mathcal{O}_{6}}}\int
dp\,d{\overline{p}}\,e^{-2i(p\cdot x_{12}+{\overline{p}}\cdot x_{34})}$ (130)
$\displaystyle\frac{B^{3652}_{\text{tree}}(-{\overline{p}},-p)^{*}\;\,\widehat{T}^{(\rho_{5})}(p)\,\widehat{T}^{(\rho_{6})}(\overline{p})\;\,B^{1564}_{\text{tree}}(p,{\overline{p}})}{(-p^{2})^{\Delta_{5}-\frac{d}{2}}(-\overline{p}^{2})^{\Delta_{6}-\frac{d}{2}}}\Bigg{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,,$

with an implicit index contraction between $B^{1564}_{\text{tree}}$ and
$B^{3652}_{\text{tree}}$ and the tensor structures $\widehat{T}^{(\rho)}$. At
this point we use (129) and absorb the $t^{(\rho)}(q)$ tensors into the
definition of the phase shifts $B_{\text{tree}}$. This means that one needs to
take it into account if one wants to relate tensor structures of CFT
correlators and phase shifts, but we will not need to do such a basis change
explicitly in this work. Using (106) we see that the double discontinuity
corresponds to the quantity $-\mathop{\mathrm{Re}}B(p,\overline{p})$ in impact
parameter space. Thus we find the following gluing formula for the impact
parameter representation, which is purely multiplicative,

$-\mathop{\mathrm{Re}}B_{\text{1-loop}}(p,\overline{p})=\frac{\pi^{2d}}{2}\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\frac{1}{\mathcal{M}_{\mathcal{O}_{5}}\mathcal{M}_{\mathcal{O}_{6}}}\,\frac{B^{3652}_{\text{tree}}(-{\overline{p}},-p)^{*}\,B^{1564}_{\text{tree}}(p,{\overline{p}})}{(-p^{2})^{\Delta_{5}-\frac{d}{2}}(-\overline{p}^{2})^{\Delta_{6}-\frac{d}{2}}}\Big{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,.$
(131)

Let us consider the case when $\mathcal{O}_{5}=\mathcal{O}_{1}$ and
$\mathcal{O}_{6}=\mathcal{O}_{3}$. In this case, it is useful to strip out a
scale factor similar to that in (118) from the impact parameter representation

$B^{jjkk}(p,\overline{p})=\frac{\mathcal{M}_{\mathcal{O}_{j}}\mathcal{M}_{\mathcal{O}_{k}}\;\mathcal{B}^{jjkk}(p,{\overline{p}})}{\left(-p^{2}\right)^{\frac{d}{2}-\Delta_{j}}\left(-\overline{p}^{2}\right)^{\frac{d}{2}-\Delta_{k}}}\,.$
(132)

Using (122) one sees that with this choice of normalization the impact
parameter representation of the MFT correlator is
$\mathcal{B}^{jjkk}_{\text{MFT}}=1$, which is necessary for the eikonalization
of the phase shift in AdS gravity. Inspired by this fact we choose the
normalization

$B^{ijkl}(p,\overline{p})=\frac{\sqrt{\mathcal{M}_{\mathcal{O}_{i}}\mathcal{M}_{\mathcal{O}_{j}}\mathcal{M}_{\mathcal{O}_{k}}\mathcal{M}_{\mathcal{O}_{l}}}\;\mathcal{B}^{ijkl}(p,{\overline{p}})}{\left(-p^{2}\right)^{\frac{d-\Delta_{i}-\Delta_{j}}{2}}\left(-\overline{p}^{2}\right)^{\frac{d-\Delta_{k}-\Delta_{l}}{2}}}\,,$
(133)

for the general case. This gives the following compact form for the optical
theorem in impact parameter space

$\displaystyle-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(p,\overline{p})=\frac{1}{2}\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\;\mathcal{B}^{3652}_{\text{tree}}(-{\overline{p}},-p)^{*}\;\mathcal{B}^{1564}_{\text{tree}}(p,{\overline{p}})\;\Bigg{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,.$
(134)

In Meltzer:2019pyl , the Regge limit of a one-loop four-point function of
scalars was studied in the large $\lambda$ regime with $S\gg\lambda\gg 1$. The
contribution of tidal excitations to the correlator is suppressed in this
regime. It corresponds to just one term in the sum on the right hand side of
(134) i.e. with $\mathcal{O}_{5}=\mathcal{O}_{1}$ and
$\mathcal{O}_{6}=\mathcal{O}_{3}$. We show in section 5.1 that this term from
our formula (134) reproduces the result from Meltzer:2019pyl  in the large
$\lambda$ or equivalently the large $\Delta_{\text{gap}}^{2}$ limit. We do not
need to discard any shadow double-trace contributions for this match. This
motivates us to assume that the only non-zero contributions to the gluing of
tree-level correlators in the Regge limit is from the physical double-traces
$[\mathcal{O}_{5}\mathcal{O}_{6}]$, and we will drop the explicit projections
henceforth. This is compatible with the intuition that there is no need to
project out shadow operators in a Lorentzian CFT optical theorem.

###  4.3 $s$-channel discontinuities in the Regge limit

Next we have to analyze the discontinuities on the right hand side of the
optical theorem (103). The discontinuity of the scalar $s$-channel block was
recently computed in general without taking the Regge limit in Caron-
Huot:2020nem  and we will review it here. The generalization to external
spinning operators is done in section 4.4, after taking the Regge limit. Let
us take the conformal partial wave expansion (29) in the $s$ channel and use
the symmetry of the integrand to extend the integration region at the cost of
a factor $1/2$

$\displaystyle A^{1234}(z,\overline{z})$
$\displaystyle=\frac{1}{2}\sum_{J}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta}{2\pi
i}\,I^{1234}(\Delta,J)\,\psi^{1234}_{\text{good},\mathcal{O}}(z,\overline{z})\,.$
(135)

Let $\psi^{1234}(z,\overline{z})$ be the partial wave $\Psi^{1234}(y_{i})$
with the prefactor $T^{1234}$ stripped off.
$\psi^{1234}_{\text{good},\mathcal{O}}(z,\overline{z})$ is the conformal
partial wave with an additional term that vanishes for integer spin but
ensures favorable properties for non-integer spin Caron-Huot:2020nem . The new
partial wave is given by

$\displaystyle\psi^{1234}_{\text{good},\mathcal{O}}(z,\overline{z})=\psi^{1234}_{\mathcal{O}}(z,\overline{z})+2\pi\,S(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}^{\dagger}])\,K_{J+d-1,1-\Delta}\,\xi_{\Delta,J}^{(a,b)}\,g^{1234}_{J+d-1,1-\Delta}(z,\overline{z})\,,$
(136)

where $g^{1234}_{\Delta,J}(z,\overline{z})$ is the usual conformal block, the
constants $a,b$ are defined below (60) and

$\displaystyle\xi_{\Delta,J}^{(a,b)}=$
$\displaystyle\left(s^{(a,b)}_{\Delta+J}-s^{(a,b)}_{\Delta+2-d-J}\right)\frac{\Gamma\big{(}-J-\tfrac{d-2}{2}\big{)}}{\Gamma(-J)}\,,$
(137) $\displaystyle s^{(a,b)}_{\beta}=$
$\displaystyle\,\frac{\sin\big{(}\pi(a+\beta/2)\big{)}\,\sin\big{(}\pi(b+\beta/2)\big{)}}{\sin(\pi\beta)}\,,$
$\displaystyle K_{\Delta,J}=$
$\displaystyle\,\frac{\Gamma(\Delta-1)}{\Gamma\big{(}\Delta-\frac{d}{2}\big{)}}\,\kappa^{(a,b)}_{\Delta+J}\,,$
$\displaystyle\kappa^{(a,b)}_{\beta}=$
$\displaystyle\,\frac{\Gamma\big{(}\frac{\beta}{2}-a\big{)}\Gamma\big{(}\frac{\beta}{2}+a\big{)}\Gamma\big{(}\frac{\beta}{2}-b\big{)}\Gamma\big{(}\frac{\beta}{2}+b\big{)}}{2\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}\,.$

With this conformal partial wave it is possible to compute the discontinuity
exactly Caron-Huot:2020nem

$\frac{\operatorname{Disc}_{14}\,\psi^{1234}_{\text{good},\mathcal{O}}(z,\overline{z})}{S(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}^{\dagger}])}=\frac{R^{1234}_{\mathcal{O}}(z,\overline{z})}{\pi
i\kappa_{\Delta,J}^{(a,b)}}\,.$ (138)

Here $R$ is the so-called Regge block

$\displaystyle R^{1234}_{\mathcal{O}}(z,\overline{z})$
$\displaystyle=g^{1234}_{1-J,1-\Delta}-\kappa^{\prime(a,b)}_{\Delta+J}g^{1234}_{\Delta,J}-\frac{\Gamma(d-\Delta-1)\Gamma\big{(}\Delta-\tfrac{d}{2}\big{)}}{\Gamma(\Delta-1)\Gamma\big{(}\tfrac{d}{2}-\Delta\big{)}}\,\kappa^{\prime(a,b)}_{d-\Delta+J}\
g^{1234}_{d-\Delta,J}+$ (139)
$\displaystyle\phantom{=}+\frac{\Gamma(J+d-2)\Gamma\big{(}-J-\tfrac{d-2}{2}\big{)}}{\Gamma\big{(}J+\tfrac{d-2}{2}\big{)}\Gamma(-J)}\,\kappa^{\prime(a,b)}_{\Delta+J}\
\kappa^{\prime(a,b)}_{d-\Delta+J}\ g^{1234}_{J+d-1,1-\Delta}\,,$

with $\kappa^{\prime(a,b)}_{\beta}$ defined as

$\displaystyle\kappa^{\prime(a,b)}_{\beta}=\frac{r^{(a,b)}_{\beta}}{r^{(a,b)}_{2-\beta}}\,,\qquad
r^{(a,b)}_{\beta}=\frac{\Gamma(\frac{\beta}{2}+a)\Gamma(\frac{\beta}{2}+b)}{\Gamma(\beta)}\,.$
(140)

$\operatorname{Disc}_{23}$ in the $3412$ OPE channel can be obtained by using
(104) on (138)

$\frac{\operatorname{Disc}_{23}\,\psi^{3412}_{\text{good},\mathcal{O}}(z,\overline{z})}{S(\mathcal{O}_{1}\mathcal{O}_{2}[\widetilde{\mathcal{O}}^{\dagger}])}=\frac{R^{3412}_{\mathcal{O}}(z,\overline{z})}{\pi
i\kappa_{\Delta,J}^{(-b,-a)}}\,,$ (141)

which was the reason to consider this channel for the correlators on the right
hand side of (103). The Regge block is dominated in the Regge limit by Caron-
Huot:2020nem ; Caron_Huot_2017 ; Cornalba:2007fs

$g^{1234}_{1-J,1-\Delta}(z,\overline{z})=\frac{4\pi^{\frac{d}{2}}\Gamma(\Delta-\frac{d}{2})}{\Gamma(\Delta-1)}\sigma^{1-J}\left(\Omega_{\Delta-\frac{d}{2}}(\rho)+O(\sigma)\right)\,.$
(142)

The $\sigma,\rho$ cross-ratios introduced here are defined as

$\sigma=\sqrt{z\overline{z}}=\sqrt{x^{2}\overline{x}^{2}}\,,\quad\cosh(\rho)=\frac{z+\overline{z}}{2\sqrt{z\overline{z}}}=-\frac{x\cdot\overline{x}}{\sqrt{x^{2}\overline{x}^{2}}}\,.$
(143)

$\Omega_{i\nu}(\rho)$ is the harmonic function on $d-1$ dimensional hyperbolic
space $H_{d-1}$ transverse to the scattering plane in $\text{AdS}_{d+1}$
Cornalba:2006xk

$\displaystyle\Omega_{i\nu}(\rho)={}$ $\displaystyle-\frac{i\nu\sin(\pi
i\nu)\Gamma(h-1+i\nu)\Gamma(h-1-i\nu)}{2^{2h-1}\pi^{h+\frac{1}{2}}\Gamma\big{(}h-\frac{1}{2}\big{)}}$
(144)
$\displaystyle{}_{2}F_{1}\left(h-1+i\nu,h-1-i\nu,h-\frac{1}{2},\frac{1-\cosh(\rho)}{2}\right)\,.$

Inserting everything into (135), we find the following expression for the
discontinuity of the correlator in the Regge limit

$\operatorname{Disc}_{14}A^{1234}(z,\overline{z})=2\pi
i\sum\limits_{J}\int\limits_{-\infty}^{\infty}d\nu\
\alpha(\nu,J)\,\sigma^{1-J}\Omega_{i\nu}(\rho)\,,$ (145)

with

$\alpha(\nu,J)=-\,\frac{\pi^{\frac{d}{2}-2}\,S\big{(}\mathcal{O}_{3}\mathcal{O}_{4}\big{[}\big{(}-i\nu-\frac{d}{2}\big{)}^{\dagger}\big{]}\big{)}\,\Gamma(i\nu)}{2\pi\kappa_{i\nu+\frac{d}{2},J}^{(a,b)}\,\Gamma\big{(}i\nu+\frac{d}{2}-1\big{)}}\,I^{1234}\\!\left(i\nu+\frac{d}{2},J\right).$
(146)

As in flat space the sum in (145) is dominated by the large $J$ contributions
in the Regge limit and only finite due to a conspiration of the coefficients
to ensure Regge boundedness. The next step is therefore to perform a
Sommerfeld-Watson resummation over $J$ to evaluate (145). Also, note that the
spectral function, as given by the Lorentzian inversion formula
Caron_Huot_2017 , is of the form

$\displaystyle I^{1234}(\nu)=I^{1234,t}(\nu)+(-1)^{J}I^{1234,u}(\nu)\,.$ (147)

Let us first consider the case of a correlator with pairwise equal external
operators i.e. $a=b=0$, $I^{1234,t}=I^{1234,u}$, where only even spins are
exchanged. Now for the resummation we replace the sum by an integral,

$2\sum\limits_{J\text{ even}}\to\int_{C}dJ\frac{e^{i\pi J}}{1-e^{i\pi J}}\,.$
(148)

The contour $C$ encloses all poles on the positive real axis (at even
integers) in a clockwise direction. The leading Regge trajectory is given by
the operators with the lowest dimension $\Delta(J)$ for every even spin $J$
and $\alpha(\nu,J)$ has poles at $i\nu=\pm(\Delta(J)-d/2)$. Defining the
inverse function $j=j(\nu)$ of the spectral function $\Delta(J)$ by

$\nu^{2}+\big{(}\Delta(j(\nu))-\tfrac{d}{2}\big{)}^{2}=0\,,$ (149)

we see that the poles in $\nu$ translate into a single pole at $J=j(\nu)$. By
deforming the $J$ contour to the left one sees that the $J$ integral is given
by the residue at $J=j(\nu)$, i.e.

$\operatorname{Disc}_{14}A^{1234}(z,\overline{z})=2\pi
i\int\limits_{-\infty}^{\infty}d\nu\
\alpha(\nu)\,\sigma^{1-j(\nu)}\Omega_{i\nu}(\rho)\,,$ (150)

where

$\alpha(\nu)=-\underset{J=j(\nu)}{\mathop{\mathrm{Res}}}i\,\frac{e^{i\pi
J}}{1-e^{-i\pi
J}}\frac{\pi^{\frac{d}{2}-2}\,S\big{(}\mathcal{O}_{3}\mathcal{O}_{4}\big{[}\big{(}-i\nu-\frac{d}{2}\big{)}^{\dagger}\big{]}\big{)}\,\Gamma(i\nu)}{\kappa_{i\nu+\frac{d}{2},J}^{(a,b)}\,\Gamma\\!\big{(}i\nu+\frac{d}{2}-1\big{)}}\,I^{1234,t}\left(i\nu+\frac{d}{2},J\right).$
(151)

When the operators are not pairwise equal, the even and odd spin operators
organize into two analytic families as evident from the Lorentzian inversion
formula Caron_Huot_2017 . To obtain the contribution of the leading Regge
trajectory we still sum over the even spin exchanges. The result is of the
same form as in (150) with $I^{1234,t}$ replaced by $\frac{1}{2}I^{1234}$ in
$\alpha(\nu)$. For the other correlator we can use (104) to see that we get an
analogous result with the complex conjugate spectral function

$\operatorname{Disc}_{23}A^{3412}(z,\overline{z})=2\pi
i\int\limits_{-\infty}^{\infty}d\nu\
\alpha(\nu)^{*}\,\sigma^{1-j(\nu)}\Omega_{i\nu}(\rho)\,.$ (152)

One can show that the corresponding impact parameter representation is given
in general by the same spectral function times a multiplicative factor which
cancels poles for the external double-trace operators Cornalba:2006xm

$\mathcal{B}(p,{\overline{p}})=2\pi i\int\limits_{-\infty}^{\infty}d\nu\
\beta(\nu)\,S^{j(\nu)-1}\Omega_{i\nu}(L)\,,$ (153)

where

$\beta(\nu)=\frac{4\pi^{2-d}(\sqrt{\mathcal{M}_{\mathcal{O}_{1}}\mathcal{M}_{\mathcal{O}_{2}}\mathcal{M}_{\mathcal{O}_{3}}\mathcal{M}_{\mathcal{O}_{4}}})^{-1}\;\alpha(\nu)}{\chi_{j(\nu)}(\nu)\,\chi_{j(\nu)}(-\nu)}\,,$
(154)

with the definition

$\chi_{j(\nu)}(\nu)=\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+j(\nu)-d/2+i\nu}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}+j(\nu)-d/2+i\nu}{2}\right).$
(155)

The impact parameter space cross-ratios, analogous to (143), are

$S=\sqrt{p^{2}{\overline{p}}^{2}},\qquad\cosh
L=-\frac{p\cdot\overline{p}}{\sqrt{p^{2}{\overline{p}}^{2}}}\,.$ (156)

In the dual AdS scattering process these cross ratios are interpreted as the
squared of the energy with respect to global time and as the impact parameter
in the transverse space $H_{d-1}$.

###  4.4 Spinning particles and the vertex function

In this section we will introduce concrete expressions for the tree amplitudes
with spinning external legs and show that the contributions of the contracted
spinning legs can be expressed in terms of a scalar function of three spectral
parameters which we call the vertex function, analogous to (87) in flat space.
We construct tensor structures in terms of differential operators, which are a
Regge limit version of weight-shifting operators that generate spinning
conformal blocks from the scalar ones Costa:2011dw ; Karateev:2017jgd . It is
convenient to work with tensor structures which are homogeneous in $p$ and
$\overline{p}$, i.e. independent of the cross-ratio $S$ in (156), such that
all tensor structures have the same large $S$ behavior in the Regge limit.
These differential operators can be constructed from the covariant derivative
on the hyperboloid $H_{d-1}$ and from
$\widehat{p}=p/|p|,\widehat{\overline{p}}=\overline{p}/|\overline{p}|$
Cornalba:2009ax ; Costa:2017twz . The possible differential operators that
generate spin for a single particle are

$\mathcal{D}^{\rho,k}_{\mathbf{m}}(p)=\widehat{p}_{m_{1}}\ldots\widehat{p}_{m_{k}}{\nabla_{p}}_{m_{k+1}}\ldots{\nabla_{p}}_{m_{|\rho|}}\,,\qquad
k=0,\ldots,|\rho|\,.$ (157)

Tree diagrams for exchange of the Pomeron then have the form

$\mathcal{B}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(p,\overline{p})=2\pi
i\int\limits_{-\infty}^{\infty}d\nu\,S^{j(\nu)-1}\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(\nu)\,\Omega_{i\nu}(L)\,.$
(158)

Here
$\mathcal{B}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(p,\overline{p})$
is defined just as in (125) and (133), but with tensor structures constructed
from $\widehat{p}$ and $\widehat{\overline{p}}$. In (158) we introduced the
following definition for the combination of spectral functions $\beta(\nu)$
and differential operators that generate different tensor structures

$\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(\nu)=\sum\limits_{k_{5}=0}^{|\rho_{5}|}\sum\limits_{k_{6}=0}^{|\rho_{6}|}\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)\mathcal{D}^{\rho_{5},k_{5}}_{\mathbf{m}}(p)\mathcal{D}^{\rho_{6},k_{6}}_{\mathbf{n}}(\overline{p})\,.$
(159)

Notice that, in contrast to flat space, we do not impose a full factorization
into three-point structures but rather allow for a separate spectral function
for each combination of three-point structures.

The next step is to derive the general functional form of (134) after the
contractions and sums have been done. We begin by inserting (158) into (134),

$\displaystyle-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(p,{\overline{p}})={}$
$\displaystyle
2\pi^{2}\sum\limits_{\Delta_{5},\Delta_{6},\rho_{5},\rho_{6}}\int\limits_{-\infty}^{\infty}d\nu_{1}d\nu_{2}\,S^{j(\nu_{1})+j(\nu_{2})-2}$
(160)
$\displaystyle\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(\nu_{1})^{*}\,\Omega_{i\nu_{1}}(L)\,\pi_{\rho_{5}}^{\mathbf{m};\mathbf{p}}\pi_{\rho_{6}}^{\mathbf{n};\mathbf{q}}\,\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{p}\mathbf{q}}(\nu_{2})\,\Omega_{i\nu_{2}}(L)\,.$

Here $\pi_{\rho}$ is the projector to the irreducible representation $\rho$ of
$SO(d)$, which is necessary because the operators (157) do not ensure the
properties of irreducible representations such as tracelessness and Young
symmetrization. Next we will show how one can replace the contractions and
derivatives in the previous equation by spectral parameters. Note first that
due to $p\cdot\nabla_{p}=0$, all contractions involving $\widehat{p}$ or
$\widehat{{\overline{p}}}$ give factors of their norm $-1$. The remaining
contractions involve only covariant derivatives. These contracted derivatives
can all be replaced by functions of the spectral parameters by using the
Laplace equation for the harmonic function

$\left(\nabla_{H_{d-1}}^{2}+\nu^{2}+(d/2-1)^{2}\right)\Omega_{i\nu}(L)=0\,.$
(161)

Using this equation, factors of $\nabla_{p}^{2}$ can directly be replaced. To
evaluate contractions between derivatives acting on different harmonic
functions we expand the product of two scalar harmonic functions as follows,

$\Omega_{i\nu_{1}}(L)\Omega_{i\nu_{2}}(L)=\int\limits_{-\infty}^{\infty}d\nu\,\Phi(\nu_{1},\nu_{2},\nu)\Omega_{i\nu}(L)\,,$
(162)

where $\Phi(\nu_{1},\nu_{2},\nu)$ was computed (for the similar case of
harmonic functions on AdSd+1) in appendix D of Penedones:2010ue .777Note that
$\Phi_{\text{here}}\Omega_{i\nu}(0)=\Phi_{\text{there}}$. By acting repeatedly
with (161) on this equation, one can determine the function $W_{k}$ that
appears in

${\nabla_{p}}_{m_{1}}\ldots{\nabla_{p}}_{m_{k}}\Omega_{i\nu_{1}}(L)\nabla_{p}^{m_{1}}\ldots\nabla_{p}^{m_{k}}\Omega_{i\nu_{2}}(L)=\int\limits_{-\infty}^{\infty}d\nu\,W_{k}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}\,\Phi(\nu_{1},\nu_{2},\nu)\,\Omega_{i\nu}(L)\,.$
(163)

$W_{k}$ is a fixed kinematical polynomial of maximal degree $k$ in its
arguments. For example, the first non-trivial case is

$\int\limits_{-\infty}^{\infty}\\!\\!d\nu\,\Phi(\nu_{1},\nu_{2},\nu)\nu^{2}\,\Omega_{i\nu}(L)=\left(\nu_{1}^{2}+\nu_{2}^{2}+(\tfrac{d}{2}-1)^{2}\right)\Omega_{i\nu_{1}}(L)\,\Omega_{i\nu_{2}}(L)-2\nabla_{\mu}\Omega_{i\nu_{1}}(L)\nabla^{\mu}\Omega_{i\nu_{2}}(L),$
(164)

from which one can read off $W_{0}$ and $W_{1}$ to be

$W_{0}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}=1\,,\qquad
W_{1}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}=\frac{1}{2}\Big{(}\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}+(d/2-1)^{2}\Big{)}.$
(165)

More generally, by acting with the Laplacian on both sides of (163) one can
derive a recursion relation of the form

$\displaystyle\int
d\nu\,W_{k+1}(\nu_{i})\,\Phi(\nu_{i})\,\Omega_{i\nu}(L)=\int
d\nu\,W_{k}(\nu_{i})\,W_{1}(\nu_{i})\,\Phi(\nu_{i})\,\Omega_{i\nu}(L)$ (166)
$\displaystyle+\frac{1}{2}\left([\nabla^{2},\nabla_{m_{1}}\dots\nabla_{m_{k}}]\Omega_{i\nu_{1}}(L)\nabla^{m_{1}}\dots\nabla^{m_{k}}\Omega_{i\nu_{2}}(L)+(\nu_{1}\leftrightarrow\nu_{2})\right)\,.$

The terms with commutators, which will vanish in the flat space limit, can be
evaluated using the fact that the commutators of covariant derivatives can be
replaced by Riemann tensors, which for the hyperboloid can be written in terms
of the metric. This means that these terms have two derivatives less than the
other terms, and will therefore produce less than maximal powers of $\nu_{i}$.
This shows that the maximal power of $\nu_{i}$ in $W_{k}$ is just given by
repeatedly multiplying $W_{1}$. Therefore we have

$W_{k}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}=\left(\frac{\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}}{2}\right)^{k}+O\\!\left(\nu_{i}^{2(k-1)}\right).$
(167)

Having shown that all derivatives can be replaced by polynomials of the
spectral parameters, we can define

$\displaystyle\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(\nu_{1})^{*}\,\Omega_{i\nu_{1}}(L)\,\pi_{\rho_{5}}^{\mathbf{m};\mathbf{p}}\pi_{\rho_{6}}^{\mathbf{n};\mathbf{q}}\,\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{p}\mathbf{q}}(\nu_{2})\,\Omega_{i\nu_{2}}(L)$
(168)
$\displaystyle=\int\limits_{-\infty}^{\infty}d\nu\,W_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}\,\Phi(\nu_{1},\nu_{2},\nu)\,\Omega_{i\nu}(L)\,.$

This gives the contribution of a given pair of intermediate states labeled by
$(\Delta_{5},\rho_{5})$ and $(\Delta_{6},\rho_{6})$ to
$-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(p,{\overline{p}})$. Now we
can define the vertex function $V(\nu_{1},\nu_{2},\nu)$, which is even in all
its arguments, in analogy to (87) as the sum over all such contributions in
(160)

$\displaystyle\sum\limits_{\Delta_{5},\Delta_{6},\rho_{5},\rho_{6},}W_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}=\beta(\nu_{1})^{*}\beta(\nu_{2})V(\nu_{1},\nu_{2},\nu)^{2}\,,$
(169)

and reach the following representation for the 1-loop amplitude

$\displaystyle-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(p,{\overline{p}})=2\pi^{2}\int\limits_{-\infty}^{\infty}d\nu
d\nu_{1}d\nu_{2}\,\beta(\nu_{1})^{*}\beta(\nu_{2})\,V(\nu_{1},\nu_{2},\nu)^{2}$
(170) $\displaystyle
S^{j(\nu_{1})+j(\nu_{2})-2}\Phi(\nu_{1},\nu_{2},\nu)\,\Omega_{i\nu}(L)\,.$

All the information about the spinning tree-level correlators and their
contractions is encoded in the vertex function $V(\nu_{1},\nu_{2},\nu)$ which
mirrors the role of its flat space analogue.

However, in order to compute the full impact parameter representation rather
than just its real part, we have to go through a detour via the Lorentzian
inversion formula, as described in Meltzer:2019pyl . We first Fourier
transform back to $\text{dDisc}_{t}\,A_{\text{1-loop}}$ from which we obtain
the $s$-channel OPE coefficients. Then we can compute
$\operatorname{Disc}_{14}A_{\text{1-loop}}$ which we can finally Fourier
transform to obtain $\mathcal{B}_{\text{1-loop}}(p,{\overline{p}})$. Since in
the Regge limit the difference between $\text{dDisc}_{t}\,A_{\text{1-loop}}$
and $\operatorname{Disc}_{14}A_{\text{1-loop}}$ is just a phase factor (see
Meltzer:2019pyl ), the same happens for the impact parameter representation

$\displaystyle\mathcal{B}_{\text{1-loop}}(p,{\overline{p}})=-4\pi^{2}\int\limits_{-\infty}^{\infty}d\nu
d\nu_{1}d\nu_{2}\,$
$\displaystyle\frac{1+e^{-i\pi(j(\nu_{1})+j(\nu_{2})-1)}}{1-e^{-2\pi
i(j(\nu_{1})+j(\nu_{2})-1)}}\,\beta(\nu_{1})^{*}\beta(\nu_{2})\,V(\nu_{1},\nu_{2},\nu)^{2}$
$\displaystyle
S^{j(\nu_{1})+j(\nu_{2})-2}\;\Phi(\nu_{1},\nu_{2},\nu)\,\Omega_{i\nu}(L)\,.$
(171)

It is important to emphasize that this provides a finite $\Delta_{gap}$
description for the one-loop correlator in the Regge limit up to the knowledge
of the vertex function $V(\nu_{1},\nu_{2},\nu)^{2}$. For CFTs that admit a
flat space limit, we will see in sections 6 and 7 how one can fix part of this
vertex function from the knowledge of its flat space analogue. In section 5
below, we make a comparison with the large $\Delta_{gap}$ limit studied in
reference Meltzer:2019pyl , and also describe the implications of (170) for
t-channel CFT data.

##  5 Constraints on CFT data

###  5.1 Comparison with the large $\Delta_{\text{gap}}$ limit

In Meltzer:2019pyl  the Regge limit of the four-point correlator of pairwise
identical scalars was studied in an expansion in $1/N$ in the limit of large
$\Delta_{\text{gap}}$. The specific limit considered was
$S\gg\Delta_{\text{gap}}^{2}\gg 1$, so the result is sensitive to all the
higher spin interactions in the leading Regge trajectory, but tidal
excitations are ignored. Since we have kept $\Delta_{\text{gap}}$ finite, we
should be able to obtain a match between the result for the one-loop
correlator in Meltzer:2019pyl  with our result (134) after dropping the tidal
excitations $\mathcal{O}_{5}\neq\mathcal{O}_{1}$ and
$\mathcal{O}_{6}\neq\mathcal{O}_{3}$.

We pick the term $\Delta_{5}=\Delta_{1}$ and $\Delta_{6}=\Delta_{3}$ that is
the sole contribution to (134) in the large $\Delta_{\text{gap}}$ limit, and
use (153) to obtain

$\displaystyle-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(S,L)=2\pi^{2}\int
d\nu_{1}d\nu_{2}d\nu\,\beta^{*}(\nu_{1})\beta(\nu_{2})\,\Phi(\nu_{1},\nu_{2},\nu)\,S^{j(\nu_{1})+j(\nu_{2})-2}\Omega_{i\nu}(L)\,\Bigg{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,.$
(172)

Now let us extract the corresponding result from Meltzer:2019pyl . Equation
(3.15) of Meltzer:2019pyl  gives the double discontinuity of the one-loop
correlator $\mathcal{G}^{(2)}$ as follows (with $\Delta_{\phi}=\Delta_{1}$ and
$\Delta_{\psi}=\Delta_{3}$)

$\displaystyle\text{dDisc}_{t}[\mathcal{G}^{(2)}(z,\overline{z})]=$
$\displaystyle\frac{\pi^{4}}{8}\int
d\nu_{1}d\nu_{2}d\nu\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(\nu)\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(-\nu)\,\mathcal{N}$
(173)
$\displaystyle\widehat{\gamma}^{(1)}(\nu_{1})\widehat{\gamma}^{(1)}(\nu_{2})\,\Phi(\nu_{1},\nu_{2},\nu)(z\overline{z})^{\frac{2-j(\nu_{1})-j(\nu_{2})}{2}}\Omega_{i\nu}\left(\frac{1}{2}\log(z/\overline{z})\right)\,,$

where $\chi_{j(\nu_{1})+j(\nu_{2})-1}(\nu)$ is defined in (155) but with
$j(\nu)$ replaced by $j(\nu_{1})+j(\nu_{2})-1$, and with
$\Delta_{2}=\Delta_{1}$ and $\Delta_{4}=\Delta_{3}$. It accounts for the
double-trace exchanges $[\mathcal{O}_{1}\mathcal{O}_{1}]$ and
$[\mathcal{O}_{3}\mathcal{O}_{3}]$ analytically continued to spin
$j(\nu_{1})+j(\nu_{2})-1$. The operators contributing to the $t$-channel
expansion in the large $\Delta_{\text{gap}}$ limit are the double-traces
$[\mathcal{O}_{1}\mathcal{O}_{3}]_{n,\ell}$ with dimensions and OPE
coefficients given by

$\displaystyle\Delta_{h,\overline{h}}$
$\displaystyle=\Delta^{(0)}_{h,\overline{h}}+\frac{1}{N^{2}}\,\gamma^{(1)}_{h,\overline{h}}+\frac{1}{N^{4}}\,\gamma^{(2)}_{h,\overline{h}}+\cdots\,,\qquad\Delta^{(0)}_{h,\overline{h}}$
$\displaystyle=\Delta_{1}+\Delta_{3}+2n+\ell\,,$ (174) $\displaystyle
P_{h,\overline{h}}$
$\displaystyle=P^{MFT}_{h,\overline{h}}\left(1+\frac{1}{N^{2}}\,\delta
P^{(1)}_{h,\overline{h}}+\frac{1}{N^{4}}\,\delta
P^{(2)}_{h,\overline{h}}+\cdots\right)\,,$

where $h,\overline{h}=\Delta\mp\ell$. The tree-level anomalous dimensions
$\gamma^{(1)}_{h,\overline{h}}$ and tree-level corrections to OPE coefficients
$\delta P^{(1)}_{h,\overline{h}}$ can be extracted respectively from
$\widehat{\gamma}^{(1)}(\nu)$ and $\widehat{\delta P}^{(1)}(\nu)$ by

$\displaystyle\gamma^{(1)}_{h,\overline{h}}$
$\displaystyle\approx\int\limits_{-\infty}^{\infty}d\nu\,\widehat{\gamma}^{(1)}(\nu)\,(h\overline{h})^{j(\nu)-1}\,\Omega_{i\nu}\big{(}\log(h/\overline{h})\big{)}\,,$
(175) $\displaystyle\delta P^{(1)}_{h,\overline{h}}$
$\displaystyle\approx\int\limits_{-\infty}^{\infty}d\nu\,\widehat{\delta
P}^{(1)}(\nu)\,(h\overline{h})^{j(\nu)-1}\,\Omega_{i\nu}\big{(}\log(h/\overline{h})\big{)}\,.$

$\widehat{\gamma}^{(1)}(\nu)$ and $\widehat{\delta P}^{(1)}(\nu)$ can be
obtained respectively from the real and imaginary parts of the phase shift,
and are related to $\beta$ by888Note that due to difference in conventions,
$\mathcal{N}\beta$ for us is equal to $\beta$, as defined in Meltzer:2019pyl .

$\displaystyle\widehat{\gamma}^{(1)}(\nu)=$ $\displaystyle\
2\,\text{Re}\beta(\nu)\,,$ (176) $\displaystyle\widehat{\delta P}^{(1)}(\nu)=$
$\displaystyle-2\pi\,\text{Im}\beta(\nu)\,.$

Taking the Fourier transform to impact parameter space on (173) and then using
(176) gives999In our conventions the Fourier transform takes
$\text{dDisc}_{t}[\mathcal{G}^{(2)}(z,\overline{z})]$ to
$-\mathcal{N}\mathop{\mathrm{Re}}\mathcal{B}$ upto scaling factors.

$\displaystyle-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(S,L)=2\pi^{2}\int
d\nu_{1}d\nu_{2}d\nu\,\text{Re}\beta(\nu_{1})\,\text{Re}\beta(\nu_{2})\,\Phi(\nu_{1},\nu_{2},\nu)S^{j(\nu_{1})+j(\nu_{2})-2}\,\Omega_{i\nu}(L)\,.$
(177)

We need to compare (172) with (177). The only difference are the real parts in
(177), however $\text{Im}\beta$ in (172) is related to tree-level corrections
to the OPE coefficients and these are suppressed at large
$\Delta_{\text{gap}}$ Meltzer:2019pyl . This can be seen for example from
(176) and using in it the explicit expression for $\alpha(\nu)$ from
Meltzer:2019pyl . The result is

$\widehat{\delta
P}^{(1)}(\nu)=\frac{-\pi\mathop{\mathrm{Im}}\left(\frac{ie^{i\pi
j(\nu)}}{1-e^{i\pi j(\nu)}}\right)}{\mathop{\mathrm{Re}}\left(\frac{ie^{i\pi
j(\nu)}}{1-e^{i\pi
j(\nu)}}\right)}\,\widehat{\gamma}^{(1)}(\nu)=-\pi\tan\Big{(}\frac{\pi}{2}j(\nu)\Big{)}\,\widehat{\gamma}^{(1)}(\nu)\,.$
(178)

The suppression is due to the tan factor, since for large $N$ theories it is
known that Brower:2006ea ; Cornalba:2007fs ; Costa:2012cb

$\displaystyle
j(\nu)=2-2\,\frac{\nu^{2}+(d/2)^{2}}{\Delta_{\text{gap}}^{2}}+O\big{(}\Delta_{\text{gap}}^{-4}\big{)}\,.$
(179)

The anomalous dimensions $\gamma^{(1)}_{h,\overline{h}}$ are order 1, while
the the corrections to the OPE coefficients $\delta P^{(1)}_{h,\overline{h}}$
are at order $\Delta_{\text{gap}}^{-2}$.

Thus we have matched our result for the Regge limit of the dDisc of a one-loop
correlator at large $\Delta_{\text{gap}}$ to that of Meltzer:2019pyl . Note
that we managed to reproduce the result without the need for any projections
to the physical double-traces. Therefore it is reasonable to assume that the
gluing of tree-level correlators in the Regge limit does not receive
contributions from the double-traces of shadows and we can use the optical
theorem (134) without the projections onto $[\mathcal{O}_{5}\mathcal{O}_{6}]$.

###  5.2 Extracting t-channel CFT data

Next we shall see how we can extract the CFT data for the double-trace
operators exchanged in the $t$-channel to order $1/N^{4}$ from the vertex
function $V(\nu_{1}^{2},\nu_{2}^{2},\nu^{2})$. To this end we follow section
3.2 of Meltzer:2019pyl  and extend the results therein by including tidal
excitations, which make our statements valid at finite $\Delta_{\text{gap}}$.
As discussed in the previous section, the only operators contributing to the
$t$-channel expansion in the large $\Delta_{\text{gap}}$ limit are the double-
traces $[\mathcal{O}_{1}\mathcal{O}_{3}]$ Li:2017lmh . The three-point
function of these double-traces with their constituent operators
$\mathcal{O}_{1}$ and $\mathcal{O}_{3}$ has the large $N$ behavior

$\langle\mathcal{O}_{1}\mathcal{O}_{3}[\mathcal{O}_{1}\mathcal{O}_{3}]\rangle\sim
1\,.$ (180)

By including tidal excitations we have to include also double-traces
$[\mathcal{O}_{5}\mathcal{O}_{6}]$ corresponding to additional double-traces
coupling to $\mathcal{O}_{1}$ and $\mathcal{O}_{3}$. These satisfy

$\langle\mathcal{O}_{1}\mathcal{O}_{3}[\mathcal{O}_{5}\mathcal{O}_{6}]\rangle\sim\frac{1}{N^{2}}\,,\qquad[\mathcal{O}_{5}\mathcal{O}_{6}]\neq[\mathcal{O}_{1}\mathcal{O}_{3}]\,,$
(181)

so that only their classical dimension and leading OPE coefficient
squared101010We are free to insert $P^{\text{MFT}}$ here, defining $\delta P$
accordingly. This will be useful below in (189).

$\Delta_{h^{\prime},\overline{h}^{\prime}}=\Delta_{\mathcal{O}_{5}}+\Delta_{\mathcal{O}_{6}}+2n+\ell\,,\qquad
P_{h^{\prime},\overline{h}^{\prime}}=\frac{1}{N^{4}}P^{\text{MFT}}_{h^{\prime},\overline{h}^{\prime}}\delta
P^{(56)}_{h^{\prime},\overline{h}^{\prime}}+\ldots\,,$ (182)

appear in the one-loop correlator. This is compatible with the large $N$
behavior for single-trace exchange in the direct channel,

$\langle\mathcal{O}_{1}\mathcal{O}_{5}\mathcal{O}_{\Delta(J)}\rangle\sim\frac{1}{N}~{},~{}\langle\mathcal{O}_{\Delta(J)}\mathcal{O}_{6}\mathcal{O}_{3}\rangle\sim\frac{1}{N}\,,$
(183)

which justifies that the OPE coefficients
$c_{\mathcal{O}_{1}\mathcal{O}_{3}[\mathcal{O}_{5}\mathcal{O}_{6}]}$ start at
order $1/N^{2}$. As explained in Meltzer:2019pyl , the cross channel expansion
of the correlator is then dominated by the terms

$\displaystyle\frac{\mathcal{A}_{\text{1-loop}}^{\circlearrowleft}(z,\overline{z})}{(z\overline{z})^{\Delta_{\phi}}}$
$\displaystyle\approx\sum_{h,\overline{h}}P^{MFT}_{h,\overline{h}}\bigg{[}i\pi\gamma^{(2)}_{h,\overline{h}}+\delta
P^{(2)}_{h,\overline{h}}+i\pi\gamma^{(1)}_{h,\overline{h}}\,\delta
P^{(1)}_{h,\overline{h}}-\frac{\pi^{2}}{2}\left(\gamma^{(1)}_{h,\overline{h}}\right)^{2}\bigg{]}g_{h,\overline{h}}(1-z,1-\overline{z})$
$\displaystyle+\sum_{h^{\prime},\overline{h}^{\prime}}P^{MFT}_{h^{\prime},\overline{h}^{\prime}}\delta
P^{(56)}_{h^{\prime},\overline{h}^{\prime}}\,g_{h^{\prime},\overline{h}^{\prime}}(1-z,1-\overline{z})\,.$
(184)

We shall now compare with our result for the one-loop correlator in the Regge
limit and use it in the light of (184) to extract CFT data. We start with the
dDisc of the correlator in the impact parameter representation as in (170).
Doing an inverse Fourier transform on this and taking out the appropriate
scale factors gives us the dDisc of the one-loop correlator in the Regge limit

$\displaystyle\text{dDisc}_{t}\mathcal{A}_{\text{1-loop}}(z,\overline{z})=\frac{\pi^{4}\mathcal{N}}{4}\int\limits_{-\infty}^{\infty}d\nu
d\nu_{1}d\nu_{2}\,\big{(}\beta^{*}(\nu_{1})\beta(\nu_{2})+\beta(\nu_{1})\beta^{*}(\nu_{2})\big{)}V(\nu_{1},\nu_{2},\nu)^{2}$
(185)
$\displaystyle\Phi(\nu_{1},\nu_{2},\nu)\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(\nu)\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(-\nu)\,\sigma^{2-j(\nu_{1})-j(\nu_{2})}\,\Omega_{i\nu}(\rho)\,,$

where we symmetrized the product of $\beta$’s by using the symmetry of the
expression under $\nu_{1}\leftrightarrow\nu_{2}$. We can now use the
Lorentzian inversion formula Caron_Huot_2017  on (185), as shown in
Meltzer:2019pyl , to obtain the one-loop correlator in the Regge limit, and
then use (176) to express it as

$\displaystyle\mathcal{A}_{\text{1-loop}}^{\circlearrowleft}(z,\overline{z})\approx-\frac{\pi^{4}\mathcal{N}}{4}\int\limits_{-\infty}^{\infty}d\nu_{1}d\nu_{2}d\nu\,\frac{1+e^{-i\pi(j(\nu_{1})+j(\nu_{2})-1)}}{1-e^{-2\pi
i(j(\nu_{1})+j(\nu_{2})-1)}}\,V(\nu_{1},\nu_{2},\nu)^{2}$ (186)
$\displaystyle\Phi(\nu_{1},\nu_{2},\nu)\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(\nu)\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(-\nu)\,\sigma^{2-j(\nu_{1})-j(\nu_{2})}\,\Omega_{i\nu}(\rho)$
$\displaystyle\left[\widehat{\gamma}^{(1)}(\nu_{1})\,\widehat{\gamma}^{(1)}(\nu_{2})+\frac{1}{\pi^{2}}\widehat{\delta
P}^{(1)}(\nu_{1})\,\widehat{\delta P}^{(1)}(\nu_{2})\right].$

We now take the $t$-channel expansion (184), and use in it (175), (162), and
the following ansatz,

$\displaystyle\gamma^{(2)}_{h,\overline{h}}$
$\displaystyle\approx\int\limits_{-\infty}^{\infty}d\nu_{1}d\nu_{2}d\nu\,\widehat{\gamma}^{(2)}(\nu_{1},\nu_{2},\nu)\,(h\overline{h})^{j(\nu_{1})+j(\nu_{2})-2}\,\Omega_{i\nu}\big{(}\log(h/\overline{h})\big{)}\,,$
(187) $\displaystyle\delta P^{(2)/(56)}_{h,\overline{h}}$
$\displaystyle\approx\int\limits_{-\infty}^{\infty}d\nu_{1}d\nu_{2}d\nu\,\widehat{\delta
P}^{(2)/(56)}(\nu_{1},\nu_{2},\nu)\,(h\overline{h})^{j(\nu_{1})+j(\nu_{2})-2}\,\Omega_{i\nu}\big{(}\log(h/\overline{h})\big{)}\,,$

to obtain

$\displaystyle(z\overline{z})^{-\Delta_{\phi}}$
$\displaystyle\mathcal{A}_{\text{1-loop}}^{\circlearrowleft}(z,\overline{z})\approx\int\limits_{-\infty}^{\infty}d\nu_{1}d\nu_{2}d\nu\Bigg{[}\sum_{h,\overline{h}}(h\overline{h})^{j(\nu_{1})+j(\nu_{2})-2}\,\Omega_{i\nu}(\log
h/\overline{h})\,P^{MFT}_{h,\overline{h}}$ (188)
$\displaystyle\bigg{[}i\pi\widehat{\gamma}^{(2)}(\nu_{1},\nu_{2},\nu)+\frac{i\pi}{2}\left(\widehat{\gamma}^{(1)}(\nu_{1})\,\widehat{\delta
P}^{(1)}(\nu_{2})+\widehat{\gamma}^{(1)}(\nu_{2})\,\widehat{\delta
P}^{(1)}(\nu_{1})\right)\Phi(\nu_{1},\nu_{2},\nu)$
$\displaystyle\qquad\qquad-\frac{\pi^{2}}{2}\,\widehat{\gamma}^{(1)}(\nu_{1})\,\widehat{\gamma}^{(1)}(\nu_{2})\Phi(\nu_{1},\nu_{2},\nu)+\widehat{\delta
P}^{(2)}(\nu_{1},\nu_{2},\nu)\bigg{]}\,g_{h,\overline{h}}(1-z,1-\overline{z})$
$\displaystyle+\sum_{h^{\prime},\overline{h}^{\prime}}P^{MFT}_{h^{\prime},\overline{h}^{\prime}}\,\widehat{\delta
P}^{(56)}(\nu_{1},\nu_{2},\nu)(h^{\prime}\overline{h}^{\prime})^{j(\nu_{1})+j(\nu_{2})-2)}\,\Omega_{i\nu}(\log
h^{\prime}/\overline{h}^{\prime})\,g_{h^{\prime},\overline{h}^{\prime}}(1-z,1-\overline{z})\Bigg{]}\,.$

We can approximate the $h,\overline{h}$ and $h^{\prime},\overline{h}^{\prime}$
sums with integrals,
$\sum_{h,\overline{h}}\rightarrow\frac{1}{2}\int_{0}^{\infty}dh\,d\overline{h}$,
and evaluate them using Bessel function integrals (see section 2.2 of
Meltzer:2019pyl ) to arrive at the result

$\displaystyle\mathcal{A}_{\text{1-loop}}^{\circlearrowleft}(z,\overline{z})$
$\displaystyle\approx\frac{\pi^{2}\mathcal{N}}{4}\int\limits_{-\infty}^{\infty}d\nu_{1}d\nu_{2}d\nu\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(\nu)\,\chi_{j(\nu_{1})+j(\nu_{2})-1}(-\nu)\,\sigma^{2-j(\nu_{1})-j(\nu_{2})}\,\Omega_{i\nu}(\rho)$
$\displaystyle\left[i\pi\widehat{\gamma}^{(2)}(\nu_{1},\nu_{2},\nu)-\frac{\pi^{2}}{2}\,\widehat{\gamma}^{(1)}(\nu_{1})\,\widehat{\gamma}^{(1)}(\nu_{2})\,\Phi(\nu_{1},\nu_{2},\nu)+\widehat{\delta
P}^{(2)}(\nu_{1},\nu_{2},\nu)\right.$ (189)
$\displaystyle\left.+\frac{i\pi}{2}\left(\widehat{\gamma}^{(1)}(\nu_{1})\,\widehat{\delta
P}^{(1)}(\nu_{2})+\widehat{\gamma}^{(1)}(\nu_{2})\,\widehat{\delta
P}^{(1)}(\nu_{1})\right)\Phi(\nu_{1},\nu_{2},\nu)+\widehat{\delta
P}^{(56)}(\nu_{1},\nu_{2},\nu)\right].$

Comparing the real parts of the coefficient of
$\chi(\nu)\chi(-\nu)\sigma^{2-j(\nu_{1})-j(\nu_{2})}\Omega_{i\nu}(\rho)$ in
the integrands of (186) and (189), and using

$\frac{1+e^{-i\pi(j(\nu_{1})+j(\nu_{2})-1)}}{1-e^{-2\pi
i(j(\nu_{1})+j(\nu_{2})-1)}}=\frac{1}{2}+\frac{i}{2}\tan\left(\frac{\pi}{2}\big{(}j(\nu_{1})+j(\nu_{2})\big{)}\right),$
(190)

we conclude that

$\displaystyle\widehat{\delta P}^{(2)}(\nu_{1},\nu_{2};\nu)+\widehat{\delta
P}^{(56)}(\nu_{1},\nu_{2};\nu)$
$\displaystyle=-\frac{1}{2}\bigg{[}\pi^{2}\Big{(}V(\nu_{1},\nu_{2},\nu)^{2}-1\Big{)}\widehat{\gamma}^{(1)}(\nu_{1})\,\widehat{\gamma}^{(1)}(\nu_{2})$
(191) $\displaystyle+V(\nu_{1},\nu_{2},\nu)^{2}\,\widehat{\delta
P}^{(1)}(\nu_{1})\,\widehat{\delta
P}^{(1)}(\nu_{2})\bigg{]}\Phi(\nu_{1},\nu_{2},\nu)\,.$

This is the general result for fixed $\Delta_{\text{gap}}$ that extracts OPE
data from the AdS vertex function. Let us now take the large
$\Delta_{\text{gap}}$ limit to make contact with Meltzer:2019pyl . In this
limit, $V(\nu_{1},\nu_{2},\nu)^{2}=1$ and $\widehat{\delta P}^{(1)}$,
$\widehat{\delta P}^{(56)}$ are suppressed with respect to
$\widehat{\gamma}^{(1)}$. Therefore $\delta{P}^{(2)}_{h,\overline{h}}=0$, as
was obtained in Meltzer:2019pyl .

Similarly, comparing the imaginary parts we have

$\displaystyle\widehat{\gamma}^{(2)}(\nu_{1},\nu_{2};\nu)$
$\displaystyle=-\frac{1}{2}\bigg{[}\left(\widehat{\gamma}^{(1)}(\nu_{1})\widehat{\delta
P}^{(1)}(\nu_{2})+\widehat{\delta
P}^{(1)}(\nu_{1})\widehat{\gamma}^{(1)}(\nu_{2})\right)+\pi\tan\left(\frac{\pi}{2}(j(\nu_{1})+j(\nu_{2}))\right)$
$\displaystyle\\!\\!V(\nu_{1},\nu_{2},\nu)^{2}\Big{(}\widehat{\gamma}^{(1)}(\nu_{1})\widehat{\gamma}^{(1)}(\nu_{2})+\frac{1}{\pi^{2}}\widehat{\delta
P}^{(1)}(\nu_{1})\widehat{\delta
P}^{(1)}(\nu_{2})\Big{)}\bigg{]}\Phi(\nu_{1},\nu_{2},\nu)\,.$ (192)

The term $\widehat{\delta P}^{(1)}(\nu_{1})\,\widehat{\delta
P}^{(1)}(\nu_{2})$ is suppressed by $\Delta_{\text{gap}}^{-4}$ with respect to
the other terms. At leading order in $\Delta_{\text{gap}}$, we can discard
this term and set $V(\nu_{1},\nu_{2},\nu)^{2}=1$. We can then use (178) to
simplify the expression to,

$\displaystyle\widehat{\gamma}^{(2)}(\nu_{1},\nu_{2};\nu)$
$\displaystyle=-\frac{1}{2}\pi\tan\left(\frac{1}{2}\pi
j(\nu_{1})\right)\tan\left(\frac{1}{2}\pi
j(\nu_{2})\right)\tan\left(\frac{1}{2}\pi(j(\nu_{1})+j(\nu_{2}))\right)$ (193)
$\displaystyle\hskip
177.06128pt\times\widehat{\gamma}^{(1)}(\nu_{1})\widehat{\gamma}^{(1)}(\nu_{2})\Phi(\nu_{1},\nu_{2},\nu)\,.$

This is the same result as obtained in Meltzer:2019pyl  for
$\widehat{\gamma}^{(2)}(\nu_{1},\nu_{2};\nu)$. More generally, knowledge of
the vertex function $V(\nu_{1},\nu_{2},\nu)^{2}$ and of the
$\langle\mathcal{O}_{1}\mathcal{O}_{3}[\mathcal{O}_{5}\mathcal{O}_{6}]\rangle$
OPE coefficients gives additional information about the one-loop CFT data of
the $[\mathcal{O}_{1}\mathcal{O}_{3}]$ double-trace operators. It would be
interesting to analyze these equations order by order in the
$1/\Delta_{\text{gap}}^{2}$ expansion.

##  6 Flat space limit

Having fixed the general form of the impact parameter representation of the
one-loop correlator from first principles in section 4, we now want to fix
part of the dynamical data by taking the flat space limit, which relates it to
the known flat space amplitudes. The prescription to achieve this was
discovered in Cornalba:2007fs , where it was applied to scalar tree-level
amplitudes. This limit is taken by sending the AdS radius $R$ to infinity
while scaling the relevant quantities in order to match them to flat space
quantities in a sensible way. The dimensionless quantities $S$ and $L$ are
sent to dimensionless combinations of $R$ with the flat space center of mass
energy $s$ and impact parameter $b$ as

$S=\frac{R^{2}s}{4}\,,\qquad L=\frac{b}{R}\,.$ (194)

Note that $L$ is the AdS impact parameter, as it describes the geodesic
distance on $H_{d-1}$ between the impact points in transverse space. If we
impose the identification of Casimir eigenvalues

$\Delta(\Delta-d)=R^{2}m^{2}\,,$ (195)

for the states on the leading Regge trajectory and take this equation off-
shell, it becomes

$\nu^{2}+\left(\frac{d}{2}\right)^{2}=R^{2}q^{2}\,,$ (196)

so for large $R$ we further impose

$\nu^{2}=R^{2}q^{2}\,.$ (197)

Our expressions in AdS are integrals in $\nu$, while in flat space we have
vector integrals in $q$, where we recall that $q$ is a vector in the
transverse space $\mathbb{R}^{D-2}$. In order to compare the expressions, it
is instructive to do the flat space angular integrals and keep only the
integral over the modulus $|q|$. In this way, the exponential is replaced by
the harmonic function $\omega_{q}(b)$ according to

$\int\limits_{\mathbb{R}^{D-2}}\frac{dq}{(2\pi)^{D-2}}\,e^{ib\cdot
q}=2\int\limits_{0}^{\infty}d|q|\,\omega_{q}(b)\,,$ (198)

so that Costa:2014kfa

$\displaystyle\omega_{q}(b)=q\int\limits_{\mathbb{R}^{D-2}}\frac{dp}{(2\pi)^{D-2}}\,e^{ib\cdot
p}\,\delta(p^{2}-q^{2})=\frac{1}{2(2\pi)^{\frac{D-2}{2}}}\frac{|q|^{\frac{D-2}{2}}}{|b|^{\frac{D-4}{2}}}J_{\frac{D-4}{2}}(|q||b|)\,,$
(199)

where $J$ denotes the Bessel $J$-function and we recall that $\omega_{q}(b)$
only depends on the modulus of the vectors $q$ and $b$. One can check that the
flat space limit of the $H_{d-1}$ harmonic function (144) yields the flat
space harmonic function

$R^{3-D}\Omega_{i\nu}(L)\to\omega_{q}(b)\,,\qquad\nu\geq 0\,.$ (200)

For even $d$ this can be checked directly, while for general $d$ it is
convenient to use an integral representation for the hypergeometric function
which under the limit is related to an integral representation for the Bessel
function Carmi:2018qzm . For even $d$ the relation is also valid for $\nu<0$.

In the context of string theory we further have the dimensionless coupling
$\lambda$ which is expressed in terms of $\alpha^{\prime}$ and $R$ as

$\sqrt{\lambda}=\frac{R^{2}}{\alpha^{\prime}}\,,$ (201)

meaning we can also express $S$ as

$S=\frac{\sqrt{\lambda}\,\alpha^{\prime}s}{4}\,.$ (202)

To summarize, the flat space limit is taken by sending the AdS radius $R$ to
infinity while replacing

$S=\frac{\sqrt{\lambda}\alpha^{\prime}s}{4},\quad
L=\frac{b}{R},\quad\nu^{2}=R^{2}q^{2},\quad\nu_{1}^{2}=R^{2}q_{1}^{2},\quad\nu_{2}^{2}=R^{2}q_{2}^{2},\quad\sqrt{\lambda}=\frac{R^{2}}{\alpha^{\prime}}\,,$
(203)

and impact parameter representations can be compared by using (200). We can
also use these relations to relate $\Delta_{gap}$ to $\lambda$ taking as
reference a string state of mass $m^{2}=4/\alpha^{\prime}$, therefore

$\Delta^{2}_{\text{gap}}=\frac{4R^{2}}{\alpha^{\prime}}=4\sqrt{\lambda}\,.$
(204)

###  6.1 Matching in impact parameter space

Let us now see what we can learn when we apply the flat space limit to the
impact parameter representations studied in section 4. We begin with the tree-
level correlator of four scalars for which the limit was originally imposed in
Cornalba:2007fs . The flat space limit of the AdS result $\mathcal{B}$ in
(153) should match the flat space impact parameter representation $i\delta$
from (85) of the amplitude (78)111111The relative factor $i$ in
$\mathcal{B}\to i\delta$ can be determined by matching the exponents in the
eikonal approximation for $\lambda\to\infty$.

$\mathcal{B}_{\text{tree}}(p,{\overline{p}})=4\pi
i\int\limits_{0}^{\infty}d\nu\,\beta(\nu)\,S^{j(\nu)-1}\,\Omega_{i\nu}(L)\
\to\
i\delta_{\text{tree}}(s,b)=2i\int\limits_{0}^{\infty}d|q|\,\beta(t)\left(\frac{\alpha^{\prime}s}{4}\right)^{j(t)-1}\omega_{q}(b)\,.$
(205)

Here and below we do not always write the overall factors of $R$ as in (200),
but they do work out correctly when including the expansion parameters from
(3) and (71) and using the relation

$\frac{1}{N^{2}}=\frac{1}{R^{D-2}}\frac{2G_{N}}{\pi}\,.$ (206)

From (200) and (202) we see that this does indeed match, provided the flat
space limit of the AdS Regge trajectory and spectral function are sent to the
flat space Regge trajectory and Pomeron propagator121212$j(\nu)$, $j(t)$ and
$\beta(\nu)$, $\beta(t)$ are different functions and not the same function
with different arguments.

$j(\nu)\,\to\,j(t)\,,\qquad\lambda^{\frac{j(\nu)-1}{2}}\beta(\nu)\,\to\,\frac{1}{2\pi}\,\beta(t)\,.$
(207)

The power of $\lambda$ in the relation of $\beta$’s is necessary to cancel the
powers of $\lambda$ in the relation between $S$ and $\alpha^{\prime}s$. It is
compatible with the expectation that each derivative in the couplings of the
spin $J$ operators forming the Pomeron comes at least with a power of
$\lambda^{-\frac{1}{4}}$.

Next we consider the optical theorem in AdS (134) and flat space (86)

$-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}=\frac{1}{2}\sum\limits_{\begin{subarray}{c}\Delta_{5},\rho_{5}\\\
\Delta_{6},\rho_{6}\end{subarray}}\mathcal{B}^{3652\,*}_{\text{tree}}\mathcal{B}^{1564}_{\text{tree}}\
\to\
\mathop{\mathrm{Im}}\delta_{\text{1-loop}}(b)=\frac{1}{2}\sum\limits_{\begin{subarray}{c}m_{5},\rho_{5},\epsilon_{5}\\\
m_{6},\rho_{6},\epsilon_{6}\end{subarray}}\delta_{\text{tree}}^{3652\,*}\delta_{\text{tree}}^{1564}\,.$
(208)

The similarity is striking, however we have to make sure the sums and summands
are in fact related by the flat space limit. The additional sums over
polarizations can be evaluated using completeness relations such as (92),
which evaluate to contractions just as in the AdS equation. We also have to
make sure that the labels $\rho$ on both sides are irreducible representations
of the same group $SO(d)$. This is indeed the case for massive particles if we
consider the flat space limit $AdS_{d+1}\to\mathbb{R}^{1,d}$, which has the
massive Little group $SO(d)$.

The next step is to match the tree-level correlators (158) and amplitudes (78)
that involve spinning particles 5 and 6. In this case the flat space limit
gives

$\displaystyle\mathcal{B}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(p,\overline{p})=4\pi
i\int\limits_{0}^{\infty}d\nu\,S^{j(\nu)-1}\,\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(\nu)\,\Omega_{i\nu}(L)\
\to$ (209) $\displaystyle\to\ $ $\displaystyle
i\delta^{(m_{5},\rho_{5}),(m_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(s,b)=i\int\limits_{\mathbb{R}^{D-2}}\frac{dq}{(2\pi)^{D-2}}\left(\frac{\alpha^{\prime}s}{4}\right)^{j(t)-1}A^{12P}_{m_{5},\rho_{5},\mathbf{m}}(q,v)\,\beta(t)A^{34P}_{m_{6},\rho_{6},\mathbf{n}}(q,v)\,e^{iq\cdot
b}$
$\displaystyle\qquad\qquad=2i\int\limits_{0}^{\infty}d|q|\left(\frac{\alpha^{\prime}s}{4}\right)^{j(t)-1}A^{12P}_{m_{5},\rho_{5},\mathbf{m}}(-i\partial_{b},v)\,\beta(t)A^{34P}_{m_{6},\rho_{6},\mathbf{n}}(-i\partial_{b},v)\,\omega_{q}(b)\,,$

where the derivative $\partial_{b}$ is with respect to the components of the
transverse vector $b$. The difference compared to (205) is that in AdS we have
differential operators that generate tensor structures, while in flat space
the tensor structures are the ones of the on-shell three-point amplitudes. As
discussed in section 3.4, these three-point amplitudes are given in terms of
the Pomeron momentum $q$ and, for massive particles, the longitudinal
polarization vector $v$, which is transverse to $q$. We will now study the
relation of these two kinds of tensor structures to the flat space limit.

We begin with the covariant derivatives and will argue that they become
derivatives in impact parameter in flat space, i.e.

$\nabla_{p}^{m}\Omega_{i\nu}(L)\ \to\ R\partial_{b}^{m}e^{ib\cdot
q}=Riq^{m}e^{ib\cdot q}\,.$ (210)

In order to show this, we will act with two contracted covariant derivatives
either on a single harmonic function or on two different ones, covering all
situations that can occur. Acting on a single harmonic function we obtain,
from (161) and (203),

$\frac{1}{\sqrt{\lambda}}\,\nabla_{p}^{2}\,\Omega_{i\nu_{1,2}}(L)\ \to\
-\alpha^{\prime}q_{1,2}^{2}\,\omega_{i\nu_{1,2}}(b)\,.$ (211)

The action of contracted covariant derivatives on two different harmonic
functions is captured by the functions $W_{k}$ in (163), which is given in the
flat space limit by the leading term (167)

$\displaystyle\frac{W_{k}\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}}{\big{(}\sqrt{\lambda}\big{)}^{k}}\
\to\ \left(\frac{\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}}{2\sqrt{\lambda}}\right)^{k}\
\to\
\left(\alpha^{\prime}\,\frac{q_{1}^{2}+q_{2}^{2}-q^{2}}{2}\right)^{k}=(\alpha^{\prime})^{k}(-q_{1}\cdot
q_{2})^{k}\,,$ (212)

where we used that $q=q_{1}+q_{2}$. This implies that the flat space limit of
(163) is

$\displaystyle\frac{1}{\big{(}\sqrt{\lambda}\big{)}^{k}}{\nabla_{p}}_{m_{1}}\ldots{\nabla_{p}}_{m_{k}}\Omega_{i\nu_{1}}(L)\nabla_{p}^{m_{1}}\ldots\nabla_{p}^{m_{k}}\Omega_{i\nu_{2}}(L)\
\to$ (213) $\displaystyle\to\ $
$\displaystyle(-\alpha^{\prime})^{k}q_{1\,m_{1}}\ldots q_{1\,m_{k}}e^{ib\cdot
q_{1}}q_{2}^{m_{1}}\ldots q_{2}^{m_{k}}e^{ib\cdot q_{2}}\,.$ (214)

We conclude that both (211) and (214) are compatible with (210). Apart from
the covariant derivative, tensor structures depend also on the direction
$\widehat{p}$, which is normal to the transverse space $H_{d-1}$ and satisfies
$\widehat{p}^{2}=-1$. In flat space the only possible direction for
polarizations that is normal to the transverse space is the unit vector $v$,
hence we have to require that in the flat space limit

$\widehat{p}^{m}\,\to\,iv^{m}\,.$ (215)

With the identifications (210) and (215), the matching in (209) works provided
that the spectral functions
$\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)$ in
(159) are such that

$\lambda^{\frac{j(\nu)-1}{2}}\mathfrak{D}^{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}_{\mathbf{m}\mathbf{n}}(\nu)\,\Omega_{i\nu}(L)\,\to\,\frac{1}{2\pi}A^{12P}_{m_{5},\rho_{5},\mathbf{m}}(-i\partial_{b},v)\,\beta(t)\,A^{34P}_{m_{6},\rho_{6},\mathbf{n}}(-i\partial_{b},v)\,\omega_{q}(b)\,.$
(216)

Using the explicit tensor structures for three-point amplitudes in (97), the
matching (216) can also be expressed for the spectral function for any given
tensor structure

$\lambda^{\frac{j(\nu)-1}{2}}\lambda^{\frac{|\rho_{5}|-k_{5}}{4}}\lambda^{\frac{|\rho_{6}|-k_{6}}{4}}\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)\
\to\
\frac{1}{2\pi}\,a_{m_{5},\rho_{5}}^{k_{5}}(t)\,a_{m_{6},\rho_{6}}^{k_{6}}(t)\,\beta(t)\,.$
(217)

The powers of $\lambda$ are again compatible with a factor of
$\lambda^{-\frac{1}{4}}$ in
$\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)$ for
each derivative in the coupling. Such a scaling is expected from the general
arguments of Costa:2017twz ; Meltzer:2017rtf . We have now shown that all
tree-level phase shifts appearing in the AdS and flat space optical theorems
can be related by the flat space limit.

Let us also compare the vertex functions that appear in both AdS and flat
space impact parameter representations of the one-loop amplitudes. The flat
space limit of (170) is given by the impact parameter transform of (90), i.e.

$-\mathop{\mathrm{Re}}\mathcal{B}_{\text{1-loop}}(p,{\overline{p}})\ \to\
\mathop{\mathrm{Im}}\delta_{\text{1-loop}}(s,b)\,,$ (218)

becomes

$\displaystyle 2\pi^{2}\int\limits_{-\infty}^{\infty}d\nu
d\nu_{1}d\nu_{2}\,\beta(\nu_{1})^{*}\beta(\nu_{2})\,V(\nu_{1},\nu_{2},\nu)^{2}S^{j(\nu_{1})+j(\nu_{2})-2}\Phi(\nu_{1},\nu_{2},\nu)\,\Omega_{i\nu}(L)\
\to$ (219) $\displaystyle\to$
$\displaystyle\frac{1}{2}\int\limits_{\mathbb{R}^{D-2}}\frac{dqdq_{1}dq_{2}}{(2\pi)^{2(D-2)}}\,\beta(t_{1})^{*}\beta(t_{2})\,V(q_{1},q_{2})^{2}\left(\frac{\alpha^{\prime}s}{4}\right)^{j(t_{1})+j(t_{2})-2}\delta(q-q_{1}-q_{2})\,e^{iq\cdot
b}\,.$

In this case we can use the delta function to write all the other scalar
functions in terms of $q_{1}^{2}$, $q_{2}^{2}$ and $q^{2}$, however we need to
do the angular integral over the delta function itself. To this end we can
define

$\int\limits_{\mathbb{R}^{D-2}}\frac{dq_{1}dq_{2}}{(2\pi)^{D-2}}\,\delta(q-q_{1}-q_{2})=4\int\limits_{0}^{\infty}d|q_{1}|d|q_{2}|\,\phi(q_{1},q_{2},q)\,.$
(220)

Using this and (198), we can compute the angular integrals in

$\int\limits_{\mathbb{R}^{D-2}}\frac{dq_{1}dq_{2}}{(2\pi)^{2(D-2)}}\,e^{ib\cdot(q_{1}+q_{2})}=\int\limits_{\mathbb{R}^{D-2}}\frac{dq_{1}dq_{2}dq}{(2\pi)^{2(D-2)}}\,\delta(q-q_{1}-q_{2})\,e^{ib\cdot
q}\,,$ (221)

to find the flat space version of (162)

$\omega_{q_{1}}(b)\,\omega_{q_{2}}(b)=2\int\limits_{0}^{\infty}d|q|\,\phi(q_{1},q_{2},q)\,\omega_{q}(b)\,.$
(222)

Using the explicit expressions for $\Phi$ and $\phi$ (which can be found for
instance in appendix E of Penedones:2010ue ), one can further check that under
the flat space limit

$R^{4-D}\Phi(\nu_{1},\nu_{2},\nu)\ \to\ \phi(q_{1},q_{2},q)\,.$ (223)

With this relation is clear that the expressions in (219) are indeed related
by the flat space limit provided that the vertex functions are related by

$V(\nu_{1},\nu_{2},\nu)\ \to\ V(q_{1},q_{2})=V(t_{1},t_{2},t)\,.$ (224)

###  6.2 Constraining AdS quantities

We saw above that all elements of the impact parameter optical theorems in AdS
and flat space are related by the flat space limit provided that $j(\nu)$,
$\beta(\nu)$,
$\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)$ and
$V\big{(}\nu_{1}^{2},\nu_{2}^{2},\nu^{2}\big{)}$ are given by their flat space
counterparts in the limit. In this subsection, we briefly review how the limit
actually constrains these functions. All of these objects depend on two
dimensionless quantities, the spectral parameters $\nu$ and the t’Hooft
coupling $\lambda$. Let us discuss this for a generic function $f(\nu)$ that
is required to satisfy the flat space limit

$f(\nu,\lambda)\,\to\,f(t)\,.$ (225)

Existence of the gravity limit requires the function to have an expansion in
negative powers of $\sqrt{\lambda}$ of the form

$f(\nu,\lambda)=\sum\limits_{n=0}^{\infty}\frac{f_{n}(\nu)}{\lambda^{n/2}}\,.$
(226)

In order for the flat space limit (203) of the function to be finite, the
functions $f_{n}(\nu)$ must have an expansion in large $\nu$ with leading
power not larger than $2n$,

$f_{n}(\nu)=a_{n,n}\nu^{2n}+a_{n,n-1}\nu^{2(n-1)}+a_{n,n-2}\nu^{2(n-2)}+\ldots\,,$
(227)

which ensures finiteness of the limit order by order in the large $\lambda$
expansion. The flat space limit of $f(\nu,\lambda)$ is then

$f(\nu,\lambda)\ \to\
\sum\limits_{n=0}^{\infty}a_{n,n}\left(\frac{\nu^{2}}{\sqrt{\lambda}}\right)^{n}\to\
\sum\limits_{n=0}^{\infty}a_{n,n}(\alpha^{\prime}q^{2})^{n}\,.$ (228)

At every order in $1/\sqrt{\lambda}$ the leading power of $\nu$ survives and
is fixed by the flat space limit, while all the other powers are subleading,
and cannot be determined from this condition. These considerations hold for
$j(\nu)$, $\beta(\nu)$,
$\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)$ and
$V(\nu_{1},\nu_{2},\nu)$, fixing part of these functions. These facts have
been explored in detail for the functions $j(\nu)$ and $\beta(\nu)$ in
Cornalba:2008qf ; Costa:2012cb .

##  7 Relating type IIB string theory in AdS and flat space

Let us now apply our general ideas to a concrete example, the scattering of
four dilatons in type IIB superstring theory on $AdS_{5}\times S^{5}$. In the
flat space limit this is related to type IIB superstring theory on
10-dimensional flat space where the kinematics is restricted to the five
dimensions arising from AdS. This happens since both the dilatons and the
Pomerons are $R$-symmetry singlets, meaning the tidal excitations they couple
to also have to be singlets. As a consequence, the Regge limit does not probe
the 10 dimensional nature of the string scattering process, as we consider
only states with the vacuum quantum numbers associated to the compact manifold
$S^{5}$. For this case the discontinuity of the (finite $\alpha^{\prime}$)
one-loop amplitude in the Regge limit was computed in Amati:1987uf  and is
precisely of the form (90) with $D=5$. The regime of validity of this
description was discussed in detail in Amati:1987uf . All we need to specify
are the four dynamic quantities that we already discussed in the previous
section. For the Regge trajectory and Pomeron propagator we have

$j(t)=2+\frac{\alpha^{\prime}}{2}\,t\,,\qquad\beta(t)=2\pi^{2}\frac{\Gamma\big{(}-\frac{\alpha^{\prime}}{4}t\big{)}}{\Gamma\big{(}1+\frac{\alpha^{\prime}}{4}t\big{)}}\,e^{-\frac{i\pi\alpha^{\prime}}{4}t}\,.$
(229)

As discussed in section 3.3 the vertex function can be obtained from the
scattering amplitude of two dilatons and two Pomerons. This amplitude was
computed in Amati:1987uf  and reads

$A^{12P_{1}P_{2}}(k,q_{1},q_{2})=-\frac{\Gamma(1+\alpha^{\prime}q_{12}/2)\,\Gamma\big{(}-\alpha^{\prime}\frac{k^{2}}{4}-\alpha^{\prime}q_{12}/2\big{)}\,\Gamma\big{(}\alpha^{\prime}\frac{k^{2}}{4}\big{)}}{2\Gamma(-\alpha^{\prime}q_{12}/2)\,\Gamma\big{(}1+\alpha^{\prime}q_{12}/2+\alpha^{\prime}\frac{k^{2}}{4}\big{)}\,\Gamma\big{(}1-\alpha^{\prime}\frac{k^{2}}{4}\big{)}}\,,$
(230)

where $q_{12}=q_{1}\cdot q_{2}$. This amplitude has poles at the masses
($m^{2}=4n/\alpha^{\prime}$) of the string states with residues

$\underset{k^{2}=-4n/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{1},q_{2})=\left(\frac{(-\alpha^{\prime}q_{12}/2)_{n}}{n!}\right)^{2}\,,\qquad
n=0,1,2,\ldots\,,$ (231)

and the resulting vertex function is given by (89)

$V(q_{1},q_{2})=\sum\limits_{n=0}^{\infty}\left(\frac{(-\alpha^{\prime}q_{12}/2)_{n}}{n!}\right)^{2}=\frac{\Gamma\big{(}1+\alpha^{\prime}\frac{t_{1}+t_{2}-t}{2}\big{)}}{\Gamma\big{(}1+\alpha^{\prime}\frac{t_{1}+t_{2}-t}{4}\big{)}^{2}}\,.$
(232)

Using the reasoning of section 6.2, this immediately fixes the leading terms
in $\nu,\nu_{i}$ of the AdS vertex function at every order in $\lambda$

$V(\nu_{1},\nu_{2},\nu)=\frac{\Gamma\Big{(}1-\frac{\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}}{2\sqrt{\lambda}}\Big{)}}{\Gamma\Big{(}1-\frac{\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}}{4\sqrt{\lambda}}\Big{)}^{2}}+\text{vanishing
in flat space limit}\,.$ (233)

Thus, the first two corrections from expanding (233) at large $\lambda$ are

$\displaystyle V(\nu_{1},\nu_{2},\nu)$
$\displaystyle=1+\Big{(}0\cdot\big{(}\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}\big{)}+c_{1,0}\Big{)}\frac{1}{\sqrt{\lambda}}$
(234)
$\displaystyle+\left(\frac{\pi^{2}}{96}\left(\nu_{1}^{2}+\nu_{2}^{2}-\nu^{2}\right)^{2}+c_{2,1}\big{(}\nu_{1}^{2}+\nu_{2}^{2}\big{)}+c^{\prime}_{2,1}\nu^{2}+c_{2,0}\right)\frac{1}{\lambda}+\ldots\,,$

where we also included the constants that are not fixed by the flat space
limit. We note that the constants multiplying the leading power of $\nu$ at
order $\big{(}\sqrt{\lambda}\big{)}^{-n}$ have a uniform transcendentality of
weight $n$, which can be seen by explicitly expanding (233). It would be
interesting to understand the relation of this property with features of
maximal transcendentality in $\mathcal{N}=4$ SYM Kotikov:2002ab ;
Kotikov:2007cy .

Finally, all the spectral functions
$\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)$ of
tree-level correlators that contribute to the optical theorem are constrained
by the flat space limit. Their flat space limit (217) is parameterized by on-
shell three-point amplitudes in flat space. These amplitudes are in principle
encoded in the result (231), which separates the contributions of particles
with different masses, but not the ones of particles in different
representations $\rho$. The attempt to expand (231) into products of three-
point amplitudes for different $\rho$ and tensor structures $k$ using (88)
shows that this does not fully fix the $a_{m_{5},\rho_{5}}^{k}(t)$ in (97)
because the equations are quadratic. However the three-point amplitudes can of
course be computed in string theory, which is what the next subsection is
about. We will start with the 10D open superstring amplitudes of a massless
vector, a Pomeron and an open string state up to mass level 2 which were
computed in DAppollonio:2013mgj  by studying string-brane scattering. These
amplitudes have to be squared to obtain closed string amplitudes. Then the
irreducible representations of the 10D massive Little group $SO(9)$ have to be
branched into irreducible representations of $SO(4)$ to match with the CFT
irreps and account for the fact that we have five non-compact dimensions.

###  7.1 Massive tree amplitudes in flat space

Now we will discuss the flat space three-point amplitudes that take part in
the process and that will fix part of the tree-level correlators with external
spinning legs in AdS via the flat space limit. The goal is to derive the
three-point amplitudes that appear in the unitarity cut (231) of the four-
point amplitude of two dilatons and two Pomerons (230). It will be convenient
to consider the more general case of two gravitons instead of dilatons, with
polarizations $\epsilon_{i}^{\mu\nu}=\epsilon_{i}^{\mu}\epsilon_{i}^{\nu}$,
and obtain the dilaton amplitudes in the very end by replacing
$\epsilon_{i}^{\mu\nu}$ with $\eta^{\mu\nu}$. By using explicitly transverse
three-point amplitudes and the completeness relation (92), we can write tree-
level unitarity (88) in the form

$\underset{k^{2}=-4n/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})=\left(\frac{(-\alpha^{\prime}q_{12}/2)_{n}}{n!}\right)^{2}(\epsilon_{1}\cdot\epsilon_{2})^{2}=\sum\limits_{\rho,i}A^{15P_{1}}_{n,\rho,i,\mathbf{m}}\pi^{\mathbf{m},\mathbf{n}}_{\rho}A^{52P_{2}}_{n,\rho,i,\mathbf{n}}\,,$
(235)

where for the massive levels, on which we will mostly focus, $\rho$ is summed
over irreducible representations of $SO(4)$ and $i$ is summed over degenerate
states in the same representation.

Our starting point will be the open string three-point amplitudes of a
massless vector, a Pomeron and an arbitrary massive state up to mass level 2
(we give some simpler explicit examples in Appendix A). These amplitudes were
computed in DAppollonio:2013mgj  by studying string-brane scattering. Since in
flat space there is no interaction between the left- and right-moving string
modes, the closed string amplitudes factorize into products of open string
amplitudes. We can indeed check that the square root of the residues (235)
matches the expansion in terms of the open string three-point amplitudes of
DAppollonio:2013mgj

$\sqrt{\underset{k^{2}=-4n/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})}=\frac{(-\alpha^{\prime}q_{12}/2)_{n}}{n!}(\epsilon_{1}\cdot\epsilon_{2})=\sum\limits_{\rho_{L}}A^{15P_{1}}_{n,\rho_{L},\bm{\alpha}}\pi^{\bm{\alpha},\bm{\gamma}}_{\rho_{L}}A^{52P_{2}}_{n,\rho_{L},\bm{\gamma}}\,.$
(236)

We did this consistency check for the first three mass levels, for which
$\rho_{L}$ is summed over the bosonic part (NS sector) of the chiral
superstring spectrum in 10 dimensions, given by Hanany:2010da

$\displaystyle n$ $\displaystyle=0:$
$\displaystyle\qquad\ydiagram{1}_{\,8}\,,$ (237) $\displaystyle n$
$\displaystyle=1:$
$\displaystyle\qquad\ydiagram{2}_{\,9}\oplus\,\ydiagram{1,1,1}_{\,9}\,,$
$\displaystyle n$ $\displaystyle=2:$
$\displaystyle\qquad\ydiagram{3}_{\,9}\oplus\,\ydiagram{2,1,1}_{\,9}\oplus\,\ydiagram{2,1}_{\,9}\oplus\,\ydiagram{1,1}_{\,9}\oplus\,\ydiagram{1}_{\,9}\,.$

In order to obtain three-point amplitudes for closed strings in $10D$, we need
to square (236) and expand again in irreducible representations. The first
step is trivial

$\underset{k^{2}=-4n/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})=\sum\limits_{\rho_{L},\rho_{R}}A^{15P_{1}}_{n,\rho_{L},\bm{\alpha}}\
A^{15P_{1}}_{n,\rho_{R},\bm{\beta}}\ \pi^{\bm{\alpha},\bm{\gamma}}_{\rho_{L}}\
\pi^{\bm{\beta},\bm{\delta}}_{\rho_{R}}\ A^{52P_{2}}_{n,\rho_{L},\bm{\gamma}}\
A^{52P_{2}}_{n,\rho_{R},\bm{\delta}}\,,$ (238)

however expanding this into irreducible representations requires some more
work. On an abstract level this is easily done in terms of the tensor product

$\rho_{L}\otimes\rho_{R}=\bigoplus\limits_{\rho_{C}}\rho_{C}\,,$ (239)

which can be computed explicitly in terms of characters using e.g. the
WeylCharacterRing implementation in SageMath sagemath . For example, the
closed string spectrum for the first two mass levels is

$\displaystyle n=0:\
\ydiagram{1}_{\,8}\otimes\,\ydiagram{1}_{\,8}=\ydiagram{2}_{\,8}\oplus\ydiagram{1,1}_{\,8}\oplus\bullet\,,$
(240) $\displaystyle n=1:\
\left(\ydiagram{2}_{\,9}\oplus\,\ydiagram{1,1,1}_{\,9}\right)^{2}=\ydiagram{2,2,2}_{\,9}\oplus\,\ydiagram{2,2,1,1}_{\,9}\oplus\,2\,\ydiagram{3,1,1}_{\,9}\oplus\,3\,\ydiagram{2,1,1,1}_{\,9}\oplus\,\ydiagram{4}_{\,9}$
$\displaystyle\oplus\,\ydiagram{3,1}_{\,9}\oplus\,2\,\ydiagram{2,2}_{\,9}\oplus\,\ydiagram{2,1,1}_{\,9}\oplus\,\ydiagram{1,1,1,1}_{\,9}\oplus\,2\,\ydiagram{2,1}_{\,9}\oplus\,3\,\ydiagram{1,1,1}_{\,9}\oplus\,2\,\ydiagram{2}_{\,9}\oplus\,2\,\ydiagram{1,1}_{\,9}\oplus\,2\,\bullet\,.$

To use the tensor product in explicit calculations requires considerably more
work and can be done by formulating (239) as an equation in terms of
projectors to irreducible representations

$\pi_{\rho_{L}}^{\bm{\alpha};\bm{\gamma}}\pi_{\rho_{R}}^{\bm{\beta};\bm{\delta}}=\sum_{\rho_{C}\subset\rho_{L}\otimes\rho_{R}}p^{\bm{\alpha}\bm{\beta}}_{\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\mu}}\pi_{\rho_{C}}^{\bm{\mu};\bm{\nu}}p^{\bm{\gamma}\bm{\delta}}_{\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\nu}}\,.$
(241)

The tensors $p_{\rho_{L}\otimes\rho_{R}\to\rho_{C}}$ are constructed from
Kronecker deltas and are uniquely determined by this equation. By inserting
(241) into (238) we find the expansion of the residue

$\underset{k^{2}=-4n/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})=\sum\limits_{\rho_{L},\rho_{R},\rho_{C}}A^{15P_{1}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\mu}}\pi^{\bm{\mu},\bm{\nu}}_{\rho_{C}}A^{52P_{2}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\nu}}\,,$
(242)

in terms of the closed string amplitudes

$A^{15P_{1}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\mu}}=A^{15P_{1}}_{n,\rho_{L},\bm{\alpha}}\
A^{15P_{1}}_{n,\rho_{R},\bm{\beta}}\
p^{\bm{\alpha}\bm{\beta}}_{\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\mu}}\,.$
(243)

The final step is to restrict the indices of the amplitudes to five dimensions
and expand once again into irreducible representations, this times for the
massive Little group $SO(4)$. In terms of representation theory, this is done
by using branching rules to expand the $SO(9)$ representations in terms of
irreps of the product $SO(4)\times SO(5)$,

$\rho_{C}=\bigoplus\limits_{(\rho,\sigma)\subset\rho_{C}}(\rho,\sigma)\,,$
(244)

where, for massive levels, $\rho$ is an irreducible representation of $SO(4)$
and $\sigma$ of $SO(5)$. Since we consider Pomeron exchange, which caries the
vacuum quantum numbers, we project onto the singlets of $SO(5)$

$\rho_{C}|_{\bullet_{5}}=\bigoplus\limits_{(\rho,\bullet)\subset\rho_{C}}(\rho,\bullet)\,.$
(245)

This step is also abstractly implemented in SageMath. For example, we have

$\ydiagram{2}_{\,9}=\Big{(}\ydiagram{2}_{\,4},\bullet_{5}\Big{)}\oplus\Big{(}\ydiagram{1}_{\,4},\ydiagram{1}_{\,5}\Big{)}\oplus\Big{(}\bullet_{4},\ydiagram{2}_{\,5}\Big{)}\oplus\Big{(}\bullet_{4},\bullet_{5}\Big{)}\,.$
(246)

and after projection to $SO(5)$ singlets

$\ydiagram{2}_{\,9}\Big{|}_{\bullet_{5}}=\ydiagram{2}_{\,4}\oplus\bullet_{4}\,.$
(247)

In this way we find the $SO(5)$ singlets for the closed string spectrum in
terms of $SO(3)$ or $SO(4)$ irreps for the first two levels

$\displaystyle n$ $\displaystyle=0:$
$\displaystyle\ydiagram{2}_{\,3}\oplus\,\ydiagram{1}_{\,3}\,\oplus\,\,2\,\bullet\,,$
$\displaystyle n$ $\displaystyle=1:$
$\displaystyle\ydiagram{4}_{\,4}\oplus\,\ydiagram{3,1}_{\,4}\oplus\,2\,\ydiagram{2,2}_{\,4}\oplus\,2\,\ydiagram{3}_{\,4}\oplus\,4\,\ydiagram{2,1}_{\,4}\oplus\,8\,\ydiagram{2}_{\,4}$
$\displaystyle\oplus\,5\,\ydiagram{1,1}_{\,4}\oplus\,10\,\ydiagram{1}_{\,4}\oplus\,9\,\bullet_{\,4}\,.$
(248)

As for the tensor product, we can rephrase (245) as an equation in terms of
projectors. In this case we get an equation for the $SO(9)$ projector with
indices restricted to the $SO(4)$ directions $a=1,\ldots,4$

$\pi_{\rho_{C}}^{\mathbf{a};\mathbf{b}}=\sum_{\rho\subset\rho_{C}|_{\bullet_{5}}}b^{\mathbf{a}}_{\rho_{C}\to\rho,\mathbf{m}}\
\pi_{\rho}^{\mathbf{m};\mathbf{n}}\
b^{\mathbf{b}}_{\rho_{C}\to\rho,\mathbf{n}}\,,$ (249)

where the tensors $b_{\rho_{C}\to\rho}$ are uniquely determined by this
equation and can be expressed in terms of Kronecker deltas and the $SO(4)$
Levi-Civita symbol. Since we are assuming the flat space limit kinematics to
be restricted to five dimensions, we can simply insert this into (242) and
obtain the residue in the anticipated form (235)

$\underset{k^{2}=-4n/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})=\sum\limits_{\rho_{L},\rho_{R},\rho_{C},\rho}A^{15P_{1}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C}\to\rho,\mathbf{m}}\pi^{\mathbf{m},\mathbf{n}}_{\rho}A^{52P_{2}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C}\to\rho,\mathbf{n}}\,,$
(250)

with the $5D$ closed string amplitudes given by

$A^{15P_{1}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C}\to\rho,\mathbf{m}}=A^{15P_{1}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C},\mathbf{a}}b^{\mathbf{a}}_{\rho_{C}\to\rho,\mathbf{m}}\,.$
(251)

####  7.1.1 Example

Let us now give a specific example of the procedure outlined above. We will
consider the following chain of expansions at mass level 1, starting from the
product of two open string massive spin 2 fields that give rise to a $5D$
scalar

$\ydiagram{2}_{\,9}\otimes\ydiagram{2}_{\,9}\to\ydiagram{2}_{\,9}\to\bullet_{4}\,.$
(252)

In this example we discard the $5D$ massive spin 2 field that also appears in
the projection (247). We alert the reader that whenever we write explicit
amplitudes they are neither appropriately symmetrized nor traceless in order
to write them more compactly. All explicit amplitudes should be understood as
objects to be contracted with the projector for the associated representation.

We start with the open string amplitude for the state $(1,\ydiagram{2}_{\,9})$
from DAppollonio:2013mgj

$A^{15P_{1}}_{1,[2]_{9},\alpha_{1}\alpha_{2}}=-\sqrt{\frac{\alpha^{\prime}}{2}}\left(\epsilon_{1\alpha_{1}}q_{1\alpha_{2}}+\frac{1}{2}(q_{1}\cdot\epsilon_{1})\,v_{\alpha_{1}}v_{\alpha_{2}}\right)\,.$
(253)

Squaring this amplitude produces the following closed string states

$\ydiagram{2}_{\,9}\otimes\ydiagram{2}_{\,9}=\ydiagram{4}_{\,9}\oplus\ydiagram{3,1}_{\,9}\oplus\ydiagram{2,2}_{\,9}\oplus\ydiagram{1,1}_{\,9}\oplus\ydiagram{2}_{\,9}\oplus\bullet_{\,9}\,.$
(254)

To construct the relation (241) we need the projectors for all the
representations in this list. They can be found in Costa:2016hju ;
Costa:2018mcg , however here we just remind the reader of two of the most
familiar ones

$\pi_{[2]_{d}}^{\mu_{1}\mu_{2},\nu_{1}\nu_{2}}=\frac{1}{2}\big{(}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\nu_{2}}+\eta^{\mu_{1}\nu_{2}}\eta^{\mu_{2}\nu_{1}}\big{)}-\frac{1}{d}\,\eta^{\mu_{1}\mu_{2}}\eta^{\nu_{1}\nu_{2}}\,,\qquad\pi_{\bullet}=1\,,$
(255)

and state that the closed string state $\ydiagram{2}_{\,9}$ comes in (241)
with the tensor

$p^{\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}}_{[2]_{9}\otimes[2]_{9}\to[2]_{9},\mu_{1}\mu_{2}}=\sqrt{\frac{36}{91}}\,\pi_{[2]_{9}}^{\alpha_{1}\alpha_{2};\gamma_{1}\gamma_{2}}\pi_{[2]_{9}}^{\beta_{1}\beta_{2};\delta_{1}\delta_{2}}\eta_{\gamma_{2}\delta_{1}}\eta_{\gamma_{1}\mu_{1}}\eta_{\delta_{1}\mu_{2}}\,,$
(256)

where we introduced additional projectors contracted with metrics in order to
have the correct index properties. This determines the following closed string
amplitude via (243)

$\displaystyle A^{15P_{1}}_{1,[2]_{9}\otimes[2]_{9}\to[2]_{9},\mu_{1}\mu_{2}}$
$\displaystyle=\frac{\alpha^{\prime}}{4\sqrt{91}}\Big{[}\big{(}q_{1,\mu_{1}}(\epsilon_{1,\mu_{2}}q_{1}\cdot\epsilon_{1}+3q_{1,\mu_{2}})$
(257)
$\displaystyle+\epsilon_{1,\mu_{1}}(q_{1,\mu_{2}}q_{1}\cdot\epsilon_{1}-3t_{1}\epsilon_{1,\mu_{2}})+v_{\mu_{1}}v_{\mu_{2}}(q_{1}\cdot\epsilon_{1}){}^{2}\big{)}\Big{]}\,.$

The branching rule for $\ydiagram{2}_{\,9}$ was already considered in (247).
Using the projectors (255) it is easy to see that we can write explicitly

$\pi_{[2]_{9}}^{a_{1}a_{2},b_{1}b_{2}}=\delta^{a_{1}}_{m_{1}}\delta^{a_{2}}_{m_{2}}\pi_{[2]_{4}}^{m_{1}m_{2},n_{1}n_{2}}\delta^{b_{1}}_{n_{1}}\delta^{b_{2}}_{n_{2}}+\frac{5}{36}\,\delta^{a_{1}a_{2}}\pi_{\bullet}\delta^{b_{1}b_{2}}\,,$
(258)

from which we read off

$b^{a_{1}a_{2}}_{[2]_{9}\to[2]_{4},m_{1}m_{2}}=\delta^{a_{1}}_{m_{1}}\delta^{a_{2}}_{m_{2}}\,,\qquad
b^{a_{1}a_{2}}_{[2]_{9}\to\bullet_{4}}=\sqrt{\frac{5}{36}}\,\delta^{a_{1}a_{2}}\,.$
(259)

Finally, inserting (257) and (259) into (251) we compute the $5D$ closed
string amplitude

$A^{15P_{1}}_{1,[2]_{9}\otimes[2]_{9}\to[2]_{9}\to\bullet_{4}}=\frac{1}{8}\sqrt{\frac{5}{91}}\alpha^{\prime}\left((q_{1}\cdot\epsilon_{1})^{2}-2t_{1}\right)$
(260)

We derive the complete list of such level 1 three-point amplitudes in
appendices B and C.

###  7.2 Constraints on spinning AdS amplitudes

In this section we use the flat-space string amplitudes to constrain the high-
energy, tree-level AdS5 amplitudes with two dilatons and two spinning
operators $\langle\phi\phi\mathcal{O}_{5}\mathcal{O}_{6}\rangle$. Since the
operators in question are of stringy nature (i.e. the bulk fields have
$m^{2}\sim 4n/\alpha^{\prime}$ for a very large AdS radius), their dimensions
grow with the ’t Hooft coupling ($\Delta\sim\lambda^{1/4}$) and transform in
the $SO(4)$ representations discussed above in the flat space case. This means
that generically $\mathcal{O}_{5}$ and $\mathcal{O}_{6}$ are in bosonic mixed-
symmetry representations.

As discussed in section 6.1, the spectral functions that determine the
spinning AdS correlators (158) via (159) are determined in the flat space
limit by the three-point amplitudes (97)

$\lambda^{\frac{j(\nu)-1}{2}}\lambda^{\frac{|\rho_{5}|-k_{5}}{4}}\lambda^{\frac{|\rho_{6}|-k_{6}}{4}}\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)\
\to\
\frac{1}{2\pi}\,a_{m_{5},\rho_{5}}^{k_{5}}(t)\,a_{m_{6},\rho_{6}}^{k_{6}}(t)\beta(t)\,.$
(261)

In other words, the leading term in $\nu$ at each order in $\lambda$ is fixed
by the flat space expression (see section 6.2)

$\lambda^{\frac{j(\nu)-1}{2}}\lambda^{\frac{|\rho_{5}|-k_{5}}{4}}\lambda^{\frac{|\rho_{6}|-k_{6}}{4}}\beta^{k_{5},k_{6}}_{(\Delta_{5},\rho_{5}),(\Delta_{6},\rho_{6})}(\nu)=\frac{1}{2\pi}a_{m_{5},\rho_{5}}^{k_{5}}(t)\,a_{m_{6},\rho_{6}}^{k_{6}}(t)\beta(t)\Big{|}_{\alpha^{\prime}t=-\frac{\nu^{2}}{\sqrt{\lambda}}}+\ldots\,,$
(262)

where $\ldots$ are terms that vanish in the flat space limit. Let us now use
this to study some specific examples. We begin with the example of section
7.1.1 where we computed an amplitude involving a graviton, a Pomeron and a
particular scalar at mass level 1 in (260). In this result $\epsilon_{1,\mu}$
is such that $\epsilon_{\mu\nu}=\epsilon_{1,\mu}\epsilon_{1,\nu}$ parametrizes
a general graviton polarization which must be replaced by the metric
$\epsilon_{\mu\nu}\rightarrow\eta_{\mu\nu}$ to obtain the dilaton amplitude

$A_{1,[2]_{9}\otimes[2]_{9}\rightarrow[2]_{9}\rightarrow\bullet_{4}}^{D5P_{1}}=a_{4/\alpha^{\prime},[2]_{9}\otimes[2]_{9}\rightarrow[2]_{9}\rightarrow\bullet_{4}}^{0}(t)=-\frac{3}{8}\sqrt{\frac{5}{91}}\,\alpha^{\prime}t\,,$
(263)

where we used that $(q_{1}\cdot\epsilon_{1})^{2}=-t_{1}$ for the dilaton case.
Thus, for the AdS correlator of this particular scalar and three dilatons we
have

$\beta^{0,0}_{(2\lambda^{\frac{1}{4}}+\ldots,\bullet),(4,\bullet)}(\nu)=\frac{3}{8}\sqrt{\frac{5}{91}}\frac{\nu^{2}}{\sqrt{\lambda}}\,\beta(\nu)+\text{vanishing
in the flat space limit}\,.$ (264)

Note that with respect to the case of four dilaton scattering we have an extra
power of $1/\sqrt{\lambda}$, so the term of order $\lambda^{0}$ is absent,
confirming that tidal excitations are suppressed at large $\lambda$, which in
turn agrees with the considerations of Meltzer:2019pyl  (we verified this fact
for all amplitudes at level 1). In particular, this is consistent with the
large $\lambda$ suppression of $c_{\phi_{1}\phi_{2}j(\nu)}$ for non-identical
scalars, since our stringy mode is certainly different from the dilaton. Such
a suppression is not a priori obvious from writing a bulk interaction between
two different scalars and a spin $J$ field (to be Sommerfeld-Watson
transformed into a Pomeron), which makes this a non-trivial realization of the
bounds derived in Costa:2017twz ; Meltzer:2017rtf . This example is
particularly simple, since there is a unique three-point structure in the case
of the scalar.

More generally, we can consider amplitudes with several tensor structures
constructed from $v_{a}$’s and $q_{a}$’s (equivalently, $\widehat{p}$ and
$\nabla_{p}$ in AdS). Let us take as a representative example, the case of the
spin 4 operator at level 1. This operator is typically used to define
$\Delta_{\text{gap}}$ and sits in the leading Regge trajectory. The
corresponding graviton-Pomeron-spin 4 amplitude was worked out in Appendices B
and C, and reads

$A_{[2]_{9}\otimes[2]_{9}\rightarrow[4]_{9}\rightarrow[4]_{4}}^{a_{1}a_{2}a_{3}a_{4}}(\epsilon_{1})=\frac{1}{8}\alpha^{\prime}\big{(}2\epsilon_{1,a_{1}}q_{1,a_{2}}+v_{a_{1}}v_{a_{2}}q_{1}\cdot\epsilon_{1}\big{)}\big{(}2\epsilon_{1,a_{3}}q_{1,a_{4}}+v_{a_{3}}v_{a_{4}}q_{1}\cdot\epsilon_{1}\big{)}\,.$
(265)

We again emphasize that it is understood that the amplitude should be
contracted with $\pi_{[4]_{4}}^{\bm{\alpha};\bm{\beta}}$. Furthermore, we are
using the simplifying transverse kinematics discussed above. This means that
upon doing the dilaton replacement, we have

$A_{[2]_{9}\otimes[2]_{9}\rightarrow[4]_{9}\rightarrow[4]_{4}}^{a_{1}a_{2}a_{3}a_{4}}=\frac{\alpha^{\prime}}{8}\big{(}-t_{1}v_{a_{1}}v_{a_{2}}v_{a_{3}}v_{a_{4}}+4v_{a_{1}}v_{a_{2}}q_{1,a_{3}}q_{1,a_{4}}\big{)}\,,$
(266)

where we used the symmetry of the indices and note that transverse kinematics
ensures that the term proportional to $\epsilon_{1,a_{1}}\epsilon_{1,a_{2}}$
gets mapped to a transverse metric which is annihilated by the projector to
$[4]_{4}$. In this case we can directly use (216) to match both tensor
structures at once

$\displaystyle\mathfrak{D}^{(2\lambda^{\frac{1}{4}}+\ldots,[4]_{4}),(4,\bullet)}_{a_{1}a_{2}a_{3}a_{4}}(\nu)={}$
$\displaystyle\frac{\beta(\nu)}{8\sqrt{\lambda}}\big{(}\nu^{2}\widehat{p}_{a_{1}}\widehat{p}_{a_{2}}\widehat{p}_{a_{3}}\widehat{p}_{a_{4}}+4\widehat{p}_{a_{1}}\widehat{p}_{a_{2}}\nabla_{p\,a_{3}}\nabla_{p\,a_{4}}\big{)}$
(267) $\displaystyle+\text{vanishing in flat space limit}\,.$

We again note that these corrections are suppressed at large $\lambda$.

##  8 Conclusions

In this work, we derived a perturbative CFT optical theorem which computes the
dDisc of a correlator in the $1/N$ expansion in terms of single
discontinuities of lower order correlators. Notably, this allows the
determination of double-trace contributions to a given one-loop holographic
correlator, even when the intermediate fields have spin, which makes them much
harder to handle using unitarity formulas in terms of the CFT data. This also
clarifies the underlying CFT principles behind cutting formulas for AdS Witten
diagrams, which so far used bulk quantities Meltzer:2019nbs ; Meltzer:2020qbr
.

Using the perturbative CFT optical theorem we fixed the form of the AdS one-
loop four-point scattering amplitude in the high-energy limit, accounting for
the physical effect of tidal excitations. This corresponds to box Witten
diagrams with two-Pomeron exchange and general string fields as intermediate
states. To do this, we transformed the optical theorem to CFT impact parameter
space, in which the loop level phase shift is obtained as a contraction of
tree-level phase shifts. Using the general structure of spinning correlators
in the s-channel Regge limit, we rewrote all the tidal excitations in terms of
a single scalar function, the AdS vertex function.

For the case of ${\cal N}=4$ SYM, dual to type IIB strings, we fixed part of
the answer by relating our expression to the flat space results of
Amati:1987wq ; Amati:1987uf ; Amati:1988tn  for high energy string scattering,
requiring consistency with the flat space limit in impact parameter space.
This procedure fixes part of the AdS vertex function and therefore also part
of the CFT correlation function at one-loop in the Regge limit. Additionally,
interpreting the previous result in terms of unitarity, we used the flat-space
behavior to constrain the spectral function for certain spinning CFT
correlators at tree level in the Regge limit.

There are several open directions and applications of this work. First, we
emphasize that the CFT optical theorem is quite general and does not rely on
AdS ingredients. Moreover, it works directly at the level of correlators
instead of having to extract the CFT data, which is very difficult to resum
into correlators. It would be interesting to test and use this formula for
more general holographic correlators and, since the expansion parameter does
not necessarily need to be $1/N$, in weakly coupled CFTs such as $\phi^{4}$
theory at the Wilson-Fisher fixed point in $4-\epsilon$ dimensions.

Another playground to apply our gluing formula is $\mathcal{N}=4$ SYM at weak
t’Hooft coupling in the Regge limit. One could try to derive the order
$1/N^{4}$ correlator explicitly at leading order in $\lambda$, using the
techniques introduced in Cornalba:2008qf . The corresponding double
discontinuity should be the square of order $1/N^{2}$ correlators with impact
factors that include the intermediate states.

In the Regge limit there are kinematical conditions in the CFT optical theorem
that simplified the integrations over Lorentzian configurations. An
interesting generalization would be to systematically study kinematic
corrections to the Regge limit. In fact, in the recent work Caron-Huot:2020nem
the authors derived a Regge expansion for the correlator valid for any boost.
It would be interesting to see how to incorporate this into our analysis, both
in a general structural way, but also potentially to impose specific
constraints from the flat space limit in a more general kinematic setup. More
generally, it would be interesting to understand the Regge limit integrations
in terms of light-ray operators Kravchuk:2018htv , and to use the more general
Lorentzian machinery of Simmons_Duffin_2018 ; Kravchuk:2018htv  to write an
intrinsically Lorentzian gluing formula in general kinematics.

A possible extension of this work is to consider a higher number of bulk
loops. This was analyzed in the large $\Delta_{\text{gap}}$ limit in
Meltzer:2019pyl . Let us give a few concrete ideas for the stringy
generalization of that analysis. The leading contribution in the Regge limit
at $k-1$ loops is expected to be $k$-Pomeron exchange, related to a $k$-fold
product of tree-level phase shifts. By repeatedly using (162) one can define a
generalization of the function $\Phi$ for such a product, so we expect that
the contribution of intermediate states can again be expressed by a vertex
function131313This corresponds to eikonalization in the operator sense of
Amati:1987uf  where the phase shift is an operator in the string Hilbert
space, with matrix elements between all possible string states.

$\displaystyle-\mathop{\mathrm{Re}}\mathcal{B}_{k-1}(S,L)=\int\limits_{-\infty}^{\infty}d\nu\left(\prod\limits_{n=1}^{k}d\nu_{n}\,\beta^{(*)}(\nu_{n})\right)$
$\displaystyle V(\nu_{1},\ldots,\nu_{k},\nu)^{2}\,S^{\sum_{m}j(\nu_{m})-k}$
(268) $\displaystyle\Phi(\nu_{1},\ldots,\nu_{k},\nu)\,\Omega_{i\nu}(L)\,,$

where the product of $\beta(\nu_{n})$ must be real, which means that the
answer is slightly different depending on whether the number of loops is even
or odd Meltzer:2019pyl . In order to find the flat space limit of this
$(k-1)$-loop vertex function we would need to consider the flat space result
for higher loops, which is known at least in integral form Amati:1987wq ;
Amati:1987uf ; Amati:1988tn

$V_{k}(q_{1},\dots,q_{k})=\int\prod_{i=1}^{k}\frac{d\sigma_{i}}{2\pi}\prod_{1\leq
j<l\leq k}|e^{i\sigma_{j}}-e^{i\sigma_{l}}|^{\alpha^{\prime}q_{l}\cdot
q_{j}}\,.$ (269)

Note that the symmetry of the integrand is such that only $k-1$ integrals are
non-trivial. For example, the one-loop case just gives

$V_{2}(q_{1},q_{2})=\int\frac{d\sigma_{1}d\sigma_{2}}{(2\pi)^{2}}\,|e^{i\sigma_{1}}-e^{i\sigma_{2}}|^{\alpha^{\prime}q_{12}}=\int\frac{d\sigma}{2\pi}|1-e^{i\sigma}\,|^{\alpha^{\prime}q_{12}}=\frac{\Gamma\big{(}1+\alpha^{\prime}\frac{t_{1}+t_{2}-t}{2}\big{)}}{\Gamma\big{(}1+\alpha^{\prime}\frac{t_{1}+t_{2}-t}{4}\big{)}^{2}},$
(270)

recovering (232).

##  9 Acknowledgements

We would like to thank David Meltzer for discussions on conformal Regge theory
and Fernando Alday for comments on the draft. This research received funding
from the Simons Foundation grants 488637 (Simons collaboration on the Non-
perturbative bootstrap). Centro de Física do Porto is partially funded by
Fundação para a Ciência e a Tecnologia (FCT) under the grant UID-04650-FCUP.
AA is funded by FCT under the IDPASC doctoral program with the fellowship
PD/BD/135436/2017. TH received funding from the Knut and Alice Wallenberg
Foundation grant KAW 2016.0129, the VR grant 2018-04438 and the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant No. 787185).

##  Appendix A Additional examples of string amplitudes

In this appendix we provide some additional examples and comments on the
chiral and closed string amplitudes.

###  A.1 Chiral Amplitudes

In the chiral case it is trivial to reproduce level 0. Here we have only the
massless particles with residue

$\sqrt{\underset{k^{2}=0}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})}=\epsilon_{1}^{\mu}A^{15P_{1}}_{\mu\alpha}\pi_{[1]_{8}}^{\alpha;\beta}A^{52P_{2}}_{\beta\nu}\epsilon_{2}^{\nu}=\epsilon_{1}^{\mu}\eta_{\mu\alpha}\eta^{\alpha\beta}\eta_{\beta\nu}\epsilon_{2}^{\nu}=\epsilon_{1}\cdot\epsilon_{2}\,,$
(271)

where we have used $A^{15P_{1}}_{\mu\alpha}=\eta_{\mu\alpha}$ and
$\pi_{[1]_{8}}^{\alpha;\beta}=\eta^{\alpha\beta}$. At higher levels we will
have non-trivial three-point functions. It will be convenient to absorb the
external polarization into the amplitude

$\epsilon_{1}^{\mu}A^{15P_{1}}_{n,\,\mu,\rho_{5},\bm{\nu}}\equiv
A^{15P_{1}}_{n,\rho_{5},\bm{\nu}}\,,$ (272)

to be more compact in writing the amplitudes (we are using the integer $n$ to
label the mass level of the state).

From the spectrum described above, we will have two amplitudes at level 1
which are $A^{15P_{1}}_{1,[1,1,1]_{9},\bm{\alpha}}$ and
$A^{15P_{1}}_{1,[2]_{9},\bm{\alpha}}$.
Here we will keep in mind the Young diagrams explained above, along with the
index symmetrization that comes with them, packaged in our boldface multi-
index notation. The explicit level 1 three point amplitudes in the IIB
superstring are

$A^{15P_{1}}_{1,[1,1,1]_{9},\bm{\alpha}}=\frac{\sqrt{6}}{m_{1}}\epsilon_{1\alpha_{1}}q_{1\alpha_{2}}v_{\alpha_{3}}~{},~{}A^{15P_{1}}_{1,[2]_{9},\bm{\alpha}}=-\sqrt{\frac{\alpha^{\prime}}{2}}\left(\epsilon_{1\alpha_{1}}q_{1\alpha_{2}}+\frac{1}{2}(q_{1}\cdot\epsilon_{1})v_{\alpha_{1}}v_{\alpha_{2}}\right)\,,$
(273)

with $m_{1}=2/\sqrt{\alpha^{\prime}}$ being the mass at level 1, and $q_{1}$
is the transverse momentum carried by the Pomeron $P_{1}$ (similarly for
$q_{2}$ and the Pomeron $P_{2}$). All the relative factors between the
different tensor structures and the overall normalization are fixed by
computing the three-point amplitudes with the correctly normalized vertex
operators for the excited NS states in IIB super string theory
DAppollonio:2013mgj . We can now contract the three-point amplitudes on each
side using the projector for the appropriate representation and check the
residue

$\displaystyle\sqrt{\underset{k^{2}=-4/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})}$
$\displaystyle=A^{15P_{1}}_{1,\bm{\alpha}}\pi_{[1,1,1]_{9}}^{\bm{\alpha};\bm{\beta}}A^{52P_{1}}_{1,\bm{\beta}}+A^{15P_{2}}_{1,\bm{\alpha}}\pi_{[2]_{9}}^{\alpha;\bm{\beta}}A^{52P_{2}}_{1,\bm{\beta}}$
(274) $\displaystyle=\frac{\alpha^{\prime}}{2}(-q_{1}\cdot
q_{2})(\epsilon_{1}\cdot\epsilon_{2}),$

where we refrained from writing the representation labels in the amplitudes
since they are contracted with a projector with the appropriate label. This
matches what we extracted from $A^{\alpha_{1}\alpha_{2}}(q_{12})$, or
equivalently from the vertex function. We can continue this procedure to the
second level, where mixed symmetry tensors appear for the first time. For
example, the $[2,1,1]_{9}$ tensor has the amplitude

$A^{15P_{1}}_{2,[2,1,1]_{9},\bm{\alpha}}=\sqrt{\frac{3}{8}}\sqrt{\frac{\alpha^{\prime}}{2}}\left(\frac{4}{m_{2}}q_{1\alpha_{2}}+2v_{\alpha_{2}}\right)\epsilon_{1\alpha_{1}}q_{1\alpha_{3}}v_{\alpha_{4}},$
(275)

where $m_{2}=\sqrt{8/\alpha^{\prime}}$ is the mass at level 2. It is important
to emphasize that the level 2 amplitude contains a term with more powers of
$\alpha^{\prime}$ than any of the level 1 amplitudes. This would lead to
further suppression in $1/\sqrt{\lambda}$ in the AdS theory.141414Clearly,
states with higher spin, which can only appear at higher levels, can have
higher powers of $\alpha^{\prime}$ leading to a spin-dependent suppression of
couplings, as is expected from the general arguments of Costa:2017twz ;
Meltzer:2017rtf .

The remaining amplitudes can be found in section 5 of DAppollonio:2013mgj .
For our purposes it is just important to know that the amplitudes satisfy

$\displaystyle\sqrt{\underset{k^{2}=-8/\alpha^{\prime}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k,q_{12})}$
$\displaystyle=\sum_{\rho\in
S}A^{15P_{1}}_{2,\rho,\bm{\alpha}}\pi_{\rho}^{\bm{\alpha};\bm{\beta}}A^{52P_{2}}_{2,\rho,\bm{\beta}}=\frac{(-\frac{\alpha^{\prime}}{2}q_{1}\cdot
q_{2})_{2}}{2!}(\epsilon_{1}\cdot\epsilon_{2})\,,$ (276) $\displaystyle S$
$\displaystyle=\left\\{[3]_{9}~{},~{}[2,1,1]_{9}~{},~{}[2,1]_{9}~{},~{}[1,1]_{9}~{},~{}[1]_{9}\right\\},$

which we explicitly checked. More generally, we can conclude that the square
root of the residue at mass level $n$ of $A^{12P_{1}P_{2}}(k,q_{12})$ can be
recovered by unitarity if we account for all the covariant SO(9)
representations corresponding to the massive NS states. This gives a
microscopic interpretation for the vertex function at a given mass level. As
already mentioned, summing over all these mass levels reconstructs the full
vertex function.

###  A.2 Closed string amplitudes

Here we consider the simple but instructive level 0 case for the closed string
amplitudes, where the little group is SO(8). The square of the residue reads

$\displaystyle
A^{15P_{1}}_{L1,\alpha}A^{15P_{1}}_{R1,\beta}\;\left(\pi_{[1]_{8}}^{\alpha;\gamma}\pi_{[1]_{8}}^{\beta;\delta}\right)\;A^{52P_{2}}_{L1,\gamma}A^{52P_{2}}_{R1\delta}$
(277)
$\displaystyle=\sum_{\rho_{C}=[2]_{8},[1,1]_{8},\bullet_{8}}A^{15P_{1}}_{L1,\alpha}A^{15P_{1}}_{R1,\beta}(p^{\alpha\beta}_{[1]_{8}\otimes[1]_{8}\to\rho_{C},\mu_{1}\mu_{2}}\pi_{\rho_{C}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}p^{\gamma\delta}_{[1]_{8}\otimes[1]_{8}\to\rho_{C},\nu_{1}\nu_{2}})A^{52P_{2}}_{L1,\gamma}A^{52P_{2}}_{R1\delta}=(\epsilon_{1}\cdot\epsilon_{2})^{2}\,,$

where we used the group theoretical tensor product identity for projectors

$\pi_{[1]_{8}}^{\alpha;\gamma}\pi_{[1]_{8}}^{\beta;\delta}\;=\sum_{\rho_{C}=[2]_{8},[1,1]_{8},\bullet_{8}}p^{\alpha\beta}_{[1]_{8}\otimes[1]_{8}\to\rho_{C},\mu_{1}\mu_{2}}\pi_{\rho_{C}}^{\mu_{1}\mu_{2};\nu_{1}\nu_{2}}p^{\gamma\delta}_{[1]_{8}\otimes[1]_{8}\to\rho_{C},\nu_{1}\nu_{2}}\,.$
(278)

We can solve this equation for the tensors $p$ by contracting with
polarization vectors for the left and right modes on both sides of the
projector ($z_{L},z_{R}$ and $\overline{z}_{L},\overline{z}_{R}$) and equating
the polynomials in scalar products of $z$’s. In practice we will always use
this procedure, or a similar one where we contract with amplitudes to fix
coefficients. In this case it is trivial to directly check that

$\displaystyle\pi_{[1]_{8}}^{\alpha;\gamma}\pi_{[1]_{8}}^{\beta;\delta}\;$
$\displaystyle=\;\eta^{\alpha\gamma}\eta^{\beta\delta}\;\equiv\;\pi_{[2]_{8}}^{\alpha\beta;\gamma\delta}+\pi_{[1,1]_{8}}^{\alpha\beta;\gamma\delta}+\frac{1}{8}\eta^{\alpha\beta}\eta^{\gamma\delta}$
(279)
$\displaystyle=\left(\frac{1}{2}(\eta^{\alpha\gamma}\eta^{\beta\delta}+\eta^{\alpha\delta}\eta^{\beta\gamma})-\frac{1}{8}\eta^{\alpha\beta}\eta^{\gamma\delta}\right)+\frac{1}{2}(\eta^{\alpha\gamma}\eta^{\beta\delta}-\eta^{\alpha\delta}\eta^{\beta\gamma})+\frac{1}{8}(\eta^{\alpha\beta}\eta^{\gamma\delta})\,,$

where, obviously
$p^{\alpha\beta}_{[1]_{8}\otimes[1]_{8}\to[2]_{8},\mu_{1}\mu_{2}}=\delta^{\alpha}_{\mu_{1}}\delta^{\beta}_{\mu_{2}}~{},~{}p^{\alpha\beta}_{[1]_{8}\otimes[1]_{8}\to[1,1]_{8},\mu_{1}\mu_{2}}=\delta^{\alpha}_{\mu_{1}}\delta^{\beta}_{\mu_{2}}$
and
$p^{\alpha\beta}_{[1]_{8}\otimes[1]_{8}\to\bullet_{8}}=\sqrt{\frac{1}{8}}\eta^{\alpha\beta}$
extracts traces, thereby projecting to a singlet state.

##  Appendix B Tensor products for projectors

In this appendix we explain how to realize the tensor product of open string
states into closed string states in terms of the corresponding
projectors/tensors. We will consider mass level $n=1$. The chiral spectrum at
this level is

$n=1:\quad\ydiagram{2}_{\,9}\oplus\,\ydiagram{1,1,1}_{\,9}\,.$ (280)

We square the irreps using the tensor product as in the main text and analyze
the decomposition term by term. For example, taking
$\rho_{L}=\rho_{R}=[2]_{9}$ corresponds to the tensor product

$\ydiagram{2}_{\,9}\otimes\,\ydiagram{2}_{\,9}=\ydiagram{4}_{\,9}\oplus\,\ydiagram{3,1}_{\,9}\oplus\,\ydiagram{2,2}_{\,9}\oplus\,\ydiagram{2}_{\,9}\oplus\,\ydiagram{1,1}_{\,9}\oplus\,\bullet_{\,9}\,,$
(281)

which we want to write in terms of SO(9) tensors as

$\displaystyle\pi_{[2]_{9}}^{\bm{\alpha};\bm{\gamma}}\pi_{[2]_{9}}^{\bm{\beta};\bm{\delta}}$
$\displaystyle=\sum_{\rho_{C}\in
S}p^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[2]_{9}\to\rho_{C},\bm{\mu}}\pi_{\rho_{C}}^{\bm{\mu};\bm{\nu}}p^{\bm{\gamma}\bm{\delta}}_{[2]_{9}\otimes[2]_{9}\to\rho_{C},\bm{\nu}}\,,$
(282) $\displaystyle S$
$\displaystyle=\left\\{[4]_{9}\,,[3,1]_{9}\,,[2,2]_{9}\,,[2]_{9}\,,[1,1]_{9}\,,\bullet_{9}\right\\}\,.$

It will be useful to manifestly symmetrize the $\bm{\alpha}$ and $\bm{\beta}$
indices of the tensors $p$, in order to write down these tensors more
compactly. Therefore, we will use

$p^{\bm{\alpha}\bm{\beta}}_{\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\mu}}\equiv\pi^{\bm{\alpha}}_{\rho_{L};\bm{\alpha}^{\prime}}\pi^{\bm{\beta}}_{\rho_{R};\bm{\beta}^{\prime}}\tilde{p}^{\bm{\alpha}^{\prime}\bm{\beta}^{\prime}}_{\rho_{L}\otimes\rho_{R}\to\rho_{C},\bm{\mu}},$
(283)

and we will present the simpler trial tensors $\tilde{p}$ for each case. In
this case $\rho_{L}=\rho_{R}=[2]_{9}$ and we used the trial tensors151515For
irreps with the same number of indices as the tensor product we will always
take
$\tilde{p}^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}\propto\delta^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}$.

$\displaystyle\tilde{p}^{\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}}_{[2]_{9}\otimes[2]_{9}\to[4]_{9},\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$
$\displaystyle=\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\alpha_{2}}_{\mu_{2}}\delta^{\beta_{1}}_{\mu_{3}}\delta^{\beta_{2}}_{\mu_{4}}\equiv\delta^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}\,,$
$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[2]_{9}\to[3,1]_{9},\bm{\mu}}$
$\displaystyle=\delta^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}\,,$ (284)
$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[2]_{9}\to[2,2]_{9},\bm{\mu}}$
$\displaystyle=\sqrt{3}\delta^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}\,,$
$\displaystyle\tilde{p}^{\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}}_{[2]_{9}\otimes[2]_{9}\to[2]_{9},\mu_{1}\mu_{2}}$
$\displaystyle=\sqrt{\frac{36}{91}}\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\beta_{2}}_{\mu_{2}}\eta^{\alpha_{2}\beta_{1}}\,,$
$\displaystyle\tilde{p}^{\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}}_{[2]_{9}\otimes[2]_{9}\to[1,1]_{9},\mu_{1}\mu_{2}}$
$\displaystyle=\sqrt{\frac{4}{11}}\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\beta_{2}}_{\mu_{2}}\eta^{\alpha_{2}\beta_{1}}\,,$
$\displaystyle\tilde{p}^{\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}}_{[2]_{9}\otimes[2]_{9}\to\bullet_{9}}$
$\displaystyle=\sqrt{\frac{1}{44}}\eta^{\alpha_{1}\beta_{1}}\eta^{\alpha_{2}\beta_{2}}\,.$

The remaining tensor products have some additional subtleties. Taking the
cross term in the tensor product

$\ydiagram{2}_{\,9}\otimes\,\ydiagram{1,1,1}_{\,9}=\ydiagram{1,1,1}_{\,9}\oplus\,\ydiagram{2,1}_{\,9}\oplus\,\ydiagram{2,1,1,1}_{\,9}\oplus\,\ydiagram{3,1,1}_{\,9}\,,$
(285)

we note that there is a 4 row tensor appearing. When contracted with the
amplitudes, this contribution will vanish, because we have only 3 independent
vectors. However, from the point of view of the projector equation, we must
still have

$\displaystyle\pi_{[2]_{9}}^{\bm{\alpha};\bm{\gamma}}\pi_{[1,1,1]_{9}}^{\bm{\beta};\bm{\delta}}$
$\displaystyle=\sum_{\rho_{C}\in
S^{\prime}}p^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[1,1,1]_{9}\to\rho_{C},\bm{\mu}}\pi_{\rho_{C}}^{\bm{\mu};\bm{\nu}}p^{\bm{\gamma}\bm{\delta}}_{[2]_{9}\otimes[1,1,1]_{9}\to\rho_{C},\bm{\nu}}\,,$
(286) $\displaystyle S^{\prime}$
$\displaystyle=\left\\{[3,1,1]_{9}\,,[1,1,1]_{9}\,,[2,1]_{9}\,,[2,1,1,1]_{9}\right\\}\,.$

with a non-vanishing
$p^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[1,1,1]_{9}\to[2,1,1,1]_{9},\bm{\mu}}$.
However, by contracting directly with the amplitudes
$A^{\bm{\alpha}}_{[2]_{9}}A^{\bm{\beta}}_{[1,1,1]_{9}}A^{\bm{\gamma}}_{[2]_{9}}A^{\bm{\delta}}_{[1,1,1]_{9}}$
we automatically eliminate the contribution of this tensor (this also avoids
the computation of a complicated 4 row projector). With this in mind, we use
again (283), with the trial projectors

$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[1,1,1]_{9}\to[3,1,1],\bm{\mu}}$
$\displaystyle=\sqrt{\frac{5}{4}}\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\alpha_{2}}_{\mu_{2}}\delta^{\beta_{1}}_{\mu_{3}}\delta^{\beta_{2}}_{\mu_{4}}\delta^{\beta_{3}}_{\mu_{5}}\equiv\sqrt{\frac{5}{4}}\delta^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}\,,$
(287)
$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[1,1,1]_{9}\to[1,1,1],\bm{\mu}}$
$\displaystyle=\sqrt{\frac{585}{352}}\eta^{\alpha_{2}\beta_{1}}\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\beta_{2}}_{\mu_{2}}\delta^{\beta_{3}}_{\mu_{3}}\,,\,\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[1,1,1]_{9}\to[2,1],\bm{\mu}}=\eta^{\alpha_{2}\beta_{1}}\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\beta_{2}}_{\mu_{2}}\delta^{\beta_{3}}_{\mu_{3}}\,.$

With these tensors and setting
$p^{\bm{\alpha}\bm{\beta}}_{[2]_{9}\otimes[1,1,1]_{9}=[2,1,1,1]_{9},\bm{\mu}}\to
0$, which suffices for our purposes, we have that the identity (286) holds,
but only when inserted between the amplitudes.

The remaining tensor product

$\displaystyle\ydiagram{1,1,1}_{\,9}\otimes\,\ydiagram{1,1,1}_{\,9}={}$
$\displaystyle\ydiagram{2,2,1,1}_{\,9}\oplus\,\ydiagram{2,1,1,1}_{\,9}\oplus\,\ydiagram{1,1,1,1}_{\,9}\oplus\,\ydiagram{2,2,2}_{\,9}$
(288)
$\displaystyle\oplus\,\ydiagram{2,1,1}_{\,9}\oplus\,\ydiagram{2,2}_{\,9}\oplus\,\ydiagram{1,1,1}_{\,9}\oplus\,\ydiagram{2}_{\,9}\oplus\,\ydiagram{1,1}_{\,9}\oplus\,\bullet_{\,9}\,,$

can be dealt with similarly, by contracting with only 3 independent
polarizations instead of six, eliminating the contributions of the complicated
4 row tensors. The only non-vanishing contributions to projectors turn out to
be

$\displaystyle\pi_{[1,1,1]_{9}}^{\bm{\alpha};\bm{\gamma}}\pi_{[1,1,1]_{9}}^{\bm{\beta};\bm{\delta}}$
$\displaystyle=\sum_{\rho_{C}\in
S^{\prime\prime}}p^{\bm{\alpha}\bm{\beta}}_{[1,1,1]_{9}\otimes[1,1,1]_{9}\to\rho_{C},\bm{\mu}}\pi_{\rho_{C}}^{\bm{\mu};\bm{\nu}}p^{\bm{\gamma}\bm{\delta}}_{[1,1,1]_{9}\otimes[1,1,1]_{9}\to\rho_{C},\bm{\nu}}\,,$
(289) $\displaystyle S^{\prime\prime}$
$\displaystyle=\left\\{[2,2,2]_{9}\,,[2,2]_{9}\,,[2]_{9}\,,\bullet_{9}\right\\}\,.$

where we used (283) and the trial projectors

$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[1,1,1]_{9}\otimes[1,1,1]_{9}\to[2,2,2]_{9},\bm{\mu}}$
$\displaystyle=\sqrt{8}\delta^{\alpha_{1}}_{\mu_{1}}\delta^{\alpha_{2}}_{\mu_{2}}\delta^{\alpha_{3}}_{\mu_{3}}\delta^{\beta_{1}}_{\mu_{4}}\delta^{\beta_{2}}_{\mu_{5}}\delta^{\beta_{3}}_{\mu_{6}}\equiv\sqrt{8}\delta^{\bm{\alpha}\bm{\beta}}_{\bm{\mu}}\,,$
(290)
$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[1,1,1]_{9}\otimes[1,1,1]_{9}\to[2,2]_{9},\bm{\mu}}$
$\displaystyle=\sqrt{\frac{12}{5}}\eta^{\alpha_{1}\beta_{1}}\delta^{\alpha_{2}}_{\mu_{1}}\delta^{\beta_{3}}_{\mu_{2}}\delta^{\alpha_{3}}_{\mu_{3}}\delta^{\beta_{2}}_{\mu_{4}}\,,$
$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[1,1,1]_{9}\otimes[1,1,1]_{9}\to[2]_{9},\bm{\mu}}$
$\displaystyle=\sqrt{\frac{3}{7}}\eta^{\alpha_{1}\beta_{1}}\eta^{\alpha_{2}\beta_{3}}\delta^{\alpha_{3}}_{\mu_{1}}\delta^{\beta_{2}}_{\mu_{2}}\,,$
$\displaystyle\tilde{p}^{\bm{\alpha}\bm{\beta}}_{[1,1,1]_{9}\otimes[1,1,1]_{9}\to\bullet_{9}}$
$\displaystyle=\sqrt{\frac{1}{84}}\eta^{\alpha_{1}\beta_{1}}\eta^{\alpha_{2}\beta_{2}}\eta^{\alpha_{3}\beta_{3}}\,,$

where the unusual index ordering is to ensure that the resulting tensor
doesn’t vanish when we act with $\pi_{[1,1,1]_{9}}$ on the trial projectors
$\tilde{p}$. This turns (289) into an identity which holds for two identical
$[1,1,1]_{9}$ tensors, as will be the case for our amplitudes.

##  Appendix C Branching relations for projectors

In this Appendix we provide a detailed account of all the branching relations
for closed string state projectors utilized in section 7.1. Let us start by
recalling the SO(9) closed string states at level 1

$\displaystyle\left(\ydiagram{2}_{\,9}\oplus\,\ydiagram{1,1,1}_{\,9}\right)^{2}=\ydiagram{2,2,2}_{\,9}\oplus\,\ydiagram{2,2,1,1}_{\,9}\oplus\,2\,\ydiagram{3,1,1}_{\,9}\oplus\,3\,\ydiagram{2,1,1,1}_{\,9}\oplus\,\ydiagram{4}_{\,9}\oplus\,\ydiagram{3,1}_{\,9}$
$\displaystyle\oplus\,2\,\ydiagram{2,2}_{\,9}\oplus\,\ydiagram{2,1,1}_{\,9}\oplus\,\ydiagram{1,1,1,1}_{\,9}\oplus\,2\,\ydiagram{2,1}_{\,9}\oplus\,3\,\ydiagram{1,1,1}_{\,9}\oplus\,2\,\ydiagram{2}_{\,9}\oplus\,2\,\ydiagram{1,1}_{\,9}\oplus\,2\,\bullet\,.$
(291)

We recall that states with more than 3 columns will not contribute as we only
have 3 different vectors to anti-symmetrize. We are going to perform the
branching

$\text{SO(9)}~{}\rightarrow~{}\text{SO(4)}\times\text{SO(5)}|_{\bullet}\,,$
(292)

where we denote the projection to singlets of SO(5) by $|_{\bullet}$. Note
that certain representations can naturally be restricted to SO(4) by simply
taking the 5d subset of 10d indices. It is obvious that

$\ydiagram{1,1}_{\,9}\,\,\rightarrow\,\,\ydiagram{1,1}_{\,4}\,,\qquad\bullet_{\,9}\,\,\rightarrow\,\,\bullet_{\,4}\,.$
(293)

Additionally, the projector for these representations is identical for SO(9)
and SO(4) (up to restriction of the indices $\alpha\to a\,,\,\beta\to
b\,,\,\mu\to m\,,\,\dots$)

$\pi_{[1,1]_{9}}^{a_{1}a_{2};b_{1}b_{2}}=\pi_{[1,1]_{4}}^{a_{1}a_{2};b_{1}b_{2}}=\frac{1}{2}\left(\eta^{a_{1}b_{1}}\eta^{a_{2}b_{2}}-\eta^{a_{1}b_{2}}\eta^{a_{2}b_{1}}\right)\,.$
(294)

Other representations admit a direct restriction, but can also give additional
irreps, by the creation of SO(5) singlets, through the contraction of indices
with legs on the compact manifold. For example the spin 2 states branch as

$\ydiagram{2}_{\,9}\,\,\rightarrow\,\,\ydiagram{2}_{\,4}\,\,\oplus\,\,\bullet_{\,4}\,.$
(295)

The spin 2 on the RHS is interpreted as the restriction of indices to the
SO(4) and the singlet as a trace over the compact space indices. In terms of
projectors the statement is simply

$\pi_{[2]_{9}}^{a_{1}a_{2};b_{1}b_{2}}=\pi_{[2]_{4}}^{a_{1}a_{2};b_{1}b_{2}}+\frac{5}{36}\eta^{a_{1}a_{2}}\eta^{b_{1}b_{2}}\,.$
(296)

Similarly, for the spin 4 case

$\ydiagram{4}_{\,9}\,\,\rightarrow\,\,\ydiagram{4}_{\,4}\,\,\oplus\,\,\ydiagram{2}_{\,4}\,\,\oplus\,\,\bullet_{\,4},$
(297)

and the projector equation is

$\pi_{[4]_{9}}^{\mathbf{a};\mathbf{b}}=\sum_{\rho=[4]_{4},[2]_{4},\bullet_{4}}b^{\mathbf{a}}_{[4]_{9}\to\rho,\mathbf{m}}\
\pi_{\rho}^{\mathbf{m};\mathbf{n}}\
b^{\mathbf{b}}_{[4]_{9}\to\rho,\mathbf{n}}\,.$ (298)

It will be again convenient to manifestly symmetrize the tensors, in order to
present them more compactly. We define

$b^{\mathbf{a}}_{\rho_{C}\to\rho,\mathbf{m}}\equiv\pi^{\mathbf{a}}_{\rho_{C},\mathbf{a}^{\prime}}\tilde{b}^{\mathbf{a^{\prime}}}_{\rho_{C}\to\rho,\mathbf{m}}\,,$
(299)

and then present a list of the simpler $\tilde{b}$. In this case we have

$\displaystyle\tilde{b}^{a_{1}a_{2}a_{3}a_{4}}_{[4]_{9}\to[4]_{4},m_{1}m_{2}m_{3}m_{4}}$
$\displaystyle=\delta^{a_{1}}_{m_{1}}\delta^{a_{2}}_{m_{2}}\delta^{a_{3}}_{m_{3}}\delta^{a_{4}}_{m_{4}}\equiv\delta^{\mathbf{a}}_{\mathbf{m}}$
(300)
$\displaystyle\tilde{b}^{a_{1}a_{2}a_{3}a_{4}}_{[4]_{9}\to[2]_{4},m_{1}m_{2}}$
$\displaystyle=\sqrt{\frac{39}{20}}\delta^{a_{1}}_{m_{1}}\delta^{a_{2}}_{m_{2}}\eta^{a_{3}a_{4}}~{},~{}\tilde{b}^{a_{1}a_{2}a_{3}a_{4}}_{[4]_{9}\to\bullet_{4}}=\sqrt{\frac{143}{280}}\eta^{a_{1}a_{2}}\eta^{a_{3}a_{4}}\,.$

The fact that the direct restriction of the irrep $[4]_{9}\rightarrow[4]_{4}$
comes with coefficient 1 is a non-trivial consistency check of the previous
procedure.

There are other irreps that don’t admit a direct restriction, because they
have more than two columns (SO(4) Young tableaux have at most two columns, and
traces can vanish by antisymmetry). For this we need to use the SO(4) Levi-
Civita tensor. We will simply write it as
$\varepsilon^{a_{1}a_{2}a_{3}a_{4}}$. The simplest case is the 3-form

$\ydiagram{1,1,1}_{\,9}\,\,\rightarrow\,\,\ydiagram{1}_{\,4}\,,$ (301)

and the corresponding projector equation is

$\pi_{[1,1,1]_{9}}^{\mathbf{a};\mathbf{b}}=\frac{1}{6}\varepsilon^{a_{1}a_{2}a_{3}}_{\,\,\,\,\qquad
m}\pi_{[1]_{4}}^{m;n}\,\varepsilon^{\,\,\,b_{1}b_{2}b_{3}}_{n}=4\,\pi_{[1,1,1,1]_{4}}^{\mathbf{a}m;\mathbf{b}n}\,\pi_{[1]_{4},m;n}\,,$
(302)

From the first equation we can read off

$b^{a_{1}a_{2}a_{3}}_{[1,1,1]_{9}\to[1]_{4},m}=\sqrt{\frac{1}{6}}\varepsilon^{a_{1}a_{2}a_{3}}_{\,\,\,\,\qquad
m}\,.$ (303)

For the second equality in (302) we used the standard identity

$\epsilon^{a_{1}a_{2}a_{3}a_{4}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}=4!\,\pi_{[1,1,1,1]_{4}}^{\mathbf{a};\mathbf{b}}\,.$
(304)

This is convenient to square the three-point amplitudes when computing the
residue of the four-point function $A^{12P_{1}P_{2}}$. Using trace
subtractions and products of epsilon tensors we can now derive branching
identities for all the relevant irreps. Let us list the remaining identities,
where we write the trace subtractions and the Levi-Civita tensors using the
trial projectors $\tilde{b}$, but then appropriately symmetrize them through
(299)

$\displaystyle\ydiagram{2,1,1}_{\,9}$
$\displaystyle\rightarrow\,\,\ydiagram{2}_{\,4}\,\,\oplus\,\,\ydiagram{1,1}_{\,4}\,,$
(305)
$\displaystyle\tilde{b}^{a_{1}a_{2}a_{3}a_{4}}_{[2,1,1]_{9}\to[2]_{4},m_{1}m_{2}}$
$\displaystyle=\sqrt{\frac{6}{4!}}\varepsilon^{a_{1}a_{3}a_{4}}_{\qquad
m_{1}}\delta^{a_{2}}_{m_{2}}~{},~{}\tilde{b}^{a_{1}a_{2}a_{3}a_{4}}_{[2,1,1]_{9}\to[1,1]_{4},m_{1}m_{2}}=\sqrt{\frac{42}{4!5}}\varepsilon^{a_{1}a_{3}a_{4}}_{\,\,\,\,\qquad
m_{1}}\delta^{a_{2}}_{m_{2}}\,,$ $\displaystyle\ydiagram{2,1}_{\,9}$
$\displaystyle\rightarrow\,\,\ydiagram{2,1}_{\,4}\,\,\oplus\,\,\ydiagram{1}_{\,4}\,,$
$\displaystyle\tilde{b}^{\mathbf{a}}_{[2,1]_{9}\to[2,1]_{4},\mathbf{m}}$
$\displaystyle=\delta^{\mathbf{a}}_{\mathbf{m}}~{},~{}\tilde{b}^{a_{1}a_{2}a_{3}}_{[2,1]_{9}\to[1]_{4},m}=\sqrt{\frac{16}{5}}\delta^{a_{1}}_{m_{1}}\eta^{a_{2}a_{3}}\,,$

$\displaystyle\ydiagram{3,1,1}_{\,9}$
$\displaystyle\rightarrow\,\,\ydiagram{3}_{\,4}\,\,\oplus\,\,\ydiagram{2,1}_{\,4}\,\,\oplus\,\,\ydiagram{1}_{\,4}\,,$
(306) $\displaystyle\tilde{b}^{\mathbf{a}}_{[3,1,1]_{9}\to[3]_{4},\mathbf{m}}$
$\displaystyle=\sqrt{\frac{36}{4!5}}\varepsilon^{a_{1}a_{4}a_{5}}_{\,\,\,\,\qquad
m_{1}}\delta^{a_{2}}_{m_{2}}\delta^{a_{3}}_{m_{3}}~{},~{}\tilde{b}^{\mathbf{a}}_{[3,1,1]_{9}\to[2,1]_{4},\mathbf{m}}=\sqrt{\frac{1152}{4!25}}\varepsilon^{a_{1}a_{4}a_{5}}_{\,\,\,\,\qquad
m_{1}}\delta^{a_{2}}_{m_{2}}\delta^{a_{3}}_{m_{3}}\,,$
$\displaystyle\tilde{b}^{a_{1}a_{2}a_{3}a_{4}a_{5}}_{[3,1,1]_{9}\to[1]_{4},m}$
$\displaystyle=\sqrt{\frac{22}{4!5}}\varepsilon^{a_{1}a_{4}a_{5}}_{\,\,\,\,\qquad
m}\eta^{a_{2}a_{3}}\,,$ $\displaystyle\ydiagram{3,1}_{\,9}$
$\displaystyle\rightarrow\,\,\ydiagram{3,1}_{\,4}\,\,\oplus\,\,\ydiagram{1,1}_{\,4}\,\,\oplus\,\,\ydiagram{2}_{\,4}\,,$
$\displaystyle\tilde{b}^{\mathbf{a}}_{[3,1]_{9}\to[3,1]_{4},\mathbf{m}}$
$\displaystyle=\delta^{\mathbf{a}}_{\mathbf{m}},\tilde{b}^{\mathbf{a}}_{[3,1]_{9}\to[1,1]_{4},\mathbf{m}}=\sqrt{\frac{11}{10}}\delta^{a_{1}}_{m_{1}}\eta^{a_{2}a_{3}}\delta^{a_{4}}_{m_{2}},\tilde{b}^{\mathbf{a}}_{[3,1]_{9}\to[2]_{4},\mathbf{m}}=\sqrt{\frac{27}{10}}\delta^{a_{1}}_{m_{1}}\eta^{a_{2}a_{4}}\delta^{a_{3}}_{m_{2}},$
$\displaystyle\ydiagram{2,2}_{\,9}$
$\displaystyle\rightarrow\,\,\ydiagram{2,2}_{\,4}\,\,\oplus\,\,\ydiagram{2}_{\,4}\,\,\oplus\,\,\bullet_{\,4}\,,$
$\displaystyle\tilde{b}^{\mathbf{a}}_{[2,2]_{9}\to[2,2]_{4},\mathbf{m}}$
$\displaystyle=\delta^{\mathbf{a}}_{\mathbf{m}}~{},~{}\tilde{b}^{\mathbf{a}}_{[2,2]_{9}\to[2]_{4},\mathbf{m}}=\sqrt{\frac{42}{5}}\delta^{a_{1}}_{m_{1}}\eta^{a_{2}a_{3}}\delta^{a_{4}}_{m_{2}}~{},~{}\tilde{b}^{\mathbf{a}}_{[2,2]_{9}\to\bullet_{4}}=\sqrt{\frac{7}{5}}\eta^{a_{1}a_{4}}\eta^{a_{2}a_{3}}\,,$
$\displaystyle\ydiagram{2,2,2}_{\,9}$
$\displaystyle\rightarrow\,\,\ydiagram{2}_{\,4}\,\,\oplus\,\,\bullet_{\,4}\,,$
$\displaystyle\tilde{b}^{\mathbf{a}}_{[2,2,2]_{9}\to[2]_{4},m_{1}m_{2}}$
$\displaystyle=\frac{\sqrt{48}}{4!}\varepsilon^{a_{1}a_{5}a_{3}}_{\,\,\,\,\qquad
m_{1}}\varepsilon^{a_{4}a_{2}a_{6}}_{\,\,\,\,\qquad
m_{2}}~{},~{}\tilde{b}^{\mathbf{a}}_{[2,2,2]_{9}\to\bullet_{4}}=\frac{\sqrt{28}}{4!}\varepsilon^{a_{1}a_{5}a_{3}}_{\,\,\,\,\qquad
m}\varepsilon^{a_{4}a_{2}a_{6}m}\,.$

Note that for the last diagram, which has more than 2 boxes in both columns,
we are forced to use two pairs of epsilon tensors. Tensors with more than
three rows aren’t allowed by the 10d kinematics, but even if they were, their
branchings do not contain singlets of SO(5) so we can simply discard them.

###  C.1 All 5d closed string amplitudes

Let us first write down in generality the 5d amplitudes using the relations
derived in the main text. We have

$A^{15P_{1}}_{n,\rho_{L}\otimes\rho_{R}\to\rho_{C}\to\rho,\mathbf{m}}=A^{15P_{1}}_{\rho_{L},\bm{\alpha}}\
A^{15P_{1}}_{\rho_{R},\bm{\beta}}\
p^{\bm{\alpha}\bm{\beta}}_{\rho_{L}\otimes\rho_{R}\to\rho_{C},\mathbf{a}}b^{\mathbf{a}}_{\rho_{C}\to\rho,\mathbf{m}}\,.$
(307)

With all the group theory identities established, we can enumerate all the
amplitudes used in the main text to reproduce the residue at the cut with mass
level 1. However, we again emphasize that we have not explicitly symmetrized
the amplitudes by contracting with the respective projector, in order to
maintain some compactness of the tables below. Namely, all amplitudes are to
be contracted with the projector to the SO(4) irrep, and furthermore,
amplitudes where
$\text{rank}(\rho)=\text{rank}(\rho_{L})+\text{rank}(\rho_{R})$ are also not
explicitly symmetrized with respect to $\rho_{L}$ and $\rho_{R}$ as in
equation (283). Additionally, for amplitudes where the $b$ tensors contain a
Levi-Civita symbol, we write the square of the amplitude

$\left(A_{\mathbf{m}}^{\rho_{L}\otimes\rho_{R}\to\rho_{C}\rightarrow\rho}\right)^{2}\equiv
A_{\mathbf{m}}^{\rho_{L}\otimes\rho_{R}\to\rho_{C}\rightarrow\rho}\pi^{\mathbf{m};\mathbf{n}}_{\rho}A_{\mathbf{n}}^{\rho_{L}\otimes\rho_{R}\to\rho_{C}\rightarrow\rho}\,,$
(308)

since the amplitude itself cannot be written nicely in terms of
$v_{\alpha},q_{\alpha},\epsilon_{\alpha}$. With these caveats in mind, we list
the amplitudes starting by the ones with the most indices

$\displaystyle\ydiagram{4}_{\,4}$ (309) $\displaystyle
A^{[2]_{9}\otimes[2]_{9}\rightarrow[4]_{9}\rightarrow[4]_{4}}_{m_{1}m_{2}m_{3}m_{4}}$
$\displaystyle=\frac{1}{8}\alpha^{\prime}\left(2\epsilon_{1,m_{1}}q_{1,m_{2}}+v_{m_{1}}v_{m_{2}}q_{1}\cdot\epsilon_{1}\right)$
$\displaystyle\times\left(2\epsilon_{1,m_{3}}q_{1,m_{4}}+v_{m_{3}}v_{m_{4}}q_{1}\cdot\epsilon_{1}\right)\,,$
$\displaystyle 2\,\ydiagram{2,2}_{\,4}$ $\displaystyle
A^{[2]_{9}\otimes[2]_{9}\rightarrow[2,2]_{9}\rightarrow[2,2]_{4}}_{m_{1}m_{2}m_{3}m_{4}}$
$\displaystyle=\frac{1}{8}\alpha^{\prime}\left(2\epsilon_{1,m_{1}}q_{1,m_{2}}+v_{m_{1}}v_{m_{2}}q_{1}\cdot\epsilon_{1}\right)$
$\displaystyle\times\left(2\epsilon_{1,m_{3}}q_{1,m_{4}}+v_{m_{3}}v_{m_{4}}q_{1}\cdot\epsilon_{1}\right)\,,$
$\displaystyle
A^{[1,1,1]_{9}\otimes[1,1,1]_{9}\rightarrow[2,2]_{9}\rightarrow[2,2]_{4}}_{m_{1}m_{2}m_{3}m_{4}}$
$\displaystyle=\frac{\alpha^{\prime}}{4\sqrt{15}}[\left(v_{m_{1}}\epsilon_{1,m_{3}}-v_{m_{3}}\epsilon_{1,m_{1}}\right)\left(q_{1}\cdot\epsilon_{1}\left(v_{m_{2}}q_{1,m_{4}}-v_{m_{4}}q_{1,m_{2}}\right)\right.$
$\displaystyle+t_{1}\left.\left(v_{m_{2}}\epsilon_{1,m_{4}}-v_{m_{4}}\epsilon_{1,m_{2}}\right)\right)$