QCD, this leads to the well known phenomenon of gluon saturation which is
quantitatively described by the Color Glass Condensate EFT. In gravity, a
parallel development has been to describe black holes as saturated self-bound
states of wee gravitons - the Black Hole Quantum N-Portrait (BHNP) framework.
The universality between quantum states in the regime where the wee parton
occupancy is of the order of the inverse of the coupling in Eq. (22) was noted
previously in [32].

As we discussed, the CGC description of saturated gluon states is highly
developed with the evolution of the nonperturbative dynamics of wee partons
described by nonlinear RG equations that are known to next-to-leading logs in
$x$ accuracy and recovers the RG equations of perturbative QCD in the UV. The
elements of the CGC framework are semi-classical fields that can be mapped on
to the language of reggeon field theory (inspired by diagrammatic
computations) with a clear correspondence established to its building blocks,
reggeization and the Lipatov vertex. Since these are the very same building
blocks in perturbative discussions of $2\rightarrow N$ scattering in gravity,
it is natural to likewise consider that whether strong field techniques
developed in the CGC EFT might be applicable and provide a complementary
framework to that developed by ACV which employs the diagrammatic language of
reggeon field theory.

A remarkable additional feature of the two theories that goes beyond the above
stated analogies and possible universality is that of the BCJ/classical double
copy that has emerged as a powerful tool for quantitative computations. We
noted that a particular Regge double copy between the Lipatov vertex in QCD
and the effective gravitational central emission vertex was established by
Lipatov over 40 years ago. A central result of our paper is that this double
copy relation can be recovered in our semi-classical approach in gravitational
shock wave collisions at large impact parameters. This derivation parallels
the extraction of the QCD Lipatov vertex in the scattering of gluon shockwaves
performed previously. Our approach also provides fresh insight from the semi-
classical perspective into how and why the QED bremsstrahlung vertex enters
into the double copy. The QCD example also suggests a useful classification of
shockwave scattering into dilute-dilute, dilute-dense and dense-dense regimes.

Our results for gravitational shockwave scattering also suggest that one can
construct dressed gravitational shockwave propagators with an analogous
structure to QCD shockwave propagators. These can then be employed in a semi-
classical derivation of graviton reggeization along the lines of gluon
reggeization. If successful, this will allow us to construct a nonlinear RG
description of multiparticle production as a function of impact parameter and
rapidity. The formation of a black hole would then be understood as a
nontrivial fixed point of the RG flow. In QCD, the nontrivial fixed point
corresponds to the classicalization and unitarization of the $2\rightarrow N$
amplitude; this is also the conjecture in the BHNP and a quantitative
derivation in the language of RG flow is desirable. A conceptual issue that
would be important to resolve in this regard is how the Goldstone picture of
asymptotic symmetries that are broken by both graviton and gluon shockwaves
applies at finite frequencies. This is relevant to understanding the emergence
of a universal area law in the dynamics of wee partons.

Not least, we addressed how our semi-classical framework relates to the large
body of work on gravitational shockwave scattering. This is of course of great
importance in understanding gravitational wave radiation in particular with
respect to uncovering possible signatures of RG evolution and black hole
formation in the radiation spectrum. We plan to undertake a quantitative study
of such signatures in future work.

###### Acknowledgements.

We thank Gia Dvali and Gabriele Veneziano for valuable discussions that have
significantly influenced this work. We thank Simon Caron-Huot and Grisha
Korchemsky for useful comments on aspects of the extensions of this work. R.V
would also like to thank Agustin Sabio Vera for clarifications of his work on
the Lipatov double copy. R.V is supported by the U.S. Department of Energy,
Office of Science under contract DE-SC0012704 and within the framework of the
SURGE Topical Theory Collaboration. He acknowledges partial support from an
LDRD at BNL. R.V was also supported at Stony Brook by the Simons Foundation as
a co-PI under Award number 994318 (Simons Collaboration on Confinement and QCD
Strings). He thanks the DFG Collaborative Research Center SFB 1225 (ISOQUANT)
at Heidelberg University for hospitality and support during the course of this
work. H.R is a Simons Foundation postdoctoral fellow at Stony Brook supported
under Award number 994318. H.R acknowledges support by the ERC Starting Grant
853507 and thanks the EIC Theory Institute at Brookhaven National Lab for its
hospitality when this project was nearing its completion.

##  Appendix A Gluon shockwave scattering in QCD

The classical Yang-Mills (YM) equations of motion in the presence of external
currents are

$\displaystyle D_{\mu}F^{\mu\nu}=J^{\nu}~{},$ (79)

where the field strength is defined as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+ig[A_{\mu},A_{\nu}]$
and the current $J^{\mu}$ is covariantly conserved: $D_{\mu}J^{\mu}=0$. The
action of the covariant derivative on an adjoint field $F$ is given by

$D_{\mu}F=\partial_{\mu}F-ig[A_{\mu},F]~{}.$ (80)

A gluon shockwave is generated by the current

$J^{\nu,a}=g\,\delta^{\nu+}\delta\left(x^{-}\right)\rho_{H}\,T^{a}\left(\boldsymbol{x}\right)~{},$
(81)

where $T^{a}$ is a generator of the color algebra. This form is analogous to
the form of the energy-momentum tensor Eq. (36) in gravity. It is
straightforward to verify that the exact solution, $\bar{A}_{-}^{a}$ , to the
YM equations with this source is given by

$\bar{A}_{-}^{a}=-g\,\delta\left(x^{-}\right)\frac{\rho_{H}\left(\boldsymbol{x}\right)}{\square_{\perp}}\,T^{a}~{}.$
(82)

This form admits a precise double copy relation to the gravitational shockwave
in the coordinate system given in Eq. (37) and Eq. (38) through a color-
kinematic replacement made precise in [80]. In this Appendix, we shall
describe the scattering of two such shockwaves in the dilute-dilute
approximation and show how the QCD Lipatov vertex arises.

###  Gluon shockwave collisions and the Lipatov vertex

The approach for solving this problem is similar to the one in gravity. One
begins by linearizing the YM equations over the shockwave background Eq. (12)
which is linear in the light cone gauge:

$A_{+}=0~{},$ (83)

In this gauge, the YM equations can be written as

$\displaystyle\begin{split}&-\partial_{+}\partial_{\mu}A^{\mu}+ig\left[A_{\mu},\partial_{+}A^{\mu}\right]=J_{+}~{},\\\
&\square
A_{-}-\partial_{-}\partial_{\mu}A^{\mu}-ig\left[\partial_{\mu}A^{\mu},A_{-}\right]-2ig\left[A^{\mu},\partial_{\mu}A_{-}\right]+ig\left[A_{\mu},\partial_{-}A^{\mu}\right]=J_{-}~{},\\\
&\square
A_{i}-\partial_{i}\partial_{\mu}A^{\mu}-ig\left[\partial_{\mu}A^{\mu},A_{i}\right]-2ig\left[A^{\mu},\partial_{\mu}A_{i}\right]+ig\left[A_{\mu},\partial_{i}A^{\mu}\right]=J_{i}~{}.\end{split}$
(84)

A straightforward linearization $A_{\mu}=\bar{A}_{\mu}+a_{\mu}$ of these
equations gives the set of equations

$\displaystyle\begin{split}&-\partial_{+}\partial_{\mu}a^{\mu}=J_{+}\\\
&\square
a_{-}-\partial_{-}\partial_{\mu}a^{\mu}-ig\left[\partial_{\mu}a^{\mu},A_{-}\right]-2ig[\bar{A}_{-},\partial_{+}a_{-}]+2ig\left[a_{i},\partial_{i}\bar{A}_{-}\right]=J_{-}~{},\\\
&\square
a_{i}-\partial_{i}\partial_{\mu}a^{\mu}-2ig\left[\bar{A}_{-},\partial_{+}a_{i}\right]=J_{i}~{}.\end{split}$
(85)

The first equation is a constraint relation which relates $a_{i}$ to $a_{-}$.
This implies that the equations for $a_{-}$ and $a_{i}$ are not independent.
The consistency of the set of equations is represented by

$\displaystyle\partial_{\mu}J^{\mu}=ig\left[\bar{A}_{-},J_{+}\right]~{}.$ (86)

which is nothing but covariant current conservation. This discussion is
entirely analogous to that in gravity.

Next, we incorporate the effect of the other incoming shockwave. This is
generated by the current

$J_{+,a}=g\delta(x^{+})\rho_{L}\left({\boldsymbol{x}}\right)T_{a}\,.$ (87)

Plugging this relation into Eq. (85) and first solving to linear order in
$\rho_{L}$ but zeroth order in $\rho_{H}$ gives us the gauge field for the
second shock in light cone gauge:

$a_{i,a}=g\,\Theta(x^{+})\frac{\partial_{i}\rho_{L}}{\square_{\perp}}T_{a}+O(\rho_{L}\,\rho_{H})~{},\qquad
a_{-}=O(\rho_{L}\,\rho_{H})~{}.$ (88)

We now need to determine the form of the current post-collision. In QCD, the
problem of finding the evolved current is considerably simpler than that in
gravity. The reason is that in QCD one can solve for the currents using simply
the conservation law in the strict eikonal approximation where the initial
current $J_{+}$ of the second shock does not develop any components along the
transverse and $-$ longitudinal direction. Then in the dilute-dilute
approximation, Eq. (86) can be solved for the correction to $J_{+}$ by
inserting the leading order results on the r.h.s. Using the Lie algebra
relation $[T_{a},T_{b}]=if_{abc}T_{c}$, we get

$\displaystyle J_{+,c}$
$\displaystyle=g^{3}\delta(x^{+})\Theta(x^{-})\frac{\rho_{H}}{\square_{\perp}}\rho_{L}T^{a}T^{b}f_{abc}~{},$
(89)

Plugging the results from Eqs. (88) and (89) into the equations of motion of
$a_{i}$ and $a_{-}$ in Eq. (85) gives us (at order $O(\rho_{L}\rho_{H})$)

$\displaystyle\square
a_{i,c}=-g^{3}\left(\Theta(x^{+})\Theta(x^{-})\partial_{i}\left(\frac{\rho_{H}}{\square_{\perp}}\rho_{L}\right)-2\delta(x^{+})\delta(x^{-})\frac{\rho_{H}}{\square_{\perp}}\frac{\partial_{i}\rho_{L}}{\square_{\perp}}\right)T^{a}T^{b}f_{abc}~{},$
(90) $\displaystyle\square
a_{-,c}=2g^{3}\Theta(x^{+})\delta(x^{-})\left(\frac{\partial_{i}\rho_{H}}{\square_{\perp}}\frac{\partial_{i}\rho_{L}}{\square_{\perp}}\right)T^{a}T^{b}f_{abc}$
(91)

Upon taking the Fourier transforms of these equations, and putting the momenta
of the emitted gluon on-shell, we find

$\displaystyle a_{i,c}(k)$ $\displaystyle=-\frac{2ig^{3}}{k^{2}+i\epsilon
k^{-}}\int\frac{d^{2}{\boldsymbol{q}}_{2}}{(2\pi)^{2}}\left(q_{2i}-k_{i}\frac{{\boldsymbol{q}}_{2}^{2}}{{\boldsymbol{k}}^{2}}\right)\frac{\rho_{H}}{{\boldsymbol{q}}_{1}^{2}}\frac{\rho_{L}}{{\boldsymbol{q}}_{2}^{2}}T^{a}T^{b}f_{abc}~{},$
(92) $\displaystyle a_{-,c}(k)$ $\displaystyle=-\frac{2ig^{3}}{k^{2}+i\epsilon
k^{-}}\int\frac{d^{2}{\boldsymbol{q}}_{2}}{(2\pi)^{2}}\frac{({\boldsymbol{q}}_{1}\cdot{\boldsymbol{q}}_{2})}{k_{+}}\frac{\rho_{A}}{{\boldsymbol{q}}_{1}^{2}}\frac{\rho_{B}}{{\boldsymbol{q}}_{2}^{2}}T^{a}T^{b}f_{abc}~{}.$
(93)

From these expressions, we can now extract the QCD Lipatov vertices in the
light cone gauge:

$\displaystyle
C_{i}=-2\left(q_{2i}-k_{i}\frac{{\boldsymbol{q}}_{2}^{2}}{{\boldsymbol{k}}^{2}}\right)~{},\qquad
C_{-}=-2\frac{({\boldsymbol{q}}_{1}\cdot{\boldsymbol{q}}_{2})}{k_{+}}~{}.$
(94)

The above result satisfies $\partial^{\mu}C_{\mu}=0$. This computation can be
straightforwardly generalized to all orders in $\rho_{H}$ \- as performed in
[62, 63].

##  Appendix B GR review and linearization of Einstein’s equations

###  B.1 GR review and conventions

The action for Einstein-Hilbert gravity is

$S_{\text{grav }}=-\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}R\,,$ (95)

where $g=\operatorname{det}\left(g_{\mu\nu}\right)$ is the determinant of the
metric tensor and $\kappa$ is proportional to the Newton constant $G$

$\kappa^{2}=8\pi G=\frac{8\pi}{M_{p}^{2}}\,.$ (96)

It has mass dimension $[\kappa]=-1$ in natural units ($\hbar=1$). The scalar
curvature $R$ is related to the Ricci curvature tensor $R_{\mu\nu}$ by the
contraction of its indices $R=R_{\mu\nu}g^{\mu\nu}$. The Ricci tensor is in
turn given by the contraction of components of the Riemann tensor:

$R_{\mu\nu}=R_{~{}\mu\sigma\nu}^{\sigma}\,.$ (97)

We are working in conventions where the Riemann curvature tensor is given by
the commutator (following the same convention as Wald’s book but opposite to
Weinberg’s convention, where there is a minus sign on the r.h.s)

$[\nabla_{\sigma},\nabla_{\rho}]V_{\nu}=V_{\mu}R^{\mu}{}_{\nu\rho\sigma}~{},$
(98)

with $V$ being an arbitrary 4-vector and $\nabla$ the covariant derivative
whose action on $V$ is defined as

$\nabla_{\mu}V_{\nu}=V_{\nu;\mu}=\partial_{\mu}V_{\nu}-\Gamma_{\mu\nu}^{\rho}V_{\rho}~{},\qquad\Gamma_{\mu\nu}^{\rho}=\frac{1}{2}g^{\rho\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu}\right)~{}.$
(99)

Eq. (98) then gives the Riemann tensor in terms of the following combination
of Christoffel symbols $\Gamma$ and its derivatives:

$R_{~{}\mu\alpha\beta}^{\sigma}=\partial_{\alpha}\Gamma_{\mu\beta}^{\sigma}-\partial_{\beta}\Gamma_{\mu\alpha}^{\sigma}+\Gamma_{\mu\beta}^{\rho}\Gamma_{\rho\alpha}^{\sigma}-\Gamma_{\mu\alpha}^{\rho}\Gamma_{\rho\beta}^{\sigma}~{}.$
(100)

Varying the Einstein-Hilbert action with respect to the metric gives the
Einstein field equations

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa^{2}T_{\mu\nu}~{},$ (101)

where the energy-momentum tensor $T_{\mu\nu}$ is included in the r.h.s to
account for possible matter contributions. Denoting the matter action by
$S_{m}$, one defines $T_{\mu\nu}$ by the metric variation of $S_{m}$:

$\delta S_{m}=\frac{1}{2}\int d^{4}x\sqrt{-g}~{}\delta
g_{\mu\nu}T^{\mu\nu}~{}.$ (102)

The Bianchi identity of the Riemann tensor
$R_{\lambda\mu\nu\kappa;\eta}+R_{\lambda\mu\eta\nu;\kappa}+R_{\lambda\mu\kappa\eta;\nu}=0$
implies that the energy-momentum tensor is covariantly conserved,

$\nabla_{\mu}T^{\mu}_{~{}\nu}=0~{}.$ (103)

The action for a massless point particle is

$S_{m}=\frac{1}{2}\int
d\lambda~{}\eta^{-1}\dot{X}^{\mu}\dot{X}^{\nu}g_{\mu\nu}(X)~{},$ (104)

where dots denote differentiation w.r.t. $\lambda$, $\eta$ is the determinant
of the metric along the worldline that can be gauged away and $g_{\mu\nu}(X)$
is the metric tensor of the background that the worldline $X(\sigma)$ is
probing. Functionally differentiating Eq. (104) with respect to the spacetime
metric $g_{\mu\nu}$ gives the energy-momentum tensor in Eq. (55).

###  B.2 Linearization of Einstein’s equations

Here we present the linearization of Einstein’s equations around the shockwave
background in Eq. (37). One begins by expanding out the metric as in Eq. (51).
We will be working throughout in the light cone gauge

$h_{\mu+}=0~{}.$ (105)

After straightforward (though lengthy) algebra we find that the linearization
of Ricci scalar is given by ($h\equiv h_{ij}\delta_{ij}$ below)

$\displaystyle\delta R$
$\displaystyle=-\bar{g}_{--}\partial_{+}^{2}h+\partial_{+}^{2}h_{--}-2\partial_{+}\partial_{i}h_{-i}+2\partial_{+}\partial_{-}h+\left(\partial_{k}\partial_{l}h_{kl}-\square_{\perp}h\right)~{},$
(106)

and the the linearization of Einstein’s field equation gives (we have
suppressed factors of $\kappa^{2}$ for clarity)

$\displaystyle\frac{1}{2}\partial_{+}^{2}h=T_{++}~{},$ (107a)
$\displaystyle\frac{1}{2}\left(\partial_{+}^{2}h_{--}-\partial_{+}\partial_{j}h_{-j}+\partial_{+}\partial_{-}h\right)-\frac{1}{2}\delta
R=T_{+-}~{},$ (107b)
$\displaystyle\frac{1}{2}\left(\partial_{+}^{2}h_{-i}-\partial_{+}\partial_{j}h_{ij}+\partial_{+}\partial_{i}h\right)=T_{+i}~{},$
(107c)
$\displaystyle\frac{1}{2}\bigg{(}\bar{g}_{--}\partial_{+}^{2}h_{ij}-2\partial_{+}\partial_{-}h_{ij}+\partial_{+}\partial_{i}h_{-j}+\partial_{+}\partial_{j}h_{-i}+\square_{\perp}h_{ij}-\partial_{k}\partial_{i}h_{kj}-\partial_{k}\partial_{j}h_{ki}+\partial_{i}\partial_{j}h\bigg{)}+\frac{1}{2}\delta_{ij}\delta
R=T_{ij}~{},$ (107d)
$\displaystyle\frac{1}{4}\bigg{(}2\bar{g}_{--}\partial_{+}^{2}h_{--}-\partial_{-}(\bar{g}_{--})\partial_{+}h+2\square_{\perp}h_{--}-4\partial_{-}\partial_{j}h_{-j}+2\partial_{-}^{2}h$
$\displaystyle+2\left(\partial_{i}\partial_{j}\bar{g}_{--}\right)h_{ij}+2\left(\partial_{i}\bar{g}_{--}\right)\partial_{j}h_{ij}+2\left(\partial_{i}\bar{g}_{--}\right)\partial_{+}h_{-i}-\partial_{i}\left(\bar{g}_{--}\right)\partial_{i}h\bigg{)}-\frac{1}{2}\bar{g}_{--}\delta
R=T_{--}~{},$ (107e)
$\displaystyle\frac{1}{2}\bigg{(}\bar{g}_{--}\partial_{+}^{2}h_{-i}+\square_{\perp}h_{-i}-\partial_{i}\partial_{j}h_{-j}+\partial_{i}\partial_{-}h-\partial_{j}\partial_{-}h_{ij}-\partial_{+}\partial_{-}h_{-i}$
$\displaystyle+\partial_{+}\partial_{i}h_{--}-\frac{1}{2}\partial_{+}h~{}\partial_{i}\bar{g}_{--}+\partial_{+}h_{ij}\partial_{j}\bar{g}_{--}\bigg{)}=T_{-i}~{}.$
(107f)

These are respectively the $++,+-,+i,ij,--,-i$ components of Einstein’s
equations. Furthermore, linearizing the covariant energy-momentum conservation
equations over the background in Eq. (38) give us

$\displaystyle\partial_{-}T_{++}+\partial_{+}T_{+-}-\partial_{i}T_{i+}={g}_{--}\partial_{+}T_{++}~{},$
(108a)
$\displaystyle\partial_{-}T_{+-}+\partial_{+}T_{--}-\partial_{i}T_{i-}={g}_{--}\partial_{+}T_{+-}+\frac{1}{2}\left(\partial_{-}{g}_{--}\right)T_{++}+\frac{1}{2}\square_{\perp}{g}_{--}\frac{1}{\partial_{+}}T_{++}~{},$
(108b)
$\displaystyle\partial_{+}T_{-i}+\partial_{-}T_{+i}-\partial_{j}T_{ji}=\frac{1}{2}\left(\partial_{i}{g}_{--}\right)T_{++}+{g}_{--}\partial_{+}T_{+i}~{}.$
(108c)

The conservation equations (108) are not independent of Einstein’s equations
(107). The latter implies the former (which, as mentioned earlier, is a
consequence of the Bianchi identity). This can be seen upon diagonalizing
(107) which we briefly outline below.

From the $++$ component we simply get

$h=\frac{2}{\partial_{+}^{2}}T_{++}~{}.$ (109)

The $+i$ equation Eq. (107c) together with Eq. (109) gives a constrain
relation

$\displaystyle\partial_{j}h_{ij}$
$\displaystyle=\partial_{+}h_{-i}+\frac{2\partial_{i}}{\partial_{+}^{2}}T_{++}-\frac{2}{\partial_{+}}T_{+i}$
(110)

Next, a simple manipulation reveals that the $+-$ equation Eq. (107b) together
with Eq. (109) and Eq. (110) gives the first energy-momentum conservation
equation Eq. (108a). Upon using Eqs. (109), (110) and (108a), the trace of
$ij$ component Eq. (107d) is found to give a second constraint relation,

$\displaystyle\partial_{+}^{2}h_{--}=\partial_{i}\partial_{j}\tilde{h}_{ij}-\frac{\square_{\perp}}{\partial_{+}^{2}}T_{++}+T+2T_{+-}-\bar{g}_{--}T_{++}$
(111)

where $\tilde{h}_{ij}$ is defined to be the traceless combination

$\tilde{h}_{ij}\equiv h_{ij}-\frac{1}{2}\delta_{ij}h~{}.$ (112)

Inserting Eq. (111) back into Eq. (107d) finally gives the diagonalized
equation for $\tilde{h}_{ij}$:

$\displaystyle\begin{split}\bar{g}_{--}\partial_{+}^{2}\tilde{h}_{ij}-2\partial_{+}\partial_{-}\tilde{h}_{ij}+\square_{\perp}\tilde{h}_{ij}=&\left(2\partial_{i}\partial_{j}-\square_{\perp}\delta_{ij}\right)\frac{1}{\partial_{+}^{2}}T_{++}-2\delta_{ij}\left(\bar{g}_{--}T_{++}-\frac{\partial_{-}}{\partial_{+}}T_{++}\right)\\\
&+2\delta_{ij}T_{+-}+2T_{ij}-\delta_{ij}T-\frac{2}{\partial_{+}}\left(\partial_{i}T_{+j}+\partial_{j}T_{+i}\right)~{}.\end{split}$
(113)

Likewise, one can get the simplified form of the equations of motion for the
remaining components $h_{i-}$ and $h_{--}$ (which couple to $\tilde{h}_{ij}$).
We record these below. For the $h_{-i}$ we get

$\displaystyle\bar{g}_{--}\partial_{+}^{2}h_{-i}+\square_{\perp}h_{-i}-2\partial_{+}\partial_{-}h_{-i}+\partial_{+}\tilde{h}_{ij}\partial_{j}\bar{g}_{--}=2T_{-i}-\frac{2\partial_{-}}{\partial_{+}}T_{+i}-\frac{\partial_{i}}{\partial_{+}}\left(\bar{g}_{--}T_{++}-\frac{2\partial_{-}}{\partial_{+}}T_{++}+T\right)$
(114)

whereas for the $h_{--}$, we get

$\displaystyle\begin{split}&\bar{g}_{--}\partial_{+}^{2}h_{--}+\square_{\perp}h_{--}-2\partial_{+}\partial_{-}h_{--}+\left(\partial_{i}\partial_{j}\bar{g}_{--}\right)\tilde{h}_{ij}+2\left(\partial_{i}\bar{g}_{--}\right)\partial_{j}\tilde{h}_{ij}\\\\[5.0pt]
&=\partial_{-}(\bar{g}_{--})\frac{1}{\partial_{+}}T_{++}-\left(\square_{\perp}\bar{g}_{--}\right)\frac{1}{\partial_{+}^{2}}T_{++}+\left(\partial_{i}\bar{g}_{--}\right)\frac{\partial_{i}}{\partial_{+}^{2}}T_{++}+\frac{2\partial_{-}}{\partial_{+}}\left(\frac{\partial_{-}}{\partial_{+}}T_{++}-\bar{g}_{--}T_{++}\right)\\\\[5.0pt]
&+\left(\bar{g}_{--}\right)^{2}T_{++}+2T_{--}+\bar{g}_{--}\left(T-2T_{+-}\right)-\frac{2\partial_{-}}{\partial_{+}}T-\left(\partial_{i}\bar{g}_{--}\right)\frac{2}{\partial_{+}}T_{+i}~{}.\end{split}$
(115)

We emphasize that the equations for $h_{i-}$ and $h_{--}$ cannot be
independent of the equation for $\tilde{h}_{ij}$ since the two constraint
relations Eq. (110) and Eq. (111) relates them. The two constraints and the
three equations of motion Eqs. (113), (114) and (LABEL:EinsteinsEq--1) should
form a consistent system of equations. One finds that this consistency is met
only when energy-momentum conservation (Eq. (108)) is satisfied. Given the
complicated form of these equations, it is useful to perform this (tedious)
sanity check to ensure that the equations obtained are free of any errors.

##  Appendix C Evolution of the energy-momentum tensor in shockwave collisions

Here we provide details behind the result Eq. (59) for the evolved EM tensor
in the dilute-dilute approximation. As mentioned in the main text, in the
point particle approximation (consistent with the dilute-dilute
approximation), thanks to the formula Eq. (55), the problem of computing the
energy-momentum tensor reduces to the problem of computing null geodesics.
These can be computed from the geodesic equation Eq. (56) with the appropriate
initial conditions. The solution given in Eq. (58) is then used to reconstruct
the energy-momentum tensor.

From Eq. (55), we find the result

$T^{\mu\nu}=\mu_{L}\dot{X}^{\mu}(X^{-})\dot{X}^{\nu}(X^{-})\delta(x^{+}-X^{+})\delta^{(2)}\left({\boldsymbol{x}}-{\boldsymbol{X}}\right)\,,$
(116)

where, from (56) we have

$\displaystyle\dot{X}^{-}=1~{},\qquad\dot{X}^{i}=-\kappa^{2}\mu_{H}\Theta(X^{-})\frac{\partial_{i}\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}~{},\qquad\dot{X}^{+}=-\kappa^{2}\mu_{H}\delta(X^{-})\frac{\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}+\frac{\kappa^{4}\mu_{H}^{2}}{2}\Theta(X^{-})\left(\frac{\partial_{i}\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}\right)^{2}\,.$
(117)

Eq. (116) is the exact result for the evolved energy-momentum tensor of
particle (2) probing the shockwave background Eq. (37) of particle (1).
However because we are working the dilute-dilute approximation, the relevant
expression for us is the expansion of Eq. (116) to order $\mu_{H}\mu_{L}$. The
nonvanishing components of $T^{\mu\nu}$ are

$\displaystyle\begin{split}T^{--}&=\mu_{L}\delta(x^{+})\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})+\kappa^{2}\mu_{H}\mu_{L}\Theta(x^{-})\left[\delta^{\prime}(x^{+})\frac{\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})+x^{-}\delta(x^{+})\frac{\partial_{i}\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}\partial_{i}\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})\right]\,,\\\\[5.0pt]
T^{-+}&=-\kappa^{2}\mu_{H}\mu_{L}\delta(x^{+})\delta(x^{-})\frac{\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})~{},\qquad
T^{-i}=-\kappa^{2}\mu_{H}\mu_{L}\delta(x^{+})\Theta(x^{-})\frac{\partial_{i}\rho_{H}({\boldsymbol{b}})}{\square_{\perp}}\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})\,.\end{split}$
(118)

All the other components are of higher order. This expansion is valid only for
large enough impact factors $b$ and small enough times $x^{-}$.

At this stage, this result can be generalized to a beam of massless particles
moving in the $-z$ direction at various impact factors by simply coarse
graining
$\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})\to\rho_{L}({\boldsymbol{x}})$.
Although this generalization misses various “shearing” and “expansion” effects
of the null geodesic congruence, for our purposes this replacement suffices.
Another limitation of the point-particle approximation is that under
generalization to extended transverse sources, it will miss terms of the form
$\rho_{H}\rho_{L}$. In the point particle approximation such a term is
proportional to
$\delta^{(2)}({\boldsymbol{x}})\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})=\delta^{(2)}({\boldsymbol{b}})\delta^{(2)}({\boldsymbol{x}}-{\boldsymbol{b}})$
which vanishes for nonvanishing impact parameters. In the generalized setup,
such terms can be present as will be seen below.

In the dilute-dilute approximation, we also need to know the corrections to
the energy-momentum tensor of particle (1) due to the gravitational influence
of particle (2). In light cone gauge, the metric that particle (1) sees is the
(linearized) metric created by particle (2) (whose leading order energy-
momentum tensor is $T^{--}=\mu_{L}\delta(x^{+})\rho_{L}$) which is given by

$\displaystyle
ds^{2}=2dx^{+}dx^{-}-\delta_{ij}dx^{i}dx^{j}+2\kappa^{2}\mu_{L}x^{+}\Theta(x^{+})\frac{\partial_{i}\partial_{j}}{\square_{\perp}}\rho_{L}~{}.$
(119)

This follows from the previously computed result in Eq. (62). To compute null
geodesics in this metric, we first compute the connection coefficients. These
are found to be

$\Gamma^{-}_{ij}=-\frac{1}{2}\partial_{+}\partial_{i}\partial_{j}g_{++}~{},~{}~{}~{}~{}\Gamma^{i}_{+j}=-\frac{1}{2}\partial_{+}\partial_{i}\partial_{j}g_{++}~{},~{}~{}~{}~{}\Gamma^{i}_{jk}=-\frac{1}{2}\partial_{i}\partial_{j}\partial_{k}g_{++}~{},\qquad
g_{++}=2\kappa^{2}\mu_{L}\delta(x^{+})\frac{\rho_{L}}{\square_{\perp}}~{}.$
(120)

The corresponding geodesic equations are

$\displaystyle\ddot{X}^{+}=0~{},\qquad\ddot{X}^{-}+\Gamma^{-}_{ij}\dot{X}^{i}\dot{X}^{j}=0~{},\qquad\ddot{X}^{i}+\Gamma^{i}_{jk}\dot{X}^{j}\dot{X}^{k}+2\Gamma^{i}_{+j}\dot{X}^{+}\dot{X}^{j}=0~{}.$
(121)

In the dilute-dilute approximation, it is sufficient for us to solve this
equation to linear order in $\rho_{L}$. By inspection, it can be seen that the
only solution with the appropriate boundary condition is
$X^{+}=\lambda,X^{-}=0,{\boldsymbol{X}}={\boldsymbol{b}}$. As such, in light
cone gauge there is no correction to the energy-momentum tensor of particle
(1) in the dilute-dilute approximation. Hence the total energy-momentum tensor
after the collision (with all components upper) is

$\displaystyle\begin{split}T^{--}&=\mu_{L}\delta(x^{+})\rho_{L}+\kappa^{2}\mu_{H}\mu_{L}\Theta(x^{-})\left[\delta^{\prime}(x^{+})\frac{\rho_{H}}{\square_{\perp}}\rho_{L}+x^{-}\delta(x^{+})\frac{\partial_{i}\rho_{H}}{\square_{\perp}}\partial_{i}\rho_{L}\right]~{},\qquad
T^{++}=\sqrt{s_{1}}\delta(x^{-})\rho_{H}\\\\[5.0pt]
T^{-+}&=-\kappa^{2}\mu_{H}\mu_{L}\delta(x^{+})\delta(x^{-})\frac{\rho_{H}}{\square_{\perp}}\rho_{L}~{},\qquad\qquad
T^{-i}=-\kappa^{2}\mu_{H}\mu_{L}\delta(x^{+})\Theta(x^{-})\frac{\partial_{i}\rho_{H}}{\square_{\perp}}\rho_{L}\,.\end{split}$
(122)

Upon lowering the components with respect to the leading order metric

$\displaystyle ds^{2}=~{}$
$\displaystyle~{}2dx^{+}dx^{-}-\delta_{ij}dx^{i}dx^{j}+2\kappa^{2}\sqrt{s_{1}}\delta(x^{-})\frac{\rho_{H}}{\square_{\perp}}(dx^{-})^{2}+2\kappa^{2}\mu_{L}x^{+}\Theta(x^{+})\frac{\partial_{i}\partial_{j}}{\square_{\perp}}\rho_{L}dx^{i}dx^{j}+O(\rho_{H}\rho_{L})~{},$
(123)

we find the result in Eq. (59) of the main text. One can check the consistency
of this result by verifying energy-momentum conservation (Eq (108)). We find
that Eq. (59) violates energy-momentum conservation by a term proportional to
$\rho_{H}\rho_{L}$. If one adds to $T_{+-}$ in Eq. (59) the term

$\kappa^{2}\mu_{H}\mu_{L}\Theta(x^{+})\Theta(x^{-})\rho_{H}\rho_{L}~{},$ (124)

energy-momentum conservation is restored even for vanishing impact parameter
(albeit a regime that cannot be probed in the approximation we are working
in). Conversely, energ-momentum conservation enforces that $T_{+-}$ contains
such a term.

##  Appendix D Details of the Lipatov vertex and Fourier transforms

Our convention for the Fourier transform is (to avoid clutter, we denote the
function and its FT transform by the same symbol)

$f(x)=\int\frac{d^{d}k}{(2\pi)^{d}}e^{ik_{\mu}x^{\mu}}f(k)~{},\qquad f(k)=\int
d^{d}x~{}e^{-ik_{\mu}x^{\mu}}f(x)\,.$ (125)

This implies $\partial_{\mu}\to ik_{\mu}$ and $\Theta(x^{\mu})\to-i/k_{\mu}$
(which follows from $\partial_{\mu}\Theta(x^{\mu})=\delta(x^{\mu})$). Finally,
the FT of a product of functions $f(x)g(x)$ is

$F[f(x)g(x)](k)\equiv\int
d^{d}x~{}e^{-ik_{\mu}x^{\mu}}f(x)g(x)=\int\frac{d^{d}q}{(2\pi)^{d}}f(k-q)g(q)~{}.$
(126)

Using these conventions we next perform the Fourier transform of equations
Eqs. (114), (LABEL:EinsteinsEq--1) in the dilute-dilute approximation. (The
analysis for Eq. (113) was presented in the main text.)

Inserting the leading order result for $\tilde{h}_{ij}$ (from Eq. (62)) and
the energy-momentum tensor, a simple algebra gives (where we restored factors
of $\kappa^{2}$ and used the freedom to insert a contact term
$\rho_{H}\rho_{L}$ in the second line below)

$\displaystyle-\square h_{-i}$
$\displaystyle=2\kappa^{4}\mu_{H}\mu_{L}\Theta(x^{+})\left[x^{+}\Theta(x^{-})\partial_{i}\left(\frac{\partial_{k}\rho_{H}}{\square_{\perp}}\partial_{k}\rho_{L}+\rho_{H}\rho_{L}\right)-2\delta(x^{-})\frac{\partial_{i}\partial_{j}\rho_{L}}{\square_{\perp}}\frac{\partial_{j}\rho_{H}}{\square_{\perp}}\right]~{},$
(127) $\displaystyle-\square h_{--}$
$\displaystyle=4\kappa^{4}\mu_{H}\mu_{L}\delta(x^{-})x^{+}\Theta(x^{+})\bigg{[}\rho_{H}\rho_{L}-\frac{\partial_{i}\partial_{j}\rho_{H}}{\square_{\perp}}\frac{\partial_{i}\partial_{j}\rho_{L}}{\square_{\perp}}\bigg{]}~{}.$
(128)

Taking the Fourier transform of these expressions (and putting the momenta of
the emitted graviton on-shell) gives

$\displaystyle h_{-i}(k)$
$\displaystyle=\frac{4\kappa^{4}\mu_{H}\mu_{L}}{k^{2}+i\epsilon
k^{-}}\int\frac{d^{2}{\boldsymbol{q}}_{2}}{\left(2\pi\right)^{2}}\frac{1}{k_{+}}\left[({\boldsymbol{q}}_{1}\cdot{\boldsymbol{q}}_{2})\left(q_{2i}-k_{i}\frac{{\boldsymbol{q}}_{2}^{2}}{{\boldsymbol{k}}^{2}}\right)-k_{i}\frac{{\boldsymbol{q}}_{1}^{2}{\boldsymbol{q}}_{2}^{2}}{{\boldsymbol{k}}^{2}}\right]\frac{\rho_{H}}{{\boldsymbol{q}}_{1}^{2}}\frac{\rho_{L}}{{\boldsymbol{q}}_{2}^{2}}~{},$
(129) $\displaystyle h_{--}(k)$
$\displaystyle=\frac{4\kappa^{4}\mu_{H}\mu_{L}}{k^{2}+i\epsilon
k^{-}}\int\frac{d^{2}{\boldsymbol{q}}_{2}}{\left(2\pi\right)^{2}}\frac{1}{k_{+}^{2}}\left[({\boldsymbol{q}}_{1}\cdot{\boldsymbol{q}}_{2})^{2}-{\boldsymbol{q}}_{1}^{2}{\boldsymbol{q}}_{2}^{2}\right]\frac{\rho_{H}}{{\boldsymbol{q}}_{1}^{2}}\frac{\rho_{L}}{{\boldsymbol{q}}_{2}^{2}}~{}.$
(130)

Eqs. (129) and (130) are the results quoted in the main text in Eq. (66).

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