$\ell$, and this is a familiar result for transitionary potentials
Frank:1971xx . It arises because the wave function of both modes are finite as
$r\rightarrow 0$ and so the phase shift is ambiguous. An additional
consequence is that the system admits an infinity of bound states with
arbitrarily negative energies Frank:1971xx  suggesting it is unstable. Either
way, this implies that the phase shift is strongly sensitive to the UV physics
that regulates the $1/r^{2}$ potential. Indeed, this aspect will occur for
essentially any field with energy $k_{v}$ in this given pp-wave background.
Since the monopole ($\ell=0$) of essentially all fields are explicitly UV
sensitive, we cannot trust the spherically symmetric background solution we
started with, and while it may not be unstable, it cannot be described within
the remit of the EFT. Consequently we regard the case $k_{v}^{2}f_{0}>1$
outside of the regime of validity of the EFT at least as long as the source is
point-like. We shall consider extended sources below.

##### Case $k_{v}^{2}f_{0}<1$ —

The phase shift of the EFT $S$-matrix defined in Section 3.6 can be computed
in the Born approximation using the known exact GR partial wave states, and
takes the form

$\displaystyle\delta_{\ell}^{\rm EFT}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\langle P_{u},\ell|\hat{H}_{\rm
EFT}|P_{u},\ell\rangle$ (247) $\displaystyle\sim$ $\displaystyle c_{\rm
GB}\int_{0}^{\infty}\mathrm{d}r\,r^{3}\,\frac{k_{v}^{2}f_{0}}{\Lambda^{2}r^{4}}\frac{1}{r^{2}}\left(J_{\nu}(\sqrt{2k_{v}P_{u}}r)\right)^{2}\,,$

which should in general be more accurate than (234). Since
$J_{\nu}(\rho)\propto\rho^{\nu}$ as $\rho\rightarrow 0$, this perturbative
computation of the EFT time delay is only valid for $\nu\geq 1$, and for
$0<\nu\leq 1$ we must remember that the EFT only makes sense for
$r\geq\Lambda^{-1}$ — this will regulate the divergence.

Let us first consider $\nu\geq 1$, for which the integral is

$\delta_{\ell}^{\rm EFT}\sim c_{\rm
GB}\frac{k_{v}^{2}f_{0}}{\Lambda^{2}}\frac{2k_{v}P_{u}}{4(\nu^{3}-\nu)}\sim
c_{\rm
GB}\frac{k_{v}^{2}f_{0}}{\Lambda^{2}r_{*}^{2}}\frac{\nu}{4(\nu^{2}-1)}\,.$
(248)

For generic $\nu\gg 1$, this scales as

$\delta_{\ell}^{\rm EFT}\sim c_{\rm
GB}\frac{k_{v}^{2}f_{0}}{\Lambda^{2}r_{*}^{2}}\frac{1}{\nu}\sim c_{\rm
GB}\frac{k_{v}\sqrt{f_{0}}}{\Lambda^{2}r_{*}^{2}}\frac{1}{\sqrt{L^{2}-1}}\,,$
(249)

which is consistent with our estimate (227). Restricting to the physical
region $k_{v}^{2}f_{0}<1$ and imposing the EFT condition $\Lambda r_{*}\gg 1$
is sufficient to render this effect unresolvable

$k_{v}|\Delta v_{\rm EFT}|\lesssim|c_{\rm GB}|\frac{\nu}{4(\nu^{2}-1)}\,$
(250)

for generic $\nu$. However, it still appears that, by sending $\nu\rightarrow
1$, we can enhance this effect arbitrarily. As we tune $L\rightarrow 1$, or
equivalently $\nu\rightarrow 0$, we are sending the GR effective potential to
zero. But at some point the approximation that the EFT potential can be
treated perturbatively breaks down. By comparing terms in the effective
potential, we require the scattering state to be valid in EFT perturbation
theory

$\langle\psi|\frac{\nu^{2}}{k_{v}\hat{r}^{2}}|\psi\rangle\gg\langle\psi|\frac{k_{v}f_{0}}{\Lambda^{2}\hat{r}^{4}}|\psi\rangle\,.$
(251)

Imposing this condition on the EFT time delay we have

$\displaystyle k_{v}|\Delta v_{\rm EFT}|$ $\displaystyle\ll$
$\displaystyle|c_{\rm
GB}|\int_{0}^{\infty}\mathrm{d}r\,r^{3}\,\frac{\nu^{2}}{r^{2}}\frac{1}{r^{2}}\left(J_{\nu}(\sqrt{2k_{v}P_{u}}r)\right)^{2}\,$
(252) $\displaystyle\ll$ $\displaystyle|c_{\rm
GB}|\nu^{2}\int_{0}^{\infty}\mathrm{d}x\,\frac{1}{x}J_{\nu}(x)^{2}=|c_{\rm
GB}|\frac{1}{2}\nu\,,$ (253)

When (253) is combined with (251) we find

$\displaystyle k_{v}|\Delta v_{\rm EFT}|\ll|c_{\rm GB}|\,\,.$ (254)

##### Extended sources: Case $k_{v}^{2}f_{0}>1$ —

As already noted, for $k_{v}^{2}f_{0}>1$ the singular nature of the potential
leads to an ill-defined $S$-matrix for low multipoles. However, if the source
is extended of width $w\gg\Lambda^{-1}$ then we may replace the GR potential
by something of the form

$-\frac{k_{v}f_{0}}{2r^{2}}\rightarrow-\frac{k_{v}f_{0}}{2(r^{2}+w^{2})}\,.$
(255)

In this situation, the centifugal term will win out at small $r$, leading to a
well-defined $S$-matrix. The monopole will scatter at $r=r_{0}\ll w$ for which

$P_{u}+\frac{k_{v}f_{0}}{2w^{2}}\approx\frac{1}{2k_{v}r_{0}^{2}}\,.$ (256)

which implies, given $P_{u}>0$, $f_{0}>0$

$r_{0}\lesssim\frac{w}{\sqrt{k_{v}^{2}f_{0}}}\,.$ (257)

Using (40b) applied at $r_{0}$ implies

$\frac{\sqrt{k_{v}^{2}f_{0}}}{\Lambda^{2}}\ll\frac{w^{2}}{k_{v}^{2}f_{0}}\,.$
(258)

Then, returning to high multipoles, to trust our previous expression (249) we
require $w\ll r_{*}$ and so

$k_{v}|\Delta v_{\rm EFT}|\sim|c_{\rm
GB}|\frac{k_{v}^{2}f_{0}}{\Lambda^{2}r_{*}^{2}}\frac{1}{\nu}\ll|c_{\text{GB}}|\frac{k_{v}^{2}f_{0}}{\Lambda^{2}w^{2}}\frac{1}{\nu}\ll|c_{\text{GB}}|\frac{1}{\sqrt{k_{v}^{2}f_{0}}}\frac{1}{\nu}\ll|c_{\text{GB}}|\,.$
(259)

Thus for fixed $k_{v}$ and $P_{u}$, demanding that the monopole scattering
lies in the regime of validity of the EFT ensures unresolvability of the high
multipole scattering.

#### Case $D>6$

For the general case $d>4$ ($D>6$), the attractive gravitational potential
dominates over the centrifugal repulsion at small $z_{r}$, so that the
effective potential has a maximum. Scattering states will be those whose
energy is less than the maximum, which means there is a minimum impact
parameter. States with energy above this spiral into the centre and do not
reach infinity. For the scattering states, in general, the integral on the RHS
of (238) is of order unity, consistent with expectations. However it is
possible to make it arbitrarily large for $d\geq 5$ by tuning the energy close
to the maximum of the potential. This maximum occurs when
$2L^{2}z_{r}^{d-4}=(d-2)$, i.e for

$L^{2}=\frac{(d-2)}{2}\,.$ (260)

For this choice, by construction, the function $h(z_{r})$ entering the square
root in (238) satisfies $h(1)=h^{\prime}(1)=1$, implying that the integral is
logarithmically divergent near $z_{r}=1$. If we tune $L$ to be exponentially
close to the maximum

$L^{2}=\frac{(d-2)}{2}+e^{-N}\,,$ (261)

then we expect

$\Delta\tilde{v}_{\rm EFT}\sim N\,,$ (262)

or that

$k_{v}|\Delta v_{\rm EFT}|\ll|c_{\rm GB}|N\,.$ (263)

Thus, it naïvely appears that, by making $N$ large enough, we can generate an
arbitrarily large EFT contribution to the time delay. Once again, there is a
caveat to this argument. In this case, the validity of the WKB approximation
for the phase shift requires there to be sufficient exponential damping in the
forbidden region that we may ignore the solution on the left of the barrier.
This requires

$\frac{1}{\text{}}\int_{z_{*}}^{1}\sqrt{\frac{L^{2}}{z_{r}^{2}}-\frac{1}{z_{r}^{d-2}}-(L^{2}-1)}\mathrm{d}z_{r}\gg
1\,,$ (264)

where $z_{*}$ is the turning point on the left-hand side of the maximum. This
sets the maximum $N$ in terms of  such that $N\sim\ln(\text{}^{-1})$. The same
condition can be obtained by demanding the WKB approximation to be valid at
the maximum of the effective potential $V^{\prime}_{\rm eff}(z_{\rm max})=0$

$\Big{|}\frac{\partial_{z}\omega}{\omega^{2}}\Big{|}_{z=z_{\rm max}}\ll 1\,,$
(265)

with

$\omega^{2}=-\frac{1}{\text{}^{2}}\left(\frac{L^{2}}{z_{r}^{2}}-\frac{1}{z_{r}^{d-2}}-(L^{2}-1)\right)\,.$
(266)

Putting this together, the maximum time delay generated in the regime of
validity of the EFT is

$k_{v}|\Delta v_{\rm EFT}|\ll|c_{\rm GB}|{\rm ln}(\text{}^{-1})\,.$ (267)

Thus, at best, we are forced to tune the quantum hierarchy  to be
exponentially small, e.g. $\text{}\sim e^{-100}\sim 10^{-44}$, to have an
observable EFT contribution to the time delay. This result is closely
analogous to the balancing shockwave result in (206). This is no surprise
since, in both cases, the time delay is being maximised by engineering an
equilibrium point in the effective potential — in the present case due to the
balancing of the centrifugal force and the gravitational attraction. In the
case of the balancing shockwaves, we argued in Section 5.2 that the
uncertainty in the time delay necessarily becomes larger at precisely the
point that the logarithm becomes large. In the present case, since we are
considering $P_{u}$ eigenstates of definite partial wave, there is no
scattering uncertainty $\delta v=0$ and so we cannot rely on that argument.

The issue in the present case is closely analogous to the $D=6$ case. To tune
$N$ to be large, say $100$, we need $\text{}\sim e^{-100}$ in order to trust
the WKB approximation. At the same time, we are tuning $L^{2}=(d-2)/2$ which
means $\ell\sim e^{100}$. However, for all lower multipoles of the same
energy, the centrifugal barrier is too low, meaning such states fall into the
origin — more precisely, they have an underdetermined $S$-matrix due to the
ambiguity in the wave functions at $r=0$, which can only be resolved by the
actual UV completion. This situation is in fact worse than $D=6$, since it is
now an exponentially large number of low multipoles which are ill-defined. To
avoid this pathology, and to have the background under control within the
regime of validity of the EFT, we need that the monopole $\ell=0$ to be
scattered. To do that within the above framework requires $\text{}\sim 1$ and
so we obtain

$k_{v}|\Delta v_{\rm EFT}|\ll|c_{\rm GB}|\,.$ (268)

##### Extended sources —

As previously discussed, the situation is improved if the source is extended,
for example in the manner

$-\frac{k_{v}f_{0}}{2r^{d-2}}\rightarrow-\frac{k_{v}f_{0}}{2(r^{2}+w^{2})^{(d-2)/2}}\,,$
(269)

so that the $S$-matrix is well-defined for low multipoles. Assuming we have
tuned $L^{2}\approx(d-2)/2$, the monopole will pass over the barrier and
scatter at the point $r=r_{0}$ for which the centrifugal term dominates

$P_{u}+\frac{k_{v}f_{0}}{2w^{(d-2)}}\approx\frac{((d-2)/2)^{2}}{2k_{v}r_{0}^{2}}\,.$
(270)

Given $P_{u}>0$ and $f_{0}>0$, this implies
$r_{0}\lesssim\frac{w^{(d-2)/2}}{\sqrt{k_{v}^{2}f_{0}}}$. Imposing (40b) at
$r=r_{0}$ implies

$\left(\frac{k_{v}^{2}f_{0}}{\Lambda^{4}}\right)^{1/d}\ll\frac{w^{(d-2)/2}}{\sqrt{k_{v}^{2}f_{0}}}\,.$
(271)

Returning to the high multipoles, to trust our previous formulae for the phase
shift we require $r_{*}\gg w$ and so

$\left(\frac{k_{v}^{2}f_{0}}{\Lambda^{4}}\right)^{1/d}\ll\frac{r_{*}^{(d-2)/2}}{\sqrt{k_{v}^{2}f_{0}}}\quad\quad\Rightarrow\quad\quad\frac{k_{v}^{2}f_{0}}{r_{*}^{d-4}}\ll(r_{*}\Lambda)^{\frac{8}{(d+2)}}\,.$
(272)

From (226) and (262) we estimate the time delay to be

$\displaystyle k_{v}|\Delta v_{\rm EFT}|$ $\displaystyle\sim$
$\displaystyle|c_{\rm GB}|\frac{1}{(\Lambda
r_{*})^{2}}\sqrt{\frac{k_{v}^{2}f_{0}}{r_{*}^{d-4}}}\ln\left(\frac{k_{v}^{2}f_{0}}{r_{*}^{d-4}}\right)$
(273) $\displaystyle\ll$ $\displaystyle|c_{\rm GB}|\frac{1}{\left(\Lambda
r_{*}\right)^{\frac{2(d-2)}{(d+2)}}}\ln(r_{*}\Lambda)\,,$ (274)

which combined with (40a), i.e. $\Lambda r_{*}\gg 1$ implies $k_{v}|\Delta
v_{\rm EFT}|\ll|c_{\rm GB}|$.

##  7 Shockwave scattering in the Born approximation

In this section we compute the scattering time delay in perturbation theory,
identifying the first few terms in the Born series. In general, the Born
series has a smaller regime of validity than either the eikonal or semi-
classical approximations, and so we do not draw any major conclusions from its
results. However, we include it because the calculation may be performed more
precisely and it is informative in terms of demonstrating the distinction
between the single source and balancing shockwaves. It is also worth
emphasising that almost all treatments of the eikonal approximation in the
literature rely on resumming the Born amplitude, and the leading Born
approximation to the time delay is identical to the eikonal result. The
exponential eikonal resummation is implicit in the definition of the time
delay operator.

The physical scenario for the two types of potential we have in mind is
illustrated in Fig. 3.

Figure 3: Snapshot at time $u=0$ of the point-source and balancing potentials
and the wave function which probes the spacetime.

Then the potential is generated by either a single point source located at
$\mathbf{x}=-\mathbf{b}$ or balancing sources at $\mathbf{x}=\pm\mathbf{b}$:

$V(u,\mathbf{x})=-\frac{k_{v}f(u)}{2}\left(\frac{\Theta_{\text{bal}}}{\left\lvert\mathbf{x}-\mathbf{b}\right\rvert^{d-2}}+\frac{1}{\left\lvert\mathbf{x}+\mathbf{b}\right\rvert^{d-2}}\right)\,,$
(275)

where we have introduced the notation

$\Theta_{\text{bal}}=\begin{cases}0,\quad\text{point source}\\\
1,\quad\text{balancing sources}\,.\end{cases}$

Note that, in order to accommodate the point source case and balancing source
case simultaneously, the location of the point source is shifted relative to
the discussion of Section 2. The interaction picture potential is

$\hat{H}_{1}^{\rm
int}(u)=-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}k_{v}f(u)\int\frac{\text{d}^{d}q}{(2\pi)^{d}}\frac{e^{i\mathbf{q}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}\cdot\mathbf{b}}}{\mathbf{q}^{2}}e^{i\mathbf{q}\cdot\left(\mathbf{\hat{x}}+\frac{u}{k_{v}}\mathbf{\hat{k}}\right)}.$
(276)

which is conveniently rewritten as

$\begin{split}\hat{H}_{1}^{\rm
int}(u)=-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}k_{v}f(u)\int&\frac{\text{d}^{d}q}{(2\pi)^{d}}\int\frac{\text{d}^{d}k}{(2\pi)^{d}}\int\text{d}^{d}x\frac{e^{i\mathbf{q}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}\cdot\mathbf{b}}}{\mathbf{q}^{2}}\\\
&\exp\left[i\mathbf{x}\cdot\left(\mathbf{q}+\mathbf{k}\right)+\frac{iu}{k_{v}}\mathbf{q}\cdot\left(\frac{\mathbf{q}}{2}+\mathbf{k}\right)\right]\ket{\mathbf{x}}\bra{\mathbf{k}}.\end{split}$
(277)

Denoting the terms in the Dyson series as

$\hat{S}=1+\underbrace{(-i)\int_{-\infty}^{+\infty}\text{d}u\,\hat{H}_{1}^{\rm
int}(u)}_{\hat{S}^{(1)}}+\underbrace{(-i)^{2}\int_{-\infty}^{+\infty}\text{d}u\int_{-\infty}^{u}\text{d}u^{\prime}\,\hat{H}_{1}^{\rm
int}(u)\hat{H}_{1}^{\rm int}(u^{\prime})}_{\hat{S}^{(2)}}+\dots,$ (278)

the first three terms in the series expansion for the time delay are

$\displaystyle\widehat{\Delta v}{}^{(1)}=$
$\displaystyle-i\frac{\partial\hat{S}^{(1)}}{\partial k_{v}},$ (279)
$\displaystyle\widehat{\Delta v}{}^{(2)}=$
$\displaystyle-i\left(\frac{\partial\hat{S}^{(2)}}{\partial
k_{v}}+\hat{S}^{(1)\dagger}\frac{\partial\hat{S}^{(1)}}{\partial
k_{v}}\right),$ (280) $\displaystyle\widehat{\Delta v}{}^{(3)}=$
$\displaystyle-i\left(\frac{\partial\hat{S}^{(3)}}{\partial
k_{v}}+\hat{S}^{(1){\dagger}}\frac{\partial\hat{S}^{(2)}}{\partial
k_{v}}+\hat{S}^{(2){\dagger}}\frac{\partial\hat{S}^{(1)}}{\partial
k_{v}}\right).$ (281)

Without ambiguity we denote the corresponding $n$-th order term in the
expectation value for the in-state (140) simply by

$\Delta v^{(n)}=\bra{\rm in}\widehat{\Delta v}{}^{(n)}\ket{\rm in}\,.$ (282)

We consider interactions over a finite time $0<u<u_{\text{max}}$. To first
order, the GR time delay is

$\displaystyle\Delta
v^{(1)}_{\text{GR}}(u)=\int_{0}^{u}\text{d}u_{1}\,f(u_{1})\left(1-u_{1}\frac{\partial}{\partial
u_{1}}\right)K^{(1)}(u_{1})\,.$ (283)

The second order GR time delay is given by

$\displaystyle\Delta v^{(2)}_{\text{GR}}(u)$ $\displaystyle=$ $\displaystyle
k_{v}\left(\prod_{i=1}^{2}\int_{0}^{u}\text{d}u_{i}\,f(u_{i})\right)\times$
$\displaystyle\bigg{[}\theta(u_{1}-u_{2})\left(1-u_{1}\frac{\partial}{\partial
u_{1}}\right)-\theta(u_{2}-u_{1})\left(1-u_{2}\frac{\partial}{\partial
u_{2}}\right)\bigg{]}K^{(2)}(u_{1},u_{2})\,,$

and the third order one by

$\displaystyle\Delta v^{(3)}_{\text{GR}}(u)$ $\displaystyle=$ $\displaystyle
k_{v}^{2}\left(\prod_{i=1}^{3}\int_{0}^{u}\text{d}u_{i}\,f(u_{i})\right)\times$
$\displaystyle\bigg{[}$
$\displaystyle\theta(u_{1}-u_{2})\theta(u_{2}-u_{3})\left(3-u_{1}\frac{\partial}{\partial
u_{1}}-u_{2}\frac{\partial}{\partial u_{2}}-u_{3}\frac{\partial}{\partial
u_{3}}\right)$ $\displaystyle-$
$\displaystyle\theta(u_{2}-u_{3})\left(2-u_{2}\frac{\partial}{\partial
u_{2}}-u_{3}\frac{\partial}{\partial
u_{3}}\right)+\theta(u_{2}-u_{1})\left(1-u_{3}\frac{\partial}{\partial
u_{3}}\right)\bigg{]}K^{(3)}(u_{1},u_{2},u_{3})\,,$

where $\theta(u)$ is the Heaviside step function and

$\displaystyle K^{(n)}(u_{1},\dots,u_{n})$ $\displaystyle=$
$\displaystyle(-i)^{n+1}\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{n}\left(\prod_{i=1}^{n}\int\frac{\text{d}^{d}q_{i}}{(2\pi)^{d}}\frac{e^{i\mathbf{q}_{i}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{i}\cdot\mathbf{b}}}{\mathbf{q}_{i}^{2}}\right)$
$\displaystyle\times$
$\displaystyle\exp\left[-\frac{\sigma^{2}}{2}\left\\{\left|\sum_{i=1}^{n}\mathbf{q}_{i}\right|^{2}+\left|\frac{1}{2k_{v}\sigma^{2}}\sum_{i=1}^{n}u_{i}\mathbf{q}_{i}\right|^{2}\right\\}+\frac{i}{2k_{v}}\sum_{i<j=1}^{n}(u_{i}-u_{j})\mathbf{q}_{i}\cdot\mathbf{q}_{j}\right]\,.$

###  7.1 The classical limit: Identifying diffusion and scattering effects

The effects of uncertainty and diffusion are captured at, every order in
$\Delta v$, by the term in curly brackets in the exponential of each $K^{(n)}$
in (7). In the classical limit, where this term is negligible, the first-order
term becomes

$\lim_{\rm classical}\Delta
v_{\text{GR}}^{(1)}(u)=-\frac{1}{k_{v}}\int_{0}^{u}\text{d}u_{1}V(u_{1},\mathbf{b})(1+\Theta_{\text{bal}}),$
(287)

which is exactly the eikonal expression of (167). This is to be expected since
the leading-order eikonal phase can be identified with the leading-order Born
approximation.

After some manipulation, the classical limit of the second-order term in the
time delay expansion can be written as

$\displaystyle\lim_{\rm classical}\Delta v^{(2)}_{\text{GR}}(u)=$
$\displaystyle-\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{2}k_{v}\int_{0}^{u}\text{d}u_{1}f(u_{1})\int_{0}^{u_{1}}\text{d}u_{2}f(u_{2})\int\frac{\text{d}^{d}q_{1}}{(2\pi)^{2}}\frac{\text{d}^{d}q_{2}}{(2\pi)^{2}}$
$\displaystyle\frac{e^{i\mathbf{q}_{1}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{1}\cdot\mathbf{b}}}{\mathbf{q}_{1}^{2}}\frac{e^{i\mathbf{q}_{2}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{2}\cdot\mathbf{b}}}{\mathbf{q}_{2}^{2}}$
(288)
$\displaystyle\left(\sin\left[\frac{1}{2k_{v}}(u_{1}-u_{2})\mathbf{q}_{1}\cdot\mathbf{q}_{2}\right]-\frac{u_{1}}{2k_{v}}\mathbf{q}_{1}\cdot\mathbf{q}_{2}\cos\left[\frac{1}{2k_{v}}(u_{1}-u_{2})\mathbf{q}_{1}\cdot\mathbf{q}_{2}\right]\right).$

In the balancing case, $\Theta_{\text{bal}}=1$, it is easy to see that the
integrand of (7.1) is odd under $\mathbf{q}_{1}\rightarrow-\mathbf{q}_{1}$ (or
$\mathbf{q}_{2}\rightarrow-\mathbf{q}_{2}$) and thus the integral vanishes. In
the point source case, there is no such anti-symmetry so the integral does not
vanish.

The classical limit of the third-order term in the time delay expansion can be
written as

$\displaystyle\lim_{\rm classical}\Delta v^{(3)}_{\text{GR}}(u)=$
$\displaystyle\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{3}k_{v}^{2}\int_{0}^{u}\text{d}u_{1}f(u_{1})\int_{0}^{u}\text{d}u_{2}f(u_{2})\int_{0}^{u}\text{d}u_{3}f(u_{3})\int\frac{\text{d}^{d}q_{1}}{(2\pi)^{2}}\frac{\text{d}^{d}q_{2}}{(2\pi)^{2}}\frac{\text{d}^{d}q_{3}}{(2\pi)^{2}}$
$\displaystyle\frac{e^{i\mathbf{q}_{1}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{1}\cdot\mathbf{b}}}{\mathbf{q}_{1}^{2}}\frac{e^{i\mathbf{q}_{2}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{2}\cdot\mathbf{b}}}{\mathbf{q}_{2}^{2}}\frac{e^{i\mathbf{q}_{3}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{3}\cdot\mathbf{b}}}{\mathbf{q}_{3}^{2}}$
$\displaystyle\left[\theta(u_{1}-u_{2})\theta(u_{2}-u_{3})\left(1-u_{1}\frac{\partial}{\partial
u_{1}}\right)\right.$
$\displaystyle-\theta(u_{2}-u_{1})\theta(u_{2}-u_{3})\left(1-u_{2}\frac{\partial}{\partial
u_{2}}\right)$ (289)
$\displaystyle\left.+\theta(u_{2}-u_{1})\theta(u_{3}-u_{2})\left(1-u_{3}\frac{\partial}{\partial
u_{3}}\right)\right]$
$\displaystyle\exp\left[\frac{i}{2k_{v}}\left((u_{1}-u_{2})\mathbf{q}_{1}\cdot\mathbf{q}_{2}+(u_{1}-u_{3})\mathbf{q}_{1}\cdot\mathbf{q}_{3}+(u_{2}-u_{3})\mathbf{q}_{2}\cdot\mathbf{q}_{3}\right)\right].$

The vanishing of this in the balancing case is less straightforward. First
note that, as the eigenvalue of an Hermitian operator, the expression (7.1)
should be real. We can verify that it is real by simultaneously exchanging

$u_{1}\longleftrightarrow
u_{3}\quad\text{and}\quad\mathbf{q}_{1}\longleftrightarrow\mathbf{q}_{3},$
(290)

in the integrand to retrieve its complex conjugate. Thus, the exponential
factor at the bottom of (7.1) may be replaced by its real part, a cosine.
Then, in the balancing case only, we have the following situation

$\displaystyle\lim_{\rm classical}\Delta
v^{(3)}_{\text{GR}}(u)=\int_{0}^{u}\text{d}u_{1}\text{d}u_{2}\text{d}u_{3}$
$\displaystyle
f(u_{1})f(u_{2})f(u_{3})\left[\theta(u_{1}-u_{2})\theta(u_{2}-u_{3})\left(1-u_{1}\frac{\partial}{\partial
u_{1}}\right)\right.$
$\displaystyle\left.-\theta(u_{2}-u_{1})\theta(u_{2}-u_{3})\left(1-u_{2}\frac{\partial}{\partial
u_{2}}\right)\right.$ (291)
$\displaystyle\left.+\theta(u_{2}-u_{1})\theta(u_{3}-u_{2})\left(1-u_{3}\frac{\partial}{\partial
u_{3}}\right)\right]G(u_{1},u_{2},u_{3})\,,$

where $G(u_{1},u_{2},u_{3})$ is a function that is invariant under any
permutation of the $u_{i}$'s. For example, the exchange $u_{1}\leftrightarrow
u_{2}$ in $G$, may be offset by the exchange of the integration variables
$\mathbf{q}_{1}\leftrightarrow\mathbf{q}_{2}$ and the change of variables
$\mathbf{q}_{3}\rightarrow-\mathbf{q}_{3}$ to leave it invariant. With this in
mind, the first and third terms in the square brackets give the same
contribution overall (by exchange of $u_{1}\leftrightarrow u_{3}$). Merging
these terms and additionally exchanging $u_{1}\leftrightarrow u_{2}$ in the
second term, we obtain:

$\displaystyle\begin{split}\lim_{\rm classical}\Delta
v^{(3)}_{\text{GR}}(u)=\int_{0}^{u}&\text{d}u_{1}f(u_{1})\int_{0}^{u_{1}}\text{d}u_{2}f(u_{2})\left(2\int_{0}^{u_{2}}\text{d}u_{3}-\int_{0}^{u_{1}}\text{d}u_{3}\right)f(u_{3})\\\
&\left(1-u_{1}\frac{\partial}{\partial
u_{1}}\right)G(u_{1},u_{2},u_{3}).\end{split}$ (292)

Now, note that

$\displaystyle\begin{split}\int_{0}^{u_{1}}\text{d}u_{2}\left(2\int_{0}^{u_{2}}\text{d}u_{3}-\int_{0}^{u_{1}}\text{d}u_{3}\right)&=\int_{0}^{u_{1}}\text{d}u_{2}\left(\int_{0}^{u_{2}}\text{d}u_{3}-\int_{u_{2}}^{u_{1}}\text{d}u_{3}\right)\\\
&=\int_{0}^{u_{1}}\text{d}u_{2}\int_{0}^{u_{2}}\text{d}u_{3}-\int_{0}^{u_{1}}\text{d}u_{3}\int_{0}^{u_{3}}\text{d}u_{2}\,,\end{split}$
(293)

and since the integrand is invariant under the exchange of
$u_{2}\leftrightarrow u_{3}$, the whole integral in (292) is zero.

It is expected that all higher-order corrections to the leading term (287) in
the balancing-source time delay vanish in the classical limit. Again, let it
be stressed that this is a reflection of the fact that there is, classically,
no scattering from the unstable equilibrium point in the balancing potential.
Thus, outside the classical limit, the higher-order terms $\Delta v^{(n>1)}$
may be regarded as the result of diffusion-induced scattering in the quantum
problem.

###  7.2 Perturbative regime of validity

The requirement that the higher-order scattering-induced terms $\Delta
v_{\text{GR}}^{(n>1)}(u)$ are under control compared to the leading-order GR
term $\Delta v^{(1)}_{\text{GR}}(u)$ imposes a time cut-off $u_{\text{max}}$
beyond which we may no longer trust perturbation theory. Below, we will
determine the parameter ``$\epsilon$" which controls the series expansion of
$\Delta v_{\text{GR}}(u)$. As we will see, this parameter will grow with time
$u$ itself — at early times, the series expansion is under better control and
$\Delta v^{(1)}_{\text{GR}}(u)$ represents a good approximation for $\Delta
v_{\text{GR}}(u)$, but at late times that approximation begins to fail. The
scattering timescale $u_{\text{pert}}$ will be defined as the time at which it
becomes large, $\epsilon(u_{\text{pert}})\sim 1$.

As a concession to simplicity, consider the limit of a large number of closely
spaced shockwaves for which we may approximate $f(u)=f_{0}$ as a constant.
With this simplification, the $u$-derivative-terms in expressions (283)-(7)
may be integrated by parts. For $u<u_{\text{max}}$, we therefore get,
explicitly, at first order

$\displaystyle\Delta
v^{(1)}_{\text{GR}}(u)=-2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)f_{0}\int_{0}^{u}\text{d}u_{1}\int\frac{\text{d}^{d}q_{1}}{(2\pi)^{d}}\frac{e^{i\mathbf{q}_{1}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{1}\cdot\mathbf{b}}}{\mathbf{q}_{1}^{2}}\text{exp}\left[-\frac{\sigma^{2}\mathbf{q}_{1}^{2}}{2}-\frac{u_{1}^{2}\mathbf{q}_{1}^{2}}{8k_{v}^{2}\sigma^{2}}\right]\,,$

at second order,

$\displaystyle\Delta v^{(2)}_{\text{GR}}(u)$ $\displaystyle=$
$\displaystyle-2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{2}k_{v}f_{0}^{2}\int_{0}^{u}\text{d}u_{1}\text{d}u_{2}\int\frac{\text{d}^{d}q_{1}}{(2\pi)^{d}}\frac{\text{d}^{d}q_{2}}{(2\pi)^{d}}$
$\displaystyle\frac{e^{i\mathbf{q}_{1}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{1}\cdot\mathbf{b}}}{\mathbf{q}_{1}^{2}}\frac{e^{i\mathbf{q}_{2}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{2}\cdot\mathbf{b}}}{\mathbf{q}_{2}^{2}}$
$\displaystyle\text{exp}\left[-\frac{\sigma^{2}}{2}\left|\mathbf{q}_{1}+\mathbf{q}_{2}\right|^{2}-\frac{1}{8k_{v}^{2}\sigma^{2}}\left|u_{1}\mathbf{q}_{1}+u_{2}\mathbf{q}_{2}\right|^{2}\right]$
$\displaystyle\text{sin}\left[\frac{1}{2k_{v}}(u_{1}-u_{2})\mathbf{q}_{1}\cdot\mathbf{q}_{2}\right]\,,$

and at third order

$\displaystyle\Delta v^{(3)}_{\text{GR}}(u)$ $\displaystyle=$ $\displaystyle
2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{3}k_{v}^{2}f_{0}^{3}\int_{0}^{u}\text{d}u_{1}\text{d}u_{2}\text{d}u_{3}\int\frac{\text{d}^{d}q_{1}}{(2\pi)^{d}}\frac{\text{d}^{d}q_{2}}{(2\pi)^{d}}\frac{\text{d}^{d}q_{3}}{(2\pi)^{d}}$
$\displaystyle\frac{e^{i\mathbf{q}_{1}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{1}\cdot\mathbf{b}}}{\mathbf{q}_{1}^{2}}\frac{e^{i\mathbf{q}_{2}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{2}\cdot\mathbf{b}}}{\mathbf{q}_{2}^{2}}\frac{e^{i\mathbf{q}_{3}\cdot\mathbf{b}}+\Theta_{\text{bal}}e^{-i\mathbf{q}_{3}\cdot\mathbf{b}}}{\mathbf{q}_{3}^{2}}$
$\displaystyle\text{exp}\left(-\frac{\sigma^{2}}{2}\left|\mathbf{q}_{1}+\mathbf{q}_{2}+\mathbf{q_{3}}\right|^{2}-\frac{1}{8k_{v}^{2}\sigma^{2}}\left|u_{1}\mathbf{q}_{1}+u_{2}\mathbf{q}_{2}+u_{3}\mathbf{q}_{3}\right|^{2}\right)$
$\displaystyle\text{cos}\left(\frac{1}{2k_{v}}\left[(u_{1}-u_{2})\mathbf{q}_{1}\cdot\mathbf{q}_{2}+(u_{1}-u_{3})\mathbf{q}_{1}\cdot\mathbf{q}_{3}+(u_{2}-u_{3})\mathbf{q}_{2}\cdot\mathbf{q}_{3}\right]\right)$
$\displaystyle\left[2\Theta(u_{1}-u_{2})\Theta(u_{2}-u_{3})-\Theta(u_{2}-u_{1})\Theta(u_{2}-u_{3})\right]\,.$

While the cross terms mean these expressions are not readily integrated, we
can estimate the parameter dependence of each $\Delta v^{(n)}_{\text{GR}}(u)$
as long as we make the following assumptions:

$\displaystyle\sigma\ll b,$ (296) $\displaystyle u\ll k_{v}\sigma b.$ (297)

Physically, (296) corresponds to the requirement that the width of the initial
Gaussian wave packet $\sigma$ is much smaller than the distance to the source
$b$. The second assumption (297) is that the timescale under consideration $u$
is earlier than the diffusion time $u_{\text{diff}}\sim k_{v}\sigma b$ (see
(144)). We also need to distinguish between the following two cases.

### Scattering before diffusion

By switching to dimensionless integration variables $w_{i}=u_{i}/u$ and
$\mathbf{p}_{i}=b\mathbf{q}_{i}$, the parameter dependence of $\Delta
v^{(n)}_{\text{GR}}(u)$ can be extracted to leading order in $\sigma/b$ and
$u/k_{v}\sigma b$:

$\displaystyle\begin{aligned} \Delta
v^{(1)}_{\text{GR}}(u)\sim\frac{fu}{b^{d-2}},\end{aligned}$ (298a)
$\displaystyle\begin{aligned} \Delta
v^{(2)}_{\text{GR}}(u)\sim\frac{f^{2}u^{3}}{b^{2d-2}}\left[1+\Theta_{\text{bal}}\left(-1+a^{(2)}\frac{\sigma^{2}}{b^{2}}+\tilde{a}^{(2)}\frac{u^{2}}{k_{v}^{2}b^{2}\sigma^{2}}\right)\right],\end{aligned}$
(298b) $\displaystyle\begin{aligned} \Delta
v^{(3)}_{\text{GR}}(u)\sim\frac{k_{v}^{2}f^{3}u^{3}}{b^{3d-6}}\left[1+\Theta_{\text{bal}}\left(-1+a^{(3)}\frac{\sigma^{2}}{b^{2}}+\tilde{a}^{(3)}\frac{u^{2}}{k_{v}^{2}b^{2}\sigma^{2}}\right)\right],\end{aligned}$
(298c)

where $a^{(n)}$ and $\tilde{a}^{(n)}$ are numbers expressed in terms of
$n(d+1)$ dimensionless integrals (given in appendix F).

The ratio of one term $\Delta v^{(n)}_{\text{GR}}(u)$ to the next $\Delta
v^{(n+1)}_{\text{GR}}(u)$ is not the same for all $n$. The even terms $\Delta
v^{(2n)}_{\text{GR}}(u)$ are artificially suppressed by the presence of an odd
sine in their integrands, compared to the even cosine in odd terms $\Delta
v^{(2n+1)}_{\text{GR}}(u)$. In other words, the third-order term $\Delta
v^{(3)}_{\text{GR}}(u)$ is not necessarily subdominant to the second-order
term $\Delta v^{(2)}_{\text{GR}}(u)$, and so on. This means that the ratio
which truly controls the expansion is not the ratio of one term to the next
$\Delta v^{(n+1)}_{\text{GR}}(u)/\Delta v^{(n)}_{\text{GR}}(u)$ but instead it
is of even terms $\Delta v^{(2n+2)}_{\text{GR}}(u)/\Delta
v^{(2n)}_{\text{GR}}(u)$ (or equally, the ratio of odd terms $\Delta
v^{(2n+1)}_{\text{GR}}(u)/\Delta v^{(2n+3)}_{\text{GR}}(u)$).

Although the explicit parameter dependence of the generic $n$-th term is not
given here, it is apparent from the general expression (7) and the examples
(298) that the series expansion of the time delay is controlled by

$\epsilon(u)\sim\frac{\Delta v^{(2n)}_{\text{GR}}(u)}{\Delta
v^{(2n+2)}_{\text{GR}}(u)}\sim\left(\frac{k_{v}fu}{b^{d-2}}\right)^{2}.$ (299)

The $S$-matrix expansion is under control when $\epsilon(u)\ll 1$ and so we
may identify the time at which perturbation theory breaks down as

$\epsilon(u_{\text{pert}})\sim 1\quad\implies\quad
u_{\text{pert}}\sim\frac{1}{k_{v}}\frac{b^{d-2}}{f}.$ (300)

Unlike the estimates (170), (187) and (202), this estimate for the cut-off
timescale depends on the $v$-momentum $k_{v}$, which is a property of the test
wave itself and not of the background. In fact

$\frac{u_{\rm pert}}{u_{s}}\sim\frac{u_{s}}{u_{\rm diff}}\,,$ (301)

and so for $u_{s}<u_{\rm diff}$ the Born series breaks down before the eikonal
approximation, which is why we regard it as less informative.

Using the expressions for the two timescales, (144) and (300), and the
approximation for the leading-order time delay (298a), we find

$u_{\text{pert}}\ll u_{\text{diff}}\quad\iff\quad k_{v}\left\lvert\Delta
v^{(1)}_{\text{GR}}(u_{\text{diff}})\right\rvert\gg 1.$ (302)

In other words, the assumption that perturbation theory breaks down before the
diffusion time is consistent, as long as the GR time delay generated by
$u_{\text{diff}}$ is resolvable.

In this formalism, the EFT contribution to the time delay can generally be
estimated as

$\Delta
v^{(1)}_{\text{EFT}}(u)\sim\frac{c_{\text{GB}}}{b^{2}\Lambda^{2}}\Delta
v^{(1)}_{\text{GR}}(u)\sim c_{\text{GB}}\frac{fu}{b^{d}\Lambda^{2}}\,.$ (303)

The total EFT time delay by the time at which perturbation theory breaks down
is

$\Delta
v^{(1)}_{\text{EFT}}(u_{\text{pert}})\sim\frac{c_{\text{GB}}}{b^{2}\Lambda^{2}}\frac{1}{k_{v}}\,,$
(304)

and, given the EFT-validity bound $b\Lambda\gg 1$, satisfies

$k_{v}|\Delta v^{(1)}_{\text{EFT}}(u_{\text{pert}})|\ll|c_{\text{GB}}|\,.$
(305)

### Diffusion before scattering

We have already seen that the assumption $u_{\text{scatt}}<u_{\text{diff}}$ is
self-consistent, and we will now show that it is actually the only regime of
interest. Assuming that we can trust the expressions for $\Delta
v^{(n)}_{\text{GR}}(u)$ given in (298) all the way up until $u\sim
u_{\text{diff}}$, we can write

$\Delta v^{(3)}_{\text{GR}}(u_{\text{diff}})\sim k_{v}^{2}\left(\Delta
v^{(1)}_{\text{GR}}(u_{\text{diff}})\right)^{3}\left[1+\Theta_{\text{bal}}\left(-1+a^{(3)}\frac{\sigma^{2}}{b^{2}}+\tilde{a}^{(3)}\right)\right].$
(306)

Of the parts that are exclusive to the balancing case, $\tilde{a}^{(3)}$ will
dominate over $a^{(3)}\left(\sigma/b\right)^{2}$, so we can write-off the
totality of the square brackets as an $\mathcal{O}(1)$ number plus suppressed
corrections. If the scattering time comes after the diffusion time then, by
definition, the first-order term $\Delta v^{(1)}_{\text{GR}}(u)$ should still
dominate the series expansion by $u=u_{\text{diff}}$. However, asking for the
third-order term to be sub-dominant, $\left\lvert\Delta
v^{(3)}_{\text{GR}}(u_{\text{diff}})\right\rvert<\left\lvert\Delta
v^{(1)}_{\text{GR}}(u_{\text{diff}})\right\rvert$ leads to the conclusion

$\left\lvert k_{v}^{2}\left(\Delta
v^{(1)}_{\text{GR}}(u_{\text{diff}})\right)^{3}\right\rvert<\left\lvert\Delta
v^{(1)}_{\text{GR}}(u_{\text{diff}})\right\rvert\quad\implies\quad
k_{v}\left\lvert\Delta v^{(1)}_{\text{GR}}(u_{\text{diff}})\right\rvert<1,$
(307)

i.e. the first-order GR time delay itself is unresolvable. As hinted at
previously, we see that if we try to slow down the effects of scattering for
long enough such that diffusion takes hold first, then the price to pay is a
potential so weak (i.e. $f$ so small) that even the leading-order GR time
delay cannot be resolved. Since the EFT contribution is further suppressed by
$1/b^{2}\Lambda^{2}\ll 1$ compared to the GR contribution, it will certainly
not be resolvable.

##  8 Conclusions

In this paper, we reviewed the formal definition of the scattering time delay
appropriate to spacetimes with a null Killing vector (namely pp-wave
geometries), showed how to compute the time delay for arbitrary initial states
in the eikonal, semi-classical, and Born approximations, and discussed their
validity and connections between them. We considered the generic EFT of
gravity in arbitrary dimensions $D>4$, focussing on the Gauss-Bonnet term as
the leading correction. Assuming a generic weakly-coupled ($\Lambda\ll M_{\rm
Pl}$) Wilsonian EFT expansion (19), we carefully computed the regime of
validity necessary to determine scattering amplitudes in a general pp-wave
background. We saw that, while it is straightforward to engineer background
configurations for which there is an arbitrarily large gravitational (Shapiro)
time delay, the corrections due to the GB-term that can be consistently
inferred in the regime of validity of the low-energy EFT are bounded in the
manner

$k_{v}|\Delta v_{\rm EFT}|\lesssim|c_{\text{GB}}|\,.$ (308)

In particular, this remains true for the rather special balancing shockwave
solutions considered in Section 5, for which scattering is minimised. To
minimise scattering, one must choose highly-localised wave packets which are
far from $S$-matrix eigenstates. As such, the quantum mechanical uncertainty
in the time delay grows, swamping any potentially observable contribution.

These conclusions may be read in two ways: (1) Assuming that
$|c_{\text{GB}}|\lesssim{\cal O}(1)$, which is expected for gravitational EFTs
with consistent UV completions based on positivity/bootstrap bounds Caron-
Huot:2022jli  as well as by EFT power counting arguments, the time delay
induced by the GB-term is generically unresolvable, or (2) Demanding infrared
causality (namely causality with respect to the locally inertial metric)
$\Delta v_{\rm EFT}\gtrsim-k_{v}^{-1}$ and accounting for the fact that the
Gauss-Bonnet term can lead to time advances for scattering of specific
polarisations, then we infer $|c_{\text{GB}}|\lesssim{\cal O}(1)$. Finally, we
note that asymptotic causality in the form $\Delta T>0$ is automatically
satisfied, since the magnitude of the EFT correction is
$|c_{\text{GB}}|/(\Lambda b)^{2}$ suppressed relative to the GR/Shapiro delay
and the validity of the EFT requires $\Lambda b\gg 1$. While satisfied, it
does not provide much insight.

Either way, these results indicate a close connection between causality via
time delays and positivity/bootstrap bounds previously observed in
Adams:2006sv ; deRham:2020zyh ; Chen:2021bvg ; CarrilloGonzalez:2022fwg ;
CarrilloGonzalez:2023cbf . Further, they show that, while asymptotic causality
is automatically satisfied, it is generically too weak a condition. This
confirms the findings of deRham:2021bll  and that, in EFTs with a consistent
UV completion, causality is maintained with respect to the local metric, which
defines the local inertial frame and to which at least one field couples to
(taking care of field redefinition ambiguity, see deRham:2019ctd  for more
details on that point.).

## Acknowledgements

This work is supported by STFC Consolidated Grant ST/T000791/1. CdR is also
supported by a Simons Investigator award 690508. CC and AM are funded by a
Presidents Scholarship.

##  Appendix A EFT field equations for metric perturbations

In lightcone gauge, where $h_{v\alpha}=0$, the EFT field equations for the
metric perturbations

$\delta\mathcal{E}_{\alpha\beta}=\delta
G_{\alpha\beta}+\frac{2c_{\text{GB}}}{\Lambda^{2}}\delta B_{\alpha\beta}=0,$
(309)

are explicitly given by

$\displaystyle-2\delta\mathcal{E}_{vv}=\partial_{v}^{2}h_{ii},$ (310a)
$\displaystyle-2\delta\mathcal{E}_{vu}=-\tilde{\Box}h_{ii}+\partial_{v}\partial_{u}h_{ii}+\partial_{i}(\partial_{v}h_{iu}+\partial_{j}h_{ij}),$
(310b)
$\displaystyle-2\delta\mathcal{E}_{vi}=\partial_{v}\partial_{i}h_{jj}-\partial_{v}(\partial_{v}h_{iu}+\partial_{j}h_{ij}),$
(310c) $\displaystyle\,\,\begin{aligned}
-2\delta\mathcal{E}_{uu}=&\,\,\tilde{\Box}h_{uu}-2\partial_{u}(\partial_{v}h_{uu}+\partial_{i}h_{iu})+\partial_{i}H\partial_{v}h_{iu}-\partial_{j}(\partial_{i}Hh_{ij})+\partial_{u}^{2}h_{ii}-H\tilde{\Box}h_{ii}\\\
&+\frac{1}{2}\partial_{j}H\partial_{j}h_{ii}-\frac{1}{2}\partial_{u}H\partial_{v}h_{ii}+H\partial_{i}(\partial_{v}h_{iu}+\partial_{j}h_{ij})+H\partial_{v}(\partial_{v}h_{uu}+\partial_{i}h_{iu})\\\
&+\frac{4c_{\text{GB}}}{\Lambda^{2}}\partial_{i}\partial_{j}H(\partial_{v}\partial_{i}h_{ju}-\partial_{v}\partial_{j}h_{iu}+\partial_{k}\partial_{k}h_{ij}-\partial_{i}\partial_{k}h_{jk}-\partial_{j}\partial_{k}h_{ik}+\partial_{i}\partial_{j}h_{kk}),\end{aligned}$
(310d) $\displaystyle\,\,\begin{aligned}
-2\delta\mathcal{E}_{ui}=&\,\,\tilde{\Box}h_{ui}+\partial_{j}H\partial_{v}h_{ij}-\partial_{i}(\partial_{v}h_{uu}+\partial_{j}h_{ju})-\partial_{u}(\partial_{v}h_{iu}+\partial_{j}h_{ij})\\\
&+\partial_{i}\partial_{u}h_{jj}-\frac{1}{2}\partial_{i}H\partial_{v}h_{jj}\\\
&-\frac{4c_{\text{GB}}}{\Lambda^{2}}\partial_{v}\left[\partial_{i}\partial_{j}H(\partial_{j}h_{kk}-\partial_{k}h_{jk})+\partial_{j}\partial_{k}H(\partial_{i}h_{jk}-\partial_{j}h_{ik})\right],\end{aligned}$
(310e) $\displaystyle\,\,\begin{aligned}
-2\delta\mathcal{E}_{ij}=&\,\,\tilde{\Box}h_{ij}-\partial_{i}(\partial_{v}h_{ju}+\partial_{k}h_{jk})-\partial_{j}(\partial_{v}h_{iu}+\partial_{k}h_{ik})+\partial_{i}\partial_{j}h_{kk}-\delta_{ij}\tilde{\Box}h_{kk}\\\
&+\delta_{ij}\partial_{k}(\partial_{v}h_{ku}+\partial_{l}h_{kl})+\delta_{ij}\partial_{v}(\partial_{v}h_{uu}+\partial_{k}h_{ku})\\\
&+\frac{4c_{\text{GB}}}{\Lambda^{2}}\partial_{v}^{2}(\partial_{i}\partial_{j}Hh_{kk}-\partial_{i}\partial_{k}Hh_{jk}-\partial_{j}\partial_{k}Hh_{ik}+\delta_{ij}\partial_{k}\partial_{l}Hh_{kl}).\end{aligned}$
(310f)

These equations are satisfied with the constraints (25) – (27) and the
equation of motion for $h_{ij}$ (LABEL:eq:_eom_for_hij).

##  Appendix B Master variables in balancing source background

Identifying the master variables for the metric perturbations on the point-
source background was a straightforward procedure described in the main text,
above equation (30). When a second source is introduced (to ``balance" the
first, as discussed in Section 5), spherical symmetry is lost and the previous
decomposition is no longer suitable. Without loss of generality, assume the
two sources are aligned along the $z$-axis. Then, we can write the transverse-
space metric in cylindrical coordinates as:

$\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}=\mathrm{d}z^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{d-2}^{2}.$
(311)

For this appendix only, $\gamma_{ab}$ will refer to the metric on the
$(d-2)$-dimensional sphere. In these coordinates, the harmonic condition on
the metric function is

$\frac{\partial^{2}}{\partial z^{2}}H(u,r,z)=-\frac{\partial^{2}}{\partial
r^{2}}H(u,r,z)-(d-2)\frac{1}{r}\frac{\partial}{\partial r}H(u,r,z).$ (312)

While most of the components of the metric perturbations — $h_{rr}$, $h_{rz}$,
$h_{zz}$, $h_{ra}$, $h_{za}$ and the diagonal components of $h_{ab}$ — are
coupled in a complicated way via (LABEL:eq:_eom_for_hij), it happens that the
off-diagonal components of $h_{ab}$ are completely decoupled from the rest and
evolve independently. These account for $(D-5)(D-4)/2$ of the total $D(D-3)/2$
propagating modes. Their equation of motion is

$\tilde{\Box}h_{ab}-8\frac{c_{\text{GB}}}{\Lambda^{2}}\frac{\partial_{r}H}{r}\partial_{v}^{2}h_{ab}=0,\quad
a\neq b\,,$ (313)

which has the same format as the equations for the modes in the spherically-
symmetric background (30).

##  Appendix C Determining the EFT regime of validity

In Section 2.4, we argue that the EFT regime of validity is given
schematically by

$\left(\frac{\nabla}{\Lambda}\right)^{m}\left(\frac{\bar{R}}{\Lambda^{2}}\right)^{n}\left(\frac{k}{\Lambda}\right)^{p}\ll
1.$ (314)

In this appendix, we will explain why only the particular contractions given
in (33) lead to non-trivial bounds on the energy of the perturbations and the
background parameters.

Firstly, contracting two momentum vectors together will lead to a weaker bound
than if they had been contracted with other components. The reason for this is
that the gravitational wave momentum is a null vector in GR, so its norm
vanishes $k_{\alpha}k^{\alpha}\approx 0$ up to EFT corrections.

Secondly, any contractions of indices on the same Riemann tensor produces a
Ricci tensor (or scalar) which is zero in vacuum. Therefore, each of the four
indices on a Riemann tensor need to be contracted with either a momentum
vector $k_{\alpha}$, a covariant derivative $\nabla_{\alpha}$ or a different
Riemann tensor. Also, it can be checked that the divergence of the Riemann
tensor vanishes in vacuum

$\nabla^{\alpha}\bar{R}_{\alpha\beta\gamma\delta}=0\quad\text{when}\quad\nabla^{2}_{d}H=0.$
(315)

Finally, by symmetry, the Riemann tensor may only enter via contractions with
at most two $k$'s, at most two $\nabla$'s and at most two other Riemann
tensors.

The expressions in (33) represent the only cases where no covariant
derivatives have been contracted with an index on a Riemann tensor. The
remaining options are to contract one covariant derivative with a Riemann
tensor index,

$(\bar{R}\cdot\nabla)^{2p}k^{4p}\ll\Lambda^{10p},$ (316)

or two covariant derivatives with Riemann tensor indices,

$(\bar{R}\cdot\nabla\nabla)^{2q}k^{4q}\ll\Lambda^{12q}.$ (317)

Both cases produce weaker bounds, which follow from those already given in
(40).

##  Appendix D Higher-dimension EFT field equations

In Section 2.5, we show that control over the higher-dimension EFT action
(2.5) amounts to exactly the regime of validity found by generic arguments in
Section 2.4. In this appendix, we provide the field equations for that action
and also identify the tensor-type master variable and its associated master
equation.

The background equation is

$\mathcal{E}_{\alpha\beta}\coloneqq
G_{\alpha\beta}+2\frac{c_{\text{GB}}}{\Lambda^{2}}B_{\alpha\beta}+2\frac{c_{\text{R3}}}{\Lambda^{4}}C_{\alpha\beta}+2\frac{c_{\text{R4}}}{\Lambda^{6}}D_{\alpha\beta}=0,$
(318)

where

$\displaystyle B_{\alpha\beta}=$ $\displaystyle
4R_{\sigma\alpha\beta\rho}R^{\sigma\rho}+2R\indices{{}_{\alpha}^{\sigma\rho\kappa}}R_{\beta\sigma\rho\kappa}-4R_{\alpha\sigma}R\indices{{}_{\beta}^{\sigma}}+2RR_{\alpha\beta}-\frac{1}{2}R_{\text{GB}}^{2}\,g_{\alpha\beta},$
(319) $\displaystyle C_{\alpha\beta}=$
$\displaystyle-\frac{1}{2}g_{\alpha\beta}(R^{3})+3(R^{3})\indices{{}_{\alpha\sigma\beta}^{\sigma}}+6\nabla_{\sigma}\nabla_{\rho}(R^{2})\indices{{}_{\alpha}^{\sigma}{}_{\beta}^{\rho}},$
(320) $\displaystyle D_{\alpha\beta}=$
$\displaystyle-\frac{1}{2}g_{\alpha\beta}(R^{4})+4(R^{4})\indices{{}_{\alpha\sigma\beta}^{\sigma}}+8\nabla_{\sigma}\nabla_{\rho}(R^{3})\indices{{}_{\alpha}^{\sigma}{}_{\beta}^{\rho}},$
(321)

and

$\displaystyle(R^{n})_{\alpha\beta\sigma\rho}=R_{\alpha\beta\gamma\delta}R^{\gamma\delta\bullet\bullet}\dots
R^{\bullet\bullet\kappa\lambda}R_{\kappa\lambda\sigma\rho},$ (322)
$\displaystyle(R^{n})=(R^{n})\indices{{}_{\alpha\beta}^{\alpha\beta}}.$ (323)

The vacuum pp-wave metric is again a solution to (318). The perturbations of
these tensors are

$\displaystyle\begin{aligned} \delta
B_{\mu\nu}=&-4R\indices{{}_{(\mu}^{\alpha}{}_{\nu)}^{\beta}}\nabla_{\lambda}\nabla_{\beta}h\indices{{}_{\alpha}^{\lambda}}+2R\indices{{}_{\mu}^{\alpha}{}_{\nu}^{\beta}}\left(\Box
h_{\alpha\beta}+\nabla_{\beta}\nabla_{\alpha}h\right)\\\
&+4R_{(\mu|\alpha\beta\rho}\nabla^{\rho}\nabla_{|\nu)}h^{\alpha\beta}-4R_{(\mu|\rho\alpha\beta}\nabla^{\rho}\nabla^{\beta}h\indices{{}_{|\nu)}^{\alpha}}\\\
&+2g_{\mu\nu}R_{\alpha\rho\beta\sigma}\nabla^{\sigma}\nabla^{\rho}h^{\alpha\beta}\end{aligned}$
(324) $\displaystyle\begin{aligned} \delta
C_{\mu\nu}=&12R\indices{{}_{(\mu|\rho\alpha\beta}}\nabla^{\rho}\Box\nabla^{\beta}h\indices{{}_{|\nu)}^{\alpha}}-12R\indices{{}_{(\mu|\rho\alpha\beta}}\nabla^{\rho}\nabla^{\beta}\nabla_{\sigma}\nabla_{|\nu)}h^{\sigma\alpha}\\\
&-12\nabla_{\sigma}R\indices{{}_{(\mu|\rho\alpha\beta}}\nabla^{\rho}\nabla^{\beta}\nabla_{|\nu)}h^{\alpha\sigma}+12\nabla_{\sigma}R\indices{{}_{(\mu|\rho\alpha\beta}}\nabla^{\rho}\nabla^{\sigma}\nabla^{\beta}h\indices{{}_{|\nu)}^{\alpha}}\\\
&+3\left(R\indices{{}_{\mu}^{\rho\lambda\alpha}}R\indices{{}_{\nu\lambda}^{\sigma\beta}}+R\indices{{}_{\mu}^{\lambda\rho\alpha}}R\indices{{}_{\nu}^{\sigma}{}_{\lambda}^{\beta}}\right)\nabla_{\rho}\nabla_{\sigma}h_{\alpha\beta}\end{aligned}$
(325) $\displaystyle\begin{aligned} \delta
D_{\mu\nu}=&-16R_{(\mu|\rho\beta\lambda}R_{|\nu)\sigma\alpha\kappa}\nabla^{\sigma}\nabla^{\rho}\nabla^{\lambda}\nabla^{\kappa}h^{\alpha\beta}-32\nabla_{\sigma}R\indices{{}_{(\mu|\rho\beta\lambda}}R\indices{{}_{|\nu)}^{\sigma}{}_{\alpha\kappa}}\nabla^{\rho}\nabla^{(\kappa}\nabla^{\lambda)}h^{\alpha\beta}\\\
&-16\nabla^{\sigma}R\indices{{}_{(\mu|}^{\rho}{}_{\alpha\kappa}}\nabla_{\rho}R\indices{{}_{|\nu)\sigma\beta\lambda}}\nabla^{\lambda}\nabla^{\kappa}h^{\alpha\beta}.\end{aligned}$
(326)

In lightcone gauge, the presence of the $R^{3}$-operator modifies the
$v$-component perturbation equations

$\displaystyle\begin{aligned}
-2\delta\mathcal{E}_{vv}=\partial_{v}^{2}h_{ii},\end{aligned}$ (327a)
$\displaystyle\begin{aligned}
-2\delta\mathcal{E}_{vu}=&-\tilde{\Box}_{v}^{2}h_{ii}+\partial_{v}\partial_{u}h_{ii}+\partial_{i}(\partial_{v}h_{iu}+\partial_{j}h_{ij})\\\
&-12\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}^{2}\partial_{i}\left[\partial_{i}\partial_{j}H(\partial_{v}h_{ju}+\partial_{k}h_{jk})+\partial_{i}\partial_{j}\partial_{k}Hh_{jk}\right],\end{aligned}$
(327b) $\displaystyle\begin{aligned}
-2\delta\mathcal{E}_{vi}=&\,\,\partial_{v}\partial_{i}h_{jj}-\partial_{v}(\partial_{v}h_{iu}+\partial_{j}h_{ij})\\\
&+12\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}^{3}\left[\partial_{i}\partial_{j}H(\partial_{v}h_{ju}+\partial_{k}h_{jk})+\partial_{i}\partial_{j}\partial_{k}Hh_{jk}\right],\end{aligned}$
(327c)

such that, while the traceless condition $h_{ii}=0$ is unchanged, the second
constraint equation (26) becomes

$\partial_{v}h_{iu}+\partial_{j}h_{ij}=12\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}^{2}\left[\partial_{i}\partial_{j}H(\partial_{v}h_{ju}+\partial_{k}h_{jk})+\partial_{i}\partial_{j}\partial_{k}Hh_{jk}\right].$
(328)

The $\delta\mathcal{E}_{ij}$ is also modified by both new operators

$\displaystyle\begin{split}-2\delta\mathcal{E}_{ij}=\,\,&\tilde{\Box}h_{ij}-\partial_{i}(\partial_{v}h_{ju}+\partial_{k}h_{jk})-\partial_{j}(\partial_{v}h_{iu}+\partial_{k}h_{ik})+\partial_{i}\partial_{j}h_{kk}-\delta_{ij}\tilde{\Box}h_{kk}\\\
&+\delta_{ij}\partial_{k}(\partial_{v}h_{ku}+\partial_{l}h_{kl})+\delta_{ij}\partial_{v}(\partial_{v}h_{uu}+\partial_{k}h_{ku})\\\
&+\frac{4c_{\text{GB}}}{\Lambda^{2}}\partial_{v}^{2}(\partial_{i}\partial_{j}Hh_{kk}-\partial_{i}\partial_{k}Hh_{jk}-\partial_{j}\partial_{k}Hh_{ik}+\delta_{ij}\partial_{k}\partial_{l}Hh_{kl})\\\
&\begin{aligned}
-12\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}^{2}&\left[\partial_{i}\partial_{k}H\tilde{\Box}h_{jk}+\partial_{i}\partial_{k}H\tilde{\Box}h_{jk}+\partial_{u}\partial_{i}\partial_{k}H\partial_{v}h_{jk}+\partial_{u}\partial_{j}\partial_{k}H\partial_{v}h_{ik}\right.\\\
&-\partial_{i}\partial_{k}H\partial_{j}(\partial_{v}h_{ku}+\partial_{l}h_{kl})-\partial_{j}\partial_{k}H\partial_{i}(\partial_{v}h_{ku}+\partial_{l}h_{kl})\\\
&\left.-\partial_{i}\partial_{k}\partial_{l}H(\partial_{j}h_{kl}-\partial_{k}h_{jl})-\partial_{j}\partial_{k}\partial_{l}H(\partial_{i}h_{kl}-\partial_{k}h_{il})\right]\end{aligned}\\\
&+16\frac{c_{\text{R4}}}{\Lambda^{6}}\partial_{i}\partial_{k}H\partial_{j}\partial_{l}H\partial_{v}^{4}h_{kl}\,.\end{split}$
(329)

The final constraint will again come from the trace of this equation. However,
the process is complicated by the presence of the $R^{3}$-operator. In EGB
theory, the second constraint (26) allowed us to exactly remove the $h_{iu}$
components, which appeared in the $ij$-equation. But now, the $h_{iu}$
components also appear on the right hand side of (328). As a result, they may
only be replaced perturbatively in inverse powers of the cut-off inside
$\delta\mathcal{E}_{ij}$. Up to corrections of $\mathcal{O}(\Lambda^{-8})$,
the third constraint equation is

$\displaystyle\begin{split}\partial_{v}h_{uu}+\partial_{i}h_{iu}=\,\,&-4\frac{d-2}{d}\frac{c_{\text{GB}}}{\Lambda^{2}}\partial_{i}\partial_{j}H\partial_{v}h_{ij}\\\
&+\frac{24}{d}\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}\left[\partial_{i}\partial_{j}H\tilde{\Box}h_{ij}+\partial_{u}\partial_{i}\partial_{j}H\partial_{v}h_{ij}-\frac{d-2}{2}\partial_{i}\partial_{j}\partial_{k}H\partial_{i}h_{jk}\right]\\\
&-\frac{16}{d}\frac{c_{\text{R4}}}{\Lambda^{6}}\partial_{i}\partial_{j}H\partial_{i}\partial_{k}H\partial_{v}^{3}h_{jk}.\end{split}$
(330)

As before, we see that the dynamical degrees of freedom are contained within
the metric perturbations in the transverse directions $h_{ij}$. Applying the
three constraint equations, their equation of motion becomes (up to
corrections of $\mathcal{O}(\Lambda^{-8})$):

$\displaystyle\tilde{\Box}h_{ij}-8\frac{c_{\text{GB}}}{\Lambda^{2}}\partial_{v}^{2}X_{ij}-24\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}^{2}Y_{ij}-16\frac{c_{\text{R4}}}{\Lambda^{6}}\partial_{v}^{4}Z_{ij}=0,$
(331)

where

$\displaystyle
X_{ij}=\frac{1}{2}\left(h_{ik}\partial_{j}\partial_{k}H+h_{jk}\partial_{i}\partial_{k}H\right)-\frac{1}{d}\delta_{ij}h_{kl}\partial_{k}\partial_{l}H,$
(332) $\displaystyle\begin{aligned}
\,Y_{ij}=\,\,&\frac{1}{2}\left(\tilde{\Box}h_{ik}\partial_{j}\partial_{k}H+\tilde{\Box}h_{jk}\partial_{i}\partial_{k}H\right)-\frac{1}{d}\delta_{ij}\tilde{\Box}h_{kl}\partial_{k}\partial_{l}H\\\
&+\frac{1}{2}\left(\partial_{v}h_{ik}\partial_{u}\partial_{j}\partial_{k}H+\partial_{v}h_{jk}\partial_{u}\partial_{i}\partial_{k}H\right)-\frac{1}{d}\delta_{ij}\partial_{v}h_{kl}\partial_{u}\partial_{k}\partial_{l}H\\\
&+\frac{1}{2}\left(\partial_{l}h_{ik}\partial_{j}\partial_{k}\partial_{l}H+\partial_{l}h_{jk}\partial_{i}\partial_{k}\partial_{l}H\right)-\frac{1}{d}\delta_{ij}\partial_{m}h_{kl}\partial_{k}\partial_{l}\partial_{m}H\\\
&+h_{kl}\partial_{i}\partial_{j}\partial_{k}\partial_{l}H,\end{aligned}$ (333)
$\displaystyle
Z_{ij}=-h_{kl}\partial_{i}\partial_{k}H\partial_{j}\partial_{l}H+\frac{1}{d}\delta_{ij}h_{lm}\partial_{k}\partial_{l}H\partial_{k}\partial_{m}H.$
(334)

###  D.1 Tensor modes

The perturbation equations (331) couple all components of $h_{ij}$ in a non-
trivial fashion when $c_{\text{R3}}\neq 0$. Even in the point source case, the
identification of master variable used in Section 2.3 for EGB theory no longer
works when the $R^{3}$-operator is included. For illustrative purposes, we
consider just the spherically symmetric point source case. Due to the
symmetry, we may perform a scalar-vector-tensor (SVT) decomposition with each
type of perturbation (scalar, vector or tensor) evolving independently of the
other types. The simplest of them are the tensor modes, which are
parameterised by

$\displaystyle h_{rr}=h_{ra}=0,\quad h_{ab}=r^{2}\Phi_{T}\mathbb{T}_{ab},$
(335)

where $\mathbb{T}_{ab}$ is a tensor spherical harmonic on the $(d-1)$-sphere
and satisfies

$\displaystyle\left(\hat{\Delta}_{d-1}+\kappa_{T}^{2}\right)\mathbb{T}_{ab}=0,$
(336a)
$\displaystyle\mathbb{T}\indices{{}^{a}_{a}}=0,\quad\hat{D}^{a}\mathbb{T}_{ab}=0.$
(336b)

We used $\kappa_{T}^{2}$ to represent the eigenvalue of $\mathbb{T}_{ab}$
instead of the traditional $k^{2}$ to avoid any possible confusion with the
momentum of the perturbations $(k_{u},k_{v},\mathbf{k})$ used elsewhere. There
is one tensor mode $\Phi_{T}$ for each tensor spherical harmonic on the
$(d-1)$-sphere. They all share a master equation, which can be derived from
the $(a,b)$-components (with $a\neq b$) of (331)

$\displaystyle\begin{split}0=\,\,&\tilde{\Box}\Phi_{T}-\frac{\kappa_{T}^{2}+2}{r^{2}}\Phi_{T}-8\frac{c_{\text{GB}}}{\Lambda^{2}}\frac{\partial_{r}H}{r}\partial_{v}^{2}\Phi_{T}+24\frac{c_{\text{R3}}}{\Lambda^{4}}\partial_{v}^{2}\left[d\frac{\partial_{r}H}{r^{2}}\left(\partial_{r}\Phi_{T}+\frac{\Phi_{T}}{r}\right)\right.\\\
&\left.-\frac{\partial_{u}\partial_{r}H}{r}\partial_{v}\Phi_{T}\right]+16\frac{c_{\text{R4}}-12c_{\text{GB}}c_{\text{R3}}}{\Lambda^{6}}\left(\frac{\partial_{r}H}{r}\right)^{2}\partial_{v}^{4}\Phi_{T}.\end{split}$
(337)

In arriving at this equation, the $\tilde{\Box}h_{ab}$ terms present in
$Y_{ab}$ have been replaced perturbatively using the lower-order equation of
motion, giving rise to the cross term $\propto c_{\text{GB}}c_{\text{R3}}$.
Once again, this equation is valid up to corrections of order
$\mathcal{O}(\Lambda^{-8})$.

##  Appendix E EFT validity bound on wave function spread

In Section 3.9, we discuss how a generic initial quantum state has a non-zero
spread $\sigma$ in transverse space and thus cannot be perfectly balanced at
an unstable equilibrium point. As a state in a low-energy EFT, this spread is
in fact also bounded below by $\Lambda^{-1}$. To demonstrate this, we must
bound a Lorentz-invariant quantity dependent on $\sigma$ (as we did for
$H(u,r)$, $k_{v}$ and $b$ in Section 2.4). The spread in momentum space is
inversely related to the spread in real space,
$\left\lvert\Delta\mathbf{k}\right\rvert\sim\sigma^{-1}$, so we would like to
bound the inner product of two opposing wave vectors
$\mathbf{k^{\pm}}=\mathbf{k^{0}}\pm\Delta\mathbf{k}$, where $\mathbf{k^{0}}$
is some central momentum. However, to construct a scalar quantity, they must
first be embedded in $D$-vectors on the full $D$-dimensional pp-wave spacetime

$\left(k^{\pm}_{\alpha}\right)=\left(k^{\pm}_{u},k^{\pm}_{v},\mathbf{k^{\pm}}\right)\,,$
(338)

with the condition that $k^{\pm}$ is null due to the GR equations of motion,

$g^{\alpha\beta}k^{\pm}_{\alpha}k^{\pm}_{\beta}=0\quad\implies\quad
k^{\pm}_{u}=\frac{F(k^{\pm}_{v})^{2}-\mathbf{k^{\pm}}^{2}}{2k^{\pm}_{v}}.$
(339)

For a fixed ``mass" $k_{v}^{+}=k_{v}^{-}=k_{v}$, the inner product reduces to

$k^{+}\cdot k^{-}=-2\Delta\mathbf{k}^{2}.$ (340)

With the Lorentz-invariant bound $k^{+}\cdot k^{-}\ll\Lambda^{2}$, we get the
desired result $\left\lvert\Delta\mathbf{k}\right\rvert\ll\Lambda$, or
$\sigma\Lambda\gg 1$.

##  Appendix F Expansion of the Wigner-Smith time delay

The expressions (298) give the parameter dependence of the first three orders
of the Wigner-Smith time delay expanded in 1) $S$-matrix perturbation theory,
2) small Gaussian width, and 3) early times. Four dimensionless numbers
$\\{a^{(2)},\tilde{a}^{(2)},a^{(3)},\tilde{a}^{(3)}\\}$ feature in these
equations and are given below in terms of integrals, where
$\hat{\mathbf{b}}=\mathbf{b}/b$ is the unit vector in the direction of
$\mathbf{b}$.

$\displaystyle\begin{aligned}
a^{(2)}=\,\,&-2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{2}\int_{0}^{1}\text{d}w_{1}\text{d}w_{2}\Theta(w_{1}-w_{2})\int\frac{\text{d}^{d}p_{1}}{(2\pi)^{d}}\frac{\text{d}^{d}p_{2}}{(2\pi)^{d}}\\\
&\frac{e^{i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{1}^{2}}\frac{e^{i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{2}^{2}}\left(\mathbf{p}_{1}+\mathbf{p}_{2}\right)^{2}(w_{1}-w_{2})\mathbf{p}_{1}\cdot\mathbf{p}_{2}\,,\end{aligned}$
(341a) $\displaystyle\begin{aligned}
\tilde{a}^{(2)}=&-2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{2}\int_{0}^{1}\text{d}w_{1}\text{d}w_{2}\Theta(w_{1}-w_{2})\int\frac{\text{d}^{d}p_{1}}{(2\pi)^{d}}\frac{\text{d}^{d}p_{2}}{(2\pi)^{d}}\\\
&\frac{e^{i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{1}^{2}}\frac{e^{i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{2}^{2}}\left(w_{1}\mathbf{p}_{1}+w_{2}\mathbf{p}_{2}\right)^{2}(w_{1}-w_{2})\mathbf{p}_{1}\cdot\mathbf{p}_{2}\,,\end{aligned}$
(341b) $\displaystyle\begin{aligned}
a^{(3)}=&2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{3}\int_{0}^{1}\text{d}w_{1}\text{d}w_{2}\text{d}w_{3}\int\frac{\text{d}^{d}p_{1}}{(2\pi)^{d}}\frac{\text{d}^{d}p_{2}}{(2\pi)^{d}}\frac{\text{d}^{d}p_{3}}{(2\pi)^{d}}\\\
&\frac{e^{i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{1}^{2}}\frac{e^{i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{2}^{2}}\frac{e^{i\mathbf{p}_{3}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{3}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{3}^{2}}\left(\mathbf{p}_{1}+\mathbf{p}_{2}+\mathbf{p}_{3}\right)^{2}\\\
&\left(2\Theta(w_{1}-w_{2})\Theta(w_{2}-w_{3})-\Theta(w_{2}-w_{1})\Theta(w_{2}-w_{3})\right)\,,\end{aligned}$
(341c) $\displaystyle\begin{aligned}
\tilde{a}^{(3)}=&2\left(-\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\right)^{3}\int_{0}^{1}\text{d}w_{1}\text{d}w_{2}\text{d}w_{3}\int\frac{\text{d}^{d}p_{1}}{(2\pi)^{d}}\frac{\text{d}^{d}p_{2}}{(2\pi)^{d}}\frac{\text{d}^{d}p_{3}}{(2\pi)^{d}}\\\
&\frac{e^{i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{1}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{1}^{2}}\frac{e^{i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{2}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{2}^{2}}\frac{e^{i\mathbf{p}_{3}\cdot\hat{\mathbf{b}}}+\Theta_{\text{bal}}e^{-i\mathbf{p}_{3}\cdot\hat{\mathbf{b}}}}{\mathbf{p}_{3}^{2}}\left(w_{1}\mathbf{p}_{1}+w_{2}\mathbf{p}_{2}+w_{3}\mathbf{p}_{3}\right)^{2}\\\
&\left(2\Theta(w_{1}-w_{2})\Theta(w_{2}-w_{3})-\Theta(w_{2}-w_{1})\Theta(w_{2}-w_{3})\right)\,.\end{aligned}$
(341d)

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