I_{1,1,0,0}=\int\frac{\textrm{d}^{4}\ell}{(2\pi)^{4}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}}=\frac{1}{16\pi^{3}}\int_{0}^{1}\textrm{d}x\int\frac{\textrm{d}^{3}\boldsymbol{\ell}}{[\boldsymbol{\ell}^{2}+x(1-x)\boldsymbol{q}_{2}^{2}]^{2}}=\frac{1}{16\sqrt{-q_{2}^{2}}}\,.$
(132)

This integral is purely real as expected because further cutting either
$\ell^{2}$ or $(\ell+q_{2})^{2}$ leads to vanishing results due to on-shell
three-point kinematics. In contrast, the other triangle master integral
$I_{1,0,1,0}$ is complex. We start with the same Feynman parameterization, and
the integration proceeds as before,

$\displaystyle
I_{1,0,1,0}=\int\frac{\textrm{d}^{4}\ell}{(2\pi)^{4}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{1})^{2}+i0]}$
$\displaystyle=\frac{1}{16\pi^{3}}\int_{0}^{1}\textrm{d}x\int\frac{\textrm{d}^{3}\boldsymbol{\ell}}{[\boldsymbol{\ell}^{2}+x(1-x)\boldsymbol{q}_{1}^{2}-xE_{q_{1}}^{2}-i0]^{2}}$
$\displaystyle=\frac{1}{8\pi\sqrt{\boldsymbol{q}_{1}^{2}}}\arcsin\left(\sqrt{\frac{\boldsymbol{q}_{1}^{2}}{\boldsymbol{q}_{1}^{2}-E_{q_{1}}^{2}}}+i0\right)$
$\displaystyle=\frac{1}{16\sqrt{\boldsymbol{q}_{1}^{2}}}+\frac{i}{8\pi\sqrt{\boldsymbol{q}_{1}^{2}}}\operatorname{arccosh}\sqrt{\frac{\boldsymbol{q}_{1}^{2}}{\boldsymbol{q}_{1}^{2}-E_{q_{1}}^{2}}}\,.$
(133)

We note that the argument of $\arcsin$ in the second line is greater than $1$
in the physical region. The analytic continuation in $+i0$ prescription is

$\displaystyle\arcsin(x+i0)=\frac{\pi}{2}+i\operatorname{arccosh}(x)\quad\text{for}\quad
x>1\,.$ (134)

By using section 6, we can uplift the result to a generic frame,

$\displaystyle I_{1,0,1,0}=\frac{1}{16\sqrt{(\bar{u}_{2}\cdot
q_{1})^{2}-q_{1}^{2}}}+\frac{i}{8\pi\sqrt{(\bar{u}_{2}\cdot
q_{1})^{2}-q_{1}^{2}}}\operatorname{arcsinh}\frac{\bar{u}_{2}\cdot
q_{1}}{\sqrt{-q_{1}^{2}}}\,,$ (135)

where we have also used the identity
$\operatorname{arccosh}\sqrt{x+1}=\operatorname{arcsinh}\sqrt{x}$.

To compute the integrals with a linear matter propagator, we use a different
parameterization to combine the denominator,

$\displaystyle\frac{1}{ab}=\int_{0}^{\infty}\frac{\textrm{d}x}{(a+xb)^{2}}\,.$
(136)

Applying eq. (136) to $I^{+}_{0,0,1,1}$, we get

$\displaystyle
I^{+}_{0,0,1,1}=\int\frac{\textrm{d}^{4}\ell}{(2\pi)^{4}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell+i0)}$
$\displaystyle=\int_{0}^{\infty}\textrm{d}x\int\frac{\textrm{d}^{4}\ell}{(2\pi)^{4}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{[(\ell-
q_{1})^{2}+2x\bar{u}_{1}\cdot\ell+i0)]^{2}}$
$\displaystyle=\frac{i}{16\pi\sqrt{y^{2}-1}}\int_{-\alpha}^{\infty}\frac{\textrm{d}x}{(x^{2}+\beta)^{1/2}}\,,$
(137)

where $\alpha=\frac{y(\bar{u}_{2}\cdot q_{1})}{y^{2}-1}$ and
$\beta=-\frac{(\bar{u}_{2}\cdot q_{1})^{2}}{(y^{2}-1)^{2}}+i0$. The
integration over $x$ is divergent at $x\rightarrow\infty$, which encodes the
IR divergence due to the linear matter propagator $2\bar{u}_{1}\cdot\ell$.
However, in the classical amplitude, the linear matter propagator only appears
in the principal-valued combination $I^{+}_{0,0,1,1}+I^{-}_{0,0,1,1}$. After
including $I^{-}_{0,0,1,1}$,

$\displaystyle
I^{-}_{0,0,1,1}=\int\frac{\textrm{d}^{4}\ell}{(2\pi)^{4}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell-i0)}$
$\displaystyle=-\int\frac{\textrm{d}^{4}\ell}{(2\pi)^{4}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+q_{1})^{2}(2\bar{u}_{1}\cdot\ell+i0)}$
$\displaystyle=-I^{+}_{0,0,1,1}\Big{|}_{q_{1}\rightarrow-q_{1}}\,$ (138)

we find that the range of $x$ gets truncated and we get a finite result,

$\displaystyle
I^{+}_{0,0,1,1}+I^{-}_{0,0,1,1}=\frac{i}{8\pi\sqrt{y^{2}-1}}\int_{0}^{\alpha}\frac{\textrm{d}x}{(x^{2}+\beta)^{1/2}}=\frac{i}{8\pi\sqrt{y^{2}-1}}\operatorname{arcsinh}\frac{\alpha}{\sqrt{\beta}}\,.$
(139)

The $+i0$ prescription in $\beta$ leads to the following analytic continuation
of the $\operatorname{arcsinh}$ function in the physical region,

$\displaystyle\operatorname{arcsinh}\frac{\alpha}{\sqrt{\beta}}=-\frac{i\pi}{2}+\operatorname{arccosh}(y)\,.$
(140)

Therefore, the final result of this triangle integral is

$\displaystyle
I^{+}_{0,0,1,1}+I^{-}_{0,0,1,1}=\frac{1}{16\sqrt{y^{2}-1}}+\frac{i}{8\pi\sqrt{y^{2}-1}}\operatorname{arccosh}(y)\,.$
(141)

We can apply the same technique to compute

$\displaystyle
I^{\pm}_{0,1,0,1}=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+q_{2})^{2}(2\bar{u}_{1}\cdot\ell\pm
i0)}\,.$ (142)

In particular, the principal value combination is given by

$\displaystyle
I^{+}_{0,1,0,1}+I^{-}_{0,1,0,1}=\frac{i}{8\pi\sqrt{y^{2}-1}}\int_{0}^{\gamma}\frac{\textrm{d}x}{(x^{2}-\gamma^{2}+i0)^{1/2}}=\frac{1}{16\sqrt{y^{2}-1}}\,,$
(143)

where $\gamma=\frac{\bar{u}_{1}\cdot q_{2}}{y^{2}-1}$.

###  6.2 Box and pentagon integrals

The box master integrals are all individually IR divergent. For $I_{1,1,1,0}$,
the IR divergence is due to the presence of a massless three-point vertex,

$\displaystyle I_{1,1,1,0}=\frac{1}{32q_{2}^{2}(\bar{u}_{2}\cdot
q_{1})}+\frac{i}{32\pi q_{2}^{2}(\bar{u}_{2}\cdot
q_{1})}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(4\pi(\bar{u}_{2}\cdot
q_{1})^{2})+2\log\frac{q_{2}^{2}}{q_{1}^{2}}\right]\,,$ (144)

The other box integrals, $I^{\pm}_{1,1,0,1}$, $I^{\pm}_{1,0,1,1}$ and
$I^{\pm}_{0,1,1,1}$, all have an IR divergence due to the linear matter
propagator $2\bar{u}_{1}\cdot\ell$. This IR divergence cancels in the
classical amplitude because the linear matter propagator always appears as a
principal value,

$\displaystyle
I^{+}_{1,1,0,1}+I^{-}_{1,1,0,1}=\frac{1}{16q_{2}^{2}\sqrt{y^{2}-1}}\,,$ (145a)
$\displaystyle
I^{+}_{1,0,1,1}+I^{-}_{1,0,1,1}=\frac{1}{16q_{1}^{2}\sqrt{y^{2}-1}}+\frac{i}{8\pi
q_{1}^{2}\sqrt{y^{2}-1}}\operatorname{arccosh}(y)\,.$ (145b)

On the other hand, $I^{+}_{0,1,1,1}+I^{-}_{0,1,1,1}$ remains IR divergent due
to the presence of the same massless three-point vertex as in $I_{1,1,1,0}$,

$\displaystyle I^{+}_{0,1,1,1}+I^{-}_{0,1,1,1}$
$\displaystyle=-\frac{1}{32(\bar{u}_{1}\cdot q_{2})(\bar{u}_{2}\cdot q_{1})}$
(146) $\displaystyle\quad-\frac{i}{32\pi(\bar{u}_{1}\cdot
q_{2})(\bar{u}_{2}\cdot
q_{1})}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(4\pi(\bar{u}_{2}\cdot
q_{1})^{2})+2\log\frac{\bar{u}_{1}\cdot q_{2}}{\bar{u}_{2}\cdot
q_{1}}\right]\,.$

We leave the derivations of these integrals to appendix B.

Scalar pentagon integrals with quadratic propagators in $d=4-2\epsilon$
dimensions can be written as a sum of box integrals and a six-dimensional
pentagon integral vanNeerven:1983vr ; Bern:1992em ; Bern:1993kr . We derive
here an analogous decomposition for our pentagon integral with linear matter
propagators one of which is cut, $I^{\pm}_{1,1,1,1}$. We first decompose the
loop momentum $\ell$ into its four-dimensional and extra-dimensional
component, $\ell^{2}=\ell_{4}^{2}+\mu^{2}$, where $\ell_{4}$ can be expressed
as a linear combination of external kinematic data,
$\ell_{4}=\alpha_{1}\bar{u}_{1}+\alpha_{2}\bar{u}_{2}+\alpha_{3}q_{1}+\alpha_{4}q_{2}$.
The coefficients $\alpha_{i}$ contain the $\ell$ dependence through scalar
products $v\cdot\ell_{4}=v\cdot\ell$ with
$v\in\\{\bar{u}_{1},\bar{u}_{2},q_{1},q_{2}\\}$. We then plug the above
relation for $\ell_{4}$ into the identity

$\displaystyle
0=\int\frac{\textrm{d}^{4-2\epsilon}\ell}{(2\pi)^{4-2\epsilon}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell)}(\ell^{2}-\ell_{4}^{2}-\mu^{2})\,,$ (147)

express all the $v\cdot\ell$ in terms of inverse propagators, and perform the
standard tensor reduction. This will lead to a linear relation that expresses
the pentagon integral in terms of the box integrals in $d=4-2\epsilon$ and
another pentagon integral $I^{\pm,d=6-2\epsilon}_{1,1,1,1}$ in
$d=6-2\epsilon$,

$\displaystyle
I^{\pm}_{1,1,1,1}=\beta_{1}I^{\pm}_{1,1,0,1}+\beta_{2}I^{\pm}_{1,0,1,1}+\beta_{3}I^{\pm}_{0,1,1,1}+\beta_{4}I_{1,1,1,0}+\beta_{5}\,\epsilon\,I^{\pm,d=6-2\epsilon}_{1,1,1,1}\,,$
(148)

where $\beta_{i}$ only depend on external Mandelstam variables. The pentagon
integral $I^{\pm,d=6-2\epsilon}_{1,1,1,1}$ comes from evaluating the $\mu^{2}$
term contribution by the dimension shift relation BernMorgan ,

$\displaystyle\int\textrm{d}^{4-2\epsilon}\ell\frac{\mu^{2}\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell)}=-\frac{\epsilon}{\pi}\int\textrm{d}^{6-2\epsilon}\ell\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell)}\,.$ (149)

In the principal value combination $I^{+}_{1,1,1,1}+I^{-}_{1,1,1,1}$, this
term is finite in $d=6-2\epsilon$ such that it contributes at most to
$\mathcal{O}(\epsilon)$. The coefficients $\beta_{i}$ in eq. 148 are
straightforward to compute. Here we just give the final result of the pentagon
integral,

$\displaystyle I^{+}_{1,1,1,1}+I^{-}_{1,1,1,1}$
$\displaystyle=\frac{1}{32q_{2}^{2}}\left(\frac{2}{q_{1}^{2}\sqrt{y^{2}-1}}-\frac{1}{(\bar{u}_{1}\cdot
q_{2})(\bar{u}_{2}\cdot q_{1})}\right)$ (150)
$\displaystyle\quad+\frac{i}{32\pi q_{2}^{2}(\bar{u}_{1}\cdot
q_{2})(\bar{u}_{2}\cdot
q_{1})}\left[\frac{1}{\epsilon_{\text{IR}}}-\log(4\pi(\bar{u}_{2}\cdot
q_{1})^{2})-2\log\frac{q_{2}^{2}}{q_{1}^{2}}\right]$
$\displaystyle\quad+\frac{i}{16\pi}\left[\frac{C_{1}}{(\bar{u}_{1}\cdot
q_{2})(\bar{u}_{2}\cdot q_{1})}\log\frac{(\bar{u}_{1}\cdot
q_{2})q_{1}^{2}}{(\bar{u}_{2}\cdot
q_{1})q_{2}^{2}}+\frac{C_{2}\operatorname{arccosh}(y)}{q_{1}^{2}\sqrt{y^{2}-1}}\right]\,,$

where $C_{1}$ and $C_{2}$ are given by

$\displaystyle C_{1}$ $\displaystyle=\frac{y(\bar{u}_{2}\cdot
q_{1})(\bar{u}_{1}\cdot q_{2})(q_{1}^{2}+q_{2}^{2})-(\bar{u}_{1}\cdot
q_{2})^{2}q_{1}^{2}-(\bar{u}_{2}\cdot q_{1})^{2}q_{2}^{2}+2(\bar{u}_{1}\cdot
q_{2})^{2}(\bar{u}_{2}\cdot q_{1})^{2}}{(\bar{u}_{2}\cdot
q_{1})^{2}q_{2}^{4}-2y(\bar{u}_{1}\cdot q_{2})(\bar{u}_{2}\cdot
q_{1})q_{1}^{2}q_{2}^{2}+(\bar{u}_{1}\cdot q_{2})^{2}q_{1}^{4}}\,,$
$\displaystyle C_{2}$ $\displaystyle=\frac{2(\bar{u}_{2}\cdot
q_{1})^{2}q_{2}^{2}-2y(\bar{u}_{1}\cdot q_{2})(\bar{u}_{2}\cdot
q_{1})q_{1}^{2}-(y^{2}-1)q_{1}^{2}(q_{1}^{2}-q_{2}^{2})}{(\bar{u}_{2}\cdot
q_{1})^{2}q_{2}^{4}-2y(\bar{u}_{1}\cdot q_{2})(\bar{u}_{2}\cdot
q_{1})q_{1}^{2}q_{2}^{2}+(\bar{u}_{1}\cdot q_{2})^{2}q_{1}^{4}}\,.$ (151)

##  7 The five-point amplitude and its infrared divergences

Having evaluated all the relevant master integrals, we can assemble the
classical amplitudes and carry our various consistency checks. When writing
down the classical amplitude we can also remove the distinction between
$\\{m_{i},p_{i},u_{i},\sigma=u_{1}\cdot u_{2}\\}$ and
$\\{\bar{m}_{i},\bar{p}_{i},\bar{u}_{i},y=\bar{u}_{1}\cdot\bar{u}_{2}\\}$, as
they differ only by positive powers $q_{1}$ and $q_{2}$, i.e. by terms with
quantum scaling in the soft expansion.

By inspecting the non-rational terms in the integrals evaluated in section 6
and appendix B it is straightforward to see that they are all linear
combinations of the functions

$\displaystyle\frac{1}{\sqrt{-q_{1}^{2}}}\,,\quad\frac{1}{\sqrt{-q_{2}^{2}}}\,,\quad\frac{1}{\sqrt{(k\cdot{u}_{2})^{2}-q_{1}^{2}}}\,,\quad\frac{1}{\sqrt{(k\cdot{u}_{1})^{2}-q_{2}^{2}}}\,,\quad\log\frac{(k\cdot{u}_{1})^{2}(k\cdot{u}_{2})^{2}}{\mu^{4}}\,,$
$\displaystyle\log\frac{q_{2}^{2}}{q_{1}^{2}}\,,\quad\log\frac{k\cdot{u}_{1}}{k\cdot{u}_{2}}\,,\quad\frac{\operatorname{arcsinh}\frac{k\cdot{u}_{2}}{\sqrt{-q_{1}^{2}}}}{\sqrt{(k\cdot{u}_{2})^{2}-q_{1}^{2}}}\,,\quad\frac{\operatorname{arcsinh}\frac{k\cdot{u}_{1}}{\sqrt{-q_{2}^{2}}}}{\sqrt{(k\cdot{u}_{1})^{2}-q_{2}^{2}}}\,,\quad\frac{\operatorname{arccosh}\sigma}{(\sigma^{2}-1)^{3/2}}\,,$
(152)

with rational coefficients. Here $\mu$ is the scale of dimensional
regularization, and this $\mu$-dependent logarithm is intimately related to IR
divergence of amplitudes.

In both GR and ${\cal N}=8$ supergravity classical two-scalar-one-graviton
five-point amplitude has the general form

$\displaystyle M_{5,\text{1 loop}}^{\text{2MPI},\text{cl.}}$
$\displaystyle=-\frac{i\,\kappa^{2}}{32\pi}(k\cdot{p}_{1}+k\cdot{p}_{2})\left(\frac{1}{\epsilon}-\frac{1}{2}\log\frac{16\pi^{2}(k\cdot{u}_{1})^{2}(k\cdot{u}_{2})^{2}}{\mu^{4}}\right)M_{5,\text{tree}}^{\text{class}}$
$\displaystyle\quad+\kappa^{5}\left(A^{R}_{\text{rat}}+\frac{A^{R}_{1}}{\sqrt{(k\cdot{u}_{2})^{2}-q_{1}^{2}}}+\frac{A^{R}_{2}}{\sqrt{(k\cdot{u}_{1})^{2}-q_{2}^{2}}}+\frac{A_{3}^{R}}{\sqrt{-q_{1}^{2}}}+\frac{A_{4}^{R}}{\sqrt{-q_{2}^{2}}}\right)$
$\displaystyle\quad+i\,\kappa^{5}\,\Bigg{(}A^{I}_{\text{rat}}+A^{I}_{1}\frac{\operatorname{arcsinh}\frac{k\cdot
u_{2}}{\sqrt{-q_{1}^{2}}}}{\sqrt{(k\cdot{u}_{2})^{2}-q_{1}^{2}}}+A^{I}_{2}\frac{\operatorname{arcsinh}\frac{k\cdot
u_{1}}{\sqrt{-q_{2}^{2}}}}{\sqrt{(k\cdot{u}_{1})^{2}-q_{2}^{2}}}+A^{I}_{3}\log\frac{q_{2}^{2}}{q_{1}^{2}}$
$\displaystyle\qquad\qquad\qquad+A^{I}_{4}\log\frac{k\cdot{u}_{1}}{k\cdot{u}_{2}}+A^{I}_{5}\frac{\operatorname{arccosh}\sigma}{(\sigma^{2}-1)^{3/2}}\Bigg{)}\
,$ (153)

where the coefficient functions $A^{R,I}_{i}$ are rational combinations of
momentum invariants and polarization vector $\varepsilon$. Their scaling in
the soft limit is homogeneous and it is such that $M_{5,\text{1
loop}}^{\text{2MPI},\text{cl.}}$ has classical $q^{-1}$ scaling. They are
gauge invariant as a consequence of the gauge invariance of the master
integral coefficients in eqs. 112 and 127. The explicit expressions of the
coefficient functions $A^{R,I}_{i}$ are included in two ancillary Mathematica-
readable files, GR_Coeffs.m and Neq8_Coeffs.m, for GR and ${\cal N}=8$
supergravity, respectively.

The first line of section 7, reproducing expectations based on Weinberg’s
theorem (57), requires a nontrivial interplay between the $c_{i}$ and
$c_{i}^{{\cal N}=8}$ coefficients determining the GR and ${\cal N}=8$
supergravity classical integrands, eqs. 112 and 127 respectively, is a check
of our calculation. As discussed in section 3.3, the definition of the time
coordinate (59) absorbing the IR divergence also eliminates the
$\log\frac{16\pi^{2}(k\cdot{u}_{1})^{2}(k\cdot{u}_{2})^{2}}{\mu^{4}}$ term by
rendering inert the ${\cal O}(\epsilon)$ terms in the soft-graviton-free
scattering amplitude. Thus, the part of the one-loop five-point amplitude that
contributes to the (spectral) waveform and the Newman-Penrose scalar is

$\displaystyle M_{5,\text{1 loop}}^{0,\text{2MPI},\text{cl.}}$
$\displaystyle=\kappa^{5}\left(A^{R}_{\text{rat}}+\frac{A^{R}_{1}}{\sqrt{(k\cdot{u}_{2})^{2}-q_{1}^{2}}}+\frac{A^{R}_{2}}{\sqrt{(k\cdot{u}_{1})^{2}-q_{2}^{2}}}+\frac{A_{3}^{R}}{\sqrt{-q_{1}^{2}}}+\frac{A_{4}^{R}}{\sqrt{-q_{2}^{2}}}\right)$
$\displaystyle+i\,\kappa^{5}\,\Bigg{(}A^{I}_{\text{rat}}+A^{I}_{1}\frac{\operatorname{arcsinh}\frac{k\cdot
u_{2}}{\sqrt{-q_{1}^{2}}}}{\sqrt{(k\cdot{u}_{2})^{2}-q_{1}^{2}}}+A^{I}_{2}\frac{\operatorname{arcsinh}\frac{k\cdot
u_{1}}{\sqrt{-q_{2}^{2}}}}{\sqrt{(k\cdot{u}_{1})^{2}-q_{2}^{2}}}+A^{I}_{3}\log\frac{q_{2}^{2}}{q_{1}^{2}}$
$\displaystyle\qquad\qquad\qquad+A^{I}_{4}\log\frac{k\cdot{u}_{1}}{k\cdot{u}_{2}}+A^{I}_{5}\frac{\operatorname{arccosh}\sigma}{(\sigma^{2}-1)^{3/2}}\Bigg{)}\,.$
(154)

We have also explored the fate of spurious singularities. The Gram determinant
$G[\bar{p}_{1},\bar{p}_{2},q_{1},q_{2}]$ arising from the decomposition (109)
of the graviton polarization tensor into a basis of external momenta cancel in
section 7 within each coefficient $A_{i}^{R,I}$ with the help of four
dimensional identities involving the vanishing Gram determinant
$G[\bar{p}_{1},\bar{p}_{2},q_{1},q_{2},\varepsilon]$. Other spurious
singularities occur for kinematic configurations that set to zero denominator
factors arising from the IBP reduction. They are solutions to the equations

$\displaystyle\Delta_{1}$ $\displaystyle=-2\sigma(u_{1}\cdot k)(u_{2}\cdot
k)q_{1}^{2}q_{2}^{2}+(u_{2}\cdot k)^{2}(q_{2}^{2}){}^{2}+(u_{1}\cdot
k)^{2}(q_{1}^{2}){}^{2}=0\ ,$ $\displaystyle\Delta_{2}$
$\displaystyle=-2\sigma(u_{1}\cdot k)(u_{2}\cdot k)+(u_{2}\cdot
k)^{2}+(u_{1}\cdot k)^{2}=0\,,$ $\displaystyle\Delta_{3}$
$\displaystyle=(q_{1}^{2}-q_{2}^{2})^{2}+4(u_{2}\cdot k)^{2}q_{2}^{2}=0\,,$
(155) $\displaystyle\Delta_{4}$
$\displaystyle=(q_{1}^{2}-q_{2}^{2})^{2}+4(u_{1}\cdot k)^{2}q_{1}^{2}=0\,.$

It is not difficult to check that when this relation is satisfied, the
logarithmic functions in eq. 152 are no longer linearly-independent.
Therefore, these singularities can cancel only in the complete expression,
which they indeed do. While all four $\Delta$’s appear in the GR amplitude,
only $\Delta_{1}$ and $\Delta_{2}$ appear in the $\mathcal{N}=8$ amplitude.

In the real part of the amplitude, the rational coefficient has a very simple
structure. For the GR amplitude, it reads

$\displaystyle
A^{R,\text{GR}}_{\text{rat}}=\frac{1}{32}\left(1-\frac{\sigma(\sigma^{2}-3/2)}{(\sigma^{2}-1)^{3/2}}\right)(k\cdot
p_{1}+k\cdot p_{2})M_{5,\text{tree}}^{\text{cl.GR}}\Big{|}_{\kappa=1}\,,$
(156)

where we set $\kappa=1$ in the tree amplitude because the overall $\kappa$
dependence has been pulled out. On the other hand, the square-root functions
in eq. 152 originate only from the triangle integrals $I_{1,1,0,0}$,
$I_{1,0,1,0}$ and their up-down flip, while their coefficients are more
complicated in GR. However, such master integrals are absent for
$\mathcal{N}=8$ supergravity, see eq. 127, and thus
$A_{1,2,3,4}^{R,\mathcal{N}=8}=0$. This makes the real part of the five-point
$\mathcal{N}=8$ amplitude much simpler than section 7 implies. The complete
real part is given by $A_{\text{rat}}^{R}$ only, which is similar to its GR
counterpart,

$\displaystyle\varepsilon^{\mu}\varepsilon^{\nu}\,{\rm Re}\left[M_{5,\text{1
loop}}^{0,\text{2MPI},\text{cl.}{\cal N}=8}\right]_{\mu\nu}$
$\displaystyle=A^{R,{\cal N}=8}_{\text{rat}}$
$\displaystyle=\frac{1}{32}\left(1-\frac{\sigma(\sigma^{2}-2)}{(\sigma^{2}-1)^{3/2}}\right)(k\cdot
p_{1}+k\cdot p_{2})M_{5,\text{tree}}^{\text{cl.}{\cal N}=8}\Big{|}_{\kappa=1}\
.$ (157)

We note that this is the complete $L=1$ that is used to compute the waveform
for $\mathcal{N}=8$ in section 3.4. Curiously, the $\sigma$-dependent factor
in section 7 vanishes in the ultrarelativistic limit,
$\sigma\rightarrow\infty$, implying that in this limit only the imaginary part
of the amplitude contributes to the waveform at ${\cal O}(G^{2})$ in ${\cal
N}=8$ supergravity according to section 3.4. Even though the real part of the
${\cal N}=8$ classical one-loop amplitude is proportional to the tree-level
amplitude, the real part of the one-loop spectral function is different from
the tree-level one because the $\omega$-dependent distributional factor is now
$\Theta(\omega)-\Theta(-\omega)$ for $L=1$, see section 3.4.

Finally, we have verified that our one-loop amplitudes reproduce the expected
soft limit under $k\rightarrow 0$. At the leading order, the imaginary part
vanishes, and the real part factorizes into the four-point amplitude and the
Weinberg soft factor,

$\displaystyle\lim_{k\rightarrow 0}M_{5,\text{1
loop}}^{0,\text{2MPI,cl.GR}}=\frac{\kappa}{2}\big{[}\varepsilon^{\mu}\varepsilon^{\nu}\mathcal{S}(k,q)_{\mu\nu}\big{]}M_{4,\text{1
loop}}^{\text{2MPI,cl.GR}}\,,$ $\displaystyle\lim_{k\rightarrow 0}M_{5,\text{1
loop}}^{0,\text{2MPI,cl.}{\cal N}=8}=0\,.$ (158)

The leading soft limit for the 2MPI amplitude of $\mathcal{N}=8$ vanishes
because at four points it is only contributed by the cut box, which is 2MPR.
Meanwhile, the $2\rightarrow 2$ amplitude in GR receives contributions from
triangles, and it is given by

$\displaystyle M_{4,\text{1
loop}}^{\text{2MPI,cl.GR}}=\frac{3\kappa^{4}m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})(5\sigma^{2}-1)}{512\sqrt{-q^{2}}}\,,$
(159)

where $q$ is the momentum transfer for this $2\rightarrow 2$ scattering. At
the leading order, we can take $q=q_{1}=-q_{2}$, such that the soft factor
$\mathcal{S}$ is given by

$\displaystyle\mathcal{S}(k,q)^{\mu\nu}$
$\displaystyle=\sum_{i=1}^{4}\frac{p_{i}^{\mu}p_{i}^{\nu}}{p_{i}\cdot
k}=-\sum_{i=1}^{2}\left[\frac{p_{i}^{\mu}q_{i}^{\nu}+p_{i}^{\nu}q_{i}^{\mu}}{p_{i}\cdot
k}-\frac{p_{i}^{\mu}p_{i}^{\nu}q_{i}\cdot k}{(p_{i}\cdot
k)^{2}}\right]+\mathcal{O}(\hbar)\,,$ (160)
$\displaystyle\varepsilon^{\mu}\varepsilon^{\nu}\mathcal{S}(k,q)_{\mu\nu}$
$\displaystyle=-\frac{2\varepsilon\cdot p_{1}\varepsilon\cdot q}{p_{1}\cdot
k}+\frac{(\varepsilon\cdot p_{1})^{2}k\cdot q}{(p_{1}\cdot
k)^{2}}+\frac{2\varepsilon\cdot p_{2}\varepsilon\cdot q}{p_{2}\cdot
k}-\frac{(\varepsilon\cdot p_{2})^{2}k\cdot q}{(p_{2}\cdot
k)^{2}}+\mathcal{O}(k^{0})\,.$

The gravitational memory only receive contribution from the leading soft
limit. Thus there is no memory effect for $\mathcal{N}=8$ at NLO, and we will
explicitly compute the memory for GR in the next section.

##  8 Leading order and next-to-leading order waveforms

Having constructed the relevant part of the one-loop five-point amplitude, we
now use the formulae summarized in section 3.4 to construct the leading order
and next-to-leading order waveform observables focusing on ${\cal N}=8$
supergravity. The time-domain leading-order asymptotic metric was first
discussed in Ref. Kovacs:1978eu  and more recently from the worldline QFT
perspective in Ref. Jakobsen:2021smu .

The last ingredient that we need are polarization vectors
$\varepsilon_{\pm}(k)$ for a massless particle with momentum $k$. We may use
spinor-helicity notations for it; for numerical evaluation, however, it is
more convenient to start with the special outgoing momentum $k_{0}$

$\displaystyle
k_{0}=(1,0,0,1)\qquad\varepsilon_{\pm}(k_{0})=-\frac{1}{\sqrt{2}}(0,1,\mp
i,0)$ (161)

and obtain the general angle-dependent polarization vector through a rotation:

$\displaystyle\varepsilon_{\pm}(k)=R(\theta,\phi)\varepsilon_{\pm}(k_{0})\qquad
R(\theta,\phi)={\tiny{\left(\begin{array}[]{cccc}1&0&0&0\cr
0&\cos\phi&-\sin\phi&0\cr 0&\sin\phi&\cos\phi&0\cr
0&0&0&1\end{array}\right)\cdot\left(\begin{array}[]{cccc}1&0&0&0\cr
0&\cos\theta&0&\sin\theta\cr 0&0&1&0\cr
0&-\sin\theta&0&\cos\theta\end{array}\right)}}\ .$ (170)

The third angle of a general rotation simply multiplies $\varepsilon$ by a
phase, so we will ignore it. This rotation also maps $k_{0}$ to
$\tilde{k}=(1,\bm{n}_{\bm{k}})$ with $\bm{n}_{\bm{k}}$ a general unit vector.
We stress that, as discussed before, the complex nature of these polarization
tensors does not change the definition of $L$ and $\mathcal{I}_{L}$ in eq. 69,
because they are stripped off before taking the real and imaginary part.
Indeed, we may decompose the outgoing polarization tensor
$\varepsilon_{--}=\varepsilon_{-}\otimes\varepsilon_{-}$ as

$\displaystyle\varepsilon_{--}=\varepsilon_{+}+i\varepsilon_{\times}\ ,$ (171)

and use separately the real polarizations $\varepsilon_{+}$ and
$\varepsilon_{\times}$ to define the real and imaginary parts of the matrix
element ${\cal B}_{-}$ and subsequently the waveforms. The asymptotic metric
$h_{\mu\nu}^{\infty}$ is defined in eq. 32. We further parametrize it as

$\displaystyle
h_{\mu\nu}^{\infty}=(\kappa\mathsf{M})\hat{h}_{\mu\nu}\,,\qquad{\hat{h}}_{\mu\nu}=\frac{\kappa^{2}\mathsf{M}}{\sqrt{-b^{2}}}{\hat{h}}^{(1)}_{\mu\nu}+\left(\frac{\kappa^{2}\mathsf{M}}{\sqrt{-b^{2}}}\right)^{2}{\hat{h}}^{(2)}_{\mu\nu}+\dots$
(172)

where $\mathsf{M}=m_{1}+m_{2}$ is the total mass. Here
$\hat{h}_{\mu\nu}^{(1)}$ and $\hat{h}_{\mu\nu}^{(2)}$ are respectively the
reduced waveform at 1PM and 2PM order. Similar to the previous section, we
will not distinguish barred and unbarred variables since their difference is
quantum.

We will perform explicit calculations in the center-of-mass (COM) frame in
which the black holes move along the $z$ axis and $b^{\mu}$ is along the $x$
axis,161616Technically, $u_{1}$ and $u_{2}$ are outgoing four-velocities in
our setup. However, in the classical limit, we can treat them as incoming
since the difference is quantum.

$\displaystyle u_{1}^{\mu}=(\gamma,0,0,-\gamma v)\,,\qquad
u_{2}^{\mu}=(\gamma,0,0,\gamma v)\,,\qquad b^{\mu}=(0,|\bm{b}|,0,0)\,,$ (173)

where $\gamma=\frac{1}{\sqrt{1-v^{2}}}$. We will plot waveforms at various
locations $(\theta,\phi)$ at the spatial infinity with different velocities
$v$.

###  8.1 Gravitational memory

By plugging eqs. 62, 63 and 64 into eq. 66, we can write the gravitational
memory as

$\displaystyle\Delta(h_{+}^{\infty}+ih^{\infty}_{\times})$
$\displaystyle=\lim_{\omega\rightarrow
0^{+}}(-i\omega)\Theta(\omega)(-i)\langle\psi_{\text{in}}|\hat{S}^{\dagger}\hat{a}_{--}(k)\hat{S}|\psi_{\text{in}}\rangle^{0}\big{|}_{k=(\omega,\omega\bm{n})}$
$\displaystyle\quad+\lim_{\omega\rightarrow
0^{-}}(-i\omega)\Theta(-\omega)(+i)\langle\psi_{\text{in}}|\hat{S}^{\dagger}\hat{a}^{\dagger}_{++}(k)\hat{S}|\psi_{\text{in}}\rangle^{0}\big{|}_{k=(|\omega|,|\omega|\bm{n})}\,.$
(174)

In the soft limit, we have seen explicitly in section 7 through one loop
order, that the matrix elements become

$\displaystyle\lim_{\omega\rightarrow
0^{+}}(-i\omega)\Theta(\omega)(-i)\langle\psi_{\text{in}}|\hat{S}^{\dagger}\hat{a}_{--}(k)\hat{S}|\psi_{\text{in}}\rangle^{0}\big{|}_{k=(\omega,\omega\bm{n})}$
$\displaystyle\qquad=-\frac{i\kappa}{4}\int\frac{\textrm{d}^{d}q}{(2\pi)^{d}}\hat{\delta}(2p_{1}\cdot
q)\hat{\delta}(2p_{2}\cdot q)e^{iq\cdot
b}\,\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}\mathcal{S}(\tilde{k},q)_{\mu\nu}M_{4}^{\text{2MPI.cl.}}\,,$
(175)

where $\tilde{k}=(1,\bm{n})$, and we have used $\Theta(0)=1/2$. The graviton
soft theorem implies this result holds at all orders. A similar calculation
can be applied to the conjugate matrix element.171717We note that in this
calculation, cancelling $\omega$ in $k$ leads to an extra minus sign because
$k=(-\omega,-\omega\bm{n})$ and $\omega<0$. This sign will be absorbed by
redefining $q\rightarrow-q$ to align the exponential factor $e^{iq\cdot b}$
because the soft factor $\mathcal{S}_{\mu\nu}$ is odd in $q$. The final result
differs from section 8.1 only by a complex conjugation on
$M_{4}^{\text{2MPI.cl.}}$. Therefore, section 8.1 becomes

$\displaystyle\Delta(h_{+}^{\infty}+ih_{\times}^{\infty})=-\frac{i\kappa}{2}\int\frac{\textrm{d}^{d}q}{(2\pi)^{d}}{\hat{\delta}}(2p_{1}\cdot
q){\hat{\delta}}(2p_{2}\cdot q)e^{iq\cdot
b}\,\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}{\cal
S}({\tilde{k}},q)_{\mu\nu}\,{\rm Re}\big{[}M_{4}^{\text{2MPI.cl}}\big{]}\,.$
(176)

The typical integral in eq. 176 is

$\displaystyle
I^{\mu}=\int\frac{\textrm{d}^{d}q}{(2\pi)^{d}}\hat{\delta}(2p_{1}\cdot
q)\hat{\delta}(2p_{2}\cdot q)e^{iq\cdot b}q^{\mu}f(q^{2})\,.$ (177)

We can fix $d=4$ and decompose $q$ in terms of Ref. Cristofoli:2021vyo

$\displaystyle
q^{\mu}=z_{1}u_{1}^{\mu}+z_{2}u_{2}^{\mu}+z_{b}b^{\mu}+z_{v}v^{\mu}\,,\quad
v^{\mu}\equiv 4\epsilon^{\mu\nu\rho\sigma}u_{1\nu}u_{2\rho}b_{\sigma}\,.$
(178)

We then integrate over the coefficients $\\{z_{1},z_{2},z_{b},z_{v}\\}$. It is
easy to see that the two delta functions fix $z_{1}=z_{2}=0$. The integration
over $z_{v}$ vanishes because the integrand is odd in $z_{v}$. As a result,
$I^{\mu}$ must be proportional to $b^{\mu}$.
$\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}{\cal S}({\tilde{k}},q)_{\mu\nu}$
can therefore be pulled out of the integral by replacing
$q^{\mu}\rightarrow-i\partial/\partial b_{\mu}$. Recalling that, up to two
loops, the constrained Fourier-transform of the real part of the 2MPI four-
point amplitude is expected to be the radial action Bern:2021yeh ;
Brandhuber:2021eyq , we find

$\displaystyle\Delta(h_{+}^{\infty}+ih_{\times}^{\infty})$
$\displaystyle=-\frac{\kappa}{2}\left[\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}{\cal
S}({\tilde{k}},\partial/\partial b)_{\mu\nu}\right]I_{r}(b)\,,$ $\displaystyle
I_{r}(b)$
$\displaystyle=\int\frac{\textrm{d}^{d}q}{(2\pi)^{d}}{\hat{\delta}}(2p_{1}\cdot
q){\hat{\delta}}(2p_{2}\cdot q)e^{iq\cdot b}\;{\rm
Re}\left[M_{4}^{\text{2MPI.cl}}\right]\,.$ (179)

The action of $\partial/\partial b_{\mu}$ on the radial action is proportional
to the scattering angle $\chi$,

$\displaystyle\frac{\partial I_{r}}{\partial
b_{\mu}}=-\frac{m_{1}m_{2}\sqrt{\sigma^{2}-1}}{\mathsf{M}\sqrt{1+2\nu(\sigma-1)}}\frac{b^{\mu}}{\sqrt{-b^{2}}}\frac{\partial
I_{r}}{\partial
J}=\frac{m_{1}m_{2}\sqrt{\sigma^{2}-1}}{\mathsf{M}\sqrt{1+2\nu(\sigma-1)}}\tilde{b}^{\mu}\chi\,,$
(180)

where $\nu=m_{1}m_{2}/\mathsf{M}^{2}$, $\tilde{b}^{\mu}=b^{\mu}/\sqrt{-b^{2}}$
and $\chi=-\partial I_{r}/\partial J$. The angular momentum $J$ is given by
$J=p_{\infty}\sqrt{-b^{2}}$, and $p_{\infty}$ is the norm of the initial COM
momentum, which is given by the prefactor in eq. 180. Therefore, the memory is
also proportional to $\chi$,

$\displaystyle\Delta(h_{+}^{\infty}+ih^{\infty}_{\times})$
$\displaystyle=-\frac{\kappa
m_{1}m_{2}\sqrt{\sigma^{2}-1}}{2\mathsf{M}\sqrt{1+2\nu(\sigma-1)}}\left(\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}{\cal
S}({\tilde{k}},\tilde{b})_{\mu\nu}\right)\chi\,,$ (181)
$\displaystyle\varepsilon^{\mu}\varepsilon^{\nu}\mathcal{S}(\tilde{k},\tilde{b})_{\mu\nu}$
$\displaystyle=-\frac{2\varepsilon\cdot
p_{1}\varepsilon\cdot\tilde{b}}{p_{1}\cdot\tilde{k}}+\frac{(\varepsilon\cdot
p_{1})^{2}\tilde{k}\cdot\tilde{b}}{(p_{1}\cdot\tilde{k})^{2}}+\frac{2\varepsilon\cdot
p_{2}\varepsilon\cdot\tilde{b}}{p_{2}\cdot\tilde{k}}-\frac{(\varepsilon\cdot
p_{2})^{2}\tilde{k}\cdot\tilde{b}}{(p_{2}\cdot\tilde{k})^{2}}\,.$

Importantly, the angular dependence of the memory is completely encoded in the
soft factor. We now plug in the 1PM and 2PM scattering angles for GR
Damour:2016gwp ; Damour:2017zjx  and $\mathcal{N}=8$ supergravity Parra-
Martinez:2020dzs ,

$\displaystyle\chi^{(1)}=\frac{\kappa^{2}\mathsf{M}}{\sqrt{-b^{2}}}\frac{\sqrt{1+2\nu(\sigma-1)}(2\sigma^{2}-X)}{16\pi(\sigma^{2}-1)}\,,$
$\displaystyle\chi^{(2)}=\left(\frac{\kappa^{2}\mathsf{M}}{\sqrt{-b^{2}}}\right)^{2}\frac{3\sqrt{1+2\nu(\sigma-1)}(5\sigma^{2}-1)X}{4096\pi(\sigma^{2}-1)}\,,$
(182)

where $X=1$ for GR and $X=0$ for $\mathcal{N}=8$ supergravity. The final
result for gravitational memory is

$\displaystyle\Delta(\hat{h}_{+}^{(1)}+i\hat{h}_{\times}^{(1)})$
$\displaystyle=-\frac{m_{1}m_{2}(2\sigma^{2}-X)}{32\pi\mathsf{M}^{2}\sqrt{\sigma^{2}-1}}\left(\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}{\cal
S}({\tilde{k}},\tilde{b})_{\mu\nu}\right)\,,$
$\displaystyle\Delta(\hat{h}_{+}^{(2)}+i\hat{h}_{\times}^{(2)})$
$\displaystyle=-\frac{3m_{1}m_{2}(5\sigma^{2}-1)X}{8192\pi\mathsf{M}^{2}\sqrt{\sigma^{2}-1}}\left(\varepsilon_{-}^{\mu}\varepsilon_{-}^{\nu}{\cal
S}({\tilde{k}},\tilde{b})_{\mu\nu}\right)\,,$ (183)

where $\hat{h}$ is defined in eq. 172. In particular, there are no memories
for $\mathcal{N}=8$ at 2PM. We note that the above relation between memory and
the soft factor was first predicted in Ref. Strominger:2014pwa .

###  8.2 Leading order (LO)

As discussed in eq. 74, the LO waveform is determined by the tree-level five-
point amplitude. From sections 3.4 and 74, we get

$\displaystyle(h_{+}^{\infty}+ih^{\infty}_{\times})(\tau,\bm{n})\Big{|}_{\text{LO}}$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}e^{-i\omega\tau}\int\frac{\textrm{d}^{d}q_{1}}{(2\pi)^{d}}\frac{\textrm{d}^{d}q_{2}}{(2\pi)^{d}}{\hat{\delta}}(2{\bar{p}}_{1}\cdot
q_{1}){\hat{\delta}}(2{\bar{p}}_{2}\cdot q_{2})e^{iq_{1}\cdot b}$
$\displaystyle\quad\times{\hat{\delta}}^{(d)}(q_{1}+q_{2}-k)M^{\text{\text{cl.}}}_{5,\text{tree}}(p_{1}p_{2}\rightarrow
p_{1}-q_{1},p_{2}-q_{2},k^{--})\big{|}_{k=\omega(1,\bm{n})}$
$\displaystyle=\int\frac{\textrm{d}^{d}q_{1}}{(2\pi)^{d}}\frac{\textrm{d}^{d}q_{2}}{(2\pi)^{d}}{\hat{\delta}}(2{\bar{p}}_{1}\cdot
q_{1}){\hat{\delta}}(2{\bar{p}}_{2}\cdot q_{2})e^{iq_{1}\cdot b}\delta(\tau-
q_{1}\cdot b)$
$\displaystyle\quad\times{\hat{\delta}}^{(d)}(q_{1}+q_{2}-\tilde{k})M^{\text{\text{cl.}}}_{5,\text{tree}}(p_{1}p_{2}\rightarrow
p_{1}-q_{1},p_{2}-q_{2},\tilde{k}^{--})\,,$ (184)

where $\tilde{k}=(1,\bm{n})$. After plugging in the explicit form of the tree
amplitudes, eq. 84 for GR and section 5 for $\mathcal{N}=8$ supergravity, we
find that the contribution to the reduced LO waveform, defined in eq. 172, has
the following unified form,

$\displaystyle(\hat{h}{}^{(1)}_{+}+i\hat{h}{}^{(1)}_{\times})(\tau,\bm{n})=-\frac{{m}_{1}{m}_{2}}{16\mathsf{M}^{2}}$
$\displaystyle\big{(}B_{1}{\cal J}^{0,0}_{1,0}+B_{2}{\cal
J}^{1,0}_{1,0}+B_{3}{\cal J}^{0,1}_{1,0}+B_{4}{\cal J}^{0,0}_{0,1}+B_{5}{\cal
J}^{1,0}_{0,1}$ $\displaystyle+B_{6}{\cal J}^{0,1}_{0,1}+B_{7}{\cal
J}^{0,0}_{1,1}+B_{8}{\cal J}^{1,0}_{1,1}+B_{9}{\cal J}^{2,0}_{1,1}\big{)}\,,$
(185)

where the integrals
$\mathcal{J}_{\beta_{1},\beta_{2}}^{\alpha_{1},\alpha_{2}}$ are defined as

$\displaystyle\mathcal{J}_{\beta_{1},\beta_{2}}^{\alpha_{1},\alpha_{2}}$
$\displaystyle=\int\frac{\textrm{d}^{d}q_{1}}{(2\pi)^{d}}\hat{\delta}(u_{1}\cdot
q_{1})\hat{\delta}(u_{2}\cdot(q_{1}-\tilde{k}))\delta(\tau/\sqrt{-b^{2}}-q_{1}\cdot\tilde{b})\frac{(\varepsilon_{-}\cdot
q_{1})^{\alpha_{1}}({\tilde{k}}\cdot
q_{1})^{\alpha_{2}}}{(q_{1}^{2})^{\beta_{1}}((q_{1}-\tilde{k})^{2})^{\beta_{2}}}\,,$
(186)

and the coefficients $B_{1,\dots,9}$ given by

$\displaystyle B_{1}$
$\displaystyle=\frac{4\sigma(\varepsilon_{-}\cdot{u}_{2})}{(\tilde{k}\cdot{u}_{2})}({u}_{1}\cdot\tilde{f}_{-}\cdot{u}_{2})$
$\displaystyle B_{2}$
$\displaystyle=2(2\sigma^{2}-X)\frac{(\varepsilon_{-}\cdot{u}_{2})}{(\tilde{k}\cdot{u}_{2})}$
$\displaystyle B_{3}$
$\displaystyle=-(2\sigma^{2}-X)\frac{(\varepsilon_{-}\cdot{u}_{2})^{2}}{(\tilde{k}\cdot{u}_{2})^{2}}$
$\displaystyle B_{4}$
$\displaystyle=\frac{4\sigma(\varepsilon_{-}\cdot{u}_{1})}{(\tilde{k}\cdot{u}_{1})}({u}_{2}\cdot\tilde{f}_{-}\cdot{u}_{1})$
$\displaystyle B_{5}$
$\displaystyle=-2(2\sigma^{2}-X)\frac{(\varepsilon_{-}\cdot{u}_{1})}{(\tilde{k}\cdot{u}_{1})}$
$\displaystyle B_{6}$
$\displaystyle=(2\sigma^{2}-X)\frac{(\varepsilon_{-}\cdot{u}_{1})^{2}}{(\tilde{k}\cdot{u}_{1})^{2}}$
$\displaystyle B_{7}$
$\displaystyle=4({u}_{2}\cdot\tilde{f}_{-}\cdot{u}_{1})^{2}$ $\displaystyle
B_{8}$ $\displaystyle=8\sigma({u}_{1}\cdot\tilde{f}_{-}\cdot{u}_{2})$
$\displaystyle B_{9}$ $\displaystyle=2(2\sigma^{2}-X)\ .$ (187)

In these expressions,
$\tilde{f}_{-}^{\mu\nu}={\tilde{k}}^{\mu}\varepsilon_{-}^{\nu}-{\tilde{k}}^{\nu}\varepsilon_{-}^{\mu}$,
and $X=1,0$ for GR and ${\cal N}=8$ supergravity, respectively.

With foresight on the integrals required by the evaluation of time-domain
observables, it is convenient to follow the strategy of Ref.
Cristofoli:2021vyo  and compute the
$\mathcal{J}_{\beta_{1},\beta_{2}}^{\alpha_{1},\alpha_{2}}$ integrals in $d=4$
by decomposing the integration variable $q_{1}$ along four orthogonal fixed
vectors as in eq. 178. The integrals over $z_{1}$, $z_{2}$, and $z_{b}$ are
localized due to the three delta functions in eq. 186. This leaves the
integral over $z_{v}$ as the only nontrivial integral. The first six integrals
in section 8.2 have been evaluated in Ref. Cristofoli:2021vyo . Using this
method, the last three integrals can be brought to a one-parameter integral
over a finite range. The background at $\tau\rightarrow-\infty$ is also
subtracted. To demonstrate the final result at LO, we plot the evolution of
the waveform at a particular location at the spatial infinity in figure 9. The
GR and $\mathcal{N}=8$ waveforms have the same qualitative features. The
difference is approximately an overall scale. Here and after, all the plots
are made with $m_{1}=m_{2}$.

Figure 9: The evolution of the LO waveform at a particular location at the
spatial infinity. We work in the COM frame with $v=1/5$ and $m_{1}=m_{2}$.

###  8.3 Next-to-leading order (NLO)

The NLO time-domain waveform follows from the $L=1$ component of section 3.4.
Here we further separate the contribution from the real and imaginary parts of
the one-loop amplitudes and write

$\displaystyle(h^{\infty}_{+}+i{h}^{\infty}_{\times})(t,\bm{n})\Big{|}_{\text{NLO}}$
$\displaystyle=\mathcal{H}_{R}+i\,\mathcal{H}_{I}\,,$ (188)

where $\mathcal{H}_{R,I}$ are obtained by combining eqs. 70, 3.4 and 3.4:

$\displaystyle\mathcal{H}_{R}$
$\displaystyle=-\frac{1}{2\pi}\int\frac{\textrm{d}^{d}q_{1}}{(2\pi)^{d}}\hat{\delta}(2\bar{p}_{1}\cdot
q_{1})\hat{\delta}\big{(}2\bar{p}_{2}\cdot(q_{1}-\tilde{k})\big{)}\,\varepsilon^{\mu}_{-}\varepsilon^{\nu}_{-}{\rm
Re}\left[M_{5,\text{1 loop}}^{0,\text{2MPI, cl}}\right]_{\mu\nu}$
$\displaystyle\hskip 85.35826pt\times\left[\frac{1}{(\tau-{q}_{1}\cdot
b+i0)^{2}}+\frac{1}{(\tau-{q}_{1}\cdot b-i0)^{2}}\right],$ (189)
$\displaystyle\mathcal{H}_{I}$
$\displaystyle=-\frac{1}{2\pi}\int\frac{\textrm{d}^{d}q_{1}}{(2\pi)^{d}}\hat{\delta}(2\bar{p}_{1}\cdot
q_{1})\hat{\delta}\big{(}2\bar{p}_{2}\cdot(q_{1}-\tilde{k})\big{)}\,\varepsilon^{\mu}_{-}\varepsilon^{\nu}_{-}{\rm
Im}\left[M_{5,\text{1 loop}}^{0,\text{2MPI, cl}}\right]_{\mu\nu}$
$\displaystyle\hskip 85.35826pt\times\left[\frac{1}{(\tau-{q}_{1}\cdot
b+i0)^{2}}-\frac{1}{(\tau-{q}_{1}\cdot b-i0)^{2}}\right].$ (190)

We note that $\mathcal{H}_{R}$ and $\mathcal{H}_{I}$ do not correspond to the
real and imaginary part of the final waveform, because the polarization
vectors in these expressions are still complex.

We deploy the same strategy as before to compute the integral over $q_{1}$ in
$d=4$ by decomposing it as in eq. 178. For both GR and $\mathcal{N}=8$
supergravity, it is easier to compute $\mathcal{H}_{I}$ due to the presence of

$\displaystyle\frac{1}{(\tau-{q}_{1}\cdot
b+i0)^{2}}-\frac{1}{(\tau-{q}_{1}\cdot b-i0)^{2}}=2\pi i\delta^{\prime}(\tau-
q_{1}\cdot b)\,.$ (191)

Thus we can use the three delta functions in $\mathcal{H}_{I}$ to localize
$z_{1}$, $z_{2}$ and $z_{b}$. The integrand is now an algebraic function of
$z_{v}$ parametrized by $\tau$. The resulting integral over $z_{v}$ is
convergent and performed numerically. The background at
$\tau\rightarrow-\infty$ is also subtracted at the end.

In contrast, there are only two delta functions in $\mathcal{H}_{R}$, which
localize $z_{1}$ and $z_{2}$. For $\mathcal{N}=8$ supergravity, the integrand
is a rational function in $z_{b}$ and $z_{v}$ according to section 7.
Therefore, we can integrate $z_{b}$ using the residue theorem, resulting in an
algebraic function of $z_{v}$, which will be integrated numerically. For
$\mathcal{N}=8$ supergravity, the subtraction of background at
$\tau\rightarrow-\infty$ can be done before or after the final $z_{v}$
integral since it does not affect the convergence.

However, for GR, after localizing $z_{1}$ and $z_{2}$, the integrand contains
square roots according to section 7. For this case, we still first integrate
over $z_{b}$. After some rescaling, the integral has the following generic
form,

$\displaystyle\int_{-\infty}^{\infty}\textrm{d}z_{b}\frac{f(z_{b})}{(z_{b}^{2}+1)^{1/2}}=\int_{0}^{\infty}\textrm{d}w\frac{f(\frac{w^{2}-1}{2w})}{w}=-\sum_{w_{i}}\text{Res}_{w=w_{i}}\frac{f(\frac{w^{2}-1}{2w})\log
w}{w}\,,$ (192)

where $f(z_{b})$ is a rational function, and we have changed the variable to
$z_{b}=\frac{w^{2}-1}{2w}$. Since the original $z_{b}$ integral is convergent,
there are no poles at $w=0$ and $w=\infty$ after the change of variable. We
can evaluate this integral by summing over all the residues, which leads to
the final answer in eq. 192. Now if we subtract the background at
$\tau=-\infty$, the final $z_{v}$ integral is convergent and we integrate it
numerically. Unlike the previous cases, the background subtraction needs to be
done before the $z_{v}$ integrals because the integral is divergent otherwise.
Alternatively, if one keeps $d=4-2\epsilon$, the integration and background
subtraction can be done in any order, the difference being at most some local
$\delta(|\bm{b}|)$ terms in the waveform and memory.

Figure 10: The NLO waveform in GR with COM velocity $v=1/5$. Figure 11: The
dependence of the NLO waveform in GR on the COM velocities at the fixed angle
$\theta=7\pi/10$ and $\phi=7\pi/5$. Figure 12: The NLO waveform in
$\mathcal{N}=8$ supergravity with the COM velocity $v=1/5$. Both
$\hat{h}_{+}^{(2)}$ and $\hat{h}_{\times}^{(2)}$ converge to zero at large
$\tau$. Figure 13: The NLO waveform in $\mathcal{N}=8$ supergravity on the COM
velocities at the fixed angle $\theta=7\pi/10$ and $\phi=7\pi/5$. Both
$\hat{h}_{+}^{(2)}$ and $\hat{h}_{\times}^{(2)}$ converge to zero at large
$\tau$.

We plot the NLO waveforms in GR observed at several angles at the spatial
infinity in figure 10. In this figure, we fix the COM velocity of the two
black holes to be $v=1/5$. Then, in figure 11, we fix the observation angle to
be $(\theta,\phi)=(\frac{7\pi}{10},\frac{7\pi}{5})$ and vary the velocities of
the black holes. At low velocities, the waveforms have a much larger magnitude
and are more spread out in the time domain; we may understand this intuitively
by noting that at lower velocities, the two particles have a smaller minimal
separation, so they experience larger accelerations and therefore radiate
more. We also note that at lower velocities, we are closer to the boundary of
the PM regime, which requires that $Jv^{2}$ be sufficiently large.

Finally, for comparison, we include in figures 12 and 13 the NLO waveforms in
$\mathcal{N}=8$ supergravity. As expected, the $\mathcal{N}=8$ waveforms do
not display any memory effect at NLO. At low velocities, for example,
$v=1/20$, the imaginary part exhibits a small residual at
$\tau\rightarrow\infty$; this is merely an artifact of our initial and final
times not being infinite. We note that the waveform profile is spread over a
larger time interval and that they have a larger amplitude than the GR
waveform. A possible explanation is that fields of the ${\cal N}=8$
supergravity yield an additional net attractive force increasing the
acceleration experienced by the two particles relative to GR.

We remark that at certain angles, the GR waveform may display features which,
for lack of a better name, we refer to as “kinks.” For example, such kinks can
be seen in figure 10 for $\hat{h}_{+}^{(2)}$ at
$(\theta,\phi)=(\frac{7\pi}{10},\frac{7\pi}{5})$ and
$(\frac{7\pi}{10},\frac{2\pi}{5})$. At these locations, several terms in the
integrand become sharply peaked (but finite), and the resulting large values
of the integral cancel over many orders of magnitude. Our numerical evaluation
further suggests that more prominent kinks exist at some other angles, such as
$(\theta,\phi)=(\frac{\pi}{2},0)$, which corresponds to an observation
direction parallel to $\bm{b}$ in the plane of the two incoming particles. It
is unclear whether these features are merely numerical artifacts due to
cancelation between large numbers, or they are physical. The definitive answer
should be provided by an analytic evaluation of the waveform, which we leave
for future work.

##  9 Conclusions

The observable-based formalism Kosower:2018adc ; Cristofoli:2021vyo  directly
links scattering amplitudes and observables of unbound binary systems in the
classical regime. This framework naturally includes both conservative and
dissipative effects, providing a means to finding local observables such as
the asymptotic gravitational waveform for a scattering event in addition to
inclusive ones such as the impulse or the energy and angular momentum loss.
With the possible exception of singular momentum configurations, they are
governed by four and five-point amplitudes Cristofoli:2021jas . We examined in
detail this connection for the scattering waveform and found that, beyond
leading order, the exponential form of amplitudes proposed in Refs.
Bern:2021dqo ; Damgaard:2021ipf  implies that only (the suitably-defined) two-
matter-particle-irreducible components of amplitudes can contribute to this
observable.

We computed these parts of the four-scalar-one-graviton amplitude at one-loop
order in GR with minimally-coupled scalars and in ${\cal N}=8$ supergravity.
We obtained the former using generalized unitarity, both by evaluating the
classical part of the complete amplitude and by using a HEFT approach which
effectively corresponds to truncating the tree amplitudes to classical order.
Comparing the two results and requiring that they agree identifies a
prescription for treating uncut matter propagators, that they be treated as
principal-valued. We obtained the corresponding integrand in $\mathcal{N}=8$
supergravity, which describes the emission of gravitational radiation from the
scattering of two half-BPS black holes, by using the double copy of the
corresponding integrand in ${\cal N}=4$ super-Yang-Mills theory and
dimensional reduction.

Upon reduction to master integrals, the five-point classical amplitudes
exhibit novel features compared to their four-point counterparts. For example,
the outgoing graviton momentum injects a scale in some of the scaleless (and
thus vanishing) integrals in the four-point amplitude, implying that they can
contribute to the five-point classical amplitude.

The one-loop master integrals are sufficiently simple so we evaluated them by
direct integration for physical kinematics.181818They may also be evaluated as
classical limits of the known one-loop master integrals that appear in the
quantum theory. A distinguishing feature of the (2MPI part of the) classical
five-point amplitude is that it exhibits an infrared divergent phase. The
structure of this phase divergence is governed by Weinberg’s classic result
Weinberg:1965nx  and the classical nature of the amplitude; we argue that at
this order – and perhaps to all orders in perturbation theory – it can be
absorbed as a suitable shift in the definition of the retarded time.
Consequently, the observation of the gravitational waveform is sensitive only
to differences of retarded times rather than the absolute time of the process.
This is consistent with the basic assumption of scattering theory that
observations are carried out at infinite times. We also gave a precise
relation between the gravitational wave memory and the soft limit of the
corresponding scattering amplitude, along the lines of Ref. Strominger:2014pwa
, relating the amplitude of the memory to the scattering angle to at least
${\cal O}(G^{3})$. Interestingly, this implies that the gravitational-wave
memory vanishes at ${\cal O}(G^{2})$ for half-BPS black holes in
$\mathcal{N}=8$ supergravity, which is confirmed by the explicit calculation.

Even though our results are analytic in frequency and momentum space and in
time-momentum space, we resorted to numerical integration to evaluate the one-
dimensional integral completing the transformation to time-impact parameter
space. A complete analytic evaluation of the waveform, which we leave for
future work, would involve obtaining an analytic expression for this last one-
dimensional integral. The result would resolve certain numerical features that
we encountered at special observation angles and would open the door to the
evaluation of interesting inclusive and local observables, such as the energy
and angular momentum loss at ${\cal O}(G^{4})$ and the radiated energy
spectrum $E(\omega)$ through traditional general relativity methods
Bonga:2018gzr ; Thorne:1980ru ; DeWitt:2011nnj ; Peters:1964zz . The classical
one-loop five-point amplitude constructed here allows the computation of
inclusive quantities at 4PM order through the KMOC formalism, along the lines
of the 3PM claculations of Refs. Herrmann:2021lqe ; Herrmann:2021tct ;
Manohar:2022dea .

While the next-to-leading order relativistic waveform is the current state of
the art, it is natural to think about higher orders. The experience with
${\cal N}$-extended supergravity loop calculations Bern:2009kd ;
Carrasco:2011mn ; Bern:2013uka ; Bern:2014sna ; Bern:2017ucb ; Bern:2018jmv
suggests that the construction of the relevant classical integrand through
double copy and generalized unitarity should scale well to higher orders and
the results will provide tests of the HEFT approach to classical amplitudes
beyond those already available. We expect, however, that the evaluation of the
resulting master integrals will benefit from advanced techniques such as
differential equations and method of regions, as was done for the full quantum
two-loop five-point amplitude of ${\cal N}=8$ supergravity in Ref.
Abreu:2019rpt . Going to higher order could also help us better understand
radiation-reaction, which already appears at one-loop Edinburgh , and other
physical phenomena that could appear.

The current analytic approach to analytic and semi-analytic binary inspiral
waveforms makes use of the effective one-body formalism Buonanno:1998gg ;
Buonanno:2000ef , whose input is an off-shell binary Hamiltonian and radiation
reaction forces. It would be interesting to identify features of gravitational
waveforms that can be analytically continued between bound and unbound
systems, in analogy with inclusive observables discussed in Refs.
Kalin:2019rwq ; Kalin:2019inp ; Cho:2021arx . We may expect that a detailed
understanding of the analytic structure of the waveforms, both in the bound
and unbound case, will be important.

###### Acknowledgements.

We thank A. Elkhidir, D. O’Connell, M. Sergola, I. Vazquez-Holm, and A.
Brandhuber, G. Brown, G. Chen, S. De Angelis, J. Gowdy, G. Travaglini for
coordination on ongoing work. We also thank N. Arkani-Hamed, R. Akhoury, J.
Berman, R. Britto, L. Dixon, H. Elvang, S. Mizera and C.-H. Shen for
stimulating discussions. RR and FT also thank Z. Bern, E. Herrmann, J. Parra-
Martinez, M. Ruf, C.-H. Shen, and M. Zeng for collaboration on related topics,
and especially M. Ruf for providing valuable comments on our draft. RR and FT
would like to thank the Kavli Institute for Theoretical Physics at the
University of California, Santa Barbara, for hospitality during the program
“High-Precision Gravitational Waves”. Their research there was supported in
part by the National Science Foundation under Grant No. NSF PHY-1748958. AH is
supported by a Rackham Predoctoral Fellowship from the University of Michigan.
RR and FT are supported by the U.S. Department of Energy (DOE) under award
number DE-SC00019066.

##  Appendix A Self-consistency of HEFT

Consistency of the HEFT construction of the classical part of amplitudes
requires that the 2MPR contributions are always super-classical and then
subtracted in the KMOC formalism. We will show that, should 2MPR terms exist
at the classical order, all such contributions require quantum information
about the HEFT tree amplitudes used in the unitarity construction. To this
end, we will use the $\hbar$ counting as introduced in eq. 2.

Consider a cut of a four-scalar amplitude containing $k$ HEFT tree amplitudes,
each with $\tilde{n}_{g,i}$ internal gravitons $n_{g,i}$ external gravitons,
and $n_{\phi,i}$ scalars; call
$\tilde{N}_{g}=\frac{1}{2}\sum_{i=1}^{k}\tilde{n}_{g,i}$ and
$n_{g}=\sum_{i=1}^{k}n_{g,i}$ respectively the the total number of internal
and external gravitons, and $N_{\phi}=\frac{1}{2}\sum_{i=1}^{k}n_{\phi,i}-2$
the total number of internal scalars. Restoring the integration measure and
the matter and graviton propagators, this cut scales as

$\displaystyle\underbrace{\Bigg{[}\prod_{i=1}^{k}\hbar^{-3(\frac{1}{2}n_{\phi,i}-1)-\frac{1}{2}\tilde{n}_{g,i}-\frac{1}{2}n_{g,i}}\Bigg{]}}_{\text{HEFT
tree
amplitudes}}\underbrace{\vphantom{\Bigg{[}}\hbar^{-2\tilde{N}_{g}-N_{\phi}}\hbar^{4L}}_{\text{propagators
and measures}}=\hbar^{-3-\frac{1}{2}n_{g}}\hbar^{\tilde{N}_{g}+1-k}\,,$ (193)

where we have used eq. 3 and the loop number $L=\tilde{N}_{g}+N_{\phi}-k+1$.
This cut scales classically if

$\tilde{N}_{g}+1-k=0\ .$ (194)

Consistency requires that if we cut any line the result should still scale
classically. Let us consider cutting a pair of matter lines. The resulting
eight-scalar amplitude scales as

$\hbar^{-9-\frac{1}{2}n_{g}}\hbar^{\tilde{N}_{g}+1-k}\hbar^{8-a}\,.$ (195)

Here we have isolated the classical scaling $\hbar^{-9}$ of an eight-scalar
amplitude. If the graph remains connected, then we remove two matter
propagators and two integral measures from eq. 193, which leads to $a=8$. On
the other hand, if the graph is now disconnected, then we only need to remove
two matter propagators and one integral measure instead, which leads to $a=4$.
Thus, if $a=8$ – i.e. if the cut of the eight-scalar amplitude is connected –
then it has classical scaling upon use of eq. 194 and the truncation to
classical order is iteratively consistent. In contrast, if $a=4$ – i.e. if the
cut of the eight-scalar amplitude is disconnected – the cut is suppressed by
$\hbar^{4}$, which means that the disconnect components contain quantum
contributions.

Repeating the calculation above for any pair of matter lines, it follows that
any 2MPR contribution with a classical scaling requires quantum information
about some amplitude contributing to it. Since the HEFT approach prescribes
that tree amplitudes be truncated to classical orders, the $\hbar$ scaling of
2MPR diagrams and cuts with classical vertices is super-classical, with one
factor of $\hbar^{-1}$ for each present two-matter-particle cut. On the other
hand, if we explicitly include some quantum operators in the Lagrangian and
keep higher-order terms in the HEFT tree amplitudes, we will get classical
2MPR contributions. However, they will be consistently subtracted in the KMOC
formalism. Thus, only 2MPI cuts – and consequently only 2PMI diagrams – should
contribute to classical scattering observables to any loop order.

##  Appendix B Evaluating box master integrals

In this appendix, we derive all the box master integrals listed in section
6.2. We will repeatedly use the bubble integral with a cut matter propagator
and a generic propagator in $d=4-2\epsilon$,

$\displaystyle\int\textrm{d}^{4-2\epsilon}\ell\frac{\delta(\bar{u}_{2}\cdot\ell)}{(\ell-
Q_{1})^{2}-m^{2}+i0}=-\frac{\pi^{(3-2\epsilon)/2}\Gamma(\frac{2\epsilon-1}{2})}{[-(\bar{u}_{2}\cdot
Q_{1})^{2}+m^{2}-i0]^{(2\epsilon-1)/2}}\,,$ (196)

where $Q_{1}$ is a generic vector.

###  B.1 $I_{1,1,1,0}$

We first compute the box integral with three graviton propagators,

$\displaystyle
I_{1,1,1,0}=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}[(\ell-
q_{1})^{2}+i0]}$
$\displaystyle=\int_{0}^{1}\textrm{d}x\int\frac{\textrm{d}^{d}{\ell}}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{1}+xk_{5})^{2}+i0]^{2}}\,,$ (197)

where we have introduced a Feynman parameter to combine the two massless
propagators $(\ell+q_{2})^{2}$ and $(\ell-q_{1})^{2}$ into a single
propagator. We next use an IBP identity to reduce the double propagator
$[(\ell-q_{1}+xk_{5})^{2}+i0]^{2}$,

$\displaystyle\int_{0}^{1}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{1}+xk_{5})^{2}+i0]^{2}}$ $\displaystyle=-\frac{d-3}{2(\bar{u}_{2}\cdot
q_{1})^{2}}\int_{0}^{1}\textrm{d}x\frac{1}{(1-x)^{2}[(1-x)q_{1}^{2}+xq_{2}^{2}]}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+xk_{5})^{2}+i0}$
$\displaystyle\quad-(d-4)\int_{0}^{1}\textrm{d}x\frac{1}{(1-x)q_{1}^{2}+xq_{2}^{2}}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{1}+xk_{5})^{2}+i0]}\,.$ (198)

The integral in the last line is finite in $d=4$,191919One can show this by
using
$\displaystyle\int\textrm{d}^{4}\ell\frac{\delta(\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
Q_{1})^{2}-m^{2}]}$ $\displaystyle=\frac{2\pi^{2}}{\sqrt{(\bar{u}_{2}\cdot
Q_{1})^{2}-m^{2}-Q_{1}^{2}}}\arcsin\sqrt{1+\frac{(\bar{u}_{2}\cdot
Q_{1})^{2}-m^{2}}{-Q_{1}^{2}}}$ for the $\ell$ integral. The remaining $x$
integral is convergent.  such that the $(d-4)$ prefactor makes this term only
contribute to $\mathcal{O}(\epsilon)$ order. Thus both the IR divergence and
the finite part are given by the first term of section B.1,

$\displaystyle I_{1,1,1,0}$ $\displaystyle=-\frac{d-3}{2(\bar{u}_{2}\cdot
q_{1})^{2}}\int_{0}^{1}\textrm{d}x\frac{1}{(1-x)^{2}[(1-x)q_{1}^{2}+xq_{2}^{2}]}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+xk_{5})^{2}}+\mathcal{O}(\epsilon)$
$\displaystyle=\frac{\pi^{5/2-\epsilon}\Gamma(\epsilon+1/2)}{(2\pi)^{4-2\epsilon}[-(\bar{u}_{2}\cdot
q_{1})^{2}-i0]^{1/2+\epsilon}}\int_{0}^{1}\frac{\textrm{d}x}{(1-x)^{1+2\epsilon}[(1-x)q_{1}^{2}+xq_{2}^{2}]}+\mathcal{O}(\epsilon)$
(199) $\displaystyle=\frac{1}{32\pi q_{2}^{2}\sqrt{-(\bar{u}_{2}\cdot
q_{1})^{2}-i0}}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(-4\pi(\bar{u}_{2}\cdot
q_{1})^{2}-i0)+2\log\frac{q_{2}^{2}}{q_{1}^{2}}\right]+\mathcal{O}(\epsilon)\,.$

The $i0$ prescription suggests the analytic continuation
$\log(-x-i0)=\log(x)-i\pi$ for $x>0$ and $\sqrt{-(\bar{u}_{2}\cdot
q_{1})^{2}-i0}=-i(\bar{u}_{2}\cdot q_{1})$, which gives

$\displaystyle I_{1,1,1,0}$
$\displaystyle=\frac{1}{32q_{2}^{2}(\bar{u}_{2}\cdot q_{1})}+\frac{i}{32\pi
q_{2}^{2}(\bar{u}_{2}\cdot
q_{1})}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(4\pi(\bar{u}_{2}\cdot
q_{1})^{2})+2\log\frac{q_{2}^{2}}{q_{1}^{2}}\right]+\mathcal{O}(\epsilon)\,.$
(200)

The result agrees with eq. 144 in the main text.

###  B.2 $I^{\pm}_{1,1,0,1}$

We next consider the box integrals with one uncut matter propagator. We start
with

$\displaystyle I^{+}_{1,1,0,1}$
$\displaystyle=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}(2\bar{u}_{1}\cdot\ell+i0)}$
$\displaystyle=\int_{0}^{\infty}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+q_{2})^{2}[(\ell+x\bar{u}_{1})^{2}-x^{2}+i0]^{2}}\,,$
(201)

where we have used the eq. 136 to combine $\ell^{2}$ and
$2\bar{u}_{1}\cdot\ell+i0$. The IBP identity can reduce the double propagator

$\displaystyle\int_{0}^{\infty}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+q_{2})^{2}[(\ell+x\bar{u}_{1})^{2}-x^{2}+i0]^{2}}$
$\displaystyle=\frac{d-3}{2(y^{2}-1)}\int_{0}^{\infty}\textrm{d}x\frac{1}{x^{2}(2x\bar{u}_{1}\cdot
q_{2}-q_{2}^{2}-i0)}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+x\bar{u}_{1})^{2}-x^{2}+i0}$
$\displaystyle\quad+(d-4)\int_{0}^{\infty}\textrm{d}x\frac{1}{2x\bar{u}_{1}\cdot
q_{2}-q_{2}^{2}}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{2}+x\bar{u}_{1})^{2}-x^{2}+i0]}\,.$ (202)

The integral proportional to $d-4$ contributes to $\mathcal{O}(\epsilon)$
order, for a reason similar to the discussion below section B.1. Now using eq.
196 for the bubble integral in the second line of section B.2, we get the IR
divergence and the finite part of this integral,

$\displaystyle I^{+}_{1,1,0,1}$
$\displaystyle=\frac{d-3}{2(y^{2}-1)}\int_{0}^{\infty}\textrm{d}x\frac{1}{x^{2}(2x\bar{u}_{1}\cdot
q_{2}-q_{2}^{2}-i0)}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+x\bar{u}_{1})^{2}-x^{2}+i0}+\mathcal{O}(\epsilon)$
$\displaystyle=-\frac{\pi^{5/2-\epsilon}\Gamma(\epsilon+1/2)}{(2\pi)^{4-2\epsilon}[-(y^{2}-1)-i0]^{1/2+\epsilon}}\int_{0}^{\infty}\frac{\textrm{d}x}{x^{1+2\epsilon}(2x\bar{u}_{1}\cdot
q_{2}-q_{2}^{2}-i0)}+\mathcal{O}(\epsilon)$ $\displaystyle=\frac{i}{32\pi
q_{2}^{2}\sqrt{y^{2}-1}}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(-\pi(y^{2}-1)-i0)-2\log\frac{\bar{u}_{1}\cdot
q_{2}-i0}{-q_{2}^{2}}\right]+\mathcal{O}(\epsilon)$
$\displaystyle=\frac{i}{32\pi
q_{2}^{2}\sqrt{y^{2}-1}}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(-\pi(y^{2}-1)-i0)-2\log\frac{\bar{u}_{1}\cdot
q_{2}}{-q_{2}^{2}}\right]+\mathcal{O}(\epsilon)\,.$ (203)

Here we can omit the $i0$ prescription in $\log\frac{\bar{u}_{1}\cdot
q_{2}}{-q_{2}^{2}}$ because the argument is positive. To compute
$I^{-}_{1,1,0,1}$, we first redefine the loop momentum to flip the sign of
$i0$,

$\displaystyle
I^{-}_{1,1,0,1}=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{2})^{2}(2\bar{u}_{1}\cdot\ell-i0)}=-\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell-
q_{2})^{2}(2\bar{u}_{1}\cdot\ell+i0)}\,.$ (204)

We can directly obtain the result of this integral by sending
$q_{2}\rightarrow-q_{2}$ in section B.2 while preserving the sign of $i0$,

$\displaystyle I^{-}_{1,1,0,1}$ $\displaystyle=-\frac{i}{32\pi
q_{2}^{2}\sqrt{y^{2}-1}}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(-\pi(y^{2}-1)-i0)-2\log\frac{-\bar{u}_{1}\cdot
q_{2}-i0}{-q_{2}^{2}}\right]+\mathcal{O}(\epsilon)$
$\displaystyle=\frac{1}{16q_{2}^{2}\sqrt{y^{2}-1}}-I^{+}_{1,1,0,1}+\mathcal{O}(\epsilon)\,.$
(205)

Therefore, the combination $I^{+}_{1,1,0,1}+I^{-}_{1,1,0,1}$ is free of IR
divergences,

$\displaystyle
I^{+}_{1,1,0,1}+I^{-}_{1,1,0,1}=\frac{1}{16q_{2}^{2}\sqrt{y^{2}-1}}\,,$ (206)

which reproduces eq. 145a.

###  B.3 $I^{\pm}_{1,0,1,1}$

We apply the same strategy to compute the last set of box integrals,

$\displaystyle
I^{\pm}_{1,0,1,1}=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell\pm i0)}\,.$ (207)

The only difference here is that the combination
$I^{+}_{1,0,1,1}+I^{-}_{1,0,1,1}$ is now complex, with the imaginary part
given by further cutting $(\ell-q_{1})^{2}$. Starting with $I^{+}_{1,0,1,1}$,
we first apply eq. 136 to combine $(\ell-q_{1})^{2}$ and
$2\bar{u}_{1}\cdot\ell+i0$, followed by using an IBP identity to get,

$\displaystyle I^{+}_{1,0,1,1}$
$\displaystyle=\int_{0}^{\infty}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{1}+x\bar{u}_{1})^{2}-x^{2}+i0]^{2}}\equiv I_{a}^{+}+I_{b}^{+}\,,$ (208)

where $I_{a}^{+}$ and $I_{b}^{-}$ are defined as

$\displaystyle I_{a}^{+}$
$\displaystyle=-\frac{d-3}{2q_{1}^{2}}\int_{0}^{\infty}\textrm{d}x\frac{1}{(y^{2}-1)x^{2}-2(\bar{u}_{2}\cdot
q_{1})yx+(\bar{u}_{2}\cdot q_{1})^{2}}$
$\displaystyle\qquad\qquad\times\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+x\bar{u}_{1})^{2}-x^{2}+i0}\,,$ $\displaystyle I_{b}^{+}$
$\displaystyle=-\frac{d-4}{q_{1}^{2}}\int_{0}^{\infty}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}[(\ell-
q_{1}+x\bar{u}_{1})^{2}-x^{2}+i0]}\,.$ (209)

Similar to eq. 204, we can rewrite $I^{-}_{1,0,1,1}$ as

$\displaystyle
I^{-}_{1,0,1,1}=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell-i0)}$
$\displaystyle=-\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{\ell^{2}(\ell+q_{1})^{2}(2\bar{u}_{1}\cdot\ell+i0)}$
$\displaystyle\equiv-(I^{-}_{a}+I^{-}_{b})\,,$ (210)

where $I^{-}_{a,b}=I^{+}_{a,b}\big{|}_{q_{1}\rightarrow-q_{1}}$. Therefore, we
have

$\displaystyle
I^{+}_{1,0,1,1}+I^{-}_{1,0,1,1}=(I^{+}_{a}-I^{-}_{a})+(I^{+}_{b}-I^{-}_{b})\,,$
(211)

such that we only need to compute the combination $I^{+}_{a}-I^{-}_{a}$ and
$I^{+}_{b}-I^{-}_{b}$.

If we use eq. 196 with $d=4$ to integrate out the bubble in $I_{a}^{\pm}$, we
will find that the resulting $x$ integral is divergent at
$x\rightarrow\infty$. However, by taking the difference $I^{+}_{a}-I^{-}_{a}$,
we get a finite result since the range of $x$ gets truncated. Interestingly,
the result is proportional to the triangle integral (141),

$\displaystyle
I_{a}^{+}-I_{a}^{-}=\frac{1}{q_{1}^{2}}(I^{+}_{0,0,1,1}+I^{-}_{0,0,1,1})=\frac{1}{16q_{1}^{2}\sqrt{y^{2}-1}}+\frac{i}{8\pi
q_{1}^{2}\sqrt{y^{2}-1}}\operatorname{arccosh}(y)\,.$ (212)

To compute $I^{+}_{b}-I^{-}_{b}$, we introduce another Feynman parameter $w$
and perform the $\ell$ integral in $d=4$. The result is finite due to the same
truncation on the range of $x$,

$\displaystyle
I^{+}_{b}-I^{-}_{b}=-\frac{d-4}{16\pi^{2}q_{1}^{2}}\int_{-\alpha}^{\alpha}\textrm{d}x\int_{0}^{1}\frac{\textrm{d}w}{[-w^{2}(x^{2}+\beta)(y^{2}-1)-w(1-w)q_{1}^{2}-i0]^{1/2}}\,,$
(213)

such that $I^{+}_{b}-I^{-}_{b}=0$ in $d=4$. Therefore, we have reproduced eq.
145b,

$\displaystyle
I^{+}_{1,0,1,1}+I^{-}_{1,0,1,1}=I^{+}_{a}-I^{-}_{a}=\frac{1}{16q_{1}^{2}\sqrt{y^{2}-1}}+\frac{i}{8\pi
q_{1}^{2}\sqrt{y^{2}-1}}\operatorname{arccosh}(y)\,.$ (214)

###  B.4 $I^{\pm}_{0,1,1,1}$

We first use the same Feynman parameterization as in eq. 197 to bring together
the two massless propagators,

$\displaystyle I^{+}_{0,1,1,1}$
$\displaystyle=\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell+q_{2})^{2}(\ell-
q_{1})^{2}(2\bar{u}_{1}\cdot\ell+i0)}$
$\displaystyle=\int_{0}^{1}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+xk_{5})^{4}(2\bar{u}_{1}\cdot\ell+i0)}\,.$ (215)

The resultant double propagator $(\ell-q_{1}+xk_{5})^{4}$ is then reduced by
the IBP relation,

$\displaystyle\int_{0}^{1}\textrm{d}x\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+xk_{5})^{4}(2\bar{u}_{1}\cdot\ell+i0)}$
$\displaystyle=\frac{d-3}{4(\bar{u}_{2}\cdot
q_{1})^{2}}\int_{0}^{1}\textrm{d}x\frac{x(\bar{u}_{1}\cdot
q_{2})+(1-x)y(\bar{u}_{2}\cdot
q_{1})}{(1-x)^{2}\mathcal{F}(x)}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+xk_{5})^{2}+i0}$
$\displaystyle\quad+\frac{d-4}{2}\int_{0}^{1}\textrm{d}x\frac{y^{2}-1}{\mathcal{G}(x)}\int\frac{\textrm{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2\bar{u}_{2}\cdot\ell)}{(\ell-
q_{1}+xk_{5})^{2}(2\bar{u}_{1}\cdot\ell+i0)}\,,$ (216)

where $\mathcal{G}(x)$ is a quadratic polynomial in $x$,

$\displaystyle\mathcal{G}(x)=x^{2}(\bar{u}_{1}\cdot
q_{2})^{2}+2x(1-x)y(\bar{u}_{1}\cdot q_{2})(\bar{u}_{2}\cdot
q_{1})+(1-x)^{2}(\bar{u}_{2}\cdot q_{1})^{2}\,.$ (217)

Under the principal value combination $I^{+}_{0,1,1,1}+I^{-}_{0,1,1,1}$, the
first integral in section B.4 is doubled while the second integral is finite
in $d=4$, which only contributes to $\mathcal{O}(\epsilon)$ due to the
prefactor. We can now reproduce eq. 146 through a direct integration,

$\displaystyle I^{+}_{0,1,1,1}+I^{-}_{0,1,1,1}$
$\displaystyle=-\frac{\pi^{5/2-\epsilon}\Gamma(\epsilon+1/2)}{(2\pi)^{4-2\epsilon}[-(\bar{u}_{2}\cdot
q_{1})^{2}-i0]^{1/2+\epsilon}}\int_{0}^{1}\textrm{d}x\frac{x(\bar{u}_{1}\cdot
q_{2})+(1-x)y(\bar{u}_{2}\cdot q_{1})}{(1+x)^{1+2\epsilon}\mathcal{F}(x)}$
$\displaystyle=-\frac{1}{32(\bar{u}_{1}\cdot q_{2})(\bar{u}_{2}\cdot q_{1})}$
(218) $\displaystyle\quad-\frac{i}{32\pi(\bar{u}_{1}\cdot
q_{2})(\bar{u}_{2}\cdot
q_{1})}\left[-\frac{1}{\epsilon_{\text{IR}}}+\log(4\pi(\bar{u}_{2}\cdot
q_{1})^{2})+2\log\frac{\bar{u}_{1}\cdot q_{2}}{\bar{u}_{2}\cdot
q_{1}}\right]\,.$

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