aainstitutetext: Perimeter Institute for Theoretical Physics, 31 Caroline St
N, Waterloo ON, N2L 2Y5, Canadabbinstitutetext: Department of Physics and
Astronomy, York University, 4700 Keele St, Toronto ON, M3J 1P3,
Canadaccinstitutetext: Chennai Mathematical Institute, H1, SIPCOT IT Park,
Siruseri, Kelambakkam 603103, India

# Soft Constraints on KMOC Formalism

Yilber Fabian Bautista<EMAIL_ADDRESS>c Alok Laddha
<EMAIL_ADDRESS>
###### Abstract

In this note, we investigate the implications of classical soft theorems for
the formalism developed by Kosower, Maybee and O’Connell (KMOC) to derive
classical observables in gauge theory and gravity from scattering amplitudes.
In particular, we show that the radiative electro-magnetic field at leading
order in the soft expansion imposes an infinite hierarchy of constraints on
the expectation value of the family of observables generated by monomials of
linear impulse. We perform an explicit check on these constraints at next to
leading order (NLO) in the coupling and as a corollary show how up to NLO,
soft radiation obtained from quantum amplitudes is consistent with the
(leading) classical soft photon theorem.

We also argue that in 4 dimensions the classical log soft theorem derived by
Saha, Sahoo and Sen generates an infinite hierarchy of constraints on the
expectation value of operators which are products of one angular momentum and
an arbitrary number of linear momenta.

##  1 Introduction

Classical soft photon (graviton) theorems Laddha:2018rle ; Laddha:2018myi ;
Sahoo:2018lxl ; Laddha:2019yaj ; Saha:2019tub  are exact statements about soft
radiation emitted during a generic electro-magnetic (gravitational) scattering
process. As shown in the seminal works by Sahoo and SenSahoo:2018lxl , Saha,
Sahoo and Sen Saha:2019tub , and Sahoo Sahoo:2020ryf , in four dimensions if
we expand the electro-magnetic (or gravitational) radiative field in the
frequency of the emitted radiation, then the following terms in the expansion
have a universal analytic form independent of the details of the scattering
dynamics or even spins of the scattering particles

$\displaystyle
A_{\mu}(\omega,\hat{n})\,=\,\frac{1}{\omega}\,A_{\mu}^{\omega^{-1}}(\hat{n})\,+\,\sum_{I=1}^{2}\,\omega^{I}(\ln\omega)^{I+1}\,A_{\mu}^{\ln^{I+1}}(\hat{n})\,+\,\cdots\,.$
(1)

Here $\hat{n}$ is a unit vector pointing towards the direction of observation,
and $\cdots$ indicate sub-sub-leading terms in the soft expansion. It was
conjectured in Sahoo:2020ryf  that even among the subn-leading terms the
coefficients of $\omega^{n}\,\ln^{n+1}\omega\ n\,\geq\,3$ are universal while
other terms in the soft expansion are non-universal and depend on the details
of the dynamics. In Ghosh:2021hsk , first such non-universal soft factor
proportional to $\omega\ln\omega$ was computed and was shown to depend on the
spin of the scattering particles.

Each coefficient in the above expansion is a function of incoming and outgoing
momenta and charges of the scattering particles. For example, the leading
coefficient $A_{\mu}^{\omega^{-1}}(\hat{n})$ is simply the Weinberg soft
photon factor PhysRev.140.B516

$\displaystyle
A_{\mu}^{\omega^{-1}}(\hat{n})\,=\,\sum_{i=1}^{N_{in}}\,Q_{i}\,\frac{1}{p_{i}\cdot
n}\,p_{i}^{\mu}\,-\,\sum_{i=1}^{N_{out}}\,Q_{i}\,\frac{1}{p_{i}\cdot
n}\,p_{i}^{\mu}\,,$ (2)

where $\\{(Q_{1},p_{1}),\,\cdots,\,(Q_{i},p_{i})\,\\}$ is the collection of
the charges and momenta of scattering particles, and
$n^{\mu}\,=\,(1,\hat{n})$. Although the exact expressions for sub-leading and
higher order log soft factors in eqn.(1) are more complicated, they are all
functions of asymptotic data, namely charges and momenta of scattering
particles.

From the perspective of scattering dynamics, these theorems are rather non-
trivial as they are non perturbative in the coupling. If we consider a class
of scattering processes which can be analysed perturbatively (such as large
impact parameter scattering so that $N_{in}=N_{out}=N$) then every outgoing
momenta admits a perturbative expansion, 111We note that from the classical
perspective, this expansion is convergent as final momenta are well defined.

$\displaystyle
p_{i\,+}^{\mu}\,=\,p_{i\,-}^{\mu}+\sum_{L=0}\,g^{2(L+1)}\,(\triangle
p^{(L)})_{i}^{\mu}\,,$ (3)

where $g$ is the coupling222We have use $g$ instead of $e$, the charge of the
electron in order to avoid confusion with the exponential function. and
$(\triangle p^{(L)})^{\mu}$ is the linear impulse evaluated at $L$-th order in
the perturbation theory ($L=0$ being the leading order (LO) impulse). We thus
see when expanded in the coupling, the Weinberg soft photon factor has a
rather intricate structure

$\displaystyle
A_{\mu}^{\omega^{-1}}(\hat{n})\,=\,\sum_{s=1}^{N}\,g\,Q_{s}\sum_{n=0}g^{2(n+1)}\,\sum_{i=0}^{n}\,V^{\mu}_{s\,\alpha_{1}\,{\cdots}\,\alpha_{i+1}}\
\smashoperator[]{\sum_{\begin{subarray}{c}L_{1},\,\cdots,\,L_{i+1}=0\\\
(L_{1}+L_{i+1})+(i+1)\,=n\end{subarray}}^{n+1-i}}\ (\,(\triangle
p^{(L_{1})})^{\alpha_{1}}\,{\cdots}\,(\triangle
p^{(L_{i+1})})^{\alpha_{i+1}}\,)\,,$ (4)

where the sum
$\sum_{L_{1},\,\cdots,\,L_{i+1}=0\,|\,(L_{1}+L_{i+1})+(i+1)\,=n}^{n+1-i}$ is
over products of impulses at $L_{1},\,\cdots,\,L_{i+1}$ orders in the coupling
respectively. In the above equation we have defined a tensor

$\displaystyle
V^{\mu}_{s\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\,=\,{\color[rgb]{0,0,0}(-1)^{i}}\,\Big{[}\,\frac{(i+1)\,!\,}{(p_{s}\cdot
k)^{i+1}}\delta^{\mu}_{(\alpha_{1}}\,k_{\alpha_{2}}\,\cdots\,k_{\alpha_{i+1})}\,-\,\frac{1}{(p_{s}\cdot
k)^{i+2}}\,p_{s}^{\mu}\,k_{\alpha_{1}}\,\cdots\,k_{\alpha_{i+1}}\,\Big{]}\,.$
(5)

Classical Soft photon (or graviton) theorem are independent of the details of
the hard scattering and are applicable to perturbative scattering at finite
impact parameter as well as collisions. However, to prove the classical soft
theorems via perturbative analysis (even in the case where hard scattering can
be treated perturbatively) is a highly non-trivial task as one has to resum
the perturbation series. But the discussion above demonstrates that, due to
their universality, classical soft theorems can serve as powerful tool for any
method which computes electro-magnetic (or gravitational) radiation using
(perturbative) scattering amplitudes. For one, it can serve as a strong
diagnostic for the perturbative results of radiation kernel and when used in
conjunction with the perturbative results (such as analytic expressions for
impulse in the Post Minkowskian (PM) expansion), it can produce interesting
insights such as providing analytical formulae for classical radiation in
terms of incoming kinematic data order by order in perturbation theory.

One such methods aforementioned, was developed by Kosower, Maybee and
O’Connell (KMOC) Kosower:2018adc , whose formalism allows to compute classical
electromagnetic (gravitational) observables from the classical limit of
quantum scattering amplitudes. These observables include the linear and
angular momentum impulse – including spin – experienced by a scattering
particle, as well as radiative field emitted in a classical scattering
Kosower:2018adc ; Maybee:2019jus ; Guevara:2019fsj ; Cristofoli:2021vyo ;
Herrmann:2021lqe . In this paper we initiate a study of the implications of
classical soft theorems for KMOC formalism. As we show, consistency with the
leading classical soft theorem imposes an infinite hierarchy of constraints on
KMOC observables. In order to state these constraints we introduce following
conventions.

The scattering process we consider is a $2\,\rightarrow\,2$ scattering process
in which two incoming charged particles with momenta $p_{1},p_{2}$ and charges
$Q_{1},\,Q_{2}$ scatter via electro-magnetic interactions as well as any other
higher derivative interaction which is long range such that the KMOC formalism
applies to this scattering. As the classical soft theorems are universal and
independent of the details of the scattering, classical limit of the radiation
kernel should generate the soft factors for _any_ perturbative amplitude
involving charged particles in the external states and a photon.

To each of the two massive particles we can associate certain classical
observables defined as follows:
(1) Let $V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i}}$ be the projection operator
(5), associated to particle $1$, that is

$\displaystyle
V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i}}\,=\,(-1)^{i+1}\,\Big{[}\,\frac{i!}{(p_{1}\cdot
k)^{i}}\,\delta^{\mu}_{(\alpha_{1}}\,k_{\alpha_{2}}\,\cdots\,k_{\alpha_{i})}\,-\frac{1}{(p_{1}\cdot
k)^{i+1}}\,p_{1}^{\mu}\,k_{\alpha_{1}}\,\cdots\,k_{\alpha_{i}}\,\Big{]}\,.$
(6)

(2) Now consider certain moments of the exchange momenta

$\displaystyle\begin{array}[]{lll}{\cal
T}^{(n)\,\alpha_{1}\,\cdots\,\alpha_{i}}\,:=\,\hbar^{\frac{3}{2}}\Big{[}i\int\,\hat{d}\mu_{q}\,q^{\alpha_{1}}\,\cdots\,q^{\alpha_{i}}\,e^{-iq{\cdot}b}\,{\cal
A}^{(n)}(p_{1},p_{2}\,\rightarrow\,p_{1}-q,p_{2}+q)\\\ \hskip
99.58464pt+\,\sum_{X=0}^{n-1}\,\int\,\prod_{m=0}^{X}d\Phi(r_{m})\,\hat{d}\mu_{q}\,\hat{d}\mu_{w,X}w^{\alpha_{1}}\,\cdots\,w^{\alpha_{i}}\,e^{-iq{\cdot}b}\\\
\hskip 128.0374pt\times\sum_{a_{1}=0}^{n-1-X}{\cal
A}^{(a_{1})}(p_{1},p_{2}\,\rightarrow\,p_{1}-w,p_{2}+w,\,r_{X})\\\ \hskip
128.0374pt\times{\cal
A}^{(n-a_{1}-X-1)\star}(p_{1}-w,p_{2}+w,\,r_{X}\,\rightarrow\,p_{1}-q,p_{2}+q)\Big{]}\,\,,\end{array}$
(11)

where $b$ is the impact parameter in the $2\to 2$ scattering process. In the
above equation we have introduced the following notations which will be used
throughout the paper.

  * •

${\cal A}^{(a)}(p_{1}^{i},\,p_{2}^{i}\,\rightarrow\,p_{1}^{f},p_{2}^{f})$ is
the (stripped) amplitude for a 4 point scattering at $a$-th loop order, and
analogously for the other amplitudes with additional momentum labels.

  * •

The integral measure $\hat{d}\mu_{q}$ is defined via

$\hat{d}\mu_{q}=\hat{d}^{4}q\hat{\delta}(2p_{1}{\cdot}q-q^{2})\hat{\delta}(2p_{2}{\cdot}q+q^{2}),$
(12)

where $\hat{d}^{4}q=\frac{d^{4}q}{(2\pi)^{4}}$, and the hat on $\delta$-fn.
indicates it is defined as

$\displaystyle\hat{\delta}(x)\,=\,-i\,[\,\frac{1}{x-i\epsilon}\,-\frac{1}{x+i\epsilon}\,]\,,$
(13)

and analogous for $\hat{d}\mu_{w,X}$,

$\hat{d}\mu_{w,X}=\hat{d}^{4}q\hat{\delta}(2p_{1}{\cdot}w-w^{2})\hat{\delta}(2p_{2}{\cdot}(w+r_{X})+(w+r_{X})^{2})$
(14)

  * •

The sum over $X$ is a sum over number of intermediate photons with momenta
$\\{r_{1}\,\cdots,\,r_{X}\,\\}$. Even though integration over the momentum
space of these photons is indicated explicitly by
$d\Phi(r_{m})=\hat{d}^{4}r_{m}\hat{\delta}^{(+)}(r_{m}^{2})$, we assume that
the sum over $X$ includes the sum over intermediate helicity states.

  * •

It is understood that for $n=0$, the second term in (11) vanishes.

As we will show in section 4.1, consistency of KMOC with the classical leading
soft photon theorem Saha:2019tub , implies that at $n$-order in perturbative
expansion we have the following identities

  * •

$\forall\,n>0$ and $\forall\,1\,\leq\,i\,\leq\,n$ :

$\begin{split}\lim_{\hbar\rightarrow\,0}\,\hbar^{m}\,V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i}}\,{\cal
T}^{(n)\,\alpha_{1}\cdots\alpha_{i}}\,=\,0\,\forall\,m\,\in\,\\{1,\cdots,n+1-i\\}\,,\end{split}$
(15)

  * •

and $\forall\,n$ and $\forall\,1\,\leq\,i\,\leq\,n\,+\,1$ :

$\lim_{\hbar\,\rightarrow\,0}\,V^{\mu}_{\alpha_{1}\,{\cdots}\,\alpha_{i}}{\cal
T}^{\alpha_{1}\,\cdots\,\alpha_{i}}_{(n)}\,=g^{2(n+1)}\,V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i}}\,\smashoperator[]{\sum_{L_{1},\,{\cdots},\,L_{i-1}=0}^{n}}\,(\triangle
p_{1}^{L_{1}})^{\alpha_{1}}\,{\cdots}\,(\triangle
p_{1}^{n{-}(L_{1}{+}\cdots{+}L_{i-1})})^{\alpha_{i}}\,.$ (16)

The first set of identities eqn.(15), arise by demanding that all the super-
classical terms in the radiation kernel vanish as mandated by consistency of
KMOC formalism. The second set on the other hand (eqn. (16)), relate the
classical limit of the (expectation value) of the monomials with perturbative
coefficients of the classical soft factor. These constraints were shown to be
satisfied at tree-level in the earlier work of Bautista:2019tdr ; Manu:2020zxl
, where at LO in the coupling, leading and sub-leading classical soft photon
theorem was derived from KMOC formula, which we will review in section 5.1 for
the leading soft result.

Analogous constraints follow for the subleading soft factor, and we briefly
comment about those in section 4.2. Let us here also stress that although in
this note we have concerned mostly with 4-dimensional electromagnetic
radiation , the higher dimensional generalization of these constraints, as
well as the equivalent formulation for gravitational radiation, should follow
in a similar manner.

This paper is organized as follows: In section 2 we review perturbative
results for classical soft photon theorem at leading and subleading orders in
the soft expansion. In section 3 we provide a short review of the KMOC
formalism in the context of radiation. We then move to the derivation of
identities (15) and (16) in section 4.1. In section section 4.2 we argue that
analogous constraints on KMOC follow from the sub-leading soft photon theorem.
In section 5 we show how the KMOC formula indeed satisfies these constraints
at leading (section 5.1) and next to leading (section 5.2) order in the
coupling. That is, contribution of the tree-level and one loop amplitudes to
the $\frac{1}{\omega}$ coefficient of the radiative field indeed matches with
eqn. (4). Finally, in section 6 we conclude with a discussion. In appendix A
we provide some computational details for section 5.2.

##  2 Soft Radiation in Classical Scattering

In this section, we analyse the classical soft photon factor up to sub-leading
order in soft expansion and next to leading order in the coupling in terms of
explicit expressions for the linear impulse. We simply use the results in
Saketh:2021sri , in conjunction with classical soft theorem to write the
radiative field at the desired order. That is, we compute
$A_{\mu}^{\omega^{-1}}(\hat{n})$ and $A_{\mu}^{\ln}(\hat{n})$ to NLO in the
coupling in a classical scattering involving two charges particles with masses
$m_{1},m_{2}$ which are interacting only via electro-magnetic interactions.

Although our primary focus in this paper from the perspective of KMOC is on
the $\frac{1}{\omega}$ soft factor, we also give an explicit verification of
the sub-leading ($\ln\omega$) soft photon theorem from explicit computation.
That is, we derive the logarithmically divergent (in asymptotic time)
contribution to the angular momentum impulse and substitute it in the “formal
sub-leading soft factor" expression (which is obtained by substituting
classical angular momentum in the angular momentum operators in Cachazo-
Strominger Cachazo:2014fwa  soft factor $\frac{J^{\mu\nu}k_{\nu}}{p\cdot k}$)
to show that the result equals the log soft photon factor at NLO in the
coupling.

###  2.1 Leading soft factor

Leading order radiation:

Let $p_{i}\,|\,i=1,2$ be the momenta for in incoming massive particles, moving
in the asymptotic free trajectories $b^{\mu}+v^{\mu}\tau$ in the far pass. If
we denote the null vector $(1,\hat{n})$ as $n^{\mu}$ we can write the leading
soft factor at tree level from formulas (4) and (5), that is

$A^{(0)\,\mu}_{\omega^{-1}}(\hat{n})=g^{3}\sum_{i=1}^{2}Q_{i}\left[\frac{\Delta
p_{i}^{(0)\,\mu}}{p_{i}{\cdot}n}-\frac{\Delta p_{i}^{(0)\,\mu}\cdot
n}{(p_{i}{\cdot}n)^{2}}p_{i}^{\mu}\right]\,,$ (17)

where the leading order linear impulse has the well known form

$\Delta p^{(0)\,\mu}_{1}=-\Delta
p^{(0)\,\mu}_{2}=-\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{2\pi\sqrt{\mathcal{D}}}\frac{b^{\mu}}{b^{2}},\,\,\,\,\,b^{2}=-\vec{b}^{2},$
(18)

with the Jacobian factor

$\mathcal{D}=\left(p_{1}{\cdot}p_{2}\right)^{2}-m_{1}^{2}m_{2}^{2}.$ (19)

This simple examples shows explicitly how the radiated field to leading order
in the soft expansion is determined only from asymptotic data, and in
particular for perturbation theory, from only incoming data since the outgoing
momenta are determined by the perturbative expansion (3), which we have
truncated at leading order in the coupling.

Sub-Leading order radiation

At next to leading order, the radiated field has a more interesting form,
since as indicated in (4) and (5), both, the leading and subleading impulse
enter into the field. Indeed, it explicitly reads

$A_{\omega^{-1}}^{(1)\,\mu}(\hat{n})=g^{5}\sum_{i=1}^{2}Q_{i}\left[\frac{\Delta
p_{i}^{(1)\,\mu}}{p_{i}{\cdot}n}-\frac{\Delta p_{i}^{(1)}\cdot
n}{(p_{i}{\cdot}n)^{2}}p_{i}^{\mu}-\frac{(\Delta p_{i}^{(0)}\cdot
n)^{2}}{(p_{i}{\cdot}n)^{2}}\Delta p_{i}^{(0)\,\mu}+\frac{(\Delta
p_{i}^{(0)}\cdot n)^{2}}{(p_{i}{\cdot}n)^{3}}p_{i}^{\mu}\right]\,.$ (20)

At this order, it is still true that $\Delta p_{1}^{(1)\,\mu}=-\Delta
p_{2}^{(1)\,\mu}$, with the NLO impulse given explicitly by Saketh:2021sri

$\Delta
p_{1}^{(1)\,\mu}=\frac{g^{4}}{8\pi|b|^{3}}\left(\frac{Q_{1}Q_{2}}{p_{1}{\cdot}p_{2}}\right)^{2}\left[\pi\sqrt{\mathcal{D}}(m_{1}+m_{2})b^{\mu}+4\frac{p_{1}{\cdot}p_{2}(p_{1}{+}p_{2})^{2}|b|}{\mathcal{D}}p^{\mu}\right]\,,$
(21)

where we have used

$p^{\mu}=\frac{m_{1}m_{2}}{(p_{1}+p_{2})^{2}}\left[\left(\frac{m_{2}}{m_{1}}+\frac{p_{1}\cdot
p_{2}}{m_{1}m_{2}}\right)p_{1}^{\mu}+\left(\frac{m_{1}}{m_{2}}+\frac{p_{1}\cdot
p_{2}}{m_{1}m_{2}}\right)p_{2}^{\mu}\right]\,.$ (22)

###  2.2 Sub-leading soft factor

In Laddha:2018myi  it was argued that the classical form of the Cachazo-
Strominger soft factor is in fact of the order $\ln\omega$ and not
$\omega^{0}$ in the soft expansion. This is due to the fact that when
scattering particles are subjected to long range electro-magnetic (or
gravitational) forces, the correction to the free trajectory
$b^{\mu}\,+\,v^{\mu}\,\tau$ is not a power series in $\frac{1}{\tau}$ but
include logarithmic tails such as $\ln\tau$. As a result of this long range
effect, the orbital angular momentum of the scattering particles is dominated
by terms proportional to $\ln\tau$ at late times. Intuitively we expect late
times to correspond to soft frequencies and hence in the frequency space, the
asymptotic orbital angular momentum is expected to be dominated by terms
proportional to $\ln\omega$. Thus, the Cachazo-Strominger soft factor which
naively is of the order $\omega^{0}$, in fact generates the so-called log soft
factor. This argument was turned into a theorem in a series of remarkable
papers by Saha, Sahoo and Sen. Sahoo:2018lxl ; Saha:2019tub . More in detail,
in the case of $2\,\rightarrow\,2$ scattering, we can compute the difference
in angular momenta at late and early times for both the scattering particles.
As we show below, this difference indeed has the form

$\displaystyle J_{\mu\nu}^{+}\,-\,J_{\mu\nu}^{-}\,=\,\ln\tau\,(\triangle
J)_{\mu\nu}^{(\ln)}\,+\,(\triangle J)_{\mu\nu}^{(0)}\,,$ (23)

where $\ln\tau$ is understood as an logarithmic divergent term as
$\tau\,\rightarrow\,\infty$.333As was argued in Laddha:2018myi  frequency
space we expect $\ln\tau$ to be replaced by $\ln\omega^{-1}$ for soft
frequency $\omega$. This divergence shows that soft expansion in $\omega$
breaks down in Four dimensions. We can now use the above argument to “guess"
the log soft photon factor.

$\displaystyle
A_{\mu}^{\ln}(\hat{n})\,=\,-\,\sum_{i=1}^{2}\,Q_{i}\,\frac{1}{p_{i}\cdot
n}\,\Big{[}\,(\triangle J)_{\mu\nu}^{\ln(0)}\,n^{\nu}\,+\,((\triangle
J)_{\mu\nu}^{(\ln)\,(1)}\,k^{\nu}\,-\,\frac{1}{p_{i}\cdot n}(\triangle
p_{i}^{(0)}\cdot n)\,(\triangle J)_{\mu\nu}^{\ln(0)})\,\Big{]}\,,$ (24)

where $(\triangle J)_{\mu\nu}^{\ln(i)}$ is the logarithmically divergent term
in angular momentum impulse at NiLO order in coupling.

In this part of the note we use the results provided in Saketh:2021sri , to
compute $\triangle J_{\mu\nu}^{\ln}$ at leading and next to leading order in
the coupling. Along with expression for $(\triangle p^{(0)})^{\mu}$, one can
then obtain the log soft factor up to $O(g^{5})$ in the coupling.

Let us see, in Saketh:2021sri , the authors compute the deflection to the free
trajectory up to N3LO order in the perturbative expansion. That is, we expand
the trajectory for the $i$-th particle with charge $Q_{i}$ and initial
momentum $p_{i-}$ as

$\displaystyle
x_{i}^{\mu}(\tau)\,=\,b_{i}^{\mu}\,+\,\frac{1}{m_{i}}\,p_{i-}^{\mu}\tau\,+\,\sum_{n=0}\,e^{2(n+1)}\,(\delta^{n}x_{i})^{\mu}(\tau)\,,$
(25)

where $(\delta^{n}x_{i})^{\mu}$ is the deflection at NnLO order. At every
order in the perturbative expansion the deflected trajectory accelerates as
$O(\frac{1}{\tau^{2}})$ at late future times $\tau_{+}$.

$\displaystyle(\delta^{n}x_{i})^{\mu}(\tau_{+})\,=\,(\delta^{n}x_{i})^{\mu}_{1}\
\tau_{+}+\,(\delta^{n}x_{i})^{\mu}_{\ln}\,\ln\tau_{+}+O(\tau_{+}^{0})\,.$ (26)

The $\ln\tau_{+}$ co-efficient of the angular momentum impulse can now be
computed as

$\displaystyle\begin{array}[]{lll}(\triangle
J_{i}^{\ln(0)})\,=\,(\delta^{0}x_{i})_{\ln}\,\wedge\,p_{i-}\,,\\\ (\triangle
J_{i}^{\ln(1)})\,=\,(\delta^{1}x_{i})_{\ln}\,\wedge\,p_{i-}\,+\,(\delta^{0}x_{i})_{\ln}\,\wedge\,(\triangle
p_{i})^{(0)}\,.\end{array}$ (29)

Both $\delta^{0}x_{\ln},\,\delta^{1}x_{\ln}$ can be read off from expressions
given by equations $(2.24)$ and $(A.25)$ in Saketh:2021sri . We focus on the
radiative field sourced by the first particle with initial momentum
$p_{1}^{\mu}$. 444We suppress the label $-$ to indicate the initial momenta as
it will not be required in the rest of the section

$\displaystyle\begin{array}[]{lll}(\delta^{0}x_{1})^{\mu}_{\ln}\,=\,Q_{1}\,Q_{2}\,\frac{1}{m_{1}(\gamma
v)^{3}}\,(-\gamma v_{1}^{\mu}+v_{2}^{\mu})\\\
(\delta^{1}x_{1})^{\mu}_{\ln}=\frac{Q_{1}^{2}Q_{2}^{2}}{m_{1}^{2}}\,\Big{[}\,-\frac{\gamma^{3}v\hat{b}^{\mu}}{|b|(\gamma
v)^{5}}\,+\frac{m_{1}}{m_{2}}\,\frac{\gamma^{2}v\,\hat{b}^{\mu}}{|b|(\gamma
v)^{5}}\,\Big{]}\,,\\\ \hskip
43.36243pt=\,\frac{Q_{1}^{2}Q_{2}^{2}}{m_{1}^{2}}\,\frac{\hat{b}^{\mu}}{|b|(\gamma
v^{2})^{2}}\Big{[}\,-1\,+\frac{m_{1}}{m_{2}\gamma}\,\Big{]}\,.\end{array}$
(33)

The leading order perturbation of the final momenta from the initial momenta
can be easily computed and is given by 555This is the same impulse given in
(18), but here we have used the normalization followed in Saketh:2021sri .,

$\displaystyle(\delta^{0}p_{1})^{\mu}\,=\,Q_{1}Q_{2}\frac{1}{v|b|}\,\hat{b}^{\mu}\,,$
(34)

where $\gamma\,=\,v_{1}\,\cdot v_{2}\,=\,\sqrt{\frac{1}{1-v^{2}}}$. One can
write the Lorentz factors in a more familiar form as,

$\displaystyle\begin{array}[]{lll}\frac{1}{\gamma
v}\,=\,\frac{m_{1}m_{2}}{{\cal\sqrt{D}}}\,.\end{array}$ (36)

Next we can use these formulae for logarithmic deflections in the far future
to compute the $\ln\sigma_{+}$ contribution to angular momentum impulse at LO
and NLO in the coupling

$\displaystyle\triangle
J^{\ln(0)}\,=\,Q_{1}\,Q_{2}\,\frac{(m_{1}m_{2})^{2}}{{\cal
D}^{\frac{3}{2}}}\,p_{2}\,\wedge\,p_{1}\,,$ (37) $\displaystyle\triangle
J^{\ln\,(1)}\,=\,(Q_{1}\,Q_{2})^{2}\,\frac{(p_{1}\cdot
p_{2})^{2}}{m_{1}^{2}m_{2}^{3}}\frac{1}{{\cal
D}^{2}\,|b|}\,\hat{b}\,\wedge\,\Big{[}\,\frac{1}{m_{1}}(\gamma-1)p_{1}+\frac{1}{m_{2}}\Big{(}\frac{p_{1}}{\gamma}-p_{2}\Big{)}\,\Big{]}^{\mu\nu}\,.$
(38)

On substituting these expressions in eqn.(24), one can explicitly verify that
up to NLO, the result precisely matches with the classical soft factor derived
by Saha, Sahoo and Sen in Saha:2019tub .

##  3  KMOC Formalism in a Nutshell

In the KMOC formalism Kosower:2018adc , the expectation value for the change
of a classical observable $\langle\mathcal{O}\rangle$666We use
$\langle...\rangle$ to imply that the classical limit for the given observable
is taken.  during a scattering process, is computed from the scattering matrix
through the formula

$\langle\mathcal{O}\rangle=\lim_{\hbar\to
0}\hbar^{\beta_{\mathcal{O}}}\left[{}_{\rm{in}}\langle\psi|S^{\dagger}\hat{\mathcal{O}}S|\Psi\rangle_{\rm{in}}-\,_{\rm{in}}\langle\psi|\hat{\mathcal{O}}|\Psi\rangle_{\rm{in}}\right]\,,$
(39)

where $-\beta_{\mathcal{O}}$ is the power of the LO-piece in the
$\hbar$-expansion of the quantities inside the square brackets, which depends
on the specific observable, as well as on the theory considered. Then, the
factor of $\hbar^{\beta_{\mathcal{O}}}$ in this formula ensures
$\langle\mathcal{O}\rangle\sim\hbar^{0}$, i.e. classical scaling. For
instance, for the radiated photon field, we have
$\beta_{\mathcal{O}}=\frac{3}{2}$.

In formula (39), $|\Psi\rangle_{in}$ corresponds to the multi-particle initial
state, describing particles as wave functions with well-defined positions and
momenta. In this note we are interested in a $2\to 3$ scattering process,
then, we take the 2-particles initial state to be given by

$|\Psi\rangle_{in}=\int\prod_{i}\big{[}\hat{d}^{4}p_{i}\hat{\delta}^{(+)}(p_{i}^{2}-m_{i}^{2})\phi_{i}(p_{i})e^{ib_{i}{\cdot}p_{i}/\hbar}\big{]}|p_{1}p_{2}\rangle$
(40)

where we have employed the notation of the original reference Kosower:2018adc
, however, unlike for the original work, and to be more general, we have move
to a frame where both particles are displaced by the positions $b_{i}$, whose
difference, $b_{2}-b_{1}=b$, corresponds then to the impact parameters, which
is the distance of closest approach between the massive particles during the
scattering process. The next task is to relate the observable to the
scattering amplitude employing the usual representation of the scattering
matrix in terms of the transition matrix $S=1+iT$, followed by the classical
limit, the latter of which is achieved by the usual $\hbar$-rescaling of the
coupling constant $g\to\hbar^{-1/2}g$, the external massless momenta
$k_{i}\to\hbar k_{i}$, the momenta miss-match
$q_{i}=p_{i}^{\prime}-p_{i}\to\hbar q_{i}$, where $p_{i}^{\prime}$ is the
momentum label for the conjugate states, the loop momenta $l_{i}\to\hbar
l_{i}$, and the position displacements $b_{i}\to\hbar^{-1}b$ (we point to the
original reference Kosower:2018adc , for a detailed analysis of the classical
limit). At this stage, the explicit dependence on the wave functions drops
away, leaving us with the desired expression for the classical observable.

In this note, the observable of interest is the radiative electro-magnetic
field $A^{(n)\,\mu}(k)$ 777The radiative field is an observable as it is
defined at null infinity where (small) gauge transformations vanish. where $k$
is the momentum of the electro-magnetic wave, at $\rm{N}^{n}\rm{LO}$ order in
the perturbative expansion. Hence, it is understood that the $n=0$ term
represents the tree level contribution, whereas for $n>0$ we pick up loop
corrections. It is a straightforward task to show that in terms of the
scattering amplitude, the radiated field can be computed from the sum of two
terms

$\langle
A^{(n)\,\mu}(k)\rangle=\mathcal{R}^{(n)\,\mu}(k)+\mathcal{C}^{(n)\,\mu}(k),$
(41)

which have the explicit recursive form

$\mathcal{R}^{(n)\,\mu}(k)=i\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\int\prod_{i=1}^{2}\hat{d}^{4}q_{i}\hat{\delta}(2p_{i}{\cdot}q_{i}-q_{i}^{2})e^{ib_{i}{\cdot}q_{i}}\hat{\delta}^{4}(q_{1}{+}q_{2}-k)\mathcal{A}^{(n)\,\mu}\left(p_{1},p_{2}{\rightarrow}p_{1}{-}q_{1},p_{2}{-}q_{2},k\right)\,,$
(42)

and

$\begin{split}\mathcal{C}^{(n)\,\mu}(k)&={\color[rgb]{0,0,0}}\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\sum_{X=0}^{n-1}\int\prod_{m=0}^{X}d\Phi(r_{m})\prod_{i=1}^{2}\hat{d}^{4}w_{i}\hat{d}^{4}q_{i}\hat{\delta}(2p_{i}{\cdot}q_{i}{-}q_{i}^{2})\hat{\delta}(2p_{i}{\cdot}w_{i}{-}w_{i}^{2})e^{ib_{i}{\cdot}q_{i}}\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\times\hat{\delta}^{4}(w_{1}{+}w_{2}{+}r_{X}{-}k)\hat{\delta}^{4}(w_{1}{+}w_{2}{+}r_{X}{+}q_{1}{+}q_{2})\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\,\times\sum_{a_{1}=0}^{n-1-X}\mathcal{A}^{(a_{1})\,\mu}(p_{1},p_{2}{\rightarrow}p_{1}{-}w_{1},p_{2}{-}w_{2},r_{X},k)\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\,\times\mathcal{A}^{(n-a_{1}-X-1)\star}(p_{1}{-}w_{1},p_{2}{-}w_{2},r_{X}{\rightarrow}p_{1}-q_{1},p_{2}-q_{2})\,,\end{split}$
(43)

where the $\star$ in one of the amplitude indicates complex conjugation. We
refer to the $\mathcal{C}^{(n)\,\mu}(k)$ term as the cut-box contribution, to
indicate that it is given by the cut of higher loop amplitudes. In this
expression, $r_{X}$ denotes the collection of momenta
$\\{r_{1},\cdots,r_{X}\\}$ carried by additional particles propagating
thorough the cut, whose momentum phase space integration has been explicitly
indicated by $d\Phi(r_{m})=\hat{d}^{4}r_{m}\hat{\delta}^{(+)}(r_{m}^{2})$. For
$n=0\,\,\rm{and}\,\,1$, no additional photons propagate thought the cut, since
they only appear starting from $\rm{N}^{2}$LO in the perturbative expansion
(i.e. two-loops).

##  4 Radiation Kernel : A Soft Expansion

In this section we will study the $\mathcal{R}$ and
$\mathcal{C}$-contributions to the radiated field in the KMOC formalism at
leading and subleading order in the soft expansion in 4-dimensions. We will
show that consistence of the KMOC formalism with the soft theorems at the
orders considered, generates a hierarchy of constraints on the expectation
value of several operators.

###  4.1 Leading soft constraints

The aim of this part of the note is to derive the set of identities (15) and
(16). Our idea now is to use KMOC formalism in conjunction with quantum soft
theorems to obtain radiation kernel in the soft limit. In other words, we
start with the exact formula for the radiation kernel. We then follow the
theme of Bautista:2019tdr ; Manu:2020zxl  where it was shown that taking the
soft limit before the classical limit generates soft expansion of radiation
kernel in KMOC formalism. That is, to a given order in soft expansion, we can
apply quantum soft photon theorems to factorise the 5-point amplitude in terms
of a 4 point amplitude and a soft factor.

At higher orders in the loop expansion, one also has to take into account the
order between loop integration and soft expansion. If we first do a soft
expansion and then loop integration, then one can use the tree-level soft
theorems to factorise the loop integrand into a soft factor and a Four point
integrand. However, as it was shown in a seminal paper by Sahoo and Sen
Sahoo:2018lxl , the two operations do not commute in Four dimensions beyond
the leading order in soft expansion. That is, the soft expansion done after
integrating over loop momenta results in $\ln$ soft factors which are absent
in the soft expansion of the loop integrand. At leading order however, this
subtlety does not enter as Weinberg soft photon theorem is a universal
statement in all dimensions.

Let us then substitute the soft photon theorem in eqns. (44, 45), and use the
momentum conserving delta functions to do the integrals in $q_{2}$ and
$w_{2}$, we get 888We only consider the radiation emitted by the first
particle, as the radiative field emitted by the outgoing particles is
additive. We will denote this contribution as ${\cal R}^{\mu}_{1}(k)$.

$\mathcal{R}_{1}^{(n)\,\mu}(k)=i\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\int\hat{d}\mu_{q}e^{-ib{\cdot}q}S^{(0)\,\mu}(p_{1},q,k)\mathcal{A}^{(n)}\left(p_{1},p_{2}{\rightarrow}p_{1}-q,p_{2}+q\right)\,.$
(44)

We have additionally defined the impact parameter by $b=b_{2}-b_{1}$, and used
(12) to rewrite the momentum measure. Analogously, for the $\mathcal{C}$-term
we have

$\begin{split}\mathcal{C}_{1}^{(n)\,\mu}(k)&={\color[rgb]{0,0,0}}\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\sum_{X=0}^{n-1}\int\prod_{m=0}^{X}d\Phi(r_{m})\hat{d}\mu_{q}\,\hat{d}\mu_{w,X}e^{-ib{\cdot}q}\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\times\sum_{a_{1}=0}^{n-1-X}S^{(0)\,\mu}(p_{1},w,k)\mathcal{A}^{(a_{1})}(p_{1},p_{2}{\rightarrow}p_{1}{-}w,p_{2}{+}(w+r_{X}),r_{X})\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\,\times\mathcal{A}^{(n-a_{1}-X-1)\star}(p_{1}{-}w,p_{2}{+}(w+r_{X}),r_{X}{\rightarrow}p_{1}-q_{1},p_{2}-q_{2})\,,\end{split}$
(45)

with $\hat{d}\mu_{w,X}$ given in (14).

The Weinberg soft factor has the following “quantum" expansion when expressed
in terms of exchange momenta. For the first particle (with charge and mass
$Q_{1},m_{1}$)

$\displaystyle
S^{(0)\mu}(p_{1},p_{1}{-}q,k)\,=\,Q_{1}\,\Big{[}\,\sum_{i=0}^{\infty}\,q^{\alpha_{1}}\,\cdots\,q^{\alpha_{i+1}}\,V_{\alpha_{1}\,\cdots\,\alpha_{i+1}}^{\mu}\,\Big{]}\,,$
(46)

where $V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i+1}}$ is defined in (6). We see
that the i-th term inside the square bracket in (46), scales as $\hbar^{i}$ in
the KMOC sense.

We will now derive the constraints proposed in eqns. (15) and (16) by
associating soft limit of radiation kernel with the classical soft factor at
all orders in the coupling. The first contribution
$\mathcal{R}_{1}^{(n)\,\mu}(k)$ can be written as

$\begin{split}\mathcal{R}_{1}^{(n)\,\mu}(k)\,&=i\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\,g\,Q_{1}\,g^{2(n+1)}\\\
&\,\,\,V_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}^{\mu}\,\sum_{i=0}^{n}\int\hat{d}\mu_{q}\,e^{-ib{\cdot}q}\,q^{\alpha_{1}}\,\cdots\,q^{\alpha_{i+1}}\,\mathcal{\bar{A}}^{(n)}\left(p_{1},p_{2}{\rightarrow}p_{1}-q,p_{2}+q\right)\,,\end{split}$
(47)

where the bar over the amplitude indicates that we have striped the coupling
constant. Notice here we have restricted the sum over $i$ at $n$ (where $n$ is
the order of the loop expansion). This can be argued using $\hbar$ scaling
arguments. The KMOC scaling implies that

$\displaystyle\begin{array}[]{lll}\hbar^{\frac{3}{2}}\,g^{2(n+1)}\,g\,\sim\,\frac{1}{\hbar^{n}}\,,\\\
V^{\mu}_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\,q^{\alpha_{1}}\,\cdots\,q^{\alpha_{i+1}}\,\sim\,\hbar^{i}\,.\end{array}$
(50)

Additionally, we now notice that at $n$-th order in the loop expansion, the
$\hbar$ scaling of the perturbative amplitude is quantified by KMOC as follows

$\begin{split}\mathcal{\bar{A}}^{(n)}\left(p_{1},p_{2}{\rightarrow}p_{1}-q,p_{2}+q\right)\,&=:\,{\cal
I}^{(n)}_{4}(p_{1},p_{2}\,\rightarrow\,p_{1}-q,p_{2}+q)\\\
&\sim\,[\,\frac{1}{\hbar^{2}}\,+\,\frac{1}{\hbar}\,+\,O(\hbar^{0})\,]\,,\\\
\hat{d}\mu_{q}\,&\sim\,\hbar^{4}[\frac{1}{\hbar^{2}}\,+O(\frac{1}{\hbar^{3}})\,]^{2}\,,\\\
b{\cdot}q&\sim\hbar^{0}\,.\end{split}$ (51)

It can be immediately verified that if the sum $\sum_{i}$ in eqn. (47) goes
beyond $i=n$, the right hand side ($\hbar\,\rightarrow\,0$ limit) vanishes. In
fact, these scaling arguments can be used to immediately verify that the
$\hbar$ expansion of the moments are

$g^{2(n+1)}\,g\,Q_{1}\,\hbar^{\frac{3}{2}}\,V^{\mu}_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\,\int\,\hat{d}^{4}\mu_{q}\,e^{-ib\cdot
q}\,(\,q^{\alpha_{1}}\,\cdots\,q^{\alpha_{i+1}}\,)\,\,\bar{{\cal
A}}^{(n)}=\,\sum_{\beta=0}^{n-i}\,\frac{1}{\hbar^{\beta}}\,{\cal
S}_{i\,\beta}^{\mu}\,+O(\hbar)\,.$ (52)

The contribution of $\mathcal{C}^{(n)\,\mu}(k)$ at leading order in the soft
limit can be analysed as in eqn.(47).

$\begin{split}\mathcal{C}_{1}^{(n)\,\mu}(k)&={\color[rgb]{0,0,0}}\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\sum_{i=1}^{n}\,V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i+1}}\,\sum_{X=0}^{n-1}\int\prod_{m=0}^{X}d\Phi(r_{m})\hat{d}\mu_{q}\,\hat{d}\mu_{w,X}e^{-ib{\cdot}q}\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\times\sum_{a_{1}=0}^{n-1-X}\,(w^{\alpha_{i_{1}}}\,\cdots\,w^{\alpha_{i+1}})\,\mathcal{A}^{(a_{1})}(p_{1},p_{2}{\rightarrow}p_{1}{-}w,p_{2}{+}(w+r_{X}),r_{X})\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\,\times\mathcal{A}^{(n-a_{1}-X-1)\star}(p_{1}{-}w,p_{2}{+}(w+r_{X}),r_{X}{\rightarrow}p_{1}-q_{1},p_{2}-q_{2})\,.\end{split}$
(53)

Once again, the scaling arguments can be used to immediately verify that the
cut-box contribution to the moments at $g^{2n+3}$ order in the perturbative
expansion can be written as,

$\displaystyle\begin{split}g^{2(n+1)}\,g\,Q_{1}\,\hbar^{\frac{3}{2}}\,&V^{\mu}_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\,\sum_{X=0}^{n-1}\int\prod_{m=0}^{X}d\Phi(r_{m})\hat{d}\mu_{q}\,\hat{d}\mu_{w,X}e^{-ib{\cdot}q}\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\times\sum_{a_{1}=0}^{n-1-X}\,(w^{\alpha_{i_{1}}}\,\cdots\,w^{\alpha_{i+1}})\,\bar{\mathcal{A}}^{(a_{1})}(p_{1},p_{2}{\rightarrow}p_{1}{-}w,p_{2}{+}(w+r_{X}),r_{X})\\\
&\hskip
56.9055pt\,\,\,\,\,\,\,\,\,\,\,\,\,\times\bar{\mathcal{A}}^{(n-a_{1}-X-1)\star}(p_{1}{-}w,p_{2}{+}(w+r_{X}),r_{X}{\rightarrow}p_{1}-q_{1},p_{2}-q_{2})\\\
&=\,\sum_{\beta=0}^{n-i}\,\frac{1}{\hbar^{\beta}}\,{\cal
S}^{\prime\,\mu}_{i\,\beta}\end{split}$ (54)

Here the sum over $X$ is constrained by the order (in the coupling) at which
we are evaluating the cut-box contribution.

We finally see that for each $i$,

$\displaystyle V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i+1}}\,{\cal
T}^{(n)\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\,=\,\sum_{\beta=0}^{n-i}\,\frac{1}{\hbar^{\beta}}\,({\cal
S}_{i\,\beta}+{\cal S}_{i\,\beta}^{\prime})^{\mu}\,+\,O(\hbar)\,.$ (55)

Thus, at a given order in the perturbative expansion
$V^{\mu}_{\alpha_{1}\,\cdots\,\alpha_{i+1}}\,{\cal
T}^{(n)\,\alpha_{1}\,\cdots\,\alpha_{i+1}}$ has a hierarchy of super-classical
terms which scales as
$\frac{1}{\hbar^{\beta}}\,|\,\beta\in\,\\{1,\,\cdots\,,n-i\\}$. As the
classical limit in KMOC formalism must be smooth, we thus conclude that to
$n$-th order in the loop expansion and for each $i$, one has a tower of
constraints which state that all the super-classical terms must vanish

$\displaystyle{\cal S}^{\mu}_{i\,\beta}+{\cal
S}^{\prime}_{i\,\beta}\,=\,0\,\forall\,\beta\,\in\,\\{1,\,\cdots,\,n-i\\}\,,\,\,\,\,{\color[rgb]{0,0,0}n>0\,|\,1\leq
i+1\leq n}.$ (56)

This is precisely the first identity (15), written in a slightly different
notation.

We now analyse the classical $\beta=0$ contribution explicitly. We can
schematically write it in a form which makes the $\hbar$ scaling of various
terms manifest. This can be done by isolating all the terms which do not have
an $\hbar$ expansion. In particular: (1) we separate the measure factor
$\hat{d}\mu_{q}\,=\,\hat{d}^{4}q\,\hat{\delta}(q)$, and (2) we isolate all the
measure factors over loop momenta and the $(n+1)$ massless propagators. As can
be checked, this implies that in the classical term,
$\hat{\delta}(q)\,\mathcal{I}^{(n)}_{4}$ should scale as
$\frac{1}{\hbar^{n+i-1}}$.

Let us illustrate this with ${\cal S}^{\mu}_{i\,\beta=0}\,=:\,{\cal
S}^{\mu}_{i}$.

$\displaystyle\begin{array}[]{lll}{\cal S}^{\mu}_{i}\,=\\\
g^{2(n+1)}\,g\,Q_{1}\,\hbar^{\frac{3}{2}}\,V^{\mu}_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\,\int\,\hat{d}^{4}q\,e^{-ib\cdot
q}\,(\,q^{\alpha_{1}}\,\cdots\,q^{\alpha_{i+1}}\,)&\\\
&\int\,\prod_{j=1}^{n}\hat{d}^{4}l_{j}\,\frac{1}{\prod_{m=1}^{n}l_{m}^{2}\,(\sum
l_{m}-q)^{2}}\,[\,\hat{\delta}(q)\,{\cal
I}^{(n)}_{4}(p_{1},\,p_{2}\,\rightarrow\,p_{1}-q,\,p_{2}+q)\,]_{\frac{1}{\hbar^{n+2+i-1}}}\,.\end{array}$
(60)

One can write such a formal expression for ${\cal S}^{\mu\prime}_{i}$
analogously.

The classical soft theorem is then a statement that $\forall\,n$ and
$\forall\,1\,\leq\,i\,\leq\,n+1$,

$\displaystyle{\cal S}^{(n)\,\mu}_{i}+{\cal
S}^{\prime(n)\,\mu}_{i}\,=\,g^{2(n+1)}\,V^{\mu}_{s\,\alpha_{1}\,{\cdots}\,\alpha_{i+1}}\hskip
28.90755pt\smashoperator[]{\sum_{\begin{subarray}{c}L_{1},\,\cdots,\,L_{i+1}=0\\\
(L_{1}+L_{i+1})+(i+1)\,=n\end{subarray}}^{n+1-i}}\,(\,(\triangle
p^{(L_{1})})^{\alpha_{1}}\,{\cdots}\,(\triangle
p^{(L_{i+1})})^{\alpha_{i+1}}\,)\,,$ (61)

which in turn recovers identity (16).

In section 5.1 and section 5.2, we verify identities (15) and (16) up to
subleading order in the perturbative expansion, i.e. $n=0$ and $n=1$.

####  4.1.1 Monomials of linear impulses

In the previous section we expressed the soft radiation kernel as sum over
certain classical moments. Classical soft theorem implies that (expectation
value) of each such moments is sum over products of linear impulses. We can
thus ask if ${\cal S}^{\mu}_{i}+{\cal S}^{\prime\mu}_{i}$ is an expectation
value of certain observable. It is easy to see that the answer is indeed
affirmative. The tensor $V^{\mu}_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}$ can be
thought of as a map from symmetric rank $i+1$ tensor to a vector. It has a
kernel spanned by $p_{1}^{\alpha_{1}}\,\cdots\,p_{1}^{\alpha_{i+1}}$. We can
hence consider following quantum operators. Let

$\displaystyle\Pi^{\mu}_{\alpha}\,=\,\delta^{\mu}_{\alpha}+\frac{1}{m_{1}^{2}}\,p_{1}^{\mu}\,p_{1\alpha}\,.$
(62)

Now consider a quantum operator,

$\displaystyle{\cal
P}_{1}^{\mu}\,=\,\Pi^{\mu}_{1\,\alpha}\,\hat{P}_{1}^{\alpha}\,,$ (63)

with $\hat{P}_{1}$ the momentum operator for particle 1. The identities (given
in eqn.(61) implied by consistency with classical soft theorem is then a
statement that

$\displaystyle\begin{array}[]{lll}\lim_{\hbar\rightarrow\,0}\,V^{\mu}_{1\,\alpha_{1}\,\cdots\,\alpha_{i+1}}\langle\langle\,{\cal
P}^{\alpha_{1}}\,\cdots\,{\cal P}^{\alpha_{i+1}}\,\rangle\rangle^{(n)}\
=V^{\mu}_{s\,\alpha_{1}\,{\cdots}\,\alpha_{i+1}}\,\smashoperator[]{\sum_{\begin{subarray}{c}L_{1},\,\cdots,\,L_{i+1}=0\\\
(L_{1}+L_{i+1})+(i+1)\,=n\end{subarray}}^{n+1-i}}\,(\,(\triangle
p^{(L_{1})})^{\alpha_{1}}\,{\cdots}\,(\triangle
p^{(L_{i+1})})^{\alpha_{i+1}}\,)\end{array}\,.$ (65)

###  4.2 Towards Sub-leading soft constraints

In this section we sketch the possible constraints that arise at the sub-
leading order in the soft expansion due to the classical log soft photon
theorem. That is, we use the quantum sub-leading soft theorem to evaluate the
5 (or higher) point amplitude in the radiation kernel and then take the
classical limit. We then equate the sub-leading (terms which scale as
$\ln\omega$) soft radiation kernel with classical log soft factor order by
order in perturbation theory and as we argue, this generates another hierarchy
of constraints on certain moments involving exchange momenta and angular
momentum operators.

A detailed proof of these constraints is outside the scope of this work and
will be pursued elsewhere. For now, we simply sketch the structural forms of
these constraints.

In four dimensions, the soft expansion and loop integration do not commute
beyond leading order due to infra-red divergences Sahoo:2018lxl . Although in
the classical limit, the loop integral is expected to only produce IR-finite
quantities, we use the soft expansion of the loop integrand to compute the
radiation kernel. At the level of loop integrand, the soft expansion is simply
the tree-level expansion and hence we can use the sub-leading soft photon
factor to proceed.

More in detail, we write the $5$ point amplitude under consideration as,

$\displaystyle\begin{array}[]{lll}{\cal
A}_{5}^{(n)\,\mu}(p_{1},p_{2}\,{\rightarrow}\,p_{1}{-}q_{1},p_{2}{-}q_{2},k)\,\delta^{4}(q_{1}{+}q_{2}{-}k)\,=\\\
\hskip
56.9055pt\int_{l_{1},{\cdots},l_{n}}\,\sum_{i=1}^{2}Q_{i}\Big{[}\,\frac{J_{i}^{\mu\nu}k_{\nu}}{p_{i}\cdot
k}\,{+}\,\frac{\tilde{J}_{i}^{\mu\nu}k_{\nu}}{\tilde{p}_{i}\cdot
k}\,\Big{]}\delta^{4}(q_{1}+q_{2})\,I_{4}(p_{1},p_{2},q_{1},q_{2},l_{1},{\cdots},l_{n})\,,\end{array}$
(68)

where $J_{i}^{\mu\nu}=(p_{i}\wedge\partial_{p_{i}})^{\mu\nu}$, is the angular
momentum operator for the incoming particles, and with similar expression for
the outgoing ones $\tilde{p}_{i}=p_{i}-q_{i}$. In addition, $I_{4}$ is the
loop integrand. Here we have to note that $J_{i}^{\mu\nu}$ acts of both, the
momentum conserving delta function, as well as the striped integrand.

Using this expansion, the $\mathcal{R}$-part of the radiated field (44) at
subleading order in the soft expansion becomes

$\begin{array}[]{lll}\mathcal{R}^{(n)\,\mu}(k)=i\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\int\prod_{i=1}^{2}\hat{d}^{4}q_{i}\hat{\delta}(2p_{i}{\cdot}q_{i}-q_{i}^{2})e^{ib_{i}{\cdot}q_{i}}\\\
\hskip
56.9055pt\times\int_{l_{1},{\cdots},l_{n}}\,\sum_{i=1}^{2}Q_{i}\Big{[}\frac{J_{i}^{\mu\nu}k_{\nu}}{p_{i}\cdot
k}+\frac{\tilde{J}_{i}^{\mu\nu}k_{\nu}}{\tilde{p}_{i}\cdot
k}\Big{]}\delta^{4}(q_{1}{+}q_{2})\,I_{4}(p_{1},p_{2},q_{1},q_{2},l_{1},{\cdots},l_{n}\,.\end{array}$
(69)

As before when we expand $\frac{1}{(p_{i}+q_{i})\cdot k}$ in $\hbar$ expansion
by expanding in powers of $\frac{q_{i}\cdot k}{p_{i}\cdot k}$, it can be
verified that the only terms that will survive in the classical limit are,

$\begin{array}[]{lll}\mathcal{R}^{(n)\,\mu}(k)&=i\lim_{\hbar\to
0}\hbar^{\frac{3}{2}}\int\prod_{i=1}^{2}\hat{d}^{4}q_{i}\hat{\delta}(2p_{i}{\cdot}q_{i}-q_{i}^{2})e^{ib_{i}{\cdot}q_{i}}\\\
&\hskip
14.22636pt\times\int_{l_{1},{\cdots},l_{n}}\,\sum_{i=1}^{2}Q_{i}\,\Big{[}\,\frac{(\tilde{J}_{i}{+}J_{i})^{\mu\nu}k_{\nu}}{p_{i}\cdot
k}{+}\sum_{m=1}^{n}\,({-}1)^{m}(\frac{q_{m}{\cdot}k}{p_{m}{\cdot}k})^{m}\,\tilde{J}_{i}^{\mu\nu}k_{\nu}\Big{]}\\\
&\hskip
85.35826pt\times\delta^{4}(q_{1}{+}q_{2})\,I_{4}(p_{1},p_{2},q_{1},q_{2},l_{1},{\cdots},l_{n})\,.\end{array}$
(70)

One can similarly apply the (tree-level) sub-leading soft photon theorem in
${\cal C}^{(n)\mu}(k)$.

Consistency with classical soft theorem then implies that in the sub-leading
soft radiation kernel: (1) all the super-classical terms vanish, and (2) there
must exist integration regions in all of which the exchange momenta lie in
$b^{-1}\,>>\,|q\,|>>\,\omega$ that generate various terms in the radiative
field proportional to $\ln\omega$. A detailed analysis of the loop integration
region is under investigation and will be pursued elsewhere.999e.g. We suspect
that at n-th order in the perturbation theory, the contribution proportional
to $(\triangle J^{\ln})^{(n)}$ may arise from
$\omega\,<<\,|l_{i}|\,<<\,|q|\,<<\,b^{-1}$. However this remains to be shown.
As we argued previously, the log-soft radiative field can be perturbatively
expanded as,

$\displaystyle\begin{array}[]{lll}A_{\mu}^{\ln(n)}\,=\sum_{m}\,Q_{m}\,\Big{[}\,\frac{1}{p_{m}\cdot
k}(\triangle J^{\ln}_{m})^{(n)}_{\mu\nu}\,k^{\nu}\,+\\\ \hskip
28.45274pt\sum_{i=1}^{n}\,\,\smashoperator[]{\sum_{\begin{subarray}{c}L_{1},{\cdots},L_{i}\\\
L_{1}{+}{\cdots}L_{i}{+}L_{i+1}{+}i{+}1{=}n\end{subarray}}^{}}\,\,\,\,\frac{(-1)^{i}}{(p_{m}{\cdot}k)^{i+1}}\,k_{\alpha_{1}}\,k_{\alpha_{i}}\,(\,(\triangle
p^{L_{1}}_{m})^{\alpha_{1}}\,\cdots\,(\triangle
p^{L_{i}}_{m})^{\alpha_{i}}\,)\,(\triangle
J^{\ln\,(L_{i+1})}_{m})_{\mu\nu}\,k^{\nu}\,\Big{]}\,,\end{array}$ (73)

where $(\triangle J^{\ln})^{(i)}$ is the logarithmic divergent term in angular
momentum impulse at NiLO order in the coupling.

Equality between the right hand side of eqn.(73) and the $\ln\omega$
contribution from the right hand side of eqn.(70) (plus the contribution from
the $\mathcal{C}$-term), provides the sub-leading soft constraints analogs of
(15-16). At leading order in the coupling (that is $n=0$), these constraints
were verified in Manu:2020zxl . At the NLO order, we expect the sub-leading
soft kernel from KMOC side to match the results in section 2.2. Baut-Lad .

##  5 Leading Soft Constraints Verification up to NLO

Let us in the remaining of this note to provide some specific tests for
identities (15) and (16), at leading ($n=0$) and subleading ($n=1$) orders in
perturbation theory.

###  5.1 Tree-level leading soft moments

At tree-level there is not superclassical term and therefore (15) does not
impose any constrain. On the other hand, the classical moment contributing at
this order is ${\cal T}^{(0)\,\alpha}$ in (11). This in turn will allow us to
recover the radiated field (17) at leading order in perturbation theory. In
other words, in the classical limit we just need to show that $\lim_{\hbar\to
0}{\cal T}^{(0)\,\alpha}=g^{2}\,\Delta p_{1}^{(0),\alpha}$ as required by
(16). We remark that this, and the equalities written bellow, are only valid
on the support of the projector $V^{\mu}_{\alpha_{1},\cdots,\alpha_{i}+1}$, as
indicated in (15-16).

For leading order radiation, as already mentioned, only the first term in (11)
contributes to the computation of the moment, since the remaining term is
subleading in the coupling. Additionally, since we are taking the classical
limit, the following expansion for the momentum measure (12) will be useful
for us.

$\displaystyle\hat{d}\mu_{q}$
$\displaystyle=\hat{d}\mu_{1\,q}+\hat{d}\mu_{2\,q}+\cdots,$ (74)
$\displaystyle\hat{d}\mu_{1\,q}$
$\displaystyle=\hat{d}^{4}q\,\hat{\delta}(2p_{1}{\cdot}q)\hat{\delta}(2p_{2}{\cdot}q),$
(75) $\displaystyle\hat{d}\mu_{2\,q}$
$\displaystyle=-\,\hat{d}^{4}q\,q^{2}\left[\hat{\delta}^{\prime}(2p_{1}{\cdot}q)\hat{\delta}(2p_{2}{\cdot}q)-\hat{\delta}(2p_{1}{\cdot}q)\hat{\delta}^{\prime}(2p_{2}{\cdot}q)\right].$
(76)

We will also need the classical piece of the tree-level 4 point amplitude,
which is given by the one photon exchange diagram

$\mathcal{A}^{(0)}(p_{1},p_{2}\to
p_{1}-q,p_{1}+q)=4g^{2}Q_{1}Q_{2}\frac{\,p_{1}{\cdot}p_{2}}{q^{2}+i\epsilon}\,.$
(77)

With all these ingredients at hand, the only non-vanishing contribution to the
moment ${\cal T}^{(0)\,\alpha}$ in the classical limit, can be obtained by
replacing (75) and (77) into (4), after which it follows

$\begin{split}\lim_{\hbar\,\rightarrow\,0}\,V^{\mu}_{\alpha}{\cal
T}^{(0)\,\alpha}&=g^{2}V^{\mu}_{\alpha}\int\hat{d}^{4}q\hat{\delta}(p_{1}{\cdot}q)\hat{\delta}(p_{2}{\cdot}q)\frac{iQ_{1}Q_{2}\,p_{1}{\cdot}p_{2}q^{\alpha}}{q^{2}+i\epsilon}e^{-iq{\cdot}b}\,,\\\
&=g^{2}V^{\mu}_{\alpha}\Delta p_{1}^{(0)\,\alpha}\,,\end{split}$ (78)

which indeed satisfies the identity (16) for $n=0$. In the second line we have
identified the integral representation for the leading order impulse

$\Delta
p^{(0)\,\mu}_{1}=\int\hat{d}^{4}q\hat{\delta}(p_{1}{\cdot}q)\hat{\delta}(p_{2}{\cdot}q)\frac{iQ_{1}Q_{2}\,p_{1}{\cdot}p_{2}q^{\mu}}{q^{2}+i\epsilon}e^{-iq{\cdot}b}.$
(79)

To finish this example, let us explicitly evaluate this integral, even thought
it has been evaluated in several previous works (see for instance
Kosower:2018adc ; Guevara:2018wpp ). We aim to introduce some conventions that
will be used in the remaining of the paper.

We start by noticing that since there are two delta functions that allow us to
evaluate the integrals in the time and longitudinal directions, the
$i\epsilon$ prescription for the propagator is irrelevant, which in turn
implies that the result for the impulse will be the same irrespective of
whether we used the Feynman or the Retarded propagator 101010However, this
will not be the case for all of the integrals that we will find in this paper
as we will see bellow. . Moving forward in the computation, we can now
decompose the momentum $q$ in terms of the massive momenta $p_{i}$, and the
transverse momentum $q_{\perp}$, as follows

$q^{\mu}=\alpha_{2}p_{1}^{\mu}+\alpha_{1}p_{2}^{\mu}+q_{\perp}^{\mu},\quad
p_{i}{\cdot}q_{\perp}=0,$ (80)

where

$\alpha_{1}=\frac{1}{\mathcal{D}}\left[p_{1}{\cdot}p_{2}x_{1}{-}m_{1}^{2}x_{2}\right],\quad\alpha_{2}=\frac{1}{\mathcal{D}}\left[p_{1}{\cdot}p_{2}x_{2}{-}m_{2}^{2}x_{1}\right].$
(81)

Here we have introduced the dimension-full quantities $x_{i}=p_{i}{\cdot}q$,
and the Jacobian factor $\mathcal{D}$, given by (19).

Notice that the decomposition (80) is generic and does not assume any
conditions on the $x_{i}$ variables. With this change of variables, the
integral measure in (79) becomes
$\hat{d}^{4}q=\frac{1}{\sqrt{\mathcal{D}}}\hat{d}^{2}q_{\perp}\hat{d}x_{1}\hat{d}x_{2}$.

In general we will have to evaluate integrals of the form

$\mathcal{I}=\frac{1}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}q_{\perp}\hat{d}x_{1}\hat{d}x_{2}\hat{\delta}^{(n)}(x_{1})\hat{\delta}^{(m)}(x_{2})f(x_{1},x_{2},q_{\perp},\sigma),$
(82)

namely, with a certain number of derivatives acting over the on-shell delta
functions. We can use integration by part multiple times in order to remove
the derivatives acting over the delta functions, transporting them to act over
the integrand function $f(x_{1},x_{2},q_{\perp},\sigma)$ 111111 Here we have
use $\sigma$ to represent additional momenta, masses and impact parameter
labels.; once we have the on-shell delta functions free of derivatives, we can
use them to evaluate the $x_{i}$-integrals. At that point, the calculation
would have been reduced to evaluate the lower-dimensional integrals of the
form

$\mathcal{I}=(-1)^{m+n}\frac{1}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}q_{\perp}\frac{\partial^{n}}{\partial
x_{1}^{n}}\frac{\partial^{m}}{\partial
x_{2}^{m}}f(x_{1},x_{2},q_{\perp},\sigma)\Bigg{|}_{x_{1}=x_{2}=0}.$ (83)

Let us go back to the computation of the leading order impulse integral (79).
For this case, $m=n=0$, and therefore the evaluation of the integrals in the
time and longitudinal directions simply reduces to set
$\alpha_{1}=\alpha_{2}=0$. We are left then with the two-dimensional integral

$\Delta
p^{(0)\,\mu}_{1}=\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}q_{\perp}e^{-iq_{\perp}{\cdot}b}\frac{iq_{\perp}^{\mu}}{q_{\perp}^{2}}\,,$
(84)

which can be evaluated by trading the momentum $q_{\perp}$ in the numerator by
a derivative w.r.t the impact parameter. Afterwards, the integral can be
evaluated in polar coordinates

$\displaystyle\Delta p^{(0)\,\mu}_{1}$
$\displaystyle=\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{\sqrt{\mathcal{D}}}\frac{1}{2\pi}\partial_{b^{\mu}}\lim_{\mu\to
0}\int_{\mu}^{\infty}\frac{dq_{\perp}}{q_{\perp}}\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{iq_{\perp}b_{\perp}\cos\theta}\,,$
(85)
$\displaystyle=\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{\sqrt{\mathcal{D}}}\frac{1}{2\pi}\partial_{b^{\mu}}\lim_{\mu\to
0}\int_{\mu}^{\infty}dq_{\perp}\frac{\mathcal{J}_{0}(q_{\perp}b_{\perp})}{q_{\perp}}\,,$
(86)
$\displaystyle=-\frac{1}{4\pi}\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{\sqrt{\mathcal{D}}}\lim_{\mu\to
0}\partial_{b^{\mu}}\ln\left(-b^{2}\mu^{2}\right)\,.$ (87)

In the second line $\mathcal{J}_{0}(x)$ is the order zero Bessel functions of
the first kind. Taking the derivative and trivially evaluating the $\mu\to 0$
limit, leads to the well know result for the leading order electromagnetic
impulse

$\Delta
p^{(0)\,\mu}_{1}=-\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{2\pi\sqrt{\mathcal{D}}}\frac{b^{\mu}}{b^{2}},\,\,\,\,\,b^{2}=-\vec{b}^{2},$
(88)

where $\vec{b}$ is the two dimensional impact parameter. This concludes the
computation for the radiated photon field at leading order in both, the soft,
and the perturbative expansions. Let us now do the analogous computation at
next to leading order in perturbation theory.

###  5.2 One-loop leading soft moments

At next to leading order in the perturbative expansion the contributing
moments are ${\cal T}^{(1)\,\alpha}$ and ${\cal T}^{(1)\,\alpha\,\beta}$. In
this section we want to show that $\lim_{\hbar\to 0}{\cal
T}^{(1)\,\alpha}=g^{4}\Delta p^{(1)\,\alpha}_{1}$, recovering the next to
leading order impulse, whereas $\lim_{\hbar\to 0}{\cal
T}^{(1)\,\alpha\,\beta}=g^{4}\Delta p^{(0)\,\alpha}_{1}\Delta
p^{(0)\,\beta}_{1}$, as suggested by the second identity (16). Combination of
these two results allow us to recover the one loop contribution to the the
radiated field given explicitly in (20).

At next to leading order, the radiated field scales as $g^{5}$ and therefore
the moments receive contributions from both the $\mathcal{R}$ and the
$\mathcal{C}$ terms, given by the first and second line of (11), respectively.
However, at this order no extra photons propagate through the cut and we can
simply set $X=0$ in (11), which also implies that
$\hat{d}\mu_{w,X}=\hat{d}\mu_{w}$ in (14). In addition, we will show that
superclassical terms give vanishing contribution as suggested by the first
identity (15). Indeed, this corresponds to a cancellation between the
$\mathcal{R}$ and the $\mathcal{C}$ contributions to the aforementioned
moments, which is analogous to that for the computation of the at 1 loop
impulse Kosower:2018adc . Since only the moment ${\cal T}^{(1)\,\alpha}$ will
have potential superclassical contributions, coming from the superclassical
piece of the 4 point amplitude at 1 loop Kosower:2018adc , we only have to
show that for $m=1$, $\lim_{\hbar\to 0}\hbar^{m}\,{\cal T}^{(1)\,\alpha}=0$,
as for higher values of $m$, identity 1 is trivially satisfied.

Let us split the computation as follows: For the potentially superclassical
contributions we will compute

$\lim_{\hbar\to 0}\hbar\,V_{\alpha}^{\mu}{\cal T}^{(1)\,\alpha}=\lim_{\hbar\to
0}\,V_{\alpha}^{\mu}\Big{[}{\cal T}^{(1)\,\alpha}_{\mathcal{R}_{0}}+{\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{0}}\Big{]}\,,$ (89)

where

$\displaystyle{\cal T}^{(1)\,\alpha}_{\mathcal{R}_{0}}$ $\displaystyle=$
$\displaystyle
i\hbar^{5/2}\int\hat{d}\mu_{1\,q}\,e^{-ib{\cdot}q}\,q^{\alpha}\,\mathcal{A}^{(1)}_{\rm{sc}}(q)\,,$
(90) $\displaystyle{\cal T}^{(1)\,\alpha}_{\mathcal{C}_{0}}$ $\displaystyle=$
$\displaystyle\hbar^{5/2}\int\hat{d}\mu_{1\,q}\,\hat{d}\mu_{1\,w}\,e^{-ib{\cdot}q}\,w^{\alpha}\mathcal{A}^{(0)\,\star}(w-q)\mathcal{A}^{(0)}(w)\,.$
(91)

Here $\mathcal{A}^{(1)}_{\rm{sc}}(q)$ is the superclassical piece of the
1-loop, 4 point amplitude, which we will write explicitly bellow. The tree
level amplitudes in the second line are given by (77), where we have removed
the massive momenta labels to alleviate notation.

Next, we will have to compute the classical contributions, from the one and
two index moment. For the former we have

$\lim_{\hbar\to 0}\,V_{\alpha}^{\mu}{\cal T}^{(1)\,\alpha}=\lim_{\hbar\to
0}\,V_{\alpha}^{\mu}\Big{[}{\cal T}^{(1)\,\alpha}_{\mathcal{R}_{1}}+{\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{2}}+{\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{1}}+{\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{2}}\Big{]}\,,$ (92)

with each term computed as follows

$\displaystyle{\cal T}^{(1)\,\alpha}_{\mathcal{R}_{1}}$ $\displaystyle=$
$\displaystyle
i\hbar^{3/2}\int\hat{d}\mu_{1\,q}\,e^{-ib{\cdot}q}\,q^{\alpha}\,\mathcal{A}^{(1)}_{\rm{c}}(q)\,,$
(93) $\displaystyle{\cal T}^{(1)\,\alpha}_{\mathcal{R}_{2}}$ $\displaystyle=$
$\displaystyle
i\hbar^{3/2}\,\int\hat{d}\mu_{2\,q}\,e^{-ib{\cdot}q}\,q^{\alpha}\mathcal{A}^{(1)}_{\rm{sc}}(q)\,,$
(94) $\displaystyle{\cal T}^{(1)\,\alpha}_{\mathcal{C}_{1}}$ $\displaystyle=$
$\displaystyle\hbar^{3/2}\int\hat{d}\mu_{1\,q}\,\hat{d}\mu_{2\,w}\,e^{-ib{\cdot}q}\,w^{\alpha}\mathcal{A}^{(0)\,\star}(w-q)\mathcal{A}^{(0)}(w)\,,$
(95) $\displaystyle{\cal T}^{(1)\,\alpha}_{\mathcal{C}_{2}}$ $\displaystyle=$
$\displaystyle\hbar^{3/2}\int\hat{d}\mu_{2\,q}\,\hat{d}\mu_{1\,w}\,e^{-ib{\cdot}q}\,w^{\alpha}\mathcal{A}^{(0)\,\star}(w-q)\mathcal{A}^{(0)}(w)\,.$
(96)

In the first line, $\mathcal{A}^{(1)}_{\rm{c}}(q)$ is the classical part of
the 1-loop 4 point amplitude, which we will write explicitly in a moment.

Finally, the classical contribution from the two-index moment will be computed
from

$\lim_{\hbar\to 0}\,V_{\alpha\,\beta}^{\mu}{\cal
T}^{(1)\,\alpha\,\beta}=\lim_{\hbar\to 0}V_{\alpha\,\beta}^{\mu}\,\Big{[}{\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{R}_{3}}+{\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}\Big{]}\,,$ (97)

with the respective terms evaluated via

$\displaystyle{\cal T}^{(1)\,\alpha\,\beta}_{\mathcal{R}_{3}}$
$\displaystyle=$ $\displaystyle
i\hbar^{3/2}\int\hat{d}\mu_{1\,q}\,e^{-ib{\cdot}q}\,q^{\alpha}\,q^{\beta}\,\mathcal{A}^{(1)}_{\rm{sc}}(q)\,,$
(98) $\displaystyle{\cal T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}$
$\displaystyle=$
$\displaystyle\hbar^{3/2}\int\hat{d}\mu_{1\,q}\,\hat{d}\mu_{2\,w}\,e^{-ib{\cdot}q}\,w^{\alpha}\,w^{\beta}\mathcal{A}^{(0)\,\star}(w-q)\mathcal{A}^{(0)}(w)\,,$
(99)

By explicit evaluation, we will show that the actual terms contributing to the
radiated photon field are (93), (94) and (98) – as suggestively written in
(52) – with the first two giving the next to leading order impulse, and the
last one giving the square of the leading order impulse. As for the remaining
contributions we show that they canceling among themselves. In what follows we
will adventure in this computation.

####  5.2.1 The superclassical terms

Let us start by computing the superclassical terms (90) and (91). Thee 4 point
amplitude at 1-loop was computed in Kosower:2018adc . The superclassical
contribution $\mathcal{A}^{(1)}_{\rm{sc}}(q)$, arises from the addition of
superclassical parts in the box $B_{-1}$, and cut-box $C_{-1}$ diagrams, given
by eq. $(5.31)$ in Kosower:2018adc ,

$\mathcal{A}^{(1)}_{\rm{sc}}(q)=\left(B_{-1}+C_{-1}\right)_{\hbar^{-1}}=2i\,g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\int\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}l)\frac{1}{l^{2}(l-q)^{2}}\,.$
(100)

Using it into (90), together with the measure (75), (90) becomes

${\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{0}}=-\frac{1}{2}g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\int\hat{d}^{4}l\hat{d}^{4}q\prod_{i}\hat{\delta}(p_{i}{\cdot}l)\hat{\delta}(p_{i}{\cdot}q)\frac{q^{\alpha}}{l^{2}(l-q)^{2}}e^{-ib{\cdot}q}\,,$
(101)

where we see that the explicit dependence in $\hbar$ drops away by using the
KMOC $\hbar$-rescaling mentioned in section 3. We can now do the change of
variables $q=l+\bar{q}$. This in turn factorizes the integrals into two
factors corresponding to a vector, and a scalar integrals; that is

$\begin{split}{\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{0}}=ig^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}S_{\alpha,\omega^{-1}}^{\mu}\left[\int\hat{d}^{4}\bar{q}\prod_{i}\hat{\delta}(p_{i}{\cdot}q)e^{-ib{\cdot}\bar{q}}\frac{i\bar{q}^{\alpha}}{\bar{q}^{2}}\right]\left[\int\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}l)\frac{e^{-ib{\cdot}q}}{l^{2}}\right],\end{split}$
(102)

where the change of variables has produced a factor of $2$ that canceled the
$\frac{1}{2}$ overall factor in (101) 121212Note that formally the change of
variables implies that we should had changed
$\hat{\delta}(p_{i}.q)\rightarrow\hat{\delta}(p_{i}.\bar{q}-p_{i}{\cdot}l),$
however, the delta functions $\hat{\delta}(p_{1}{\cdot}q)$ allow us to set
$p_{i}{\cdot}l\rightarrow 0$.. In the integral on the left, we recognize the
leading order impulse $(\ref{eq:LO_impulse_integral})$, whereas for the
integral on the right, we obtain an IR-divergent expression, which can be
evaluated along similar steps used for the computation of the leading order
impulse (87), obtaining

$I_{1}=\int\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}l)\frac{e^{-ib{\cdot}l}}{l^{2}}=-\frac{1}{4\pi\sqrt{\mathcal{D}}}\ln\left(-\mu^{2}b^{2}\right)\,.$
(103)

Here we have introduced the IR-regulator $\mu$. Then, the first superclassical
contribution becomes

${\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{0}}=-ig^{4}\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{4\pi\sqrt{\mathcal{D}}}\,\Delta
p_{1}^{(0)\,\alpha}\,\ln\left(-\mu^{2}b^{2}\right),$ (104)

Let us now evaluate the cut-box contribution (91). For that we just need the
tree-level 4 point amplitude (77), as well as the measure factors (75); we
arrive at

${\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{0}}=16g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}\,\int\hat{d}^{4}q\hat{d}^{4}l\prod_{i}\hat{\delta}(2p_{i}{\cdot}q)\hat{\delta}(2p_{i}{\cdot}l)e^{-ib{\cdot}q}\frac{l^{\alpha}}{l^{2}(q-l)^{2}}\,.$
(105)

After doing the same change of variables $q=l+\bar{q}$, we can analogously
identify the leading order impulse from the $l$-integral, whereas the
$\bar{q}$-integral will result into the IR-divergent expression (103). We
finally get

${\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{0}}=ig^{4}\frac{Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}}{4\pi\sqrt{\mathcal{D}}}\,\Delta
p_{1}^{(0)\,\alpha}\,\ln\left(-\mu^{2}b^{2}\right),$ (106)

which is equal to (104) but with opposite sign. This explicitly shows that the
r.h.s of (89) evaluates to zero, as demanded from the first identity (15).

####  5.2.2 Classical one-index moment at 1-loop

Let us move to evaluate the classical contribution from the one-index moment
(92). We start from term (93). For that, we need the classical contribution to
4 point amplitude at 1-loop. Likewise for the superclassical term, we obtain
it from the sum $(B_{0}+C_{0})+\left(B_{-1}+C_{-1}\right)+T_{12}+T_{21}$,
where the different components where evaluated in eqs. $(5.21)$ and $\,(5.34)$
in Kosower:2018adc . This gives

$\displaystyle\mathcal{A}_{\textrm{c}}^{(1)}(q)$
$\displaystyle=2g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\int\frac{\hat{d}^{4}l}{l^{2}(l{-}q)^{2}}\Bigg{\\{}l{\cdot}(l{-}q)\left[\frac{\hat{\delta}(p_{2}{\cdot}l)}{(p_{1}{\cdot}l{+}i\epsilon)^{2}}{+}\frac{\hat{\delta}(p_{1}{\cdot}l)}{(p_{2}{\cdot}l{-}i\epsilon)^{2}}\right]$
$\displaystyle+\frac{1}{\left(p_{1}{\cdot}p_{2}\right)^{2}}\left[m_{2}^{2}\hat{\delta}(p_{2}{\cdot}l){+}m_{1}^{2}\hat{\delta}(p_{1}{\cdot}l)\right]\Bigg{\\}}+Z,$
(107)

with

$Z=ig^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\int\frac{\hat{d}^{4}l}{l^{2}(l{-}q)^{2}}(2l{\cdot}q{-}l^{2})\left[\hat{\delta}^{\prime}(p_{1}{\cdot}l)\hat{\delta}(p_{2}{\cdot}q){-}\hat{\delta}(p_{1}{\cdot}l)\hat{\delta}^{\prime}(p_{2}{\cdot}l)\right].$
(108)

Before we proceed with the computation, let us first remember the definition
for the electromagnetic impulse at 1-loop Kosower:2018adc  :

$\Delta
p^{(1)\,\mu}_{1}=\frac{i}{4}\int\hat{d}^{4}q\prod_{i}\hat{\delta}(p_{i}{\cdot}q)e^{-ib{\cdot}q}\left[\mathcal{I}_{1}^{\mu}+\mathcal{I}_{2}^{\mu}+\mathcal{I}_{3}^{\mu}\right],$
(109)

where the $\mathcal{I}_{i}^{\mu}$ integrals resemble the contributions to the
4 point amplitude from the different Feynman diagrams. The firs one comes from
the contribution from the triangle diagrams

$\mathcal{I}_{1}^{\mu}=2\left(Q_{1}Q_{2}\right)^{2}q^{\mu}\sum_{i}\int\hat{d}^{4}l\frac{m_{i}^{2}\hat{\delta}(p_{i}{\cdot}l)}{l^{2}(l-q)^{2}}\,.$
(110)

Next we have the contribution coming from the Boxes, which once by canceling
the term $Z$ in $(\ref{eq:4pt 1L classical explicit})$, using the cut-Box
(here it is the actual 4 point cut-Box diagram) reads

$\mathcal{I}_{2}^{\mu}=2\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}q^{\mu}\sum_{i,j\,i\neq
j}\int\hat{d}^{4}l\frac{l{\cdot}(l-q)}{l^{2}(l-q)^{2}}\frac{\hat{\delta}(p_{j}{\cdot}l)}{(p_{i}{\cdot}l+i\epsilon)^{2}},$
(111)

Finally, we have the 4 point cut-Box contribution

$\mathcal{I}_{3}^{\mu}=-2i\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}\int\hat{d}^{4}l\frac{l{\cdot}(l-q)l^{\mu}}{l^{2}(l-q)^{2}}\left[\hat{\delta}^{\prime}(p_{1}{\cdot}l)\hat{\delta}(p_{2}{\cdot}l)-\hat{\delta}(p_{1}{\cdot}l)\hat{\delta}^{\prime}(p_{2}{\cdot}l)\right].$
(112)

By introducing all these definitions we can check that the computation of
${\cal T}^{\alpha}_{(1),\mathcal{R}_{1}}$ in in (93), toghether with the
measure (75), can be rearrange to give exactly the next to leading order
impulse (109) plus an additional contribution coming from adding and
subtracting the 4-pt cut-Box diagram

${\cal T}^{(1)\,\alpha}_{\mathcal{R}_{1}}=g^{4}\Delta
p^{(1)\,\alpha}_{1}+{\rm[cut{-}box]^{(1)\,\alpha},}$ (113)

with the extra contribution ${\rm[cut{-}box]^{\mu}}$ given by

${\rm[cut{-}box]^{(1)\,\alpha}}=-g^{4}\left[p_{1,\beta}p_{1}^{[\beta}p_{2}^{\alpha]}-p_{2,\beta}p_{2}^{[\beta}p_{1}^{\alpha]}\right]\frac{\left(\Delta
p^{(0)}_{1}\right)^{2}}{\mathcal{D}}\,,$ (114)

The proof of this statement is lengthy and we therefore postpone it to be
discussed in Appendix A.1. For the moment, let us notice that the first term
of eq. (113) gives exactly the expected result from the second identity (16).
Therefore, to conclude the proof we simply need to show that the remaining
terms in (92) together with (114), add up to zero. In fact, also in Appendix
A.1 we will show that

${\cal T}^{(1)\,\alpha}_{\mathcal{R}_{2}}+{\rm[cut{-}box]^{(1)\,\alpha}}=0\,,$
(115)

whereas ${\cal T}^{(1)\,\alpha}_{\mathcal{C}_{1}}$ and ${\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{2}}$ evaluate to zero individually.

####  5.2.3 Classical two-index moment at 1-loop

The remaining task to complete the proof of identity (16) at 1-loop is to
evaluate two-index moment (97). Similar to previous computation, we start from
its first term, given by (98), and after inserting the measure (75), and the
superclassical amplitude (100), we arrive at

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{R}_{3}}=-\frac{1}{2}g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\int\hat{d}^{4}q\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}q)\hat{\delta}(p_{i}{\cdot}l)e^{-ib{\cdot}q}\frac{q^{\alpha}q^{\beta}}{l^{2}(l-q)^{2}}\,.$
(116)

Next we can do our usual change of variables $q=l+\bar{q}$

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{R}_{3}}=-\frac{1}{2}g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\int\hat{d}^{4}\bar{q}\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}\bar{q})\hat{\delta}(p_{i}{\cdot}l)e^{-ib{\cdot}\bar{q}}e^{-ib{\cdot}l}\frac{\left(l^{\alpha}{+}\bar{q}^{\alpha}\right)\left(l^{\beta}{+}\bar{q}^{\beta}\right)}{l^{2}\bar{q}^{2}}\,.$
(117)

We recognize the square of the leading order impulse (79) coming from the
crossed terms. On the other hand, the non-crossed terms give us the product of
two integrals, one is them is the usual IR-divergent integral $I_{1}$ in
$(\ref{eq:scalar integral I1})$, whereas the second one corresponds to the
derivative of the leading order impulse w.r.t. the impact parameter; notice
there is a factor of two for each case, which cancels the overall $1/2$
factor. That is

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{R}_{3}}=g^{4}{\color[rgb]{0,0,0}}\Delta
p^{(0)\,\alpha}_{1}\Delta
p^{(0)\,\beta}_{1}-g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)I_{1}\partial_{b^{\alpha}}\Delta
p^{(0)\,\beta}_{1}\,.$ (118)

The change of the sign for the first term comes from inserting a factor of
$i^{2}$ both, in the numerator and denominator, and absorb it for the former,
to complete the the square of the leading order impulse. Using
$(\ref{eq:scalar integral I1})$ and the derivative of the leading order
impulse

$\partial_{b^{\alpha}}\Delta
p^{(0)\,\beta}_{1}=-\frac{Q_{1}Q_{2}p_{1}{\cdot}p_{2}}{2\pi\sqrt{\mathcal{D}}}\left(b^{2}\eta^{\alpha\beta}-2b^{\alpha}b^{\beta}\right)\frac{1}{b^{4}}\,,$
(119)

and drooping the term proportional to $\eta^{\alpha\beta}$ , using the on-
shell condition for the photon momentum and gauge invariance, we finally
arrive at

${\cal T}^{(1)\,\alpha\,\beta}_{\mathcal{R}_{3}}=g^{4}\Delta
p_{1}^{(0)\,\alpha}\Delta
p_{1}^{(0)\,\beta}+\mathcal{J}_{3}^{(1)\,\alpha\beta}\,,$ (120)

where

$\mathcal{J}_{3}^{(1)\,\alpha\,\beta}=-2g^{4}\,\Delta
p_{1}^{(0)\,\alpha}\Delta p_{1}^{(0)\,\beta}\ln\left(-\mu^{2}b^{2}\right)\,.$
(121)

Similar to the previous subsection, to complete the proof of the second
identity for the two-index moment at 1-loop, we simple need to show that the
second term in (97) added to (121) evaluates to zero

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}+\mathcal{J}_{3}^{(1)\,\alpha\,\beta}=0\,.$
(122)

We leave the proof of this equation for Appendix A.2.

With this we have concluded the proof of identity (16) at next to leading
order in the perturbative expansion.

##  6 Discussion

The classical Soft theorems discovered by Weinberg, Saha, Sahoo and Sen
provide us with universal and exact formulae for radiative electromagnetic and
gravitational fields in a relativistic scattering. These formulae are for
certain coefficients in the soft expansion of the radiative field. They only
depend on the asymptotic kinematics of the incoming and outgoing bodies.
Namely their mass, charge and momenta and are independent of the details of
the scattering.131313Of course given only the incoming data, the outgoing
momenta are determined by the hard scattering, but the essence of classical
soft theorem is in the observation that given the initial kinematics, the
measurement of the momenta and charges of the outgoing particles in the
asymptotic future can be used to determine the soft radiation. In fact, the
theorems do not rely on validity of perturbation theory in the scattering
region and provide us with formulae for memory effect, tail to the memory
effect and so on produced in collisions of astrophysical objects
Strominger:2014pwa ; Laddha:2018vbn ; Sahoo:2021ctw ; Pasterski:2015tva .

In this paper, we have tried to analyse implications of classical soft
theorems for KMOC formalism through which radiative field can be computed
using on-shell techniques. As we have argued, classical soft theorems impose a
tower of an infinite hierarchy of constraints on expectation values of a class
of composite operators in the KMOC formalism. At leading order in the soft
expansion, these operators are constructed from Monomials of Momentum
operators.

At leading order in perturbation theory, these constraints were verified in
Bautista:2019tdr ; Manu:2020zxl  at leading and sub-leading order in the soft
expansion. In this paper, we have also verified them at NLO in the coupling
and at leading order in the soft expansion in scalar QED with no higher
derivative interactions. We note that addition of other interactions will not
change the structure of classical soft factor but will change the analytic
expressions for the out going momenta in terms of incoming kinematics and
impact parameter. Verifying the sub-leading soft constraints at higher orders
in the perturbative expansion requires a deeper investigation into the
integration regions involving the loop momenta. This analysis is under
progress and will be reported elsewhere.

In this paper we have solely focused on soft electro-magnetic radiation. We
believe that the leading soft constraints can be generalised to gravitational
interactions directly at NLO. Beyond NLO order, classical soft graviton factor
will receive contribution from finite energy gravitational flux. On the other
hand if we take classical limit after applying Weinberg soft theorem inside
the radiation kernel, the result will be once again turn out to be in terms of
monomials of linear impulses. We believe that this result once again should be
equated to the contribution to the classical soft graviton factor only from
outgoing massive particles. However this remains to be shown. As KMOC
naturally takes into account the dissipative effects in computation of linear
impulse, we expect this procedure to be consistent. 141414We thank Ashoke Sen
for discussion on this issue. 151515The generalisation of the sub-leading soft
constraints may be even more subtle as the classical log soft factor in
gravity has an additional contribution effect of space-time curvature on soft
radiation. These terms may not simply arise from sub-leading soft graviton
theorem for the integrands Sahoo:2018lxl .

It will be interesting to prove the leading and sub-leading soft constraints
within KMOC formalism for perturbative scattering with large impact parameter.
Universality of classical soft theorems imply that the proof is likely to
involve ideas along the lines of the classical proof in Saha:2019tub , in
which it was only assumed that the interactions outside a “hard scattering
region" (which can be parametrized as a space-time region bounded in spatial
and temporal directions by $\pm\,t_{0}$ for some sufficiently large $t_{0}$)
are simply the Coulombic interactions. However formulating the quantum
dynamics in this fashion may require use of the time-ordered perturbation
theory Sterman1993  which has in fact also been adopted to hard-soft
factorisation in the seminal paper by Schwartz and Hannesdottir
Hannesdottir:2019opa .

###### Acknowledgements.

We are grateful to Alfredo Guevara, Athira P V, Akavoor Manu, Partha Paul,
Ashoke Sen and Justin Vines for enlightening discussions. YFB acknowledges
financial support by the Natural Sciences and Engineering Research Council of
Canada (NSERC). Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Economic Development, Job Creation and Trade.

##  Appendix A Computational details NLO radiation

In this appendix we walk through the computational details of several
integrals given in section section 5.2

###  A.1 Cancellations in the classical one-index moment at 1-loop

In this section we fill in the computational details for the cancellations
announced by the end of section 5.2.2.

  * •

Proof of eq. (114)

Let us begin by walking through the proof of the stamen of equation (114). We
start from

${\rm[cut{-}box]^{(1)\,\alpha}}=\frac{1}{4}\int\hat{d}^{4}q\prod_{i}\hat{\delta}(p_{i}{\cdot}q)e^{-ib{\cdot}q}\left(\mathcal{I}_{3}^{\alpha}-q^{\alpha}Z=U^{\alpha}\right)\,,$
(123)

where the integrand has the explicit form

$U^{\alpha}=-2g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}\int\frac{\hat{d}^{4}l}{l^{2}(l{-}q)^{2}}\left[\hat{\delta}^{\prime}(p_{1}{\cdot}l)\hat{\delta}(p_{2}{\cdot}l){-}\hat{\delta}(p_{1}{\cdot}l)\hat{\delta}^{\prime}(p_{2}{\cdot}l)\right]\left[l{\cdot}(l-q)l^{\alpha}{+}\frac{q^{\alpha}}{2}(2l{\cdot}q{-}l^{2})\right]\,.$
(124)

To evaluate this integral we can expand the momentum $l$ in an analogous way
to the $q$ momentum in (80 \- 81), with say $\alpha_{i}\to\beta_{i}$, and
$x_{i}\to y_{i}=p_{i}\cdot l$. The resulting integrand takes the form (82),
and therefore we can evaluate the time and longitudinal components using
integrating by parts one time (83). That is, we can write

$U^{\alpha}=\frac{1}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}q_{\perp}\hat{d}y_{1}\hat{d}y_{2}\left[\hat{\delta}^{(1)}(y_{1})\hat{\delta}^{(0)}(y_{2})-\hat{\delta}^{(0)}(y_{1})\hat{\delta}^{(1)}(y_{2})\right]f_{u}^{\alpha}(y_{1},y_{2},l_{\perp},\sigma),$
(125)

with the identification of the integrand function

$\displaystyle
f_{u}^{\alpha}(y_{1},y_{2},l_{\perp},\sigma)=-2g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}\Bigg{[}$
$\displaystyle\frac{(\beta_{2}p_{1}{+}\beta_{1}p_{2}+l_{\perp})^{\alpha}\left((\beta_{2}p_{1}{+}\beta_{1}p_{2})^{2}{+}l_{\perp}^{2}{-}l_{\perp}{\cdot}q_{\perp}\right)}{\left((\beta_{2}p_{1}{+}\beta_{1}p_{2})^{2}{+}l_{\perp}^{2}\right)\left((\beta_{2}p_{1}{+}\beta_{1}p_{2})^{2}{+}(l_{\perp}{-}q_{\perp})^{2}\right)}$
$\displaystyle{-}\frac{\left((\beta_{2}p_{1}{+}\beta_{1}p_{2})^{2}{+}l_{\perp}^{2}-2l_{\perp}{\cdot}q_{\perp}\right)\frac{q_{\perp}^{\alpha}}{2}}{\left((\beta_{2}p_{1}{+}\beta_{1}p_{2})^{2}{+}l_{\perp}^{2}\right)\left((\beta_{2}p_{1}{+}\beta_{1}p_{2})^{2}{+}(l_{\perp}{-}q_{\perp})^{2}\right)}\Bigg{]}\,.$
(126)

where in addition to the $l$-expansion, we have used the expansion for the
$q$-momentum (80), and set $x_{i}\to 0$ using the support of the delta
function $\hat{\delta}(p_{i}{\cdot}q)$ in (123). Next, to use (83) after
integration by parts we need to evaluate the derivatives of the form

$\frac{\partial}{\partial
y_{i}}f_{u,ij}^{\alpha}\Bigg{|}_{y_{i}=y_{j}=0}=\frac{4g^{4}}{\mathcal{D}}\left(Q_{i}Q_{j}p_{i}{\cdot}p_{j}\right)^{2}p_{j,\beta}p_{j}^{[\beta}p_{i}^{\alpha]}\frac{l_{\perp}{\cdot}(l_{\perp}-q_{\perp})}{l_{\perp}^{2}(l_{\perp}-q_{\perp})^{2}}\,,$
(127)

where one can check that only the first line of (126) contributed to (127).
With all the tools at hand, it is then direct to show that the integral (123)
simplifies to

${\rm[cut{-}box]^{(1)\,\alpha}}=g^{4}\frac{\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}}{\mathcal{D}^{2}}\left[p_{2,\beta}p_{2}^{[\beta}p_{1}^{\alpha]}{-}p_{1,\beta}p_{1}^{[\beta}p_{2}^{\alpha]}\right]\int\hat{d}^{2}q_{\perp}\hat{d}^{2}l_{\perp}e^{-ib{\cdot}q_{\perp}}\frac{l_{\perp}{\cdot}(l_{\perp}{-}q_{\perp})}{l_{\perp}^{2}(l_{\perp}{-}q_{\perp})^{2}}.$
(128)

Next we do the usual change of variables
$q_{\perp}=\bar{q}_{\perp}+l_{\perp},$ so that

${\rm[cut{-}box]^{(1)\,\alpha}}=\frac{g^{4}}{\mathcal{D}}\left[p_{2,\beta}p_{2}^{[\beta}p_{1}^{\alpha]}{-}p_{1,\beta}p_{1}^{[\beta}p_{2}^{\alpha]}\right]\left[\frac{Q_{1}Q_{2}p_{1}{\cdot}p_{2}}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}q_{\perp}\hat{d}^{2}l_{\perp}e^{-ib{\cdot}q_{\perp}}i\frac{\bar{q}_{\perp}}{\bar{q}_{\perp}^{2}}\right]^{2}.$
(129)

in the big bracket we recognize the Leading order impulse
$(\ref{eq:LO_impulse_interm})$, which in turn allow us to recover the
announced result $(\ref{eq:cut-box final}).$

${\rm[cut{-}box]^{(1)\,\alpha}}=-g^{4}\left[p_{1,\beta}p_{1}^{[\beta}p_{2}^{\alpha]}-p_{2,\beta}p_{2}^{[\beta}p_{1}^{\alpha]}\right]\frac{\left(\Delta
p^{(0)}_{1}\right)^{2}}{\mathcal{D}}\,,$ (130)

  * •

Proof of eq. (115)

Next we move to prove the cancellation (115). For that we still need to
compute ${\cal T}^{\alpha}_{(1),\mathcal{R}_{2}}$ starting from (94), and
using (76) for the integral measure, and (77) for the 4 point amplitude. We
get

$\begin{split}{\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{2}}=\frac{1}{4}g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}S_{1,\beta}^{\mu}&\int\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}l)\hat{d}^{4}qe^{-ib{\cdot}q}\,\\\
&\times\left[\hat{\delta}^{\prime}(p_{1}{\cdot}q)\hat{\delta}(p_{2}{\cdot}q){-}\hat{\delta}(p_{1}{\cdot}q)\hat{\delta}^{\prime}(p_{2}{\cdot}q)\right]\frac{q^{2}q^{\alpha}}{l^{2}(l-q)^{2}}\,,\end{split}$
(131)

where we have used
$\delta^{\prime}(2p_{i}{\cdot}q)=\frac{1}{4}\delta^{\prime}(p_{i}{\cdot}q)$.
To proceed in the calculation, we follow the philosophy of the previous
subsection for the computation of integrals involving derivatives of the Dirac
delta function, i.e. using integration by parts. Doing the change of variables
(80) (and the analogous change for $q\to l$ and $x_{i}\to y_{i}$ ), and
evaluating the integrals in $(p_{i}{\cdot}l)=y_{i}$, using the corresponding
delta function, we arrive at

${\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{2}}=\frac{1}{\mathcal{D}}\int\hat{d}^{2}q_{\perp}\frac{\hat{d}^{2}l_{\perp}}{l_{\perp}^{2}}\prod_{i}\hat{d}x_{i}\left[\hat{\delta}^{\prime}(x_{i})\hat{\delta}(x_{2}){-}\hat{\delta}(x_{1})\hat{\delta}^{\prime}(x_{2})\right]Y^{\alpha}\,,$
(132)

where we have defined

$Y^{\alpha}=\frac{1}{4}g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}e^{-iq_{\perp}{\cdot}b}\frac{((\alpha_{2}p_{1}{+}\alpha_{1}p_{2})^{2}+q_{\perp}^{2})(\alpha_{2}p_{1}{+}\alpha_{1}p_{2}{+}q_{\perp})^{\alpha}}{(\alpha_{2}p_{1}{+}\alpha_{1}p_{2})^{2}{+}(l_{\perp}-q_{\perp})^{2}}\,,$
(133)

recalling that $\alpha_{i}$ are function of $x_{i}$. We then get an integral
of the form (82). Using

$\frac{\partial}{\partial
x_{i}}Y_{ij}^{\alpha}\Bigg{|}_{x_{1}=x_{2}=0}=-\frac{1}{2}g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}\frac{q_{\perp}^{2}}{\mathcal{D}(l_{\perp}-q_{\perp})^{2}}p_{j,\nu}p_{j}^{[\nu}p_{i}^{\alpha]}e^{-iq_{\perp}{\cdot}b},$
(134)

for doing the integration by parts procedure, we can write (132) as follows

${\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{2}}={-}\frac{1}{2}\frac{g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}}{\mathcal{D}^{2}}\left(p_{1,\beta}p_{1}^{[\beta}p_{2}^{\alpha]}{-}p_{2,\beta}p_{2}^{[\beta}p_{1}^{\alpha]}\right)\left[\int\hat{d}^{2}q_{\perp}\hat{d}^{2}l_{\perp}\frac{q_{\perp}^{2}e^{-iq_{\perp}{\cdot}b}}{l_{\perp}^{2}(l_{\perp}-q_{\perp})^{2}}{=}\mathcal{J}_{2}\right]\,.$
(135)

Let us now evaluate the integral $\mathcal{J}_{2}$ in the square brackets.
Doing the usual shift $q_{\perp}=\bar{q}_{\perp}+l_{\perp},$ so that

$\mathcal{J}_{2}=\int\hat{d}^{2}\bar{q}_{\perp}\hat{d}^{2}l_{\perp}\frac{(\bar{q}_{\perp}^{2}+l_{\perp}^{2}+2l_{\perp}{\cdot}\bar{q}_{\perp})e^{-i\bar{q}_{\perp}{\cdot}b}e^{-il_{\perp}{\cdot}b}}{l_{\perp}^{2}\bar{q}_{\perp}^{2}}\,.$
(136)

From the crossed terms we identify the integral representation for the leading
order impulse, whereas the remaining terms are contact integrals which we drop
assuming $b\neq 0$. We finally get

$\mathcal{J}_{2}=-2\frac{\left(\Delta
p^{(0)}\right)^{2}\mathcal{D}}{g^{4}\left(Q_{1}Q_{2}\,p_{1}{\cdot}p_{2}\right)^{2}}\,,$
(137)

which can be replace back into (135) to finally give

${\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{2}}=g^{4}\left[p_{1,\beta}p_{1}^{[\beta}p_{2}^{\alpha]}{-}p_{2,\beta}p_{2}^{[\beta}p_{1}^{\alpha]}\right]\frac{\left(\Delta
p^{(0)}\right)^{2}}{\mathcal{D}}\,.$ (138)

We note that this simply gives ${\cal
T}^{(1)\,\alpha}_{\mathcal{R}_{2}}=-{\rm[cut{-}box]^{(1)\,\alpha}}$, and
therefore this concludes the proof of (115).

  * •

Vanishing of ${\cal T}^{(1)\,\alpha}_{\mathcal{C}_{1}}$

This is a very simple proof since this term give us a contact integral. Our
starting point is the definition (95), and using (75) and (76) for the
integral measure in $q$ and $l$ respectively, and (77) for the 4 point
amplitude, we get

$\begin{split}{\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{1}}=16g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}&\int\hat{d}^{4}q\hat{d}^{4}le^{-ib{\cdot}q}\prod_{i}\hat{\delta}(2p_{i}{\cdot}q)\\\
&\times\left[\hat{\delta}^{\prime}(2p_{1}{\cdot}l)\hat{\delta}(2p_{2}{\cdot}l){-}\hat{\delta}(2p_{1}{\cdot}l)\hat{\delta}^{\prime}(2p_{2}{\cdot}l)\right]\frac{\cancel{l^{2}}l^{\alpha}}{\cancel{l^{2}}(q-l)^{2}}\end{split}\,,$
(139)

which gives us indeed a contact integral for the $l$-variable, unimportant for
long classical scattering, since $b\neq 0$.

  * •

Vanishing of ${\cal T}^{(1)\,\alpha}_{\mathcal{C}_{2}}$

Here we carry out the final piece of the computation for the one-index moment.
As usual, we start from the definition (96), and use (76) and (75) for the
integral measure in $q$ and $l$ respectively, and (77) for the 4 point
amplitude. This gives

${\cal
T}^{(1)\,\alpha}_{\mathcal{C}_{2}}=\int\hat{d}^{4}q\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}l)\left[\hat{\delta}^{\prime}(p_{1}{\cdot}q)\hat{\delta}(p_{2}{\cdot}q){-}\hat{\delta}(p_{1}{\cdot}q)\hat{\delta}^{\prime}(p_{2}{\cdot}q)\right]X^{\alpha}\,,$
(140)

where we have defined

$X^{\alpha}=-\frac{1}{2}g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}\frac{q^{2}l^{\alpha}}{l^{2}(q{-}l)^{2}}e^{-ib{\cdot}q}\,.$
(141)

Showing that this integral gives zero contribution is a straightforward task.
We do the usual change of variables (80), and doing the integrals in $y_{i}$
using the delta functions. However, since we will use integration by parts, we
need to evaluate the derivative of $X^{\alpha}$ w.r.t. $x_{1}$ or $x_{2}$,
which after evaluating $x_{1}=x_{2}=0$, vanish identically. We therefore
conclude that ${\cal T}^{\alpha}_{(1),\mathcal{C}_{2}}=0$, as announced in
section 5.2.2

###  A.2 Cancellations in the classical two-index moment at 1-loop

In the final part of this appendix we proof the cancellations announced in
(122). Recall we already obtained $\mathcal{J}_{3}$ in (121). All that is left
is to compute explicitly ${\cal T}^{\alpha\,\beta}_{(1),\mathcal{C}_{3}}$. As
usual we start from the definition (99). Use (75) for the integral measure in
both $q$ and $l$ variables, and (77) for the 4 point amplitude, to get

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}=g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}\int\hat{d}^{4}q\hat{d}^{4}l\prod_{i}\hat{\delta}(p_{i}{\cdot}q)\hat{\delta}(p_{i}{\cdot}l)\frac{l^{\alpha}l^{\beta}}{l^{2}(q{-}l)^{2}}e^{-ib{\cdot}q}\,.$
(142)

Doing our usual shift $q=\bar{q}+l,$ allows us to factorize out the IR-
divergent integral - for the $l$ variable - (103). This becomes

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}=g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)^{2}I_{1}\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{1}{\cdot}\bar{q})\hat{\delta}(p_{2}{\cdot}\bar{q})\frac{l^{\alpha}l^{\beta}}{\bar{q}^{2}}e^{-ib{\cdot}\bar{q}}\,,$
(143)

which can be further rewritten as

${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}=g^{4}\left(Q_{1}Q_{2}p_{1}{\cdot}p_{2}\right)I_{1}\partial_{b^{\alpha}}\Delta
p^{(0)\,\beta}_{1}\,.$ (144)

The computation of the derivative of the leading order impulse w.r.t. the
impact parameter was given in (119). Using it leads to

${\cal T}^{\alpha\,\beta}_{(1),\mathcal{C}_{3}}=2g^{4}\Delta
p^{(0)\,\alpha}_{1}\Delta p^{(0)\,\beta}_{1}\ln\left(-\mu^{2}b^{2}\right),$
(145)

which is nothing but $\mathcal{J}_{3}^{(1)\,\alpha\,\beta}$ as given in (121)
but with opposite sign. Thus we simply conclude that ${\cal
T}^{(1)\,\alpha\,\beta}_{\mathcal{C}_{3}}+\mathcal{J}_{3}^{(1)\,\alpha\,\beta}=0\,,$
as required from (122).

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