11institutetext: School of Mathematics and Maxwell Institute for Mathematical
Sciences, University of Edinburgh, EH9 3FD, Scotland22institutetext: School of
Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, College
Green, Dublin 2, Ireland33institutetext: Higgs Centre for Theoretical Physics,
School of Physics and Astronomy, The University of Edinburgh, EH9 3FD,
Scotland44institutetext: Centre for Theoretical Physics, School of Physical
and Chemical Sciences, Queen Mary University of London, 327 Mile End Road,
London E1 4NS, United Kingdom

# The Uncertainty Principle and Classical Amplitudes

Andrea Cristofoli 2 Riccardo Gonzo 2,3 Nathan Moynihan 3 Donal O’Connell 3
Alasdair Ross 3 Matteo Sergola 4 Chris D. White

###### Abstract

We study the variance in the measurement of observables during scattering
events, as computed using amplitudes. The classical regime, characterised by
negligible uncertainty, emerges as a consequence of an infinite set of
relationships among multileg, multiloop amplitudes in a momentum-transfer
expansion. We discuss two non-trivial examples in detail: the six-point tree
and the five-point one-loop amplitudes in scalar QED. We interpret these
relationships in terms or a coherent exponentiation of radiative effects in
the classical limit which generalises the eikonal formula, and show how to
recover the impulse, including radiation reaction, from this generalised
eikonal. Finally, we incorporate the physics of spin into our framework.

††preprint: SAGEX-21-31-E, QMUL-PH-21-56

##  1 Introduction

Scattering amplitudes are quantum mechanical objects. They are matrix elements
of the time evolution operator $S=U(\infty,-\infty)$ from the far past to the
far future; on their distributional support, amplitudes evaluate to a complex
number whose square can be interpreted as a probability. Yet recent research
has revealed that scattering amplitudes also have a useful role to play in
classical physics.

Amplitudes have found this new role for three basic reasons. One is the double
copy Kawai:1985xq ; Bern:2008qj ; Bern:2010ue ; Bern:2010yg , which simplifies
the computation of amplitudes in general relativity allowing us to go to
higher loops and legs, and therefore to higher theoretical precision
Bern:2018jmv ; Bern:2019crd ; Carrasco:2021otn . The second reason is the
detection LIGOScientific:2016aoc  of gravitational waves, leading to a
vigorous programme of research into their phenomenology. This programme
demands a high precision understanding of gravitational wave physics. Finally,
we have learned how to extract classical physics from the quantum mechanical
amplitudes Donoghue:1993eb ; Donoghue:1994dn ; Bjerrum-Bohr:2002fji ;
Holstein:2004dn ; Neill:2013wsa ; Bjerrum-Bohr:2013bxa ; Cachazo:2017jef ;
Guevara:2017csg ; Cheung:2018wkq ; KMOC ; Maybee:2019jus , allowing the double
copy to be applied in astrophysically relevant situations Bern:2019nnu ;
Antonelli:2019ytb .

Of course there are many reasons to be interested in scattering amplitudes
besides gravitational wave physics. Amplitudes are particularly important in
collider physics. They are also interesting in their own right because of
their internal structure. In many theories, including general relativity,
amplitudes have uniqueness properties which allow us (in principle) to
determine the entire $S$ matrix from basic knowledge of the helicities of the
interacting particles Benincasa:2007xk ; Arkani-Hamed:2017jhn . In this way
amplitudes provide a way to _define_ a quantum field theory. Since amplitudes
are intrinsically quantum-mechanical objects this definition hard-wires key
aspects of quantum mechanics, such as the uncertainty principle, into the
physics.

Taking the viewpoint that scattering amplitudes define the physics, an old
question reappears. How do we understand the classical limit? It is more
common to define quantum field theories in terms of path integrals: then
(thanks to Feynman) it is clear how the classical limit arises — it comes from
stationary phase in the path integral. But amplitudes, as a quantum-first
definition of a quantum field theory, do not have as clear a link to classical
physics.

Our interest is in this question. How is the classical limit encoded in the
quantum-first definition of field theory through scattering amplitudes? To
answer this question we take another look at the link between amplitudes and
classical physics.

In fact by now there are several methods available for converting amplitudes
into classical quantities. It has long been understood that four-point
amplitudes are closely connected to classical potentials Donoghue:1993eb ;
Donoghue:1994dn ; Bjerrum-Bohr:2002fji ; Holstein:2004dn ; Bjerrum-
Bohr:2013bxa , and it is possible to deduce interaction potentials from four-
point multiloop amplitudes. More generally, effective Lagrangians are often
used to compute amplitudes — and this procedure can be reversed, allowing
Wilson coefficients in effective Lagrangians to be extracted from amplitudes
Neill:2013wsa ; Cheung:2018wkq ; Bern:2019crd ; Bern:2019nnu . The versatility
of effective Lagrangians allows information and the potential, spin effects
Bern:2020buy ; delaCruz:2021gjp , etc to be readily extracted from amplitudes
and applied to bound gravitational systems.

In this article we will be particularly interested in two other links between
amplitudes and classical physics. The first of these was introduced by
Kosower, Maybee and one of the authors (KMOC) KMOC ; Maybee:2019jus . This
method is based on computing observables from scattering amplitudes. The basic
idea is that if the observable is well-defined in both the classical and
quantum-mechanical cases, then in the correspondence regime the result of a
quantum calculation must agree with the classical calculation. But it is worth
emphasising that the methods of KMOC apply equally well to quantum observables
which do not make sense in the classical theory (for example, the number of
photons radiated during a scattering event). In this sense KMOC is a quantum-
first method: it takes the quantum field theory as the basic starting point,
making contact with classical physics only at the end of the computation.

The eikonal approximation is the second link between amplitudes and classical
physics of particular interest to us. Eikonal physics has a long history,
going back to nineteenth century work on the relation between geometric and
wave optics. The “eikonal analogy” was a guide in the early development of
quantum mechanics. In its long history the precise meaning of the word
“eikonal” seems to have undergone some drift. We are interested in the eikonal
approximation to (especially four-point) massive amplitudes in quantum field
theory PhysRev.143.1194 ; Abarbanel:1969ek ; Cheng:1969eh ; PhysRev.186.1611 ;
WALLACE1973190 ; PhysRev.186.1656 ; Wallace:1977ae ; Cheng1981ConsequencesOT .
In this approximation, the incoming $s$-channel energy is large compared to
momentum transfers. The key fact in this limit is that the amplitude can be
separated into two factors. One of these factors is the exponent of an eikonal
function which we will denote as $\chi$, divided by $\hbar$. The $\chi$ is
analogous to a classical action, and this part of the amplitude describes the
classical physics. The remaining terms in the amplitude, which do not
exponentiate, are quantum mechanical.

The quintessential difference between classical and quantum physics is
uncertainty. Quantum observables naturally have a variance which is absent in
classical physics. We begin our study by computing the variance in important
classical observables using scattering amplitudes, following the methods of
KMOC. In the classical limit the variance must be negligible. We find that
this condition of negligible variance becomes an infinite set of relations
which involve different amplitudes. More specifically, in the correspondence
regime we can expand our amplitudes in powers of momentum transfer over
centre-of-mass energy. This is a semi-classical expansion because the momentum
transfers in KMOC are order $\hbar$111Alternatively one can think of the
expansion as a large-mass expansion because the centre-of-mass energy is
dominated by the particle masses.. The zero-variance conditions relate certain
terms in this “transfer” expansion between amplitudes with different numbers
of loops and legs. For example, five-point amplitudes in the transfer
expansion are related to lower loop five-point and four-point amplitudes.

At four points, the zero-variance conditions are very familiar: they are the
relations required for eikonal exponentiation. Our work therefore shows that
there is a generalisation of the eikonal formula beyond four points. We
outline the structure of the generalisation, which involves a coherent
radiative state entangled with the four-point dynamics.

Throughout our work, we consider semiclassical scattering events involving two
point-like particles. The particles may interact electromagnetically,
gravitationally, or via classical Yang-Mills forces. We begin our work in
section 2 with a discussion of a basic requirement for a successful quantum
description of such a classical event: negligible variance in a measurement of
the field strength. As we will see, this requirement becomes a non-trivial
constraint on scattering amplitudes. The leading obstruction is given by the
six-point tree amplitude; this amplitude must be suppressed relative to the
corresponding six-point one-loop amplitude. As a check we compute the tree
amplitude, demonstrating explicitly that it has the required suppression. We
build on this observation in section 3 to find an infinite series of
constraints on multi-loop, multi-leg scattering amplitudes, verifying the
first non-trivial constraint on the five-point one-loop amplitude in section
3.2. In section 4 we interpret these zero-variance conditions in eikonal
terms, arguing that radiation exponentiates in a manner analogous to the
conservative terms. We propose a formula for the $S$ matrix acting on our
state which is a product of a coherent radiative state entangled with the more
traditional eikonal (conservative) dynamics. Section 5 provides a partial
derivation of our final state proposal in the soft case using path integral
methods. We turn to particles with spin in section 6 where we show that a
simple generalisation of our final state neatly encodes the dynamics of
spinning particles. The main body of our article concludes in section 7 with a
discussion.

### Note added

As our work developed, we became aware of independent progress in a similar
direction by Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo and Gabriele
Veneziano. This group’s work was presented most recently as a series of
seminars carlosTalk ; rodolfosTalk ; gabrielesTalk  at Saclay, and will soon
appear in print. The themes most in common between this work and that of Di
Vecchia et al concern the factorisation of five-point one-loop amplitudes,
which we discuss in section 3.2, and the extension of the eikonal to the
radiative case in our section 4.3.

##  2 Negligible Uncertainty

In classical electrodynamics, a key role is played by the field strength
$F_{\mu\nu}(x)$. This object is a complete gauge-invariant characterisation of
the field; once it is known, quantities such as the energy-momentum radiated
to infinity and the field angular momentum are easily determined. In a quantum
description, the field becomes an operator $\mathbbm{F}_{\mu\nu}(x)$. In a
semiclassical situation, the expectation value of this operator on a state
$|\psi\rangle$ should equal the classical field, up to negligible quantum
corrections:

$\displaystyle\langle\psi|\mathbbm{F}_{\mu\nu}(x)|\psi\rangle=F_{\mu\nu}(x)+\mathcal{O}(\hbar)\,.$
(1)

Note that we have schematically indicated the presence of small, order
$\hbar$, quantum corrections. More precisely, these corrections must be
suppressed by dimensionless ratios involving Planck’s constant; the precise
ratios depend on the actual physical context.

Since in the quantum theory a single-valued field is replaced by the
expectation value of an operator, we must address the quintessentially quantum
mechanical issue of uncertainty. The uncertainty can be characterised by the
variance

$\displaystyle\langle\psi|\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle-\langle\psi|\mathbbm{F}_{\mu\nu}(x)|\psi\rangle\langle\psi|\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle\,.$
(2)

In the domain of validity of the classical approximation, this variance must
be negligible.

Precisely the same remarks hold in a quantum mechanical approach to GR. The
curvature tensor $R_{\mu\nu\rho\sigma}(x)$ in the classical theory is replaced
by the expectation value of the curvature operator
$\mathbbm{R}_{\mu\nu\rho\sigma}(x)$. The variance

$\displaystyle\langle\psi|\mathbbm{R}_{\mu\nu\rho\sigma}(x)\mathbbm{R}_{\alpha\beta\gamma\delta}(y)|\psi\rangle-\langle\psi|\mathbbm{R}_{\mu\nu\rho\sigma}(x)|\psi\rangle\langle\psi|\mathbbm{R}_{\alpha\beta\gamma\delta}(y)|\psi\rangle$
(3)

must be negligible.

This section is devoted to an investigation of this condition of negligible
uncertainty. Working in the KMOC formalism at lowest order in perturbation
theory, we will see that the expectations
$\langle\psi|\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle$
and
$\langle\psi|\mathbbm{R}_{\mu\nu\rho\sigma}(x)\mathbbm{R}_{\alpha\beta\gamma\delta}(y)|\psi\rangle$
are determined by tree-level _six_ -point amplitudes while
$\langle\psi|\mathbbm{F}_{\mu\nu}(x)|\psi\rangle$ and
$\langle\psi|\mathbbm{R}_{\mu\nu\rho\sigma}(x)|\psi\rangle$ are determined by
_five_ -point tree amplitudes. We must then face the question of how it can be
that the variance is negligible.

###  2.1 Field strength expectations

We begin by reviewing the evaluation of single field-strength observables in
KMOC. Although this material appears elsewhere in the literature, the
calculation will give us an opportunity to introduce notation which we will
use in the remainder of our paper. We will discuss the electromagnetic case in
some detail. The gravitational case is completely analogous to the
electromagnetic case, so we only quote key results.

The electromagnetic field operator is

$\displaystyle\mathbbm{A}_{\mu}(x)=\frac{1}{\sqrt{\hbar}}\sum_{\eta=\pm}\int\mathrm{d}\Phi(k)\left[a_{\eta}(k)\,\varepsilon^{(\eta)*}_{\mu}(k)\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]\,.$ (4)

This is a sum over helicities $\eta$; the notation h.c. indicates that the
hermitian conjugate of the explicit term should be included. The measure is
the Lorentz-invariant phase space measure, namely

$\displaystyle\mathrm{d}\Phi(k)\equiv\hat{\mathrm{d}}^{4}k\,\hat{\delta}(k^{2})\,\theta(k^{0})\,,$
(5)

where the carets indicate factors of $2\pi$:

$\displaystyle\hat{\delta}(k)\equiv(2\pi)\delta(k),\quad\hat{\mathrm{d}}k\equiv\frac{dk}{2\pi}\,.$
(6)

Furthermore, we follow reference KMOC  by introducing the notation $\bar{k}$
for the wavevector associated with the momentum $k$. These quantities differ
only by a factor of $\hbar$:

$\displaystyle k=\hbar\bar{k}\,.$ (7)

As we will soon see, it can at times be useful to keep factors of $\hbar$
explicit — and we will do so throughout this article. Notice that the
polarisation vectors $\varepsilon^{(\eta)}(k)$ are homogeneous in the
momentum, so they can equally be taken to be functions of the momentum or of
the wavenumber. We normalise creation and annihilation operators so that

$\displaystyle=\delta_{\eta\eta^{\prime}}\hat{\delta}_{\Phi}(k-k^{\prime})\,,$
(8)

where $\hat{\delta}_{\Phi}(k)$ is

$\displaystyle\hat{\delta}_{\Phi}(k)\equiv
2k^{0}\,\hat{\delta}^{3}(\bm{{k}}-\bm{{k}}^{\prime})\,.$ (9)

Although we could choose to discuss the variance of the field itself, we focus
on the gauge-invariant variance in the field strength operator. The explicit
expression for this operator follows from eq. (4) and can be written as

$\displaystyle\mathbbm{F}_{\mu\nu}(x)=\frac{1}{\sqrt{\hbar}}\sum_{\eta=\pm}\int\mathrm{d}\Phi(k)\left[-ia_{\eta}(k)\,\bar{k}_{[\mu}^{\vphantom{\mu}}\varepsilon^{(\eta)*}_{\nu]}(k)\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]\,.$ (10)

Because we frequently encounter antisymmetric operators like this, we have
defined the antisymmetrisation symbol

$\displaystyle a_{[\mu}b_{\nu]}\equiv a_{\mu}b_{\nu}-a_{\nu}b_{\mu}$ (11)

without any factor of $1/2$.

In classical GR, we define the spacetime metric to be

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}\,.$ (12)

The constant $\kappa$ is given in terms of Newton’s constant $G$ by222We
retain this definition even in units where $\hbar\neq 1$. In that case the
dimensions of the fields $A_{\mu}$ and $h_{\mu\nu}$ are both
$\sqrt{\textrm{mass}/\textrm{length}}$.

$\displaystyle\kappa\equiv\sqrt{32\pi G}\,.$ (13)

In a quantum description, the corresponding field operator is

$\displaystyle\mathbbm{h}_{\mu\nu}(x)=\frac{1}{\sqrt{\hbar}}\sum_{\eta=\pm}\int\mathrm{d}\Phi(k)\left[a_{\eta}(k)\,\varepsilon^{(\eta)*}_{\mu}(k)\varepsilon^{(\eta)*}_{\nu}(k)\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]\,.$ (14)

Notice that we have written the polarisation tensor that conventionally
appears in the graviton operator as an explicit outer product of two
polarisation vectors; this is always possible. We are using the same symbol
(namely, $a_{\eta}$) for annihilation operators in both electromagnetism and
gravity; we hope context will make clear which operator is relevant. The
linearised Riemann tensor operator follows from conventional definitions and
is given by333It could be worth remarking that the Riemann tensor is gauge-
invariant in linearised theory.

$\displaystyle\mathbbm{R}_{\mu\nu\rho\sigma}(x)$
$\displaystyle=\frac{\kappa}{2}\left(\partial_{\sigma}\partial_{\,[\mu}\mathbbm{h}_{\nu]\,\rho}-\partial_{\rho}\partial_{\,[\mu}\mathbbm{h}_{\nu]\,\sigma}\right)$
(15)
$\displaystyle=-\frac{\kappa}{2}\frac{1}{\sqrt{\hbar}}\sum_{\eta=\pm}\int\mathrm{d}\Phi(k)\left[a_{\eta}(k)\,\bar{k}_{[\mu}^{\vphantom{\mu}}\varepsilon^{(\eta)*}_{\nu]}(k)\,\bar{k}_{[\sigma}^{\vphantom{\mu}}\varepsilon^{(\eta)*}_{\rho]}(k)\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]\,.$

The basic idea of KMOC KMOC  is to take two stable incoming particles,
described by a quantum state, which are very distant in the far past. We
scatter these particles off one another gently, so that they do not approach
too closely (and so that we can describe the scattering, if we choose, using
classical field theory). The particles nevertheless radiate. We assume that
the particles remain unbound throughout the process.

We are therefore interested in classical point-like particles which should by
located at some position and should have some velocity. In a quantum
description, this can be achieved by placing the particles in appropriate
wavefunctions which localise the particles both in position and in momentum
space. We may write the initial state as

$\displaystyle|\psi\rangle=\int\mathrm{d}\Phi(p_{1})\mathrm{d}\Phi(p_{2})\,\phi_{1}(p_{1})\phi_{2}(p_{2})e^{ib\cdot
p_{1}/\hbar}\,|p_{1},p_{2}\rangle\,.$ (16)

This is a two-particle state; the impact parameter $b$ is introduced by
translating particle one with respect to the other. The Lorentz invariant
phase-space for a massive particle is

$\displaystyle\mathrm{d}\Phi(p_{1})=\hat{\mathrm{d}}^{4}p_{1}\,\hat{\delta}(p_{1}^{2}-m_{1}^{2})\,\theta(p_{1}^{0})\,.$
(17)

The wavefunctions $\phi_{i}(p_{i})$ for each particle have the property that
they localise the particle’s position with an uncertainty $\ell_{w}$ which is
characteristic of the wavepacket444We could choose different uncertainties
$\ell_{w}^{(i)}$ for each particle at the expense of a slightly more
complicated notation. There is no reason for us to exploit this freedom in our
article, so we choose the simpler case of one common $\ell_{w}$.. At the same
time, the wavepackets must localise the momenta of the particles to
uncertainty $\Delta p$ of order $\hbar/\ell_{w}$. In the classical limit, we
require that these uncertainties are negligible compared to the distance, of
order $b$, between the particles

$\displaystyle\ell_{w}\ll b\,,$ (18)

and also require that the momentum-space uncertainty is negligible compared to
the masses of the particles:

$\displaystyle\frac{\hbar}{\ell_{w}}\ll m\Rightarrow l_{c}\ll\ell_{w}\,,$ (19)

where $l_{c}\equiv\hbar/m$ is a measure of the Compton wavelength of the
particle. Thus we have a set of inequalities on the initial parameters:

$\displaystyle l_{c}\ll\ell_{w}\ll b\,.$ (20)

More generally, if $\ell_{s}$ is the distance of closest approach of the
particles during scattering, we require that

$\displaystyle l_{c}\ll\ell_{w}\ll\ell_{s}\,.$ (21)

Small-angle scattering has the property that $\ell_{s}\simeq b$.

Since we will encounter these wavepackets rather frequently, it is sometimes
convenient to write the two-particle momentum-space wavefunction as

$\displaystyle\phi_{b}(p_{1},p_{2})\equiv\phi(p_{1},p_{2})e^{ib\cdot
p_{1}/\hbar}\quad,\quad\phi(p_{1},p_{2})\equiv\phi_{1}(p_{1})\phi_{2}(p_{2})\,.$
(22)

We will further use the short-hand notation

$\displaystyle\mathrm{d}\Phi(p_{1},p_{2},p_{3},\ldots)\equiv\mathrm{d}\Phi(p_{1})\mathrm{d}\Phi(p_{2})\mathrm{d}\Phi(p_{3})\cdots\,.$
(23)

for integrals over the phase space of various particles.

Now that we have an incoming state, we may write the outgoing state in terms
of the time evolution operator $U(+\infty,-\infty)$, which is equal to the $S$
matrix:

$\displaystyle|\psi\rangle_{\textrm{out}}$ $\displaystyle=S|\psi\rangle$ (24)
$\displaystyle=\int\mathrm{d}\Phi(p_{1},p_{2})\,\phi_{b}(p_{1},p_{2})\,S|p_{1},p_{2}\rangle\,.$

Since the $S$ matrix can be written in terms of scattering amplitudes, the
outgoing state itself can be written in terms of integrals over amplitudes and
the incoming state. This simple fact will be a recurring theme in this paper.

Let us now consider the leading-order field strength in our situation. In the
far past, we have

$\displaystyle\langle\psi|\mathbbm{F}_{\mu\nu}(x)|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{\hbar}}\sum_{\eta}\int\mathrm{d}\Phi(k)\left[\langle\psi|-ia_{\eta}(k)|\psi\rangle\,\bar{k}_{[\mu}^{\vphantom{\mu}}\varepsilon^{(\eta)*}_{\nu]}(k)\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]$ (25) $\displaystyle=0\,.$

The expectation value vanishes because there are no photons in the initial
state: $a_{\eta}(k)|\psi\rangle=0$. Classically, the interpretation is that
the initial state contains no incoming radiation. Notice that the expectation
value is not sensitive to the Coulomb fields of the incoming particles;
instead, we are computing the asymptotic value of the field at infinity,
namely the coefficient of the $1/\textrm{distance}$ piece of the field
strength.

In the far future, the expectation value is

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{F}_{\mu\nu}(x)$ $\displaystyle
S|\psi\rangle$ (26)
$\displaystyle=\frac{1}{\sqrt{\hbar}}\sum_{\eta}\int\mathrm{d}\Phi(k)\left[-i\langle\psi|S^{\dagger}a_{\eta}(k)S|\psi\rangle\,\bar{k}_{[\mu}^{\vphantom{\mu}}\varepsilon^{(\eta)*}_{\nu]}(k)\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]\,.$

This no longer vanishes. We may evaluate it at lowest perturbative order by
writing the $S$ matrix in terms of the transition matrix $T$ as $S=1+iT$. The
matrix elements of $T$ on momentum eigenstates are the scattering amplitudes,
which we may organise (as usual in perturbation theory) in terms of the
coupling $e$ or $\kappa$ depending on whether we are interested in
electrodynamics or gravity. Generically we will denote the perturbative
coupling as $g$. Thus we have

$\displaystyle\langle\psi|S^{\dagger}a_{\eta}(k)S|\psi\rangle$
$\displaystyle=i\langle\psi|(a_{\eta}(k)T-T^{\dagger}a_{\eta}(k))|\psi\rangle+\langle\psi|T^{\dagger}a_{\eta}(k)T|\psi\rangle$
(27)
$\displaystyle=i\langle\psi|a_{\eta}(k)T|\psi\rangle+\langle\psi|T^{\dagger}a_{\eta}(k)T|\psi\rangle$
$\displaystyle\simeq i\langle\psi|a_{\eta}(k)T|\psi\rangle\,.$

In the middle line above, we used the fact that $a_{\eta}(k)|\psi\rangle=0$;
in the last line we neglected the term involving two $T$ matrices which does
not contribute at lowest order by counting powers of $g$.

Further expanding the state using eq. (16), and taking advantage of the short-
hand notation of eqs. (22) and (23), we may write

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{F}_{\mu\nu}(x)S|\psi\rangle$
$\displaystyle=2\operatorname{Re}\frac{1}{\sqrt{\hbar}}\sum_{\eta}\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2},k)\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})\times$
(28) $\displaystyle\hskip 120.0pt\times\langle
k^{\eta},p_{1}^{\prime},p_{2}^{\prime}|T|p_{1},p_{2}\rangle\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\,e^{-i\bar{k}\cdot
x}\,.$

The matrix element $\langle
k^{\eta},p_{1}^{\prime},p_{2}^{\prime}|T|p_{1},p_{2}\rangle$ is, at lowest
order, a five-point tree amplitude so it is proportional to $g^{3}$. This is
consistent with a classical analysis of the outgoing radiation field.

In GR, the equivalent expression is

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{R}_{\mu\nu\rho\sigma}(x)S|\psi\rangle$
$\displaystyle=2\operatorname{Re}\frac{-i}{\sqrt{\hbar}}\frac{\kappa}{2}\sum_{\eta}\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2},k)\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})\times$
(29) $\displaystyle\hskip 80.0pt\times\langle
k^{\eta},p_{1}^{\prime},p_{2}^{\prime}|T|p_{1},p_{2}\rangle\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\,\bar{k}^{\vphantom{\mu}}_{[\sigma}\varepsilon^{(\eta)*}_{\rho]}\,e^{-i\bar{k}\cdot
x}\,.$

$p_{1}$$p_{2}$$p_{1}+q_{1}$$p_{2}+q_{2}$$k$ Figure 1: The kinematic
configuration we choose for the five-point amplitude which determines the
leading-order radiation field.

It will be useful for us to simplify these expressions further, following
again the discussion of reference KMOC  at leading order. The matrix element
$\langle k^{\eta},p_{1}^{\prime},p_{2}^{\prime}|T|p_{1},p_{2}\rangle$ is the
amplitude times a momentum-conserving delta function; our expectation value
instructs us to integrate over all momenta in the amplitude. We may relabel
these external momenta as shown in fig. 1. The measure can then be written as

$\displaystyle\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2},k)=\mathrm{d}\Phi(p_{1},p_{2},k)\hat{\mathrm{d}}^{4}q_{1}\hat{\mathrm{d}}^{4}q_{2}\hat{\delta}(2p_{1}\cdot
q_{1}+q_{1}^{2})\hat{\delta}(2p_{2}\cdot q_{2}+q_{2}^{2})\,.$ (30)

In this form, the overall momentum-conserving delta function reads
$\hat{\delta}^{4}(q_{1}+q_{2}+k)$. Now, in the classical regime the photon
momentum is of order $\hbar$, as are the momentum mismatches $q_{1}$ and
$q_{2}$. The $q_{i}^{2}$ terms in the delta functions above are therefore
small shifts compared to the width (or order $1/\ell_{w}$) of the
wavefunctions in the expectation values. As the observable has the structure
of a convolution of the sharply-peaked wavefunctions multiplied by delta
functions and otherwise smooth functions, we may neglect the $q_{i}^{2}$
shifts555As we will see later in this article, care must be taken with these
delta functions at higher order. Here we work at leading order, so the
situation is simple.. Similarly, the wavefunctions themselves are

$\displaystyle\phi_{b}^{*}(p_{1}+q_{1},p_{2}+q_{2})\phi_{b}(p_{1},p_{2})=\phi^{*}(p_{1}+q_{1},p_{2}+q_{2})\phi(p_{1},p_{2})e^{-iq_{1}\cdot
b/\hbar}\,.$ (31)

We may neglect the $q_{i}$ shifts in the wavefunctions
$\phi_{i}^{*}(p_{i}+q_{i})$ because the momenta $q_{i}$ are negligible
compared to the width of the wavefunctions. Thus the field strength becomes

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{F}_{\mu\nu}(x)S|\psi\rangle$
$\displaystyle=2\operatorname{Re}\frac{1}{\sqrt{\hbar}}\sum_{\eta}\int\mathrm{d}\Phi(p_{1},p_{2},k)|\phi(p_{1},p_{2})|^{2}\,\hat{\mathrm{d}}^{4}q_{1}\hat{\mathrm{d}}^{4}q_{2}\,\hat{\delta}_{\sigma}(2p_{1}\cdot
q_{1})\hat{\delta}_{\sigma}(2p_{2}\cdot q_{2})\,$ (32)
$\displaystyle\times\,\mathcal{A}_{5,0}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})\,\hat{\delta}^{4}(k+q_{1}+q_{2})\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\,e^{-i(\bar{k}\cdot
x+\bar{q}_{1}\cdot b)}\,.$

The notation $\hat{\delta}_{\sigma}(x)$ indicates that the delta functions
have been “broadened” so that their width is of order
$\sigma\sim\hbar/\ell_{w}$. In our point-particle treatment, we neglect this
scale in the rest of this paper. The amplitude
$\mathcal{A}_{5,0}(p_{1}p_{2}\rightarrow p_{1}+q_{1},p_{2}+q_{2},k^{\eta})$ is
a five-point, tree amplitude. More generally, we will denote $n$ point $L$
loop amplitudes as $\mathcal{A}_{n,L}$ (in gauge theory) or
$\mathcal{M}_{n,L}$ (in gravity).

The integral in eq. (32) still depends on the wavefunctions of the particles.
But now the role of the wavefunctions is transparent: they are steeply-peaked
functions of the momenta $p_{1}$ and $p_{2}$ which allows us to simply replace
these variables of integration with the incoming classical momenta
$m_{1}v_{1}$ and $m_{2}v_{2}$. We thus write the field expectation value as

$\displaystyle\langle\psi|S^{\dagger}$
$\displaystyle\mathbbm{F}_{\mu\nu}(x)S|\psi\rangle=2\operatorname{Re}\hbar^{7/2}\sum_{\eta}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})\,$
(33) $\displaystyle\times\,\mathcal{A}_{5,0}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})\,\hat{\delta}^{4}(\bar{k}+\bar{q}_{1}+\bar{q}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\,e^{-i(\bar{k}\cdot
x+\bar{q}_{1}\cdot b)}\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

The double-angle brackets are shorthand notation that instructs us to evaluate
the momenta at their classical values, and remind us to take care of $q^{2}$
shifts in the delta functions. Recalling that $k$, $q_{1}$ and $q_{2}$ are all
of order $\hbar$, we have scaled out all the $\hbar$ dependence except that of
the scattering amplitude itself. Similarly, in gravity, one finds

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{R}_{\mu\nu\rho\sigma}(x)S|\psi\rangle=-2\operatorname{Re}\hbar^{7/2}\frac{i\kappa}{2}\sum_{\eta}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})\,$
(34) $\displaystyle\times\mathcal{M}_{5,0}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})\hat{\delta}^{4}(\bar{k}+\bar{q}_{1}+\bar{q}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\,\bar{k}^{\vphantom{\mu}}_{[\sigma}\varepsilon^{(\eta)*}_{\rho]}\,e^{-i(\bar{k}\cdot
x+\bar{q}_{1}\cdot b)}\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

For these expressions to make sense classically, it better be that the overall
$\hbar$ dependence of the amplitudes cancels that of the observable. Indeed we
may write

$\displaystyle\mathcal{A}_{5,0}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})$
$\displaystyle=\hbar^{-7/2}\mathcal{A}_{5,0}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})+\mathcal{O}(\hbar)\,$ (35)
$\displaystyle\mathcal{M}_{5,0}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})$
$\displaystyle=\hbar^{-7/2}\mathcal{M}_{5,0}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})+\mathcal{O}(\hbar)\,,$

where the quantities $\mathcal{A}_{5,0}^{(0)}$ and $\mathcal{M}_{5,0}^{(0)}$
are independent of $\hbar$, as was noticed in KMOC . We will return to this
structure below.

The physical interpretation of these expectation values is that they compute
the radiative part of the field at large distances. To see this explicitly,
the $\bar{k}$ integral needs to be performed taking advantage of the large
distance between the point of measurement $x$ and the particles. The
integration can be performed using textbook methods and was recently reviewed
in detail in ref. Cristofoli:2021vyo . The question of central interest to us
in this section, however, is to compute the uncertainty in the field strength;
to do so, we turn to computing the expectation of two field strengths.

###  2.2 Expectation of two field strengths

It will be quite straightforward for us to compute expectations of products of
operators using precisely the methods of the previous subsection. In
electrodynamics, we need to compute

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)S|\psi\rangle=-\frac{1}{\hbar}\sum_{\eta,\eta^{\prime}}\int$
$\displaystyle\mathrm{d}\Phi(k^{\prime},k)\langle\psi|S^{\dagger}\left[-ia_{\eta}(k)\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\,e^{-i\bar{k}\cdot
x}+\textrm{h.c.}\right]$ (36) $\displaystyle\hskip
20.0pt\times\left[-ia_{\eta^{\prime}}(k^{\prime})\,\bar{k}^{\prime}_{[\rho}\varepsilon^{\prime(\eta^{\prime})*}_{\sigma]}\,e^{-i\bar{k}^{\prime}\cdot
y}+\textrm{h.c.}\right]S|\psi\rangle\,.$

Working at lowest order, and taking advantage of the fact that
$a_{\eta}(k)|\psi\rangle=0$, the expectation simplifies to

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)S|\psi\rangle=-\frac{2}{\hbar}\operatorname{Re}\sum_{\eta,\eta^{\prime}}\int$
$\displaystyle\mathrm{d}\Phi(k^{\prime},k)\langle\psi|a_{\eta}(k)a_{\eta^{\prime}}(k^{\prime})iT|\psi\rangle$
(37) $\displaystyle\hskip
30.0pt\times\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\bar{k}^{\prime}_{[\rho}\varepsilon^{\prime(\eta^{\prime})*}_{\sigma]}\,e^{-i(\bar{k}\cdot
x+\bar{k}^{\prime}\cdot y)}\,,$

up to a purely quantum single-photon effect Cristofoli:2021vyo . Expanding the
wavefunctions, we encounter the matrix element $\langle
p_{1}^{\prime},p_{2}^{\prime}|a_{\eta}(k)a_{\eta^{\prime}}(k^{\prime})T|p_{1},p_{2}\rangle$:
a six-point tree amplitude. The classical limit is determined precisely as in
the previous section with the result

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)S|\psi\rangle$
$\displaystyle=-2\hbar^{5}\operatorname{Re}\sum_{\eta,\eta^{\prime}}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k}^{\prime},\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})$
(38)
$\displaystyle\times\,i\mathcal{A}_{6,0}\,\hat{\delta}^{4}(\bar{k}+\bar{k}^{\prime}+\bar{q}_{1}+\bar{q}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\bar{k}^{\prime}_{[\rho}\varepsilon^{\prime(\eta^{\prime})*}_{\sigma]}\,e^{-i(\bar{k}\cdot
x+\bar{k}^{\prime}\cdot y+\bar{q}_{1}\cdot
b)}\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

The amplitude is shown in figure 2.

Similarly, in gravity, we find

$\displaystyle\langle\psi|S^{\dagger}$
$\displaystyle\mathbbm{R}_{\mu\nu\rho\sigma}(x)\mathbbm{R}_{\alpha\beta\gamma\delta}(y)S|\psi\rangle$
(39)
$\displaystyle=-2\hbar^{5}\operatorname{Re}\left(-i\frac{\kappa}{2}\right)^{2}\sum_{\eta,\eta^{\prime}}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k}^{\prime},\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})\,\mathcal{M}_{6,0}$
$\displaystyle\times\,\hat{\delta}^{4}(\bar{k}+\bar{k}^{\prime}+\bar{q}_{1}+\bar{q}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\bar{k}_{[\rho}\varepsilon^{(\eta)*}_{\sigma]}\,\bar{k}^{\prime}_{[\alpha}\varepsilon^{\prime(\eta^{\prime})*}_{\beta]}\bar{k}^{\prime}_{[\gamma}\varepsilon^{\prime(\eta^{\prime})*}_{\delta]}\,e^{-i(\bar{k}\cdot
x+\bar{k}^{\prime}\cdot y+\bar{q}_{1}\cdot
b)}\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

In both cases, the expectation of two field strengths is given to leading
order in $g$ by a tree-level six-point amplitude.

$p_{1}$$p_{2}$$p_{1}+q_{1}$$p_{2}+q_{2}$$k$$k^{\prime}$ Figure 2: The
kinematic configuration we choose for the six-point amplitude appearing at
leading-order expectation of a pair of field strength operators.

###  2.3 Negligible variance?

We have now seen explicitly that the expectation of a single field strength is
determined by a five-point amplitude, while the expectation of two field
strengths is a six-point tree amplitude at lowest order in the coupling $g$.
But for the uncertainty in the field strength to be negligible, we need the
variance to be negligible. How can this happen?

Let us count powers of the coupling in the electromagnetic variance. The
product of two field-strength expectations is

$\displaystyle\langle\psi|\mathbbm{F}_{\mu\nu}(x)|\psi\rangle\langle\psi|\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle\sim(\mathcal{A}_{5,0})^{2}\sim(g^{3})^{2}\,.$
(40)

while the expectation of two field strengths is

$\displaystyle\langle\psi|\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle\sim\mathcal{A}_{6,0}\sim
g^{4}\,.$ (41)

Thus the situation seems to very bad: the variance is dominated by the
expectation of two field-strength operators! For the classical limit to emerge
as expected, we need the six-point tree amplitude to be suppressed somehow.

One possibility is that it is suppressed by powers of $\hbar$, but naively
that is not the case. Consider, for example, the 6-point Feynman diagram shown
in figure 3. We can count the powers of $\hbar$ associated to the diagram as
follows. Each vertex contributes a factor $\hbar^{-\frac{1}{2}}$; this is
because the dimensionless coupling is $e/\sqrt{\hbar}$ or
$\kappa/\sqrt{\hbar}$. Each messenger propagator contributes a factor
$\hbar^{-2}$ because messenger momenta are of order $\hbar$. Meanwhile each
massive propagator contributes a factor $\hbar^{-1}$; this arises since the
momenta flowing through these propagators are a sum of a massive on-shell
momentum $p$ and a messenger momentum $k$, so that the propagator denominator
is $(p+k)^{2}-m^{2}\simeq 2\hbar\,p\cdot\bar{k}$. We conclude that the six-
point tree amplitude contains terms of order $\hbar^{-6}$. Referring back to
eq. (38) or eq. (39) for the expectation of two field strengths, we see that
the observables contribute a total of $\hbar^{+5}$. Based on this counting,
the observable seems to scale as $\hbar^{-1}$, which would be a serious
obstruction to the emergence of a classical limit. Evidently there is more to
understand here: the powers of $\hbar$ don’t make sense.

$\hbar^{-\frac{1}{2}}$$\hbar^{-\frac{1}{2}}$$\hbar^{-\frac{1}{2}}$$\hbar^{-\frac{1}{2}}$$\hbar^{-2}$$\hbar^{-1}$$\hbar^{-1}$
Figure 3: A sample Feynman diagram contributing to the six-point tree
amplitude, indicating powers of $\hbar$ assigned by naive power counting to
the propagators and vertices.

It is a familiar story that power counting Feynman diagrams can be misleading:
upon combining diagrams to evaluate an amplitude, there can be cancellations.
In fact this already happens in the case of the five-point tree; there, naive
power counting suggests that the amplitude scales as $\hbar^{-9/2}$ but in
fact the leading term in the amplitude is of order $\hbar^{-7/2}$ Luna:2017dtq
666In reference Luna:2017dtq , the analysis of the five-point tree amplitude
was performed in gravity using the large mass expansion, which is equivalent
to expanding in small $\hbar$.. The question, then, of the fate of the six-
point tree amplitude in the expectation value of two field strengths becomes a
question of the overall $\hbar$ scaling of six-point tree amplitudes in QED
and gravity. We will shortly demonstrate explicitly that the QED amplitude in
fact scale as $\hbar^{-4}$; the gravitational case will be discussed in
reference Britto:2021xxx . Two powers of $\hbar$ cancel; consequently the
contribution of the six-point tree to the variance is entirely at the quantum
level.

It is amusing that at next-to-leading order in the perturbative coupling $g$,
namely order $g^{6}$, the expectation value of two field strengths is
sensitive to one-loop six-point amplitudes and to products of two tree five-
point amplitudes777We learned from G. Veneziano that the use of a cut 6-point
function to compute observables which are quadratic in the fields (such as
number or energy densities in inclusive cross-sections) goes back to Mueller’s
generalized optical theorem from the seventies (see e.g. PhysRevD.4.906  and
references therein).. These products of tree-level five-point amplitudes can
be viewed as the cut of a six-point one-loop amplitude. It is easy to check
that these scale as $\hbar^{-5}$, so in this sense they are _enhanced_
relative to the six-point tree amplitude. This is as desired for negligible
uncertainty:

$\displaystyle\langle\psi|$
$\displaystyle\mathbbm{F}_{\mu\nu}(x)|\psi\rangle\langle\psi|\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle\sim(\mathcal{A}_{5,0})^{2}\sim(g^{3})^{2}\,;$
(42) $\displaystyle\langle\psi|$
$\displaystyle\mathbbm{F}_{\mu\nu}(x)\mathbbm{F}_{\rho\sigma}(y)|\psi\rangle\sim\mathcal{A}_{6,1}\sim(\mathcal{A}_{5,0})^{2}\sim(g^{3})^{2}\,.$

###  2.4 Explicit six-point tree amplitudes

We now compute the leading contribution in $\hbar$ of the six-point tree
amplitude. We will discuss the computation explicitly in electromagnetism, and
explain only the mechanism for cancellation of apparent excess powers of
$\hbar$ in gravity (see also Britto:2021xxx ).

Suppose particle 1 has charge $Q_{1}$ while particle 2 has charge $Q_{2}$.
Then there are three gauge-invariant six-point tree partial amplitudes:

$\displaystyle\mathcal{A}_{6,0}(p_{1}+q_{1},p_{2}+q_{2}\rightarrow
p_{1},p_{2},k)=Q_{1}^{3}Q_{2}\,{A}_{(3,1)}+Q_{1}^{2}Q_{2}^{2}\,{A}_{(2,2)}+Q_{1}Q_{2}^{3}\,{A}_{(1,3)}\,.$
(43)

The “charge-ordered” partial amplitudes $A_{(3,1)}$, $A_{(2,2)}$, and
${A}_{(1,3)}$ are analogues of color-ordered amplitudes in gauge theory, which
motivates our choice of notation.

Evidently there can be no cancellation of powers of $\hbar$ between these
partial amplitudes because of the different powers of the charges. Thus the
problem reduces to computing the leading-in-$\hbar$ terms in these amplitudes.
There are two partial amplitudes to consider, since ${A}_{(1,3)}$ can be
obtained from ${A}_{(3,1)}$ by trivially swapping the labels 1 and 2.

We have performed two separate computations of these partial amplitudes.
Firstly, we made use of standard automated tools to directly compute the
amplitude in full detail. Specifically, we used FeynArt Hahn:2000kx  to create
a model for scalar QED, extracting the Feynman rules directly from the
Lagrangian. FeynCalc Shtabovenko:2016sxi ; Shtabovenko:2020gxv  can
automatically generate all the topologies relevant for the calculation of
$\mathcal{A}_{6,0}$. There are 42 diagrams to be computed, and FeynCalc
provided direct automatic expressions for each of these. We processed our
expressions further in Mathematica with the tensor package xAct xAct , which
help to extract the classical limit. The final result for $\mathcal{A}_{6,0}$
at leading order in the $\hbar$ expansion is provided as an attachment to this
paper and is indeed of order $\hbar^{-4}$.

To gain further insight we performed a separate computation of the partial
amplitudes $A_{(3,1)}$ and $A_{(2,2)}$ in a convenient gauge which greatly
reduced the labour necessary to see that two powers of $\hbar$ cancel. The
gauge we chose (referring to the momentum routing in equation (43)) is

$\displaystyle p_{1}\cdot\varepsilon(k_{i})=0\,\,\text{for}\,\,i=1,2,$ (44)

where $p_{1}$ is the momentum of particle 1 while $k_{i}$ is the outgoing
momentum of photon $i$.

The effect of this choice is twofold; it removes many diagrams from the
calculation and those that remain get an $\hbar$ enhancement from each
emission vertex. For example, consider $A_{(3,1)}$: in this case, the emitted
photon is radiated from particle 1. With our momentum labelling convention the
emission vertices will produce factors of the form
$(2p_{1}+\hbar\bar{Q}))\cdot\varepsilon(k_{i})$, where $\bar{Q}$ is some
combination of wavenumbers. The first part vanishes leaving only the $\hbar$
enhanced $\varepsilon(k_{i})\cdot\bar{Q}$ term. Any diagram with a photon
emitted from the outgoing line of particle 1 vanishes, as it is proportional
to $(2p_{1}+\hbar\bar{k}_{i})\cdot\varepsilon(k_{i})=0$. In what follows we
will write $\varepsilon(k_{i})=\varepsilon_{i}$. We will also suppress the
$i\epsilon$ factors in the propagators888The $i\epsilon$ factors are often
important — and will play an important role in section 3.2 — but in this
computation they are spectators.; they all implicitly come with $+i\epsilon$.
Finally we will refer to each diagram contributing to the amplitude by, for
example, $D{(3,1)}$ so that

$\displaystyle iA_{(3,1)}=iD_{(3,1)cubic}+iD_{(3,1)quartic}.$ (45)

Some of the sub-amplitudes are given by a single diagram, whereas others are
made up of multiple diagrams.

Our gauge choice is most powerful in the case of $A_{(3,1)}$, so we discuss
this case in most detail. The Feynman diagrams that constitute this amplitude
can be split into 3 classes: the first involves single photon emissions coming
from cubic vertices, the second has precisely one photon emitted into the
final state from a quartic vertex, while the third class has two photons
emitted from the same quartic vertex. These classes are shown in figure 4,
after removing diagrams which vanish by gauge choice. The first diagram is an
example of the first class, the second an example of the second class and the
last two diagrams are in the third class.

Figure 4: Diagrams in $Q_{1}^{3}Q_{2}$ sector.

We choose a particular ordering of $k_{1}$ and $k_{2}$ for the calculation,
and include the permuted case by swapping $k_{1}\leftrightarrow k_{2}$. For
the first class there is a single diagram to compute after gauge fixing, and
it is trivial to write down the leading term in the amplitude and see that it
has the desired scaling. This diagram is

$\displaystyle D_{(3,1)}|_{cubic}=$
$\displaystyle\frac{1}{\hbar^{4}\bar{q}_{2}^{2}}\left[\frac{2(\varepsilon_{2}\cdot\bar{q}_{1})(\varepsilon_{1}\cdot(2\bar{q}_{1}-\bar{k}_{2}))(2p_{2}+\hbar\bar{q}_{2})\cdot(2p_{1}-\hbar\bar{q}_{2})}{(-2p_{1}\cdot\bar{q}_{2}+\hbar\bar{q}_{2}^{2})(2p_{1}\cdot(\bar{q}_{1}-\bar{k}_{1})+\hbar(\bar{q}_{1}-\bar{k}_{1})^{2})}\right]$
(46) $\displaystyle=$
$\displaystyle\frac{1}{\hbar^{4}\bar{q}_{2}^{2}}\left[\frac{4p_{1}\cdot
p_{2}(\varepsilon_{2}\cdot\bar{q}_{1})(\varepsilon_{1}\cdot(2\bar{q}_{1}-\bar{k}_{2}))}{2(-p_{1}\cdot\bar{q}_{2})(p_{1}\cdot(\bar{q}_{1}-\bar{k}_{1})}+\mathcal{O}(\hbar)\right]$
$\displaystyle=$ $\displaystyle\frac{2p_{1}\cdot
p_{2}}{\hbar^{4}\bar{q}_{2}^{2}}\left[\frac{(\varepsilon_{2}\cdot\bar{q}_{1})(\varepsilon_{1}\cdot(\bar{q}_{1}-\bar{q}_{2}))}{(p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2}))(p_{1}\cdot\bar{k}_{1})}\right]+\mathcal{O}(\hbar^{-3}).$

We used momentum conservation $q_{1}+q_{2}=k_{1}+k_{2}$ to write our
expressions in terms of just the $k_{i}$ or the $q_{i}$. The choice here is
most natural for obtaining a similarly simple expression for the amplitude
with $k_{1}$ and $k_{2}$ swapped.

The second class is actually tractable without fixing a gauge, as there is
only a single cancellation to show. However with our gauge fixing this class
becomes trivial, and there is only a single diagram to compute. This is

$\displaystyle
D_{(3,1)}|_{cubic/quartic}=\frac{4(p_{2}\cdot\varepsilon_{2})(\bar{q}_{1}\cdot\varepsilon_{1})}{\hbar^{4}\bar{q}_{2}^{2}p_{1}\cdot\bar{k}_{1}}.$
(47)

The final class is unaffected by our choice of gauge as it is proportional to
$\varepsilon_{1}\cdot\varepsilon_{2}$. After gauge fixing this is naively
$\hbar^{-5}$, so we must find a single cancellation. The mechanism of the
cancellation is identical to the case of the five-point tree amplitude. This
is done in Luna:2017dtq ; KMOC , but we shall review it here for completeness.
The key step is to make use of the on-shell conditions

$\displaystyle(p_{i}+q_{i})^{2}=p_{i}=m_{i}^{2},\,\,\,\,i=1,2,$ (48)

which allows us, after the $\hbar$ rescaling, to replace
$2p_{i}\cdot\bar{q}_{i}\rightarrow-\hbar\bar{q}_{i}^{2}$ in the propagators.
Lastly, we Taylor expand. There are two diagrams to compute which are,

$\displaystyle D_{(3,1)}|_{quartic,1}=$
$\displaystyle-\frac{2\varepsilon_{1}\cdot\varepsilon_{2}}{\hbar^{5}\bar{q}_{2}^{2}}\left[\frac{(2p_{1}+\hbar(2\bar{q}_{1}+\bar{q}_{2}))\cdot(2p_{2}+\hbar\bar{q}_{2})}{2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})+\hbar(\bar{k}_{1}+\bar{k}_{2})^{2}}\right]$
(49) $\displaystyle=$
$\displaystyle-\frac{2\varepsilon_{1}\cdot\varepsilon_{2}}{\hbar^{5}\bar{q}_{2}^{2}}\left[\frac{4p_{1}\cdot
p_{2}}{2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})}\right.$
$\displaystyle\qquad\qquad\quad\left.+\hbar\left(\frac{4p_{2}\cdot\bar{q}_{1}+2p_{1}\cdot\bar{q}_{2}}{2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})}-\frac{4p_{1}\cdot
p_{2}(\bar{k}_{1}+\bar{k}_{2})^{2}}{(2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2}))^{2}}\right)\right]$

and

$\displaystyle D_{(3,1)}|_{quartic,2}=$
$\displaystyle-\frac{2\varepsilon_{1}\cdot\varepsilon_{2}}{\hbar^{5}\bar{q}_{2}^{2}}\left[\frac{4p_{1}\cdot
p_{2}}{-2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})}\right.$ (50)
$\displaystyle\qquad\qquad\quad+\left.\hbar\left(\frac{4p_{1}\cdot\bar{q}_{2}}{-2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})}+\frac{4p_{1}\cdot
p_{2}((\bar{k}_{1}+\bar{k}_{2})^{2}-\bar{q}_{1}^{2})}{(2p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2}))^{2}}\right)\right].$

Notice the most singular terms are equal up to a sign, and so cancel.
Combining the remaining terms we obtain

$\displaystyle
D_{(3,1)}|_{quartic}=-\frac{\varepsilon_{1}\cdot\varepsilon_{2}}{\hbar^{4}\bar{q}_{2}^{2}}\left[\frac{4(p_{1}\cdot
p_{2})(\bar{q}_{2}\cdot(\bar{k}_{1}+\bar{k}_{2}))}{(p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2}))^{2}}+\frac{4p_{2}\cdot\bar{q}_{1}+2p_{1}\cdot\bar{q}_{2}}{p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})}\right]\,.$
(51)

These can be combined as $A_{(3,1)}=D_{(3,1)cubic}+D_{(3,1)quartic}$ yielding,

$\displaystyle A_{(3,1)}=$
$\displaystyle\frac{1}{\hbar^{4}\bar{q}_{2}^{2}}\Bigg{[}\frac{4p_{1}\cdot
p_{2}(\varepsilon_{2}\cdot\bar{q}_{1})(\varepsilon_{1}\cdot(\bar{q}_{1}-\bar{q}_{2}))}{2(p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2}))(p_{1}\cdot\bar{k}_{1})}+\frac{4p_{1}\cdot
p_{2}(p_{2}\cdot\varepsilon_{2})(\bar{q}_{1}\cdot\varepsilon_{1})}{p_{1}\cdot\bar{k}_{1}}$
(52)
$\displaystyle-(\varepsilon_{1}\cdot\varepsilon_{2})\left(\frac{4(p_{1}\cdot
p_{2})(\bar{q}_{2}\cdot(\bar{k}_{1}+\bar{k}_{2}))}{(p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2}))^{2}}+\frac{4p_{2}\cdot\bar{q}_{1}+2p_{1}\cdot\bar{q}_{2}}{p_{1}\cdot(\bar{k}_{1}+\bar{k}_{2})}\right)\Bigg{]}$
$\displaystyle+(k_{1}\leftrightarrow k_{2}).$

The story is very similar for $A_{(2,2)}$. We can split into the same 3
classes, use gauge fixing to get rid of one factor of $\hbar$ and then massage
using the on-shell constraints to show the final cancellation. Here we just
quote the result and give details in the appendix. The result is

$\displaystyle A_{(2,2)}=$
$\displaystyle\frac{4}{\hbar^{4}(\bar{q}_{2}-\bar{k}_{2})^{2}}\Bigg{[}4\varepsilon_{1}\cdot\varepsilon_{2}+\frac{(\varepsilon_{1}\cdot
p_{2})(\varepsilon_{2}\cdot\bar{q}_{2})}{p_{2}\cdot\bar{k}_{2}}-\frac{(\varepsilon_{2}\cdot
p_{2})(\varepsilon_{1}\cdot(\bar{q}_{2}+\bar{q}_{1}))}{2p_{2}\cdot\bar{k}_{2}}$
(53) $\displaystyle\frac{p_{1}\cdot
p_{2}(\varepsilon_{1}\cdot\bar{q}_{1})(\varepsilon_{2}\cdot\bar{q}_{2})}{(p_{2}\cdot\bar{k}_{2})(p_{1}\cdot\bar{k}_{1})}-\frac{(p_{1}\cdot\bar{k}_{2})(\varepsilon_{1}\cdot\bar{q}_{1})(\varepsilon_{2}\cdot
p_{2})}{(p_{2}\cdot\bar{k}_{2})(p_{1}\cdot\bar{k}_{1})}$
$\displaystyle-\frac{(\varepsilon_{1}\cdot p_{2})(\varepsilon_{2}\cdot
p_{2})(\bar{q}_{2}\cdot\bar{k}_{2})}{(p_{2}\cdot\bar{k}_{2})^{2}}-\frac{p_{1}\cdot
p_{2}(\varepsilon_{1}\cdot\bar{q}_{1})(\varepsilon_{2}\cdot
p_{2})\bar{q}_{2}\cdot\bar{k}_{2}}{(p_{2}\cdot\bar{k}_{2})^{2}(p_{1}\cdot\bar{k}_{1})}\Bigg{]}$
$\displaystyle+(k_{1}\leftrightarrow k_{2}).$

Finally $A_{(1,3)}$ can be obtained by swapping the labels $1\leftrightarrow
2$ in the expression for $A_{(3,1)}$ (written in the gauge where
$p_{2}\cdot\varepsilon_{i}=0$. Of course the partial amplitudes themselves are
gauge-invariant.) In all cases, the $\hbar$ scaling is as required from
negligible variance.

##  3 Transfer relations

Our discussion so far reveals that scattering amplitudes, viewed as Laurent
series in $\hbar$, obey certain properties which permit the emergence of a
classical limit through negligible uncertainty. This Laurent expansion can
also be viewed as an expansion in small momentum transfers divided by the
centre-of-mass energy $\sqrt{s}$. In fact, the emergence of the classical
limit imposes an _infinite_ set of these relationships, which we will call
“transfer relations” on scattering amplitudes. In this section we will
describe the origin of these relations, and explicitly demonstrate a non-
trivial example at one loop and five points.

###  3.1 Mixed variances

To see where these relationships are coming from, recall that the double
field-strength expectation (38) depends on a six-point amplitude. We have seen
that the dominant term is actually the six-point one-loop amplitude, occurring
at next-to-leading order in the expansion in $g$. At this order an additional
term contributes to the double field-strength expectation; this term is the
product of two five-point amplitudes. Now, negligible uncertainty demands that
the complete double field-strength expectation must be the product of two
single field-strength expectations. At leading order in the coupling, and
leading non-trivial order in $\hbar$, we conclude that there must exist a
relationship between the leading-in-$\hbar$ six-point one-loop amplitude and
the product of two five-point trees.

Further examples of relationships between amplitudes can be obtained by
considering expectations of three (or more) field strengths, leading to
relationships between seven- (or higher-) point loop amplitudes and products
of three (or more) five-point amplitudes.

Yet more relationships occur by considering expectations of products of
operators including field strengths and momenta. For example, consider the
variance

$\displaystyle
V_{\mu\nu\rho}\equiv\langle\psi|S^{\dagger}\mathbbm{F}_{\nu\rho}(x)S\mathbbm{P}_{\mu}|\psi\rangle-\langle\psi|S^{\dagger}\mathbbm{F}_{\nu\rho}(x)S|\psi\rangle\langle\psi|\mathbbm{P}_{\mu}|\psi\rangle\,.$
(54)

This is the variance in a measurement of the initial momentum and the future
field strength; it must be negligible in the classical regime. In a quantum-
first approach, however, this variance will not vanish. Indeed it need not be
real:

$\displaystyle
V_{\mu\nu\rho}^{*}=\langle\psi|\mathbbm{P}_{\mu}S^{\dagger}\mathbbm{F}_{\nu\rho}(x)S|\psi\rangle-\langle\psi|S^{\dagger}\mathbbm{F}_{\nu\rho}(x)S|\psi\rangle\langle\psi|\mathbbm{P}_{\mu}|\psi\rangle\neq
V_{\mu\nu\rho}.$ (55)

We can derive an interesting constraint on the five-point one-loop amplitude
by demanding that imaginary part of this variance vanishes in the classical
approximation. We therefore define

$\displaystyle\mathcal{O}_{\mu\nu\rho}$
$\displaystyle=i(V_{\mu\nu\rho}^{*}-V_{\mu\nu\rho})$ (56)
$\displaystyle=i\langle\psi|\mathbbm{P}_{\mu}S^{\dagger}\mathbbm{F}_{\nu\rho}(x)S-S^{\dagger}\mathbbm{F}_{\nu\rho}(x)S\mathbbm{P}_{\mu}|\psi\rangle\,.$

Expanding the states as usual, we easily find

$\displaystyle\mathcal{O}_{\mu\nu\rho}=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2})$
$\displaystyle\,\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})i(p^{\prime}_{1\mu}-p_{1\mu})$
(57) $\displaystyle\times\langle
p^{\prime}_{1}p^{\prime}_{2}|i(\mathbbm{F}_{\nu\rho}(x)T-T^{\dagger}\mathbbm{F}_{\nu\rho}(x))+T^{\dagger}\mathbbm{F}_{\nu\rho}(x)T|p_{1}p_{2}\rangle\,.$

The factor $i(p^{\prime}_{1\mu}-p_{1\mu})$ is important here: working at
leading perturbative order, this factor is of order $\hbar$. It is also worth
noting that we may write the expectation of the field strength itself as

$\displaystyle\langle\psi|\mathbbm{F}_{\nu\rho}|\psi\rangle=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2})$
$\displaystyle\,\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})$
(58) $\displaystyle\times\langle
p^{\prime}_{1}p^{\prime}_{2}|i(\mathbbm{F}_{\nu\rho}(x)T-T^{\dagger}\mathbbm{F}_{\nu\rho}(x))+T^{\dagger}\mathbbm{F}_{\nu\rho}(x)T|p_{1}p_{2}\rangle\,.$

Thus the $i(p^{\prime}_{1\mu}-p_{1\mu})\sim\hbar$ factor in the variance is
the key distinction between the variance, which vanishes classically, and the
field strength which of course should not vanish classically. As we have
already seen that the field strength is related to five-point amplitudes, it
is now clear that the condition of vanishing $\mathcal{O}_{\mu\nu\rho}$ will
become a condition on five-point amplitudes.

It is useful to break the variance $\mathcal{O}_{\mu\nu\rho}$ up into two
structures:

$\displaystyle\mathcal{O}_{\mu\nu\rho}^{(1)}=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2})$
$\displaystyle\,\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})i(p^{\prime}_{1\mu}-p_{1\mu})$
(59) $\displaystyle\times\langle
p^{\prime}_{1}p^{\prime}_{2}|i(\mathbbm{F}_{\nu\rho}(x)T-T^{\dagger}\mathbbm{F}_{\nu\rho}(x))|p_{1}p_{2}\rangle\,,$

and

$\displaystyle\mathcal{O}_{\mu\nu\rho}^{(2)}=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2})$
$\displaystyle\,\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})i(p^{\prime}_{1\mu}-p_{1\mu})$
(60) $\displaystyle\times\langle
p^{\prime}_{1}p^{\prime}_{2}|T^{\dagger}\mathbbm{F}_{\nu\rho}(x)T|p_{1}p_{2}\rangle\,.$

Both of these objects are real, which is convenient in terms of keeping the
expressions simple.

We may simplify these structures using the explicit expression for the field
strength given in equation (10). For $\mathcal{O}^{(1)}$ we find

$\displaystyle\mathcal{O}_{\mu\nu\rho}^{(1)}$
$\displaystyle=2\operatorname{Re}\frac{1}{\sqrt{\hbar}}\sum_{\eta}\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2},k)\phi_{b}^{*}(p_{1}^{\prime},p_{2}^{\prime})\phi_{b}(p_{1},p_{2})\times$
(61) $\displaystyle\hskip 100.0pt\times i(p_{1\mu}^{\prime}-p_{1\mu})\langle
k^{\eta},p_{1}^{\prime},p_{2}^{\prime}|T|p_{1},p_{2}\rangle\,\bar{k}^{\vphantom{\mu}}_{[\nu}\varepsilon^{(\eta)*}_{\rho]}\,e^{-i\bar{k}\cdot
x}\,,$

which should be compared to equation (28). Again we see that the crucial new
ingredient is a factor $i(p_{1\mu}^{\prime}-p_{1\mu})$. In the classical
regime, we may write this term as

$\displaystyle\mathcal{O}_{\mu\nu\rho}^{(1)}$
$\displaystyle=2\operatorname{Re}\hbar^{9/2}\sum_{\eta}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})\,$
(62) $\displaystyle\hskip 10.0pt\times
i\bar{q}_{\mu}\mathcal{A}_{5}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})\,\hat{\delta}^{4}(\bar{k}+\bar{q}_{1}+\bar{q}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\nu}\varepsilon^{(\eta)*}_{\rho]}\,e^{-i(\bar{k}\cdot
x+\bar{q}_{1}\cdot b)}\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

Referring back once more to equation (33), the additional $\hbar$ suppression
is now manifest.

In order to control the $\hbar$ expansion of scattering amplitudes, it is
useful to introduce some further notation. Let us write the amplitudes as
explicit Laurent series in $\hbar$ building on equation (35). For example, at
five-points we may write

$\displaystyle\mathcal{A}_{5,0}(i\rightarrow
f)=\hbar^{-7/2}\left(\mathcal{A}_{5,0}^{(0)}(i\rightarrow
f)+\hbar\mathcal{A}_{5,0}^{(1)}(i\rightarrow f)+\cdots\right)\,,$ (63)
$\displaystyle\mathcal{A}_{5,1}(i\rightarrow
f)=\hbar^{-9/2}\left(\mathcal{A}_{5,1}^{(0)}(i\rightarrow
f)+\hbar\mathcal{A}_{5,1}^{(1)}(i\rightarrow f)+\cdots\right)\,.$

We have scaled out the dominant (inverse) power of $\hbar$; the quantities
$\mathcal{A}_{n,L}^{(p)}$ are $\hbar$-independent gauge-invariant sub-
amplitudes; it is precisely these quantities that are related by our
reasoning. This expansion defines an infinite set of objects
$\mathcal{A}_{n,L}^{(p)}$ which could in principle be reassembled into the
full amplitude. They are a kind of partial amplitude, but distinct from the
usual use of this term. We will therefore refer to them as “fragmentary
amplitudes”, or simply as “fragments.”

Since $\hbar$ is dimensionful, it is useful to view these fragmentary
amplitudes in a slightly different way. Amplitudes are functions of Mandelstam
invariants; in the semi-classical region, we are expanding in powers of
momentum transfers, such as $q^{2}=\hbar^{2}\bar{q}^{2}$ at four points,
divided by Mandelstam $s=(p_{1}+p_{2})^{2}$. The semiclassical expansion is an
expansion in powers of $\hbar\sqrt{-\bar{q}^{2}/s}$. More general amplitudes
involve a richer set of momentum transfers $\bar{q}_{ij}^{2}$; our expansion
is in powers of $\hbar\sqrt{-\bar{q}_{ij}^{2}/s}$. We only consider amplitudes
with two incoming massive particles.

We may also view the expansion as being in (inverse) powers of the large mass
of the scattering particles Luna:2017dtq ; Damgaard:2019lfh ; Aoude:2020onz ;
Haddad:2020tvs . This makes contact with effective field theory, especially
heavy quark effective theory or, more generally, heavy particle effective
theories as has been emphasised in references Damgaard:2019lfh ; Aoude:2020onz
; Haddad:2020tvs . Our fragmentary amplitudes correspond in this context to
the standard HQET expansion in inverse powers of heavy masses. It seems likely
that a study of the properties of amplitudes in these theories would
illuminate the structure of the fragmentary amplitudes.

Notice that this expansion is analogous, but different, to a soft expansion.
In the soft expansion we take the momentum of an individual particle soft. In
this transfer expansion we take the momenta in all messenger lines to be of
the same order, and small compared to the incoming centre of mass energy. It
is possible to perform the transfer expansion and then, in a second stage, to
single out some line, say an outgoing photon, and take its momentum to be
softer than all other messenger lines. This yields the soft limit of the
transfer expansion. It corresponds to the low-frequency limit in the classical
approximation. Interesting classical physics, including memory effects, appear
in this region Laddha:2018rle ; Laddha:2018myi ; Laddha:2018vbn ;
Sahoo:2018lxl ; Bautista:2019tdr ; Laddha:2019yaj ; Saha:2019tub ;
Sahoo:2021ctw ; Bautista:2021llr .

Now at classical order ($\hbar^{0}$) the tree level amplitude
$\mathcal{A}_{5,0}$ does not appear in $\mathcal{O}_{\mu\nu\rho}^{(1)}$ on
account of the explicit factor $\hbar^{9/2}$ in equation (62). The leading in
$g$, non-trivial, classical contribution arises from the fragment
$\mathcal{A}_{5,1}^{(0)}$. We conclude then that

$\displaystyle\mathcal{O}_{\mu\nu\rho}^{(1)}$
$\displaystyle=2\operatorname{Re}\sum_{\eta}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})\,\times$
(64) $\displaystyle\times
i\bar{q}_{\mu}\mathcal{A}_{5,1}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})\,\hat{\delta}^{4}(\bar{k}+\bar{q}_{1}+\bar{q}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\nu}\varepsilon^{(\eta)*}_{\rho]}\,e^{-i(\bar{k}\cdot
x+\bar{q}_{1}\cdot b)}\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

The relevant fragmentary amplitude is the leading-in-$\hbar$ five-point one-
loop amplitude, sometimes known as the “superclassical” part of the one-loop
amplitude. Of course in this context this fragment of the amplitude is
contributing precisely at classical order.

Now the full $\mathcal{O}_{\mu\nu\rho}$ should vanish at classical order.
Since $\mathcal{O}_{\mu\nu\rho}^{(1)}\neq 0$, it must be that the second
structure $\mathcal{O}_{\mu\nu\rho}^{(2)}$ cancels the contribution of
equation (64). We find that

$\displaystyle\mathcal{O}_{\mu\nu\rho}^{(2)}$
$\displaystyle=2\operatorname{Re}\sum_{\eta}\biggl{\langle}\\!\\!\\!\biggl{\langle}\int\mathrm{d}\Phi(\bar{k})\,\hat{\mathrm{d}}^{4}\bar{q}_{1}\hat{\mathrm{d}}^{4}\bar{q}_{2}\hat{\mathrm{d}}^{4}\bar{w}_{1}\hat{\mathrm{d}}^{4}\bar{w}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}\cdot\bar{q}_{2})\hat{\delta}(2p_{1}\cdot\bar{w}_{1})\hat{\delta}(2p_{2}\cdot\bar{w}_{2})$
(65)
$\displaystyle\times\bar{q}_{\mu}\,\hat{\delta}^{4}(\bar{k}+\bar{q}_{1}+\bar{q}_{2})\hat{\delta}^{4}(\bar{q}_{1}+\bar{q}_{2}-\bar{w}_{1}-\bar{w}_{2})\,\bar{k}^{\vphantom{\mu}}_{[\nu}\varepsilon^{(\eta)*}_{\rho]}\,e^{-i(\bar{k}\cdot
x+\bar{q}_{1}\cdot b)}$
$\displaystyle\times\mathcal{A}_{5,0}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+w_{1},p_{2}+w_{2},k^{\eta})\mathcal{A}_{4,0}^{(0)}(p_{1}+w_{1},p_{2}+w_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2})\biggr{\rangle}\\!\\!\\!\biggr{\rangle}\,.$

Comparing equations (64) and (65), the condition for vanishing $\mathcal{O}$
is

$\displaystyle i$ $\displaystyle\mathcal{A}_{5,1}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k^{\eta})$ (66)
$\displaystyle=-\int\hat{\mathrm{d}}^{4}\bar{w}_{1}\hat{\mathrm{d}}^{4}\bar{w}_{2}\,\hat{\delta}(2p_{1}\cdot\bar{w}_{1})\hat{\delta}(2p_{2}\cdot\bar{w}_{2})\hat{\delta}^{4}(\bar{q}_{1}+\bar{q}_{2}-\bar{w}_{1}-\bar{w}_{2})$
$\displaystyle\hskip 10.0pt\times\mathcal{A}_{5,0}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+w_{1},p_{2}+w_{2},k^{\eta})\mathcal{A}_{4,0}^{(0)}(p_{1}+w_{1},p_{2}+w_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2})\,.$

Thus the dominant part of the five-point one-loop amplitude is given by the
tree five-point and tree four-point amplitudes; we will check this relation
explicitly in the next subsection.

Clearly this explicit example is one among an infinite set of relationships.
Variances involving one field strength operator and two momenta will lead to
relationships among two-loop five-point amplitudes and the product of one
five-point tree and two four-point trees. We can continue, in principle, as
far as we wish generating similar relations. These negligible uncertainty
relations generalise the well-known relations between multiloop four-point
amplitudes required for eikonal exponentiation. Indeed consideration of
expectations such as

$\displaystyle\langle\psi|S^{\dagger}\mathbbm{P}_{\mu_{1}}\mathbbm{P}_{\mu_{2}}\cdots\mathbbm{P}_{\mu_{n}}S|\psi\rangle\simeq\langle\psi|S^{\dagger}\mathbbm{P}_{\mu_{1}}|\psi\rangle\langle\psi|S^{\dagger}\mathbbm{P}_{\mu_{2}}|\psi\rangle\cdots\langle\psi|S^{\dagger}\mathbbm{P}_{\mu_{n}}|\psi\rangle\,,$
(67)

shows that there must be a relationship between the $n-1$ loop four-point
amplitude and the product of $n$ tree amplitudes.

Thus we find a remarkable abundance of relationships between multiloop,
multileg amplitudes, considered as Laurent series in $\hbar$, forced on us by
the _absence_ of uncertainty in the classical regime. In the next section of
this article we will interpret these relationships in terms of a radiative
generalisation of the eikonal exponentiation.

As well as finding explicit relations between different fragmentary
amplitudes, we can use similar ideas to determine the $\hbar$ scaling
associated with fragments in the transfer expansion. The kinds of multiple
cancellations of $\hbar$ powers we saw at six points must continue to occur at
higher points. The reason again follows from considering expectations of
products of more than two field-strength operators.

The arguments are based simply on counting powers of coupling and $\hbar$. We
know that for the single expectation, at leading order, we have

$\displaystyle\langle\mathbbm{F}_{\mu\nu}\rangle\sim g^{3}.$ (68)

This means that we must also have
$\langle\mathbbm{F}^{n}\rangle\sim(g^{3})^{n}$. Now we perform the KMOC
analysis of $\langle\mathbbm{F}^{n}\rangle$. Following the steps of the
calculation earlier in section 2.2 we find, schematically, that

$\displaystyle\langle\mathbbm{F}^{n}\rangle\sim\hbar^{3n/2+2}\int\mathcal{A}_{4+n}.$
(69)

These relations allow us to deduce two things. Firstly the relevant fragment
of the complete amplitude $\mathcal{A}_{4+n}$ must scale as $\hbar^{-3n/2-2}$.
Secondly this fragment must have $3n$ powers of the coupling $g$ — this
corresponds to having $n-1$ loops. From this we can also infer the scaling of
all other loop and tree amplitudes, in the classical limit, since each loop
contributes an extra factor of $\hbar^{-1}$. In particular the tree scaling
will be

$\displaystyle\mathcal{A}_{4+n,0}\sim\hbar^{-n/2-3}.$ (70)

This is consistent with the scaling we computed above for six points $(n=2)$,
and we have also checked explicitly at seven points. It is interesting to see
how these two very simple power counting arguments have completely constrained
the $\hbar$ scaling of all $2\rightarrow 2+n$ amplitudes.

###  3.2 One loop factorisation

We now turn to verifying equation (66). For simplicity we focus on the case of
scalar QED, though the general nature of our arguments indicates that the
result should also hold in gravity and in Yang-Mills theory. In order to keep
the computational labour to the minimum necessary, we take advantage of
lessons we learned in the context of the six-point tree amplitude in section
2.4. First, we note that the scalar QED five-point amplitudes can be reduced
to gauge-invariant partial amplitudes analogous to colour-ordered amplitudes
in Yang-Mills theory. In particular we write

$\displaystyle\mathcal{A}_{4,0}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+w_{1},p_{2}+w_{2})$ $\displaystyle=Q_{1}Q_{2}A_{(1,1)}\,,$ (71)
$\displaystyle\mathcal{A}_{5,0}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k)$
$\displaystyle=Q_{1}Q_{2}^{2}A_{(1,2)}+Q_{1}^{2}Q_{2}A_{(2,1)}\,,$
$\displaystyle\mathcal{A}_{5,1}^{(0)}(p_{1}p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2},k)$
$\displaystyle=Q_{1}^{2}Q_{2}^{3}A_{(2,3)}+Q_{1}^{3}Q_{2}^{2}A_{(3,2)}\,.$

In view of the symmetry between $A_{2,3}$ and $A_{3,2}$ we may compute just
one choice: we choose to focus on the charge sector $Q_{1}^{2}Q_{2}^{3}$.

Second, we find it useful to choose an explicit gauge, namely

$\displaystyle\varepsilon_{\eta}(\bar{k})\cdot p_{2}=0\,.$ (72)

This choice drastically reduces the relevant number of terms in the $\hbar$
expansion. It is trivial to determine the tree partial amplitudes in this
gauge, which are

$A_{(1,1)}=e^{2}\frac{4p_{1}\cdot{p}_{2}}{\bar{w}_{1}^{2}}\,,$ (73)

and

$\displaystyle A_{(1,2)}=\frac{4e^{3}}{\bar{q}_{1}^{2}}$
$\displaystyle\left(p_{1}\cdot\varepsilon_{\eta}+\frac{p_{1}\cdot
p_{2}\,\varepsilon_{\eta}\cdot\bar{q}_{1}}{p_{2}\cdot\bar{k}}\right)\,.$ (74)

The anatomy of (66) is that the leading-in-$\hbar$ fragment of the one-loop
five-point amplitude $A_{(2,3)}$ will organise itself into a product of
$A_{(1,1)}$ times the four terms that constitute $A_{(1,2)}$. Our strategy
will be to isolate these terms one by one. Let us start gathering the relevant
diagrams of $A_{(2,3)}$. At one loop, and in the classical limit, we need to
consider the transfer expansion of the following five-point diagrams with a
photon emitted in the final state:

$p_{1}+q_{1}$$p_{2}+q_{2}$$k$$p_{1}$$p_{2}$$=$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+\,\cdots$

The ellipsis indicate purely quantum diagrams which are not relevant for us.

As in 2.4 we tidy our expressions up by making use of the on-shell conditions.
Using the momentum labelling in the figure above these read

$p_{1}\cdot\bar{q}_{1}=p_{2}\cdot\bar{q}_{2}=\mathcal{O}(\hbar).$ (75)

Keeping this in mind we start to compute the diagrams. It is helpful to
compute the first six diagrams in the figure above, which involve only three-
point vertices, separately from the rest of the diagrams (which involved
contact four-point vertices). We therefore write

$\displaystyle A_{(2,3)}=A_{(2,3)}^{\textrm{3pt}}+A_{(2,3)}^{\textrm{4pt}}\,.$
(76)

We focus first on the six diagrams which constitute
$A_{(2,3)}^{\textrm{3pt}}$. For clarity, let us begin by describing the
contribution from the pentagon diagram, in our gauge (72) in detail. We choose
the momentum routing shown in figure 5. On the support of the momentum-
conserving delta function $\hat{\delta}^{4}(q_{1}+q_{2}+k)$, its contribution
to $A_{(2,3)}$ is

$ie^{5}\int\frac{\hat{{\mathrm{d}}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}\frac{(4p_{1}\cdot
p_{2})^{2}(-2\,\varepsilon_{\eta}\cdot\bar{l})}{(2p_{1}\cdot\bar{l})(-2p_{2}\cdot\bar{l})(-2p_{2}\cdot(\bar{k}+\bar{l}))}.$
(77)

$p_{1}+q_{1}$$p_{1}$$p_{2}$$p_{2}+q_{2}$$l$$l-q_{1}$$p_{2}-l$$p_{1}+l$$p_{2}-l-k$
Figure 5: Momentum routing for the pentagon contribution to the five-point
one-loop amplitude.

The sum of the cubic diagrams yields

$\begin{split}A_{(2,3)}^{\textrm{3pt}}&=ie^{5}\int\frac{\hat{{\mathrm{d}}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}(4p_{1}\cdot
p_{2})^{2}\times\\\
&\left(\frac{-2\varepsilon_{\eta}\cdot\bar{l}}{(2p_{1}\cdot\bar{l})(-2p_{2}\cdot\bar{l})(-2p_{2}\cdot(\bar{k}+\bar{l}))}\right.-\frac{2\varepsilon_{\eta}\cdot\bar{q}_{1}}{(2p_{1}\cdot\bar{l})(-2p_{2}\cdot\bar{l})(2p_{2}\cdot\bar{k})}\\\
&+\frac{2\varepsilon_{\eta}\cdot(\bar{l}-\bar{q}_{1})}{(2p_{1}\cdot\bar{l})(2p_{2}\cdot(\bar{k}+\bar{l}))(2p_{2}\cdot\bar{l})}-\left.\frac{2\varepsilon_{\eta}\cdot\bar{q}_{1}}{(2p_{1}\cdot\bar{l})(2p_{2}\cdot(\bar{k}+\bar{l}))(2p_{2}\cdot\bar{k})}\right).\end{split}$
(78)

Note that the signs in the linearised propagators are important! We use them
to indicate the hidden $i\epsilon$’s,

$\displaystyle\frac{1}{\pm p\cdot\bar{l}}\equiv\frac{1}{\pm
p\cdot\bar{l}+i\epsilon}\neq\pm\frac{1}{p\cdot\bar{l}}=\frac{1}{\pm
p\cdot\bar{l}\pm i\epsilon}.$ (79)

which allows us to make use of the following identity

$\displaystyle-i\hat{\delta}(p\cdot\bar{l})=\frac{1}{p\cdot\bar{l}}+\frac{1}{-p\cdot\bar{l}}.$
(80)

In order to make use of this identity we apply a change of variables
$l\to\bar{q}_{1}-l$ in the last two terms in (78). This, along with the on-
shell conditions, allows pairs of terms to take an almost identical form —
denominators differ only by a sign in the $p_{1}\cdot l$ term which is
precisely what is needed to apply (80). The amplitude then reduces to

$\begin{split}A_{(2,3)}^{\textrm{3pt}}&=4e^{5}(p_{1}\cdot
p_{2})^{2}\int\frac{\hat{{\mathrm{d}}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}\frac{\hat{\delta}(p_{1}\cdot\bar{l})}{-p_{2}\cdot\bar{l}}\left(-\frac{\varepsilon_{\eta}\cdot\bar{q}_{1}}{p_{2}\cdot\bar{k}}-\frac{\varepsilon_{\eta}\cdot\bar{l}}{-p_{2}\cdot(\bar{k}+\bar{l})}\right)\,.\end{split}$
(81)

It is possible to expose a second delta function in this expression by writing
$-\varepsilon_{\eta}\cdot\bar{q}_{1}=\varepsilon_{\eta}\cdot(\bar{l}-\bar{q}_{1})-\varepsilon_{\eta}\cdot\bar{l}$.
The two terms involving $\varepsilon_{\eta}\cdot\bar{l}$ under the integral
sign can be simplified by a partial fraction, yielding

$\displaystyle A_{(2,3)}^{\textrm{3pt}}$ $\displaystyle=4e^{5}(p_{1}\cdot
p_{2})^{2}\int\frac{\hat{{\mathrm{d}}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}\frac{\hat{\delta}(p_{1}\cdot\bar{l})}{p_{2}\cdot\bar{k}}\left(\frac{\varepsilon_{\eta}\cdot(\bar{l}-\bar{q}_{1})}{-p_{2}\cdot\bar{l}}-\frac{\varepsilon_{\eta}\cdot\bar{l}}{-p_{2}\cdot(\bar{l}-\bar{q}_{1})}\right)$
(82) $\displaystyle=4ie^{5}(p_{1}\cdot
p_{2})^{2}\int\frac{\hat{{\mathrm{d}}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}\hat{\delta}(p_{1}\cdot\bar{l})\hat{\delta}(p_{2}\cdot(\bar{l}-\bar{q}_{1}))\frac{\varepsilon_{\eta}\cdot\bar{l}}{p_{2}\cdot\bar{k}}\,.$

To obtain the second of these equalities, we redefined the variable of
integration to $\bar{l}^{\prime}=-(\bar{l}-\bar{q}_{1})$ in the first term,
and dropped the prime.

Next, we address the remaining diagrams contributing to $A_{(2,3)}$ which now
involve four-point vertices. After a straightforward computation, we find

$\displaystyle A_{(2,3)}^{\textrm{4pt}}$ $\displaystyle={4ie^{5}}p_{1}\cdot
p_{2}\,\varepsilon_{\eta}\cdot
p_{1}\int\frac{\hat{\mathrm{d}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}\hat{\delta}(p_{1}\cdot\bar{l})\hat{\delta}(p_{2}\cdot(\bar{l}-\bar{q}_{1}))\,.$
(83)

At this stage we can see the structure of the required factorisation — we have
exposed the delta functions present in equation (66).

Combining the contact terms of equation (83) with the rest of the diagrams,
equation (82), we find that the total expression for the amplitude fragment is

$\displaystyle A_{(2,3)}$ $\displaystyle={4ie^{5}}\,p_{1}\cdot
p_{2}\int\frac{\hat{\mathrm{d}}^{4}\bar{l}}{\bar{l}^{2}(\bar{l}-\bar{q}_{1})^{2}}\hat{\delta}(p_{1}\cdot\bar{l})\hat{\delta}(p_{2}\cdot(\bar{l}-\bar{q}_{1}))\left(\varepsilon_{\eta}\cdot
p_{1}+p_{1}\cdot
p_{2}\frac{\varepsilon_{\eta}\cdot\bar{l}}{p_{2}\cdot\bar{k}}\right)\,.$ (84)

The final step is to compare this result with the prediction for $A_{(2,3)}$
from equation (66). Using the amplitudes of equation (71), it is easy to see
that the prediction is

$\displaystyle A_{(2,3)}$
$\displaystyle=\int\hat{\mathrm{d}}^{4}\bar{w}_{1}\hat{\mathrm{d}}^{4}\bar{w_{2}}\,\hat{\delta}(2p_{1}\cdot\bar{w}_{1})\hat{\delta}(2p_{2}\cdot\bar{w}_{2})\hat{\delta}^{4}(\bar{q}_{1}+\bar{q}_{2}-\bar{w}_{1}-\bar{w}_{2})$
(85) $\displaystyle\hskip
40.0pt\times\frac{4e^{3}}{\bar{w}_{1}^{2}}\left(p_{1}\cdot\varepsilon_{\eta}+\frac{p_{1}\cdot
p_{2}\,\varepsilon_{\eta}\cdot\bar{w}_{1}}{p_{2}\cdot\bar{k}}\right)\frac{4e^{2}}{(\bar{q}_{1}-\bar{w}_{1})^{2}}p_{1}\cdot
p_{2}\,.$

Upon performing the integral over $\bar{w}_{2}$ using the four-fold delta
function, relabelling $\bar{w}_{1}=\bar{l}$ and recognising that
$\bar{k}=-\bar{q}_{1}-\bar{q}_{2}$, we immediately recover equation (84).

##  4 Generalising the Eikonal

We have now seen that the uncertainty, or the variance, in the measurement of
a scattering observable can be computed in terms of amplitudes and, moreover,
that the classical absence of uncertainty leads to an infinite set of
relationships among fragments of amplitudes expanded in powers of momentum
transfer, which is a Laurent series in $\hbar$. In a purely conservative
limit, these relationships can be understood in terms of eikonal
exponentiation. Our goal now is to review the eikonal formula, emphasising its
connection to final state dynamics. We will then build on this eikonal state
to incorporate radiative dynamics as a kind of coherent state so that the
variance is naturally small.

###  4.1 Eikonal final state

Eikonal methods have long been used to extract classical physics from quantum
mechanics. Recent years have seen a renewed surge of interest in this
approach, especially in the context of gravitational scattering Saotome:2012vy
; Akhoury:2013yua ; Ciafaloni:2015xsr ; Bjerrum-Bohr:2016hpa ; Luna:2016idw ;
Bjerrum-Bohr:2018xdl ; Ciafaloni:2018uwe ; DiVecchia:2019kta ;
DiVecchia:2019myk ; KoemansCollado:2019ggb ; Bern:2020gjj ; Parra-
Martinez:2020dzs ; DiVecchia:2020ymx ; Mogull:2020sak ; DiVecchia:2021bdo ;
DiVecchia:2021ndb ; Jakobsen:2021smu ; Shi:2021qsb ; Heissenberg:2021tzo ;
Haddad:2021znf ; Herrmann:2021tct ; Bjerrum-Bohr:2021din ; Damgaard:2021ipf ;
Emond:2021lfy , though this has roots in earlier work Amati:1990xe ;
Kabat:1992tb ; Amati:2007ak . Originally born out of the study of high
energy/Regge scattering PhysRev.143.1194 ; Abarbanel:1969ek ; Cheng:1969eh ;
PhysRev.186.1611 ; WALLACE1973190 ; PhysRev.186.1656 ; Wallace:1977ae ;
Cheng1981ConsequencesOT  where the Feynman diagrammatics dramatically
simplify, eikonal physics now have much wider application. The simplification
in this regime allows diagrams, expressed in impact parameter space rather
than momentum space, to be summed exactly to an exponential form. This
exponential depends on the $2\rightarrow 2$ scattering amplitude, and contains
information about classical quantities such as the deflection angle. There are
rich connections to soft/IR physics and Wilson lines Korchemskaya:1994qp ;
Laenen:2008gt ; Laenen:2010uz ; White:2011yy ; Melville:2013qca ;
White:2015wha  which lead to a formal proof of the exponentiation quite
generally Laenen:2008gt . Nowadays the exponentiation is taken as a starting
point and applied to various scattering regimes.

In this section our goal is to explain the link between eikonal methods and
the KMOC approach. Firstly it is worth noting that the small $\hbar$ expansion
in the KMOC formalism is essentially the same as the soft expansion in the
eikonal literature; the $\hbar$ scaling counts the order of softness. The key
connection is to compute the final state using the methods of KMOC instead of
computing observables directly. We will see that this final, outgoing, state
is controlled by the usual eikonal function. In this section we restrict to a
purely conservative scattering scenario: then eikonal exponentiation is exact.
We take two incoming particles and (since the scattering is conservative)
assume that the outgoing state is also an element of the two-particle Hilbert
space.

We begin with the standard definition of the eikonal as the transverse Fourier
transform of the four-point amplitude

$\displaystyle
e^{i\chi(x;\,s)/\hbar}(1+i\Delta(x;\,s))-1=i\int\hat{\mathrm{d}}^{4}q\,\hat{\delta}(2p_{1}\cdot
q)\hat{\delta}(2p_{2}\cdot q)e^{-iq\cdot x/\hbar}\mathcal{A}_{4}(s,q^{2})\;,$
(86)

where $\chi(x;\,s)$ is the eikonal function and $\Delta(x;\,s)$ is the so-
called quantum remainder which takes into account contributions that do not
exponentiate (see for example DiVecchia:2021bdo ). This remainder is important
for computing the eikonal function — but it will play no role in the remainder
of our article, so we will omit it. Meanwhile the two Dirac delta functions
appearing in equation (86) ensure that we integrate only over the components
of $q$ transverse to the momenta. This is often just written instead as
$\hat{\mathrm{d}}^{D-2}q_{\perp}$ (times a Jacobian factor). Indeed, the
parameter $x$ should be thought of as an element of the $D-2$ dimensional
spatial slice perpendicular to $p_{1}$ and $p_{2}$: this is most evident on
the right-hand-side of the equation, where the Dirac delta functions project
away any components of $q$ in the (timelike) $p_{1}$ and $p_{2}$ directions.
Consequently no components of $x$ in the space spanned by $p_{1}$ and $p_{2}$
enter the dot product $q\cdot x$.

The eikonal can be written as an expansion in powers of the generic coupling
$g$

$\displaystyle\chi(x;\,s)=\sum_{n=0}^{\infty}\chi_{n}(x;\,s)\quad,\quad\chi_{n}(x;\,s)\sim
g^{2n}\,,$ (87)

and we will, as others have (see for example KoemansCollado:2018hss ;
KoemansCollado:2019ggb ; DiVecchia:2019myk ; Bern:2020gjj ; Parra-
Martinez:2020dzs ; DiVecchia:2021bdo ), assume that this holds to all orders.
The structure has been formally proven at leading order, (see for example
Laenen:2008gt ; Akhoury:2013yua ), however an all orders proof has not been
given (to the best of our knowledge).

Starting with the in state (16) from section 2.2, we obtain the final state by
acting with the $S$ matrix. Writing $S=1+iT$ and inserting a complete set of
states, we have

$\displaystyle S|\psi\rangle=$
$\displaystyle|\psi\rangle+i\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2})\,\phi_{b}(p_{1},p_{2})|p_{1}^{\prime},p_{2}^{\prime}\rangle\langle
p_{1}^{\prime},p_{2}^{\prime}|T|p_{1},p_{2}\rangle$ (88) $\displaystyle=$
$\displaystyle|\psi\rangle+i\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime},p_{1},p_{2})\,\phi_{b}(p_{1},p_{2})\mathcal{A}_{4}(s,q^{2})\delta^{(4)}(p_{1}\\!+\\!p_{2}\\!-\\!p_{1}^{\prime}\\!-\\!p_{2}^{\prime})|p_{1}^{\prime},p_{2}^{\prime}\rangle\,.$

Notice that we made explicit use of our assumption of conservative scattering
by restricting the complete set of states to the two-particle Hilbert space.

$p_{1}^{\prime}$$p^{\prime}_{2}$$p_{1}=p^{\prime}_{1}-q$$p_{2}=p^{\prime}_{2}+q$
Figure 6: Momentum labelling at four points

With the momentum labelling in figure 6 we can convert the $p_{1}$ and $p_{2}$
phase space integrals to integrals over $q$. Doing so, we may write

$\displaystyle S|\psi\rangle=|\psi\rangle$ (89)
$\displaystyle\quad+i\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,\hat{\mathrm{d}}^{4}q\,\hat{\delta}(2p_{1}^{\prime}\\!\cdot\\!q\\!-\\!q^{2})\hat{\delta}(2p_{2}^{\prime}\\!\cdot\\!q\\!+\\!q^{2})\,\phi_{b}(p_{1}^{\prime}\\!-\\!q,p_{2}^{\prime}\\!+\\!q)\mathcal{A}_{4}(s,q^{2})|p_{1}^{\prime},p_{2}^{\prime}\rangle.$

At this point, the $q$ integral is tantalising similar to the $q$ integral in
the eikonal formula (86). However there is a key difference in the nature of
the delta functions: those in equation (89) involve $q^{2}$ terms which are
absent in equation (86). This issue appeared recently in refs.
Brandhuber:2021kpo ; Brandhuber:2021eyq : there the authors proceeded using a
“HEFT” phase, which is analogous to yet distinct from the eikonal phase. We
will instead continue with the eikonal phase.

It may be worth remarking that the $q^{2}$ terms in these delta functions are
suppressed in specific examples. One such example is the leading order impulse
KMOC . Nevertheless the impulse at NLO does indeed involve the full delta
functions KMOC .

To incorporate the full delta functions, we follow the route described in
Parra-Martinez:2020dzs . We introduce new momentum variables $\tilde{p}$ as

$\displaystyle\tilde{p}_{1}=p_{1}^{\prime}-\frac{q}{2}\qquad\tilde{p}_{2}=p^{\prime}_{2}+\frac{q}{2}.$
(90)

Now, rather than using the eikonal equation (86) directly we can take
advantage of its inverse Fourier transform in the following form999As noted
above, we dropped the quantum remainder $\Delta$ which plays no role in our
analysis.

$\displaystyle i\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot
q)\mathcal{A}_{4}(s,q^{2})=\frac{1}{\hbar^{4}}\int\mathrm{d}^{4}x\,e^{iq\cdot
x/\hbar}\left\\{e^{i\chi(x_{\perp};\,s)/\hbar}-1\right\\},$ (91)

where we have written $x_{\perp}$ as one of the arguments of the eikonal
function $\chi(x_{\perp};\,s)$ to emphasise that $\chi(x_{\perp};\,s)$ only
depends on components of $x$ which are orthogonal to the space spanned by
$\tilde{p}_{1}$ and $\tilde{p}_{2}$. Indeed, integrating over the two
components of $x$ which _are_ in the space spanned by $\tilde{p}_{1}$ and
$\tilde{p}_{2}$, one recovers the two Dirac delta functions on the left-hand-
side of equation (91). In this way, we find that the final state is

$\displaystyle
S|\psi\rangle=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,|p_{1}^{\prime},p_{2}^{\prime}\rangle\frac{1}{\hbar^{4}}\int\hat{\mathrm{d}}^{4}q\,\mathrm{d}^{4}x\,e^{iq\cdot
x/\hbar}\,e^{i\chi(x_{\perp};\,s)/\hbar}\,{\phi}_{b}(p_{1}^{\prime}-q,p_{2}^{\prime}+q)\,.$
(92)

It is worth emphasising once again that $x_{\perp}$ is perpendicular to
$\tilde{p}_{i}$, rather than to $p_{i}$, so that

$\displaystyle\tilde{p}_{1}\cdot x_{\perp}=0=\tilde{p}_{2}\cdot x_{\perp}.$
(93)

In particular, $x_{\perp}$ depends on $q$.

###  4.2 The impulse from the eikonal

In this subsection, we will recover one beautiful result from the literature
on the eikonal function: the scattering angle can be extracted from the
eikonal function using a stationary phase argument. As our interest is not so
much in this conservative case but rather in its radiative generalisation
(which we discuss below), we wish to emphasise that it is, in fact, possible
to extract the _full_ final momentum from the eikonal using stationary phase.
Later, in section 4.5, we will use the same ideas to extract the final
momentum in the case of radiation — with radiation, of course, knowledge of
the direction of the final momentum is insufficient to recover the full
momentum.

The impulse is the observable

$\displaystyle\Delta p_{1}^{\mu}$
$\displaystyle\equiv\langle\psi|S^{\dagger}\mathbbm{P}_{1}^{\mu}S|\psi\rangle-\langle\psi|\mathbbm{P}_{1}^{\mu}|\psi\rangle$
(94)
$\displaystyle=\langle\psi|S^{\dagger}[\mathbbm{P}_{1}^{\mu},S]|\psi\rangle\,.$

It is convenient to focus on

$\displaystyle|\psi\rangle\,.$ (95)

As we shall see, in essence the operator $[\mathbbm{P}_{1}^{\mu},S]$ pulls out
a factor of the momentum transfer multiplying $S|\psi\rangle$. We will
evaluate $[\mathbbm{P}_{1}^{\mu},S]|\psi\rangle$ by stationary phase; it is
then trivial to determine $\langle\psi|S^{\dagger}$ in the same way.

We begin our expression for the final-state wavepacket, equation (92), quickly
finding

$\displaystyle[\mathbbm{P}_{1}^{\mu},S]|\psi\rangle=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,e^{ib\cdot
p_{1}^{\prime}/\hbar}$
$\displaystyle|p_{1}^{\prime},p_{2}^{\prime}\rangle\frac{1}{\hbar^{4}}\int\hat{\mathrm{d}}^{4}q\,\mathrm{d}^{4}x\,e^{iq\cdot
x/\hbar}e^{-ib\cdot q/\hbar}$ (96) $\displaystyle\times
e^{i\chi(x_{\perp};\,s)/\hbar}\,\,{\phi}(p_{1}^{\prime}-q,p_{2}^{\prime}+q)\,q^{\mu}\,.$

To obtain formulae for the impulse, we apply the stationary phase
approximation to the $x$ and $q$ integrals. (Our approach is very similar to
that of Ciafaloni and Colferai Ciafaloni:2014esa  who previously discussed
wavepacket dynamics, the eikonal, and stationary phase.)

The stationary phase condition for $x$ is101010One can find this expression
also by promoting the eikonal phase to an operator $\hat{\chi}(x_{\perp},s)$:
the impulse on particle $i$ in the transverse plane $\hat{Q}^{\mu}_{\perp i}$
is then be related to the standard commutator $[\hat{Q}^{\mu}_{\perp
i},e^{i\hat{\chi}(\hat{x}_{\perp};\,s)/\hbar}]=e^{i\hat{\chi}(\hat{x}_{\perp};\,s)/\hbar}\partial^{\mu}_{\perp
i}\hat{\chi}(\hat{x}_{\perp},s)$ and the result follows from unitarity.

$\displaystyle q_{\mu}=-\frac{\partial}{\partial x^{\mu}}\chi(x_{\perp},s)\,.$
(97)

One thing to note immediately is that this $q^{\mu}$ is _not_ of order
$\hbar$: it is a classical momentum, of order $g^{2}$. Further, it is useful
to note that $\chi$ is actually a function of $x_{\perp}^{2}$ (since the four-
point function is a function of $s$ and $q^{2}$.) Therefore we may write the
momentum transfer as

$\displaystyle q^{\mu}=-2\chi^{\prime}(x_{\perp}^{2},s)x_{\perp}^{\mu}\,,$
(98)

where $\chi^{\prime}(x_{\perp}^{2},s)$ is the derivative of $\chi$ with
respect to its first argument. Since $\tilde{p}_{1}\cdot
x_{\perp}=0=\tilde{p}_{2}\cdot x_{\perp}$, it now follows that
$\tilde{p}_{1}\cdot q=0=\tilde{p}_{2}\cdot q$: thus the on-shell delta
functions in equation (91) have reappeared, now as “equations of motion”
following from the stationary phase conditions111111Since some of the $x$
integrations may be performed exactly to yield delta functions, we are
slightly abusing terminology by referring to all of the conditions on $x$ and
$q$ as “stationary phase conditions”. Some of the conditions arise
approximately by stationary phase, and some are exact conditions due to the
delta functions. Nevertheless it is most convenient to treat all the
conditions as one set..

The second stationary phase condition, associated with $q$, is

$\displaystyle x^{\mu}-b^{\mu}+\frac{\partial}{\partial
q_{\mu}}\chi(x_{\perp}^{2},s)=0\,.$ (99)

The $q$ derivative is non-vanishing because $x_{\perp}$ depends on $q$. It is
often useful to introduce a particular notation for the variables $q$ and $x$
when they satisfy the stationary phase conditions: we will denote these by
$q_{*}$ and $x_{*}$. Armed with this notation, we may use equation (97) we may
write equation (99) as

$\displaystyle x^{\mu}=b^{\mu}+q_{*\nu}\frac{\partial}{\partial
q_{\mu}}x_{\perp}^{\nu}\,.$ (100)

Performing the derivative is straightforward, but requires some notation which
we relegate to appendix B. The result may be expressed in the form

$\displaystyle
x_{*\perp}^{\mu}=b^{\mu}-\tilde{N}_{q}(p_{1}^{\mu}-p_{2}^{\mu})-\tilde{N}_{0q}(p_{1}^{\mu}+p_{2}^{\mu})\,,$
(101)

where $\tilde{N}_{q}$ and $\tilde{N}_{0q}$ can be interpreted as Lagrange
multipliers ensuring that
$x_{\perp}\cdot\tilde{p}_{1}=0=x_{\perp}\cdot\tilde{p}_{2}$.

Our result for $x_{*\perp}$ takes a familiar form in the CM frame. Then
$x_{*\perp}^{0}=0$ and, using, $\bm{{p_{2}}}=-\bm{{p_{1}}}$, we have

$\displaystyle\bm{{x}}_{*\perp}$ $\displaystyle=\bm{{b}}-2N_{q}\bm{{p}}$ (102)
$\displaystyle\Rightarrow\bm{{b}}\cdot\bm{{x}}_{*\perp}$
$\displaystyle=\bm{{b}}^{2}\,.$

Denoting the scattering angle by $\Psi$, we may write this result121212In this
context, the quantity $x_{\perp}$ is sometimes referred to as the “eikonal”
impact parameter. as

$\displaystyle|\bm{{b}}|=|\bm{{x}}_{*\perp}|\cos(\Psi/2)\,,$ (103)

where $\Psi$ is the scattering angle.

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\bm{p}}{\color[rgb]{0.96,0.65,0.14}{}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}}{}}{\color[rgb]{0.96,0.65,0.14}\definecolor[named]{pgfstrokecolor}{rgb}{0.96,0.65,0.14}}$${\color[rgb]{0.96,0.65,0.14}\definecolor[named]{pgfstrokecolor}{rgb}{0.96,0.65,0.14}\bm{b}}$$\bm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}p^{\prime}}{\color[rgb]{0.96,0.65,0.14}{}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}}{}}{\color[rgb]{0.96,0.65,0.14}\definecolor[named]{pgfstrokecolor}{rgb}{0.96,0.65,0.14}}}$$\Psi/2$$\Psi/2$$\bm{{\color[rgb]{0.29,0.56,0.89}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.56,0.89}q}}{\color[rgb]{0.29,0.56,0.89}{}_{*}}$${\color[rgb]{0.82,0.01,0.11}\definecolor[named]{pgfstrokecolor}{rgb}{0.82,0.01,0.11}\varpropto}{\color[rgb]{0.82,0.01,0.11}\definecolor[named]{pgfstrokecolor}{rgb}{0.82,0.01,0.11}\bm{e}}{\color[rgb]{0.82,0.01,0.11}{}_{q}{}}{\color[rgb]{0.96,0.65,0.14}\definecolor[named]{pgfstrokecolor}{rgb}{0.96,0.65,0.14}}$${\color[rgb]{0.49,0.83,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{0.49,0.83,0.13}\bm{x}}{\color[rgb]{0.49,0.83,0.13}{}_{\perp}}$
Figure 7: Geometry of eikonal scattering.

In this way, we have performed the integrals over $q$ and $x$. The result has
been to evaluate the factor $q^{\mu}$ as a derivative of the eikonal function.
To obtain the full expectation, we simply evaluate $\langle\psi|S^{\dagger}$
using stationary phase in precisely the same way: the only difference (apart
from the obvious Hermitian conjugation) is the absence of the $q^{\mu}$
factor. We then exploit unitarity to conclude that131313We discuss the details
of unitarity more explicitly in appendix C, performing all the remaining
integrals.

$\displaystyle\Delta p_{1}^{\mu}=q^{\mu}=-\partial^{\mu}\chi(x_{\perp},s)\,,$
(104)

where all quantities are defined on using the solution of the stationary phase
conditions.

Notice that we have determined the _complete_ impulse four-vector, not just
the scattering angle. The distinction between these quantities is obviously
unimportant at the level of conservative dynamics, but it is important when
radiation occurs. The key aspect of our argument which leads to the full
impulse rather than the scattering angle is the presence of a perpendicular
projector. To see how this works, let us discuss an explicit example: the
impulse at next-to-leading order in gravitational fast scattering.

Focusing on the scattering between two massive bodies in general relativity,
the eikonal phase at next-to-leading order in $G$ is

$\displaystyle\chi=-2Gm_{1}m_{2}\left(\frac{\left(2\gamma^{2}-1\right)}{\sqrt{\gamma^{2}-1}}\log|x_{\perp}|-\frac{3\pi}{8}\frac{\left(5\gamma^{2}-1\right)}{\sqrt{\gamma^{2}-1}}\frac{G\left(m_{1}+m_{2}\right)}{|x_{\perp}|}\right)\
,$ (105)

where we have defined $\gamma=u_{1}\cdot u_{2}$ as the scalar product between
the four-velocities of the particles. Using equation (104) and straightforward
differentiation, we obtain the following expression in terms of
$x_{\perp}^{\mu}$,

$\displaystyle\Delta
p_{1}^{\mu}=\frac{2Gm_{1}m_{2}x^{\mu}_{\perp}}{\left|x^{2}_{\perp}\right|}\left(\frac{\left(2\gamma^{2}-1\right)}{\sqrt{\gamma^{2}-1}}+\frac{3\pi}{8}\frac{\left(5\gamma^{2}-1\right)}{\sqrt{\gamma^{2}-1}}\frac{G\left(m_{1}+m_{2}\right)}{|x_{\perp}|}\right)\
,$ (106)

where the four-velocities in $\gamma$ can now be identified with the incoming
four-velocities of the particles due to the integrals over the wavepackets
that took place to arrive at (104). At this point, it is important to remember
that $x^{\mu}_{\perp}$ is not quite $b^{\mu}$. It is trivial to show that
$x^{\mu}_{\perp}$ coincides with $b^{\mu}$ at leading order in the
gravitational coupling, but this is no longer the case at next-to-leading
order; then instead

$\displaystyle
x^{\mu}_{\perp}=b^{\mu}-\frac{G(2\gamma^{2}-1)}{(\gamma^{2}-1)^{3/2}}\left(u^{\mu}_{1}(m_{2}+\gamma
m_{1})-u^{\mu}_{2}(m_{1}+\gamma m_{2})\right)\ ,$ (107)

where

$\displaystyle|x_{\perp}^{2}|=|b^{2}|\ .$ (108)

We can now express the impulse in terms of the impact parameter $b^{\mu}$. At
the order we are interested in we find,

$\displaystyle\Delta
p_{1}^{\mu}=\frac{2Gm_{1}m_{2}b^{\mu}}{\left|b^{2}\right|}\left(\frac{\left(2\gamma^{2}-1\right)}{\sqrt{\gamma^{2}-1}}+\frac{3\pi}{8}\frac{\left(5\gamma^{2}-1\right)}{\sqrt{\gamma^{2}-1}}\frac{G\left(m_{1}+m_{2}\right)}{|b|}\right)$
(109)
$\displaystyle\quad-\frac{G^{2}m_{1}m_{2}\left(2\gamma^{2}-1\right)^{2}\left(\left(\gamma
m_{1}+m_{2}\right)u_{1}^{\mu}-\left(\gamma
m_{2}+m_{1}\right)u_{2}^{\mu}\right)}{\left(\gamma^{2}-1\right)^{2}\left|b^{2}\right|}\
,$

in perfect agreement with the literature Herrmann:2021tct .

From the perspective of this paper, the key achievement of the eikonal
resummation is that negligible variance becomes automatic in the stationary
phase argument. Indeed since the stationary phase condition (97) sets the
momentum transfer to a specific, classical, value it is clear that the
expectation value of any polynomial in the momentum operator will evaluate to
the classical expectation.

###  4.3 Coherent radiative state

In the context of radiation, another exponentiation is well known to lead to
small or negligible variance. This occurs when the radiative state is a
coherent state with large occupation number. Indeed, in this context the
requirement of minimal uncertainty is sometimes referred to as as complete
coherence Glauber:1963fi ; Glauber:1963tx . Coherent states have a long
history in applications to semi-classical physics Yaffe ; KlauderJohnR1985Cs:a
as well as more recent interest delaCruz:2020bbn ; Bern:2020buy ;
Monteiro:2020plf ; Aoude:2021oqj ; Cristofoli:2021vyo . In this subsection, we
describe how these states naturally describe the radiation in the final state,
working in electrodynamics for simplicity.

Consider the state

$\displaystyle|\alpha^{\eta}\rangle=\mathcal{N}_{\alpha}\exp\left(\int\mathrm{d}\Phi(k)\alpha^{\eta}(k)a^{\dagger}_{\eta}(k)\right)|0\rangle\,.$
(110)

It is not difficult to verify that $|\alpha^{\eta}\rangle$ is indeed a
coherent state satisfying the usual properties: in particular, the minimal
uncertainty relation (2) is obeyed explicitly if the occupation number is
large Cristofoli:2021vyo . Normalising these states to unity requires

$\displaystyle\mathcal{N}_{\alpha}=\exp\left(-\frac{1}{2}\int\mathrm{d}\Phi(k)|\alpha^{\eta}(k)|^{2}\right)\,.$
(111)

We would like to understand the final particle distribution for the photons
emitted in a scattering process in the classical limit. In particular, we want
to show here that for classical radiation the factorization property of
expectation values of physical observables is directly connected to coherence.
To simplify our discussion, we work in a spacetime box of finite dimension $V$
with periodic boundary conditions so that there will be a finite number
$|V^{D}|$ of allowed momenta $\\{k_{i}\\}_{i\in V^{D}}$ in the dual momentum
lattice $V^{D}$. At the end of the discussion, we will take $V\to+\infty$. In
this framework, there is a coherent state for each harmonic oscillator with
momentum $\\{k_{i}\\}$

$\displaystyle|\alpha^{\eta}_{k_{i}}\rangle=\mathcal{N}_{\alpha}\exp\left(\alpha^{\eta}(k_{i})a^{\dagger}_{\eta}(k_{i})\right)|0\rangle,\qquad\forall\,k_{i}\in
V^{D}\,.$ (112)

It is known that we can write every classical radiation density matrix as a
probability distribution in the coherent state space, as proved by Glauber and
Sudarshan Glauber:1963tx ; Glauber:1963fi ; Sudarshan:1963ts ; Zhang:1999is

$\displaystyle\hat{\rho}_{\text{out}}=\sum_{\eta=\pm}\int\prod_{l_{i}\in
V^{D}}\mathrm{d}^{2}\alpha^{\eta}_{l_{i}}\,\mathcal{P}^{\eta}(\alpha)|\alpha^{\eta}_{l_{i}}\rangle\langle\alpha^{\eta}_{l_{i}}|\qquad\mathrm{d}^{2}\alpha^{\eta}_{l}:=(\mathrm{d}\operatorname{Re}\alpha^{\eta}_{l}\mathrm{d}\operatorname{Im}\alpha^{\eta}_{l})/\pi$
(113)

where $\mathcal{P}^{\eta}(\alpha)$ is a separable function of
$\alpha^{\eta}_{l_{i}}$ with $i\in V^{D}$

$\displaystyle\mathcal{P}^{\eta}(\alpha):=\prod_{l_{i}\in
V^{D}}\mathcal{P}^{\eta}(\alpha(l_{i})).$ (114)

For the classical case $\mathcal{P}^{\eta}(\alpha)\geq 0$, and this is what
allows to talk about probability distribution in the standard mathematical
sense. Let us stress here that in the notation $\sum_{l_{i}\in V^{D}}$ we
include not only the summation over the dual lattice vectors but also the
appropriate finite-volume on-shell phase-space normalization.

What is the implication of the exact classical factorization on the final
radiation density matrix? Based on the previous discussion of negligible
uncertainty we expect that the expectation value
$\langle\,\cdot\,\rangle_{\rho_{\text{out}}}$ in the density matrix (113)
gives

$\displaystyle\langle\mathbb{F}_{\mu\nu}(x)\mathbb{F}_{\rho\sigma}(y)\rangle_{\rho_{\text{out}}}\stackrel{{\scriptstyle\hbar\to
0}}{{=}}\langle\mathbb{F}_{\mu\nu}(x)\rangle_{\rho_{\text{out}}}\langle\mathbb{F}_{\rho\sigma}(y)\rangle_{\rho_{\text{out}}}\,.$
(115)

We use the on-shell mode expansion for the field strength operator

$\displaystyle\mathbb{F}_{\mu\nu}(x)=\frac{-i}{\sqrt{\hbar}}\sum_{\eta=\pm}\sum_{k\in
V^{D}}\,\bigl{[}a_{\eta}(k)\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}e^{-i\bar{k}\cdot
x}-\textrm{h.c.}\bigr{]}\,,$ (116)

and by taking advantage of the completeness relation in the Hilbert space of
photons141414For $n=0$, the first term in the sum has to be interpreted as
$|0\rangle\langle 0|$. This will be implicitly assumed in the rest of the
argument.

$\displaystyle\sum_{n=0}^{+\infty}\frac{1}{n!}\sum_{\eta_{1},...,\eta_{n}=\pm}\sum_{l_{1},...,l_{n}\in
V^{D}}|l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}\rangle\langle
l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}|=1\,,$ (117)

we have,

$\displaystyle\text{Tr}_{\rho_{\text{out}}}(\mathbb{F}_{\mu\nu}(x))$
$\displaystyle=\sum_{n=0}^{+\infty}\frac{1}{n!}\sum_{\eta_{1},...,\eta_{n}=\pm}\sum_{l_{1},...,l_{n}\in
V^{D}}\langle
l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}|\mathbb{F}^{\mu\nu}(x)\rho_{\text{out}}|l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}\rangle$
$\displaystyle=\sum_{n,m=0}^{+\infty}\frac{1}{n!m!}\sum_{\eta_{1},\eta^{\prime}_{1},...,\eta_{n},\eta^{\prime}_{m}=\pm}\sum_{l_{1},l^{\prime}_{1},...,l_{n},l^{\prime}_{m}\in
V^{D}}$ $\displaystyle\hskip 15.0pt\times\langle
l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}|\mathbb{F}^{\mu\nu}(x)|(l^{\prime}_{1})^{\eta_{1}}...(l^{\prime}_{m})^{\eta^{\prime}_{m}}\rangle\langle(l^{\prime}_{1})^{\eta_{1}}...(l^{\prime}_{m})^{\eta^{\prime}_{m}}|\rho_{\text{out}}|l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}\rangle$
$\displaystyle=-i\hbar^{\frac{3}{2}}\sum_{n=0}^{+\infty}\frac{1}{n!}\sum_{\eta_{1},...,\eta_{n}=\pm}\sum_{\bar{l}_{1},...,\bar{l}_{n}\in\bar{V}^{D}}$
$\displaystyle\hskip
15.0pt\times\sum_{\bar{k}\in\bar{V}^{D}}\bigl{[}\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\langle
l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}k^{\eta}|\rho_{\text{out}}|l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}\rangle
e^{-i\bar{k}\cdot x}-\textrm{h.c.}\bigr{]}\,.$

where in the last line we have restored also the $\hbar$ scaling implicit in
the finite volume phase-space normalization. Further manipulations show that

$\displaystyle\sum_{n=0}^{+\infty}\frac{1}{n!}$
$\displaystyle\sum_{\eta_{1},...,\eta_{n}=\pm}\sum_{\bar{l}_{1},...,\bar{l}_{n}\in\bar{V}^{D}}\langle
l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}k^{\eta}|\rho_{\text{out}}|l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}\rangle$
$\displaystyle=\sum_{n=0}^{+\infty}\frac{1}{n!}\sum_{\eta_{1},...,\eta_{n}=\pm}\sum_{\bar{l}_{1},...,\bar{l}_{n}\in\bar{V}^{D}}\prod_{i=1}^{n}\left[\mathcal{N}_{\alpha_{l_{i}}}^{2}\int\mathrm{d}^{2}\alpha^{\eta}_{l_{i}}\,|\alpha^{\eta_{i}}(l_{i})|^{2}\,\mathcal{P}^{\eta_{i}}(\alpha(l_{i}))\right]\alpha^{\eta}(k)$
$\displaystyle=\text{Tr}_{\rho_{\text{out}}}(1)\alpha^{\eta}(k)$
$\displaystyle=\alpha^{\eta}(k)$

where we have used the fact that the density matrix is normalized
$\text{Tr}_{\rho_{\text{out}}}(1)=1$. If we demand the expectation value of
$\text{Tr}_{\rho_{\text{out}}}(\mathbb{F}_{\mu\nu}(x))$ to be classical,
restoring powers of $\hbar$ requires the waveshape to scale as determined in
Cristofoli:2021vyo

$\displaystyle\alpha^{\eta}(k)\to\hbar^{-\frac{3}{2}}\bar{\alpha}^{\eta}(k)\,.$
(118)

Using this condition and the classical scaling of the normalization of the
final state, we get

$\displaystyle\text{Tr}_{\rho_{\text{out}}}(\mathbb{F}_{\mu\nu}(x))=-i\sum_{\bar{k}\in\bar{V}^{D}}\sum_{\eta=\pm}\int\mathrm{d}^{2}\bar{\alpha}^{\eta}_{k}\,\mathcal{P}^{\eta}(\bar{\alpha}_{k})$
$\displaystyle\bigl{[}\bar{k}^{\vphantom{\mu}}_{[\mu}\varepsilon^{(\eta)*}_{\nu]}\bar{\alpha}^{\eta,*}(k)e^{-i\bar{k}\cdot
x}-\textrm{h.c.}\bigl{]}\,.$ (119)

Using a similar argument, for the expectation value of the product we obtain

$\displaystyle\text{Tr}_{\rho_{\text{out}}}(\mathbb{F}_{\mu\nu}(x)\mathbb{F}_{\rho\sigma}(y))$
$\displaystyle=\sum_{n=0}^{+\infty}\frac{1}{n!}\sum_{\eta_{1},...,\eta_{n}=\pm}\sum_{\bar{l}_{1},...,\bar{l}_{n}\in\bar{V}^{D}}\langle
l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}|\mathbb{F}^{\mu\nu}(x)\mathbb{F}^{\rho\sigma}(y)\rho_{\text{out}}|l_{1}^{\eta_{1}}...l_{n}^{\eta_{n}}\rangle$
$\displaystyle=-\sum_{\bar{k}_{1},\bar{k}_{2}\in\bar{V}^{D}}\sum_{\eta_{1},\eta_{2}=\pm}\int\mathrm{d}^{2}\bar{\alpha}^{\eta_{1}}_{k_{1}}\,\mathrm{d}^{2}\bar{\alpha}^{\eta_{2}}_{k_{2}}\mathcal{P}^{\eta_{1},\eta_{2}}(\bar{\alpha}_{k_{1}},\bar{\alpha}_{k_{2}})\,$
$\displaystyle\hskip
20.0pt\times\bigl{[}\bar{k}^{\vphantom{\mu}}_{1,[\mu}\varepsilon^{(\eta_{1})*}_{\nu]}\bar{k}^{\vphantom{\mu}}_{2,[\rho}\varepsilon^{(\eta_{2})*}_{\sigma]}\bar{\alpha}^{\eta_{1},*}(k_{1})\bar{\alpha}^{\eta_{2},*}(k_{2})e^{-i\bar{k}_{1}\cdot
x-i\bar{k}_{2}\cdot y}$ $\displaystyle\hskip
40.0pt\quad+\bar{k}^{\vphantom{\mu}}_{1,[\mu}\varepsilon^{(\eta_{1})}_{\nu]}\bar{k}^{\vphantom{\mu}}_{2,[\rho}\varepsilon^{(\eta_{2})*}_{\sigma]}\bar{\alpha}^{\eta_{1}}(k_{1})\bar{\alpha}^{\eta_{2},*}(k_{2})e^{i\bar{k}_{1}\cdot
x-i\bar{k}_{2}\cdot y}+\textrm{h.c.}\bigl{]}\,,$

up to the commutator term which can be neglected in the $\hbar\to 0$ limit.

Therefore we are effectively asking whether the product of the averages is
equal to the average of the product over a distribution
$\mathcal{P}^{\eta}(\alpha)$ for all $k\in V^{D}$, i.e. we are asking the
distribution to have _zero variance_ in the coherent state space. But
distributions of zero variance are degenerate because it means that the random
variable $\alpha^{\eta}(k)$151515Technically this is a random variable
$\alpha^{\eta}(k)$ for each value of the momentum in the dual lattice, i.e.
there is an harmonic oscillator for each quanta of radiation. But since they
are all independent, we can promote this statement to the full expression. is
almost surely constant for each $k\in V^{D}$ (see page 173 in blitzstein2019
). Therefore the distribution has support in a lower-dimensional space, and
since we can apply this argument independently both for the real and for the
imaginary part of $\alpha^{\eta}(k)$ we have

$\displaystyle\mathcal{P}^{\eta}(\alpha)=\prod_{k\in
V^{D}}\sum_{j=1}^{+\infty}c^{\eta}_{j}\delta^{2}(\alpha^{\eta}(k)-\alpha^{\eta}_{j}(k))\,.$
(120)

What this is essentially saying is that we get a (possibly infinite) sum of
discrete distributions with a constant value $\alpha^{\eta}_{j}(k)$. But then,
making use of a crucial result due to Hillery HILLERY1985409 , we get

$\displaystyle\mathcal{P}^{\eta}(\alpha)=\prod_{k\in
V^{D}}\delta^{2}(\alpha^{\eta}(k)-\alpha^{\eta}_{\star}(k))\,,$ (121)

which means that we can describe the final state only with one coherent state
for each helicity and for each momentum $k$ in the dual momentum lattice. At
this point we can take the large volume limit and what this calculation
implies is that we can describe the final state with a single coherent state
(112), which takes naturally into the account the infinite-dimensional
superposition of harmonic oscillators of momentum $k$.

This state only describes the gauge particles (photons in this case), but we
would like to use this to describe a state with the two massive scattering
particles as well as the radiation emitted during the interaction. This is the
situation one would generally expect after some scattering event. We will do
exactly this later and determine the form of $\alpha$ at leading order. We
will find that $\alpha$ depends on the 5-point amplitude, much as the eikonal
depends on the 4-point.

###  4.4 Extension with coherent radiation

The exponentiated eikonal final state of equation (92) beautifully describes
semiclassical conservative dynamics, leading to a transparent method for
extracting the impulse (or scattering angle) from amplitudes in a manner which
automatically enforces minimal uncertainty. We have also seen that coherent
states naturally enforce minimal uncertainty for radiation. Now let us put
these two ideas together to form a proposal for an eikonal-type final state in
the fully dynamical, radiative, case.

It is very natural to consider a modification of the eikonal formula which
includes radiation, and indeed this idea has received attention
Ciafaloni:2018uwe  in the literature. Given that our motivation is to extend
the eikonal while maintaining its minimal uncertainty property, an obvious way
to proceed is to include an additional factor in the eikonal formula which has
the structure of a coherent state like (112). If this radiative part of the
state has large occupation number, expectations of products of field-strength
operators will naturally factorise into products of expectations of the
operators.

We will simply propose one possibility for the structure of this final state,
depending on a coherent waveshape parameter $\alpha^{(\eta)}$ (of helicity
$\eta$) in addition to an eikonal function $\chi$. We believe there is strong
evidence in favour of the basic structure of our proposal, and in particular
in the idea that two objects $\chi$ and $\alpha^{(\eta)}$ suffice to define
it; however, it seems possible to implement the idea in somewhat different
ways. We will discuss the basic virtues of our proposal in the remainder of
this section, leaving it to future work to determine further details. Since we
are primarily interested in classical effects, we will continue to neglect the
quantum remainder $\Delta$ in this discussion161616Indeed radiative quantum
effects will require $\Delta$ to be upgraded to an operator..

To describe our proposal, we begin with the eikonal final state (92). With an
eye towards a situation where momentum is lost to radiation, we need a
description in which the sum of the momenta of the two final particles differs
from the initial momenta. A first step, then, is to Fourier transform the
wavepacket to position space:

$\displaystyle
S|\psi\rangle|_{\textrm{conservative}}=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}$
$\displaystyle\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}$ (122) $\displaystyle\times
e^{i[q\cdot(x-x_{1}+x_{2})+\chi(x_{\perp};s)]/\hbar}\,|p_{1}^{\prime},p_{2}^{\prime}\rangle\,.$

Our proposal is now very straightforward: we simply incorporate a coherent
state by assuming that

$\displaystyle S|\psi\rangle=\int$
$\displaystyle\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}$ (123)
$\displaystyle\times\,e^{i[q\cdot(x-x_{1}+x_{2})+\chi(x_{\perp};s)]/\hbar}\exp\left[\sum_{\eta}\int\mathrm{d}\Phi(k)\alpha^{(\eta)}(k,x_{1},x_{2})a^{\dagger}_{\eta}(k)\right]\,|p_{1}^{\prime},p_{2}^{\prime}\rangle\,.$

This is a minimal proposal: more generally, one could imagine that the
coherent waveshape parameter $\alpha^{(\eta)}$ depends on other variables, for
example $x$ or $q$ which appear in the eikonal dynamics. We will nevertheless
restrict throughout this article to our minimal proposal. However, it is
important that the state is not merely an outer product of a conservative
eikonal state with a radiative factor. Some entanglement is necessary so that
the radiation can backreact on the motion. In the case of the present
proposal, the integrals over the variables $x_{1}$, $x_{2}$ and $x$ perform
this role. We will present a partial derivation of this proposal in section 5,
showing that the leading low-frequency classical radiation does indeed
exponentiate as anticipated. There is a connection between our proposal here
and recent work Damgaard:2021ipf  on the exponential structure of the $S$
matrix.

Following the discussion in section 4.3, we know that the waveshape should be
proportional to $\hbar^{-3/2}$ so we may also write the state as

$\displaystyle S|\psi\rangle$
$\displaystyle=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}$ (124)
$\displaystyle\times\,e^{i[q\cdot(x-x_{1}+x_{2})+\chi(x_{\perp};s)]/\hbar}\exp\left[\frac{1}{\hbar^{3/2}}\sum_{\eta}\int\mathrm{d}\Phi(k)\bar{\alpha}^{(\eta)}(k,x_{1},x_{2})a^{\dagger}_{\eta}(k)\right]\,|p_{1}^{\prime},p_{2}^{\prime}\rangle\,.$

In this expression, the classical waveshape $\bar{\alpha}^{(\eta)}$ is
_independent_ of $\hbar$, just as the eikonal function $\chi$ is independent
of $\hbar$.

In order to determine $\alpha^{(\eta)}$, we follow the same steps as in
sections 2.2 and 4.1; we act on the incoming state with the $S$ matrix, and
then expand in terms of integrals of amplitudes. To isolate the waveshape, we
consider the overlap of our proposed final state with the bra $\langle
p_{1}^{\prime}\,p_{2}^{\prime}\,k^{\eta}|$:

$\displaystyle\langle
p_{1}^{\prime}\,p_{2}^{\prime}\,k^{\eta}|S|\psi\rangle=\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}$
$\displaystyle\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}$ (125) $\displaystyle\times
e^{i[q\cdot(x-x_{1}+x_{2})+\chi(x_{\perp};s)]/\hbar}\,\alpha^{(\eta)}(k,x_{1},x_{2})\,.$

Next we expand the $S$ matrix as

$\displaystyle\langle p_{1}^{\prime}\,p_{2}^{\prime}\,k^{\eta}|S|\psi\rangle$
$\displaystyle=\int\mathrm{d}\Phi(p_{1},p_{2})\,\phi_{b}(p_{1},p_{2})\,\langle
p_{1}^{\prime}\,p_{2}^{\prime}\,k^{\eta}|S|p_{1}p_{2}\rangle$ (126)
$\displaystyle=\int\mathrm{d}\Phi(p_{1},p_{2})\int\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}\cdot
x_{1}+p_{2}\cdot x_{2})/\hbar}$ $\displaystyle\hskip 50.0pt\times
i\mathcal{A}_{5}(p_{1}p_{2}\rightarrow
p_{1}^{\prime}p_{2}^{\prime}k^{\eta})\,\hat{\delta}^{4}(p_{1}+p_{2}-p_{1}^{\prime}-p_{2}^{\prime}-k)\,.$

We note that the five-point amplitude appearing here could in principle
include disconnected components beginning at order $g$. This order $g$
disconnected term would involve exactly zero-energy photons, and does not
contribute to observables such as the radiated momentum or the asymptotic
Newman-Penrose scalar. We therefore omit this term in the remainder of this
paper.

To continue, it is useful to perform a change of variable in the phase space
measures, taking $q_{1}\equiv p_{1}-p_{1}^{\prime}$ and $q_{2}\equiv
p_{2}-p_{2}^{\prime}$ as variables of integration. Neglecting Heaviside theta
functions (which will always be unity in the domain of validity of our
calculation) we find

$\displaystyle\langle p_{1}^{\prime}\,p_{2}^{\prime}\,k^{\eta}|S|\psi\rangle=$
$\displaystyle\int\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p^{\prime}_{1}\cdot
x_{1}+p^{\prime}_{2}\cdot x_{2})/\hbar}$ (127)
$\displaystyle\times\int\hat{\mathrm{d}}^{4}q_{1}\,\hat{\mathrm{d}}^{4}q_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot
q_{1}+q_{1}^{2})\hat{\delta}(2p_{2}^{\prime}\cdot
q_{2}+q_{2}^{2})\,e^{i(q_{1}\cdot x_{1}+q_{2}\cdot x_{2})/\hbar}$
$\displaystyle\hskip 50.0pt\times
i\mathcal{A}_{5}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})\,\hat{\delta}^{4}(q_{1}+q_{2}-k)\,.$

Requiring equations (125) and (127) to be equal for any (appropriately
classical) initial wavepacket $\tilde{\phi}_{b}(x_{1},x_{2})$ we deduce that

$\displaystyle\alpha^{(\eta)}(k,x_{1},x_{2})$
$\displaystyle=i\int\hat{\mathrm{d}}^{4}q_{1}\,\hat{\mathrm{d}}^{4}q_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot
q_{1}+q_{1}^{2})\hat{\delta}(2p_{2}^{\prime}\cdot
q_{2}+q_{2}^{2})\hat{\delta}^{4}(q_{1}+q_{2}-k)e^{i(\bar{q}_{1}\cdot
x_{1}+\bar{q}_{2}\cdot x_{2})}$ (128)
$\displaystyle\times\mathcal{A}_{5}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})\left[\int\hat{\mathrm{d}}^{4}q\,\mathrm{d}^{4}x\,e^{iq\cdot
x}e^{iq\cdot(x_{2}-x_{1})/\hbar}e^{i\chi(x_{\perp};s)/\hbar}\right]^{-1}.$

It is easy to use the eikonal equation (86) to show that equivalently we may
write the waveshape as

$\displaystyle\alpha^{(\eta)}(k,x_{1},x_{2})=i\int\hat{\mathrm{d}}^{4}q_{1}\,\hat{\mathrm{d}}^{4}q_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot
q_{1}+q_{1}^{2})\hat{\delta}(2p_{2}^{\prime}\cdot
q_{2}+q_{2}^{2})\hat{\delta}^{4}(q_{1}+q_{2}-k)e^{i(\bar{q}_{1}\cdot
x_{1}+\bar{q}_{2}\cdot x_{2})}$ (129)
$\displaystyle\times\mathcal{A}_{5}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})\left[1+\int\hat{\mathrm{d}}^{4}q\,\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot
q)\,e^{iq\cdot(x_{2}-x_{1})/\hbar}i\mathcal{A}_{4}(s,q^{2})\right]^{-1}\,.$

This last expression makes the physical meaning transparent: the waveshape is
obtained by removing iterated contributions of four-point amplitudes from the
five-point amplitude.

To see this in more detail, it is instructive to expand the $\alpha^{(\eta)}$
order-by-order in perturbation theory. We again consider a generic coupling
$g$ and expand the waveshape as

$\displaystyle\alpha^{(\eta)}=\alpha^{(\eta)}_{0}+\alpha^{(\eta)}_{1}+\cdots\,.$
(130)

The leading order term, $\alpha^{(\eta)}_{0}$, follows immediately from
equation (129):

$\displaystyle\alpha^{(\eta)}_{0}(k,x_{1},x_{2})=i\int\hat{\mathrm{d}}^{4}q_{1}\,\hat{\mathrm{d}}^{4}q_{2}$
$\displaystyle\hat{\delta}(2p_{1}^{\prime}\cdot
q_{1}+q_{1}^{2})\hat{\delta}(2p_{2}^{\prime}\cdot
q_{2}+q_{2}^{2})\hat{\delta}^{4}(q_{1}+q_{2}-k)$ (131) $\displaystyle\times
e^{i(q_{1}\cdot(x_{1}+b)+q_{2}\cdot
x_{2})/\hbar}\mathcal{A}_{5,0}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})\,.$

It is determined by the tree-level five-point amplitude $\mathcal{A}_{5,0}$,
so it is of order $g^{3}$ in gauge theory and gravity. The fact that the
leading-order classical radiation field is intimately related to five-point
amplitudes was already discussed in Luna:2017dtq ; KMOC ; Bautista:2019tdr ;
Cristofoli:2021vyo . The basic structure of this leading-order waveshape is
strikingly reminiscent of a coherent state which describes the _static_
Coulomb/Schwarzschild background Monteiro:2020plf  on analytic continuation
from Minkowski signature to $(2,2)$ signature $(+,+,-,-)$.

More precisely, $\alpha^{(\eta)}_{0}$ is really determined by
$\mathcal{A}_{5,0}^{(0)}$. This follows by counting powers of $\hbar$. Indeed
extracting dominant powers of $\hbar$ using equation (63), we find

$\displaystyle\alpha^{(\eta)}_{0}(k,x_{1},x_{2})=\frac{i}{\hbar^{3/2}}\int\hat{\mathrm{d}}^{4}\bar{q}_{1}$
$\displaystyle\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}^{\prime}\cdot\bar{q}_{2})\hat{\delta}^{4}(\bar{q}_{1}+\bar{q}_{2}-\bar{k})$
(132) $\displaystyle\times e^{i(\bar{q}_{1}\cdot x_{1}+\bar{q}_{2}\cdot
x_{2})}\mathcal{A}_{5,0}^{(0)}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})\,.$

Note that the factor $\hbar^{-3/2}$ arises as expected on general grounds. The
conclusion is that the leading-in-$\hbar$ part of the five-point tree
amplitude determines the radiation. The amplitude itself contains higher order
terms in $\hbar$; rather than arising from the radiative factor in our
proposal (124), these terms arise from a generalised quantum remainder.

The next-to-leading order correction to the waveshape following from equation
(129) is

$\displaystyle\alpha^{(\eta)}_{1}(k,x_{1},x_{2})=i\int\hat{\mathrm{d}}^{4}q_{1}\,\hat{\mathrm{d}}^{4}q_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot
q_{1}+q_{1}^{2})\hat{\delta}(2p_{2}^{\prime}\cdot
q_{2}+q_{2}^{2})\hat{\delta}^{4}(q_{1}+q_{2}-k)$ (133) $\displaystyle\hskip
100.0pt\times e^{i(q_{1}\cdot x_{1}+q_{2}\cdot
x_{2})/\hbar}\mathcal{A}_{5,1}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})$
$\displaystyle-\alpha^{(\eta)}_{0}(k,x_{1},x_{2})\int\hat{\mathrm{d}}^{4}q\,\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot
q)\,e^{iq\cdot(x_{2}-x_{1})/\hbar}i\mathcal{A}_{4,0}(s,q^{2})\,.$

This correction involves the five-point one-loop amplitude, after subtracting
an iteration term. To understand the role of the subtraction, it is
instructive to extract the leading-in-$\hbar$ part of $\alpha^{(\eta)}_{1}$:

$\displaystyle\alpha^{(\eta)}_{1}(k,x_{1},x_{2})=\frac{i}{\hbar^{5/2}}\int\hat{\mathrm{d}}^{4}\bar{q}_{1}\,\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}^{\prime}\cdot\bar{q}_{2})\hat{\delta}^{4}(\bar{q}_{1}+\bar{q}_{2}-\bar{k})$
(134) $\displaystyle\hskip 100.0pt\times e^{i(\bar{q}_{1}\cdot
x_{1}+\bar{q}_{2}\cdot
x_{2})}\mathcal{A}_{5,1}^{(0)}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})$
$\displaystyle-\frac{1}{\hbar^{5/2}}\bar{\alpha}^{(\eta)}_{0}(k,x_{1},x_{2})\int\hat{\mathrm{d}}^{4}\bar{q}\,\hat{\delta}(2p^{\prime}_{1}\cdot\bar{q})\hat{\delta}(2p^{\prime}_{2}\cdot\bar{q})\,e^{i\bar{q}\cdot(x_{2}-x_{1})}i\mathcal{A}_{4,0}^{(0)}(s,q^{2})$
$\displaystyle+\mathcal{O}(\hbar^{-3/2})\,.$

At this stage it seems that there is an unwanted order $\hbar^{-5/2}$ term in
the NLO waveshape! Consistency with our proposal therefore demands

$\displaystyle\int$
$\displaystyle\hat{\mathrm{d}}^{4}\bar{q}_{1}\,\hat{\mathrm{d}}^{4}\bar{q}_{2}\,\hat{\delta}(2p_{1}^{\prime}\cdot\bar{q}_{1})\hat{\delta}(2p_{2}^{\prime}\cdot\bar{q}_{2})\hat{\delta}^{4}(\bar{q}_{1}+\bar{q}_{2}-\bar{k})e^{i(\bar{q}_{1}\cdot
x_{1}+\bar{q}_{2}\cdot
x_{2})}\mathcal{A}_{5,1}^{(0)}(p_{1}^{\prime}+q_{1},p_{2}^{\prime}+q_{2}\to
p_{1}^{\prime},p_{2}^{\prime},k^{\eta})$ (135)
$\displaystyle=-\bar{\alpha}^{(\eta)}_{0}(k,x_{1},x_{2})\int\hat{\mathrm{d}}^{4}\bar{q}\,\hat{\delta}(2p^{\prime}_{1}\cdot\bar{q})\hat{\delta}(2p^{\prime}_{2}\cdot\bar{q})\,e^{i\bar{q}\cdot(x_{2}-x_{1})}\mathcal{A}_{4,0}^{(0)}(s,q^{2})\,.$

Since $\bar{\alpha}^{(\eta)}_{0}$ is determined by $\mathcal{A}^{(0)}_{5,0}$,
this requirement relates $\mathcal{A}^{(0)}_{5,1}$ to
$\mathcal{A}^{(0)}_{5,0}$ and $\mathcal{A}_{4,0}^{(0)}$. The requirement is
nothing but a Fourier transform of the zero-variance relation (66) which we
encountered in section 3.1. So we see that the zero-variance relations retain
their importance in the context of this eikonal/coherent resummation: their
validity admits the possibility of exponentiation.

In the same vein, it is interesting to project our proposal onto a two-photon
final state:

$\displaystyle\langle
p_{1}^{\prime},p_{2}^{\prime},k_{1}^{\eta_{1}},k_{2}^{\eta_{2}}|S|\psi\rangle$
$\displaystyle=\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}$ (136) $\displaystyle\times
e^{i[q\cdot(x-x_{1}+x_{2})+\chi(x_{\perp};s)]/\hbar}\,\alpha^{(\eta_{1})}(k_{1},x_{1},x_{2})\alpha^{(\eta_{2})}(k_{2},x_{1},x_{2})\,.$

Since the waveshape is at least of order $g^{3}$, it follows that this overlap
begins at order $g^{6}$. However by expanding the $S$ matrix out directly, we
encounter a six-point amplitude. The conclusion is that our proposal does not
populate the (order $g^{4}$) tree-level six-point amplitude. Of course this is
as it should be: we saw that the six-point tree is suppressed in the classical
region in section 2.4. Similarly the seven-point tree and one-loop amplitudes
are suppressed, etc.

As a final remark, note that we did not introduce any normalisation factor in
our proposal. This may be surprising in view of the necessity to normalise the
coherent state as in equation (110). The origin of the difference is simply
that unitarity must already guarantee the normalisation of the final state. As
is by now well understood, at two loops the eikonal function $\chi$ ceases to
be real in the radiative case. Instead the imaginary part of $\chi$ is related
to $\sum_{\eta}|\alpha^{(\eta)}|^{2}$; this supplies the necessary
normalisation.

###  4.5 Radiation reaction

Once there is radiation, there must also be radiation reaction: the particle’s
motion must change in the radiative case relative to the conservative case to
account for the loss of momentum to radiation. In this section we will see
that the waveshape indeed contributes to the impulse of a particle in the
manner required.

We begin by acting on our conjectural final state, equation (123), with the
momentum operator of the field corresponding to particle 1:

$\displaystyle\mathbbm{P}^{\mu}_{1}S|\psi\rangle$
$\displaystyle=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\tilde{\phi}_{b}(x_{1},x_{2})\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}$ (137)
$\displaystyle\times\,e^{i[q\cdot(x-x_{1}+x_{2})+\chi(x_{\perp};s)]/\hbar}\exp\left[\sum_{\eta}\int\mathrm{d}\Phi(k)\alpha^{(\eta)}(k,x_{1},x_{2})a^{\dagger}_{\eta}(k)\right]\,p_{1}^{\prime\mu}|p_{1}^{\prime},p_{2}^{\prime}\rangle\,.$

The operator simply inserts a factor $p_{1}^{\prime\mu}$. We proceed by
rewriting this factor in terms of a derivative $-i\hbar\partial/\partial
x_{1\mu}$ acting on the exponential factor in the first line of equation
(137), and then integrating by parts. Neglecting the boundary term, the result
is

$\displaystyle\mathbbm{P}^{\mu}_{1}$ $\displaystyle
S|\psi\rangle=\int\mathrm{d}\Phi(p_{1}^{\prime},p_{2}^{\prime})\,\int\hat{\mathrm{d}}^{4}\bar{q}\,\mathrm{d}^{4}x\,\mathrm{d}^{4}x_{1}\,\mathrm{d}^{4}x_{2}\;\,e^{i(p_{1}^{\prime}\cdot
x_{1}+p_{2}^{\prime}\cdot x_{2})/\hbar}e^{i(q\cdot
x+\chi(x_{\perp};\,s))/\hbar}$ (138) $\displaystyle\times
i\hbar\partial_{1}^{\mu}\left(\tilde{\phi}_{b}(x_{1},x_{2})e^{iq\cdot(x_{2}-x_{1})/\hbar}\exp\left[\sum_{\eta}\int\mathrm{d}\Phi(k)\,\alpha^{(\eta)}(k,x_{1},x_{2})a^{\dagger}_{\eta}(k)\right]\right)\,|p_{1}^{\prime},p_{2}^{\prime}\rangle\,.$

Expanding out the derivative, we encounter three terms. In the first, the
derivative acts on the spatial wavefunction: as usual in quantum mechanics,
this term will evaluate (in the expectation value of the final momentum) to
the contribution of the initial momentum. The second term arises when the
derivative operator acts on $e^{iq\cdot(x_{2}-x_{1})/\hbar}$, which inserts a
factor of $q^{\mu}$. This term is familiar from equation (96) in section 4.2,
and contributes to the impulse as a (suitably projected) derivative of the
eikonal function. Only the final term is new: it involves the waveshape, and
must then be the origin of radiation reaction in our approach.

Since we have discussed the conservative impulse in detail in section 4.2, we