# Final state interactions for high energy scattering off atomic electrons

Ryan Plestid  Mark B. Wise  Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

We consider the scattering of high energy leptons off bound atomic electrons
focusing primarily on final state interactions between the outgoing energetic
electron, and the heavy residual charged “debris” in the final state. These
effects are inherently absent from calculations for a free electron at rest.
Coulomb exchanges are enhanced by the large number of electrons in the atomic
debris, and are unsuppressed by non-relativistic velocities in the debris. We
find that these exchanges can be resummed using operator methods, and cancel
at the level of the cross section until at least $O(\alpha^{3})$. Furthermore,
we argue that both final and initial state Coulomb exchanges (enhanced by the
number of electrons in the atom) do not affect the cross section until at
least $O(\alpha^{3})$. Transverse photon couplings to non relativistic
electrons are proportional to their small velocities, and rotational
invariance suppresses their contribution to $O(\alpha^{3})$. Our results are
relevant for precision experiments involving neutrinos, electrons, positrons,
and muons scattering off of atomic electrons in a fixed target.

††preprint: CALT-TH/2024-026

##  I Introduction

Fixed target experiments play an important role in high energy physics at the
intensity frontier. Examples include neutrino experiments Nunokawa  _et al._
(2008); Abe  _et al._ (2014); Diwan  _et al._ (2016); Acero  _et al._ (2019);
Abe  _et al._ (2021); Abi  _et al._ (2020), searches for feebly interacting
particles Berlin  _et al._ (2019); Nardi  _et al._ (2018); Marsicano  _et al._
(2018); Arias-Aragón  _et al._ (2024); Andreev  _et al._ (2023), rare
processes Perrevoort (2019); Bernstein (2019); Ahn  _et al._ (2021); Baldini
_et al._ (2021); Cortina Gil  _et al._ (2021); Elam  _et al._ (2022), and
precision measurements Anthony  _et al._ (2005); Benesch  _et al._ (2014);
Abbiendi  _et al._ (2017); Aguilar-Arevalo  _et al._ (2015); Altmannshofer
_et al._ (2022). It is often implicitly assumed that atomic electrons can be
treated as free and at rest when they participate in a scattering interaction
with an incident high energy beam. This is a good leading approximation,
however for high precision experiments sub-leading corrections must be
understood theoretically.

In recent work Plestid and Wise (2024) we have studied the corrections due to
atomic binding arising both from phase space, and the initial state (many-
body) wavefunction of the atom. These corrections stem from the finite three-
momentum of electrons in bound atomic orbitals, and the shifted kinematics
from binding energies. Both of these effects enter at the same order and they
must be treated simultaneously. They account for the non-perturbative bound-
state dynamics i.e., from iterated Coulomb exchange in the initial state.

Further corrections arise from perturbative photon exchange between “hard”
leptons, and “soft” (i.e., non-relativistic) electrons and nuclei. These
effects are inherently absent from calculations for a free electron at rest.
For example, for neutrino scattering off an atomic electron, the outgoing
$\sim{\rm GeV}$ electron can exchange photons with the ionized “atomic
debris”. Incident charged lepton scattering (e.g., $\mu e\rightarrow\mu e$)
involves both initial and final state photon exchange with atomic electrons or
nuclei.

In this work, we focus mostly on perturbative final-state interactions
involving the outgoing energetic electron (in Section IV we also comment on
the “shake-up” of the initial state from Coulomb exchange with an incoming
charged lepton). We specifically focus on ladder-graphs involving photon
exchanges with the final state debris. If we track the charge of the debris,
$Q_{B}$, and the charge of the electron, $Q_{e}$, separately then the (gauge
invariant) class of corrections we consider are of order111We explicitly
neglect “mixed” corrections such as those of order $O(Q_{B}Q_{e}^{3})$.
$(Q_{B}Q_{e})^{n}$. Although $Q_{B}=+e$, matrix elements involving the debris
are enhanced for multi-electron atoms. This enhancement occurs for momentum
transfers on the order of the atomic scale (i.e., the inverse Bohr radius)
when the struck electron is able to see a partially (but not fully) screened
atomic nucleus. To match the order studied in Ref. Plestid and Wise (2024),
requires working to order $(Q_{B}Q_{e})^{2}$.

These corrections are a necessary input for an extraction of hadronic vacuum
polarization from $\mu e\rightarrow\mu e$ scattering Carloni Calame  _et al._
(2015) as is being pursued by the MuonE collaboration Abbiendi  _et al._
(2017); Abbiendi (2022). For neutrino electron scattering (relevant for
neutrino flux normalization measurements Marshall  _et al._ (2020)), we find
these corrections are much smaller than the uncertainty from the 1-loop
radiative corrections computed in Ref. Tomalak and Hill (2020) (roughly $\sim
0.5\%$).

We use a non-relativistic power counting for photon interactions with the
atomic system, and standard perturbative counting for the highly relativistic
electron in the final state. We find a number of interesting results. First,
Coulomb corrections cancel in the differential cross section upon integration
over the final state phase space of the atomic debris; this result holds to
all orders in perturbation theory [i.e., resumming graphs of order
$(Q_{B}Q_{e})^{n}$] in the limit of small energy splittings of the debris. For
hydrogen, the cancellation can be understood in terms of a final-state
wavefunction, whereas for multi-electron atoms it stems from an operator-level
identity. Second, sub-leading corrections due to a single transverse photon
exchange in are also shown to cancel. Finally, a class of corrections which
are sensitive to the energy splittings of the atomic debris and appear at
order $e^{2}$ can be shown to have vanishing interference with the leading
order matrix element. Taken together these results imply that final state
interactions (as defined above) do not alter the predictions of Ref. Plestid
and Wise (2024) up to and including $O(Q_{e}^{2}Q_{B}^{2})$ in our power
counting. A similar result holds for interactions that involve the initial
atomic state.

In everything that follows, we will omit factors of $Q_{e}$ and $Q_{B}$ and
simply count powers of $\alpha$. This includes factors of $\alpha$ that stem
from bound-state dynamics i.e., that the typical velocity of an electron in an
atom is $O(\alpha)$. We aim to work to an accuracy of $O(\alpha^{2})$.

The rest of the paper is organized as follows: In Section II we discuss the
case of scattering neutrinos from hydrogen. We find that Coulomb distortion of
the final state cancels at the level of the cross section after integrating
over the final-state proton’s phase space. Next in Section III we show how to
generalize our analysis to multi-electron atoms where the corrections
similarly cancel at the level of the cross section. We find an operator level
expression for Coulomb photons, and also discuss sub-leading corrections such
as transverse photon exchange with non-relativistic electrons in the final
state debris. Transverse photons are discussed in Section III.3. We find the
non-relativistic nature of the electrons combined with rotational invariance
of the initial atom implies that transverse photons first enter at
$O(\alpha^{3})$. In Section IV we describe how to apply our results at the
level of a measurable cross section. We show that the relevant sum-rules that
control cross sections (inclusive with respect to the debris) are unaffected
by Coulomb exchange through $O(\alpha^{2})$. We discuss how this result
applies both to neutrino electron scattering and muon electron scattering.
Finally in Section V we provide a summary and outlook.

##  II Hydrogen

Let us first focus on neutrino scattering off hydrogen as the simplest example
where only final state interaction are present,

$\nu(\mathbf{k})+~\\!\\!{}^{1}{\rm H}({\bf\it
0})\rightarrow\nu(\mathbf{k}^{\prime})+e^{-}(\mathbf{p}^{\prime})+p^{+}(\mathbf{h}^{\prime})~{}.$
(1)

The outgoing proton is heavy, and couples dominantly to Coulomb modes with
couplings to transverse photons suppressed by
$|\mathbf{h}^{\prime}|/m_{p}\sim\alpha m_{e}/m_{p}$. While discussing
hydrogen, we will therefore focus exclusively on the exchange of Coulomb
photons in the final state.

Consider an outgoing scattering state
$\ket{f}=\ket{e^{-}(\mathbf{p}^{\prime})p^{+}(\mathbf{h}^{\prime})}$ labeled
by the asymptotic momentum of the electron and proton. Including the iterated
Coulomb potential, the state $\ket{f}$ satisfies a Lippmann-Schwinger equation

$\ket{f}=\ket{e^{-}(\mathbf{p}^{\prime})}_{0}\ket{p^{+}(\mathbf{h}^{\prime})}_{0}+\frac{1}{E_{f}-\hat{H}_{0}-\mathrm{i}0}\hat{V}\ket{f}~{},$
(2)

where
$\ket{e^{-}(\mathbf{p}^{\prime})}_{0}\ket{p^{+}(\mathbf{h}^{\prime})}_{0}$ is
the free-particle state. Here $\hat{V}$ is the Coulomb interactions,
$\hat{H}_{0}$ is the free-Hamiltonian and $E_{f}=\sqrt{\mathbf{p}^{\prime
2}+m_{e}^{2}}+\sqrt{\mathbf{h}^{\prime 2}+m_{p}^{2}}$ is the exact energy of
$\ket{f}$; the causal regulator ($-\mathrm{i}0$) is fixed by demanding that
$\ket{f}$ be an out-state. The final state can be written in terms of its
valence free-particle Fock states, as222Strictly speaking the Dirac
wavefunction should be spin-dependent, but we suppress spin indices for
simplicity. The answer is unaffected in the kinematic regime of interest Tjon
and Wallace (2006) and Eq. 3 is correct as written.

$\begin{split}\ket{f(\mathbf{p}^{\prime},\mathbf{h}^{\prime})}=\sqrt{2E_{e}({\mathbf{p}^{\prime}})}\sqrt{2E_{p}({\mathbf{h}^{\prime}})}&\int\frac{\differential^{3}q}{(2\pi)^{3}}\frac{\tilde{\phi}_{\mathbf{p}^{\prime},\mathbf{h}^{\prime}}(\mathbf{q})}{\sqrt{2E_{e}({\mathbf{p}^{\prime}+\mathbf{q}})}\sqrt{2E_{p}({\mathbf{h}^{\prime}-\mathbf{q}})}}\ket{e^{-}(\mathbf{p}^{\prime}+\mathbf{q})}_{0}\ket{p^{+}(\mathbf{h}^{\prime}-\mathbf{q})}_{0}~{}.\end{split}$
(3)

Note that when $\hat{V}=0$, the scattering wavefunction is
$\tilde{\phi}_{\mathbf{p}^{\prime},\mathbf{h}^{\prime}}(\mathbf{q})=(2\pi)^{3}\delta^{(3)}(\mathbf{q})$.
The two-particle scattering state is normalized according to

$\begin{split}\innerproduct{f(\mathbf{p}^{\prime},\mathbf{h}^{\prime})}{f(\mathbf{p},\mathbf{h})}=2&E_{e}(\mathbf{p})(2\pi)^{3}\delta^{(3)}(\mathbf{p}-\mathbf{p}^{\prime})\\\
&\times
2E_{p}(\mathbf{h})(2\pi)^{3}\delta^{(3)}(\mathbf{h}-\mathbf{h}^{\prime})~{}.\end{split}$
(4)

The scattering state’s wavefunction depends only on the asymptotic relative
momentum $\mathbf{f}^{\prime}\equiv\mathbf{p}^{\prime}-\mathbf{h}^{\prime}$ so
that
$\tilde{\phi}_{\mathbf{p}^{\prime},\mathbf{h}^{\prime}}(\mathbf{q})=\tilde{\phi}_{\mathbf{f}^{\prime}}(\mathbf{q})$.
The scattering state wavefunction is normalized according to

$\begin{split}\int&\frac{\differential^{3}q}{(2\pi)^{3}}\tilde{\phi}_{\mathbf{f}^{\prime}}^{*}(\mathbf{q}+\mathbf{f}^{\prime})\tilde{\phi}_{\mathbf{f}}(\mathbf{q}+\mathbf{f})=(2\pi)^{3}\delta^{(3)}(\mathbf{f}^{\prime}-\mathbf{f})~{}.\end{split}$
(5)

It is readily checked that Eq. 5 can be substituted into Eq. 3 to obtain Eq.
4. In coordinate space Eq. 5 becomes,

$\begin{split}\int&\differential^{3}x~{}\mathrm{e}^{\mathrm{i}(\mathbf{f}-\mathbf{f}^{\prime})\cdot\mathbf{x}}~{}\phi_{\mathbf{f}^{\prime}}^{*}(\mathbf{x})\phi_{\mathbf{f}}(\mathbf{x})=(2\pi)^{3}\delta^{(3)}(\mathbf{f}^{\prime}-\mathbf{f})~{}.\end{split}$
(6)

We similarly write the initial hydrogen atom in terms of free-particle valence
states,

$\begin{split}\\!\ket{~\\!\\!{}^{1}{\rm H}({\bf\it
0})}=\\!\int\\!\\!\\!\frac{\differential^{3}p}{(2\pi)^{3}}&\frac{\sqrt{2M_{H}}}{\sqrt{2E_{e}(\mathbf{p})}\sqrt{2E_{p}(\mathbf{p})}}\\\
&\hskip
43.36464pt\times\tilde{\psi}(\mathbf{p})\ket{e^{-}(\mathbf{p})}_{0}\ket{p^{+}(-\mathbf{p})}_{0}~{}.\end{split}$
(7)

The bound state’s wavefunction is normalized according to

$\int\frac{\differential^{3}q}{(2\pi)^{3}}|\tilde{\psi}(\mathbf{q})|^{2}=1~{}.$
(8)

The relevant matrix element is

$\mathcal{M}=\matrixelement{f(\mathbf{p}^{\prime},\mathbf{h}^{\prime})\nu(\mathbf{k}^{\prime})}{\mathcal{H}_{W}}{~\\!\\!{}^{1}{\rm
H}(\mathbf{0})\nu(\mathbf{k})}~{},$ (9)

where $\mathcal{H}_{W}$ is the weak Hamiltonian density evaluated at the
origin of spacetime. Using the above definitions, and
$\\!{}_{0}\innerproduct{p^{+}(\mathbf{h}^{\prime}-\mathbf{q})}{p^{+}(\mathbf{-p})}_{0}=2E_{p}(2\pi)^{3}\delta^{(3)}(\mathbf{q}-(\mathbf{h}^{\prime}+\mathbf{p}))$,
we arrive at

$\begin{split}\mathcal{M}&=\int\frac{\differential^{3}p}{(2\pi)^{3}}\sqrt{\frac{2M_{H}2E_{p}}{2E_{e}}}\tilde{\phi}_{\mathbf{p}^{\prime}-\mathbf{h}^{\prime}}^{*}(\mathbf{p}+\mathbf{h}^{\prime})\tilde{\psi}(\mathbf{p})\\\
&\hskip 151.76964pt\times{\sf
M}(\mathbf{p}^{\prime}+\mathbf{p}+\mathbf{h}^{\prime},\mathbf{k}^{\prime};\mathbf{p},\mathbf{k})\\\
&\simeq\int\frac{\differential^{3}p}{(2\pi)^{3}}\sqrt{\frac{2M_{H}2E_{p}}{2E_{e}}}\tilde{\phi}_{\mathbf{p}^{\prime}-\mathbf{h}^{\prime}}^{*}(\mathbf{p}+\mathbf{h}^{\prime})\tilde{\psi}(\mathbf{p})\\\
&\hskip 195.12767pt\times{\sf
M}(\mathbf{p}^{\prime},\mathbf{k}^{\prime};\mathbf{p},\mathbf{k})~{},\end{split}$
(10)

where in the second line we have neglected the
$\mathbf{p}+\mathbf{h}^{\prime}$ (both small) relative to
$\mathbf{p}^{\prime}$ (large); the error due to this approximation is very
small $\sim|\mathbf{p}|/|\mathbf{p}^{\prime}|$ Plestid and Wise (2024). The
free-electron matrix element is defined as

${\sf
M}(\mathbf{p}^{\prime},\mathbf{k}^{\prime};\mathbf{p},\mathbf{k})=\\!~{}_{0}\langle
e^{-}(\mathbf{p}^{\prime})\nu(\mathbf{k}^{\prime})|\mathcal{H}_{W}|e^{-}(\mathbf{p})\nu(\mathbf{k})\rangle_{0}~{}.$
(11)

As mentioned above, in the absence of Coulomb exchanges
$\tilde{\phi}_{\mathbf{p}^{\prime}-\mathbf{h}^{\prime}}(\mathbf{q})\rightarrow(2\pi)^{3}\delta^{(3)}(\mathbf{q})$,
and Eq. 10 reduces to Eq. (14) of Ref. Plestid and Wise (2024).

###  II.1 Perturbation theory

Figure 1: Diagramatic representation of the $O(\alpha)$ correction to the
final state $\ket{f}$ from the exchange of a Coulomb mode. One computes the
tree-level diagram to the right of the dashed line and folds it against the
wavefunction to the left of the dashed line. The proton is drawn as a double
line, the electron as a single line, the neutrino-electron vertex as a grey
box, and the photon as a wavy line.

It is instructive to compute the matrix element perturbatively. This both
serves to illustrate the size of Coulomb corrections at the level of the
amplitude and the connection between diagramatics and wavefunctions.
Furthermore, by analyzing different regions of loop momenta we learn which
characteristic scales dominate the Coulomb corrections.

The $O(\alpha)$ correction involves an integral of the form (including a
photon mass, $\mu_{\gamma}$, to regulate IR divergences)

$I^{(1)}=\\!\int\\!\\!\frac{\differential^{3}\mathbf{q}}{(2\pi)^{3}}\tilde{\psi}(\mathbf{h}^{\prime}-\mathbf{q})\frac{\gamma_{0}(\not{p}^{\prime}+\not{q}+m)}{2p^{\prime}\cdot
q+q^{2}+\mathrm{i}0}\frac{e^{2}}{q^{2}-\mu_{\gamma}^{2}+\mathrm{i}0}~{},$ (12)

where contraction against $\bar{u}(\mathbf{p}^{\prime})$ is implicit. The
energy transfer is small,
$q_{0}\simeq(2\mathbf{h}^{\prime}\cdot\mathbf{q}+\mathbf{q}^{2})/(2m_{p})$. We
will suppress the causal regulators ($+\mathrm{i}0$) in what follows.

Let us focus on a $1s$ orbital of hydrogen for concreteness,

$\tilde{\psi}_{\rm
1s}(\mathbf{h}^{\prime}-\mathbf{q})\simeq\frac{8\sqrt{\pi}\Lambda^{5/2}}{\quantity(|\mathbf{h}^{\prime}-\mathbf{q}|^{2}+\Lambda^{2})^{2}}~{},$
(13)

where we have used the non-relativistic approximation and introduced
$\Lambda=1/a_{0}=\alpha m_{e}$ with $a_{0}$ the Bohr radius.

It is interesting to first understand the limit of large momentum transfers.
These may occur in two qualitatively different regions of phase space. First
consider the final state proton’s momentum to be small
$|\mathbf{h}^{\prime}|\sim\Lambda$. Then, if the Coulomb photon mediates a
large momentum transfer we must have a large initial momentum for the proton
which is suppressed by the hydrogenic wavefunction; in this case we find a
power suppression of $\sim\Lambda^{4}/|\mathbf{q}|^{\prime 4}$. Alternatively,
one can consider a large final state proton momentum with
$\mathbf{h}^{\prime}\sim\mathbf{q}$. In this case the integral factorizes into
a tree-level matrix element (for a proton at rest) multiplied by the
wavefunction at the origin $\psi_{\rm 1s}(\mathbf{x}=0)$ (which is suppressed
by powers of $\alpha$).

Hence the integral in Eq. 12 is dominated by $|\mathbf{q}|\sim\Lambda$ where
one finds

$I^{(1)}\sim\alpha\Lambda^{-3/2}\sim\alpha\tilde{\psi}_{\rm 1s}~{},$ (14)

which is $O(\alpha)$ relative to the leading expression. We may therefore
approximate $I^{(1)}$ by expanding the integrand counting
$|\mathbf{h}^{\prime}|\sim|\mathbf{q}|\ll E^{\prime}_{e}$,

$I^{(1)}\simeq e^{2}\int\frac{\differential^{3}q}{(2\pi)^{3}}\
\tilde{\psi}_{1s}(\mathbf{h}^{\prime}-\mathbf{q})\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}}\frac{1}{\mathbf{q}^{2}+\mu_{\gamma}^{2}}~{},$
(15)

where $\mathbf{v}^{\prime}=\mathbf{p}^{\prime}/E^{\prime}$; this corresponds
to the eikonal approximation. To get a sense of the size of the one-Coulomb-
exchange correction, we may evaluate at the kinematic point
$\mathbf{h}^{\prime}=0$. Then the integral is

$\begin{split}I^{(1)}\rvert_{\mathbf{h}^{\prime}=0}&\simeq
e^{2}\int\frac{\differential^{3}q}{(2\pi)^{3}}\frac{8\sqrt{\pi}\Lambda^{5/2}}{\quantity(\mathbf{q}^{2}+\Lambda^{2})^{2}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}}\frac{1}{\mathbf{q}^{2}+\mu_{\gamma}^{2}}\\\
&=\frac{\mathrm{i}\alpha}{|\mathbf{v}^{\prime}|}\quantity[\log\quantity(\frac{\mu_{\gamma}^{2}}{\Lambda^{2}})+1]\times\tilde{\psi}(\mathbf{h}^{\prime}=\mathbf{0})~{}.\end{split}$
(16)

As expected the correction is controlled by
$\alpha/|\mathbf{v}^{\prime}|\sim\alpha$ and is naively much larger than the
binding corrections computed in Plestid and Wise (2024). At the special
kinematic point $\mathbf{h}^{\prime}=\mathbf{0}$ the correction is purely
imaginary, but is in general complex333Equation 15 has the form of a
convolution and can be written in coordinate space
$\int\differential^{3}x\mathrm{e}^{-\mathrm{i}\mathbf{h}^{\prime}\cdot\mathbf{x}}f(\mathbf{x})\psi_{\rm
1s}(\mathbf{x})$. If for $\mathbf{h}^{\prime}=\mathbf{0}$ this is purely
imaginary, then for non-zero $\mathbf{h}^{\prime}$ it has both real and
imaginary parts. (with non-zero real and imaginary parts) for kinematics such
that $\mathbf{h}^{\prime}\neq 0$.

If we keep the next corrections from the electron’s propagator we find terms
suppressed by $\Lambda/E^{\prime}$ or $\Lambda/m_{p}$ (coming from the energy
transfer $q_{0}=\mathbf{q}^{2}/2m_{p}$). As a concrete example, consider the
correction proportional to $\mathbf{q^{2}}/2m_{p}$. Expanding
$1/(\mathbf{v}^{\prime}\cdot\mathbf{q}+\mathbf{q}^{2}/2m_{p})$ leads to the
correction,

$\begin{split}\delta
I^{(1)}\rvert_{\mathbf{h}^{\prime}=0}&\simeq\frac{e^{2}}{2m_{p}}\int\frac{\differential^{3}q}{(2\pi)^{3}}\frac{8\sqrt{\pi}\Lambda^{5/2}}{\quantity(\mathbf{q}^{2}+\Lambda^{2})^{2}}\quantity(\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}-\mathrm{i}0})^{2}\\\
&=\mathrm{i}\frac{\alpha}{|\mathbf{v}^{\prime}|^{2}}\quantity(\frac{\Lambda}{4m_{p}})\times\tilde{\psi}(\mathbf{h}^{\prime}=0)~{},\end{split}$
(17)

which is suppressed by $(\Lambda/m_{p})$ relative to the leading result.444In
the case of a multi-electron atom, the final state debris can contain
electrons with kinetic energy $T\sim\mathbf{q}^{2}/m_{e}$. In this case the
suppression is $\Lambda/m_{e}\sim O(\alpha)$.

Equation 16 is suppressed by the explicit perturbative factor of $\alpha$, but
is not suppressed by any hierarchy of scales. Coulomb corrections must
therefore be controlled to at least two-loop order. As we discuss below, the
Coulomb corrections actually exponentiate and cancel after integrating over
the phase space of the outgoing proton. The exponentiation of Coulomb
corrections can be seen transparently in coordinate space and we therefore
return to the description in terms of final state wavefunctions.

###  II.2 Coordinate space representation

For non-zero recoil momentum, $\mathbf{h}^{\prime}\neq 0$, we can obtain a
useful representation of the integrals by noticing they have the form of a
convolution. At $n^{\rm th}$-order in $(e^{2})$ we encounter integrals of the
form,

$I^{(n)}=\int\differential^{3}x~{}E_{\mathbf{f}}^{*(n)}(\mathbf{x})\mathrm{e}^{-\mathrm{i}\mathbf{h}^{\prime}\cdot\mathbf{x}}\psi(\mathbf{x})~{}.$
(18)

The function $E_{\mathbf{f}}^{*(n)}(\mathbf{x})$ is a series of nested
convolutions. For example $E_{\mathbf{f}}^{*(1)}$ is given by,

$E_{\mathbf{f}}^{*(1)}(\mathbf{x})=e^{2}\int\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\mathrm{e}^{\mathrm{i}\mathbf{q}_{1}\cdot\mathbf{x}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}}\frac{1}{|\mathbf{q}_{1}|^{2}}~{},$
(19)

and $E_{\mathbf{f}}^{*(2)}$ by

$\displaystyle E_{\mathbf{f}}^{*(2)}(\mathbf{x})=e^{4}\int$
$\displaystyle\frac{\differential^{3}q_{2}}{(2\pi)^{3}}\mathrm{e}^{\mathrm{i}\mathbf{q}_{2}\cdot\mathbf{x}}\int\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{2}}$
(20) $\displaystyle\hskip
43.36464pt\times\frac{1}{|\mathbf{q}_{1}-\mathbf{q}_{2}|^{2}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}}\frac{1}{|\mathbf{q}_{1}|^{2}}~{}.$

Inserting Eq. 19 and Eq. 20 into Eq. 18, writing
$\psi(\mathbf{x})=\int\differential^{3}p/(2\pi)^{3}\mathrm{e}^{\mathrm{i}\mathbf{p}\cdot\mathbf{x}}\tilde{\psi}(\mathbf{p})$,
and carrying out the integral over $\differential^{3}x$ one easily arrives at
the momentum space representation of $I^{(1)}$ and $I^{(2)}$ [cf. Eq. 15].

Next, summing over $n$ we find

$\sum_{n}I^{(n)}=\int\differential^{3}x~{}\psi(\mathbf{x})\mathrm{e}^{-\mathrm{i}\mathbf{h}^{\prime}\cdot\mathbf{x}}E_{\mathbf{f}}^{*}(\mathbf{x})~{},$
(21)

with

$E_{\mathbf{f}}^{*}(\mathbf{x})=\sum_{n}E_{\mathbf{f}}^{*(n)}(\mathbf{x})~{}.$
(22)

This resummation of diagrams is equivalent to solving the Lippmann-Schwinger
equation and $E_{\mathbf{f}}(\mathbf{x})$. The function
$E_{\mathbf{f}}(\mathbf{x})$ is the eikonal approximation to
$\phi_{\mathbf{f}}(\mathbf{x})$. For both relativistic and non-relativistic
electrons, the eikonal wavefunction satisfies Tjon and Wallace (2006)

$\quantity[\mathrm{i}\mathbf{v}^{\prime}\cdot{\bf\it\nabla}+V(\mathbf{x})]E_{\mathbf{f}}(\mathbf{x})=0~{},$
(23)

with $V(\mathbf{x})$ the Coulomb potential. When approximating
$\phi_{\mathbf{f}}$ with $E_{\mathbf{f}}$, Eqs. 5 and 6 hold up to corrections
suppressed by555In the rest frame of the atom the momentum
$\mathbf{f}\simeq\mathbf{p}^{\prime}$, and therefore $\mathbf{p}^{\prime}$
labels the state. $\Lambda/|\mathbf{p}^{\prime}|$ as can be checked usign the
method of stationary phase. The solution to Eq. 23 exponentiates in coordinate
space,

$E_{\mathbf{f}}(\mathbf{x})=\exp[-\frac{\mathrm{i}}{|\mathbf{v}^{\prime}|}\int_{z}^{\infty}\differential
s~{}V(|\mathbf{x}_{\perp}|,s)]\equiv\mathrm{e}^{\mathrm{i}\varphi(\mathbf{x})}~{},$
(24)

where $|\mathbf{x}_{\perp}|$ is the the spatial coordinate perpendicular to
$\mathbf{v}^{\prime}$, and $z$ labels the coordinate direction parallel to
$\mathbf{v}^{\prime}$.

Equation 10 can be written in coordinate space rather than momentum space. One
then finds,

$\mathcal{M}=\sqrt{\frac{2M_{H}2E_{p}}{2E_{e}}}\int\differential^{3}x~{}\mathrm{e}^{-\mathrm{i}\mathbf{h}^{\prime}\cdot\mathbf{x}}\mathrm{e}^{-\mathrm{i}\varphi(\mathbf{x})}{\sf
M}(-\mathrm{i}{\bf\it\nabla})\psi(\mathbf{x})~{},$ (25)

where we have abbreviated ${\sf
M}(\mathbf{p}^{\prime},\mathbf{k}^{\prime};\mathbf{p},\mathbf{k})={\sf
M}(\mathbf{p})$ and transformed to coordinate space where the matrix element
should be interpreted as an operator ${\sf M}(-\mathrm{i}{\bf\it\nabla})$.

Let us now consider the matrix element squared integrated over phase space.
Importantly, in the $m_{p}\rightarrow\infty$ limit, the energy conserving
delta function is independent of the proton’s recoil momentum. Furthermore,
corrections to the phase space due to a finite $\mathbf{h}^{\prime}$ are
suppressed by $\mathbf{h}^{\prime}/E_{e}^{\prime}$ and are negligible. We
therefore have an unconstrained integral over $\mathbf{h}^{\prime}$,

$\displaystyle\int\frac{\differential^{3}h^{\prime}}{(2\pi)^{3}}|\mathcal{M}|^{2}=$
$\displaystyle\int\frac{\differential^{3}h^{\prime}}{(2\pi)^{3}}\int\differential^{3}x\int\differential^{3}y~{}\mathrm{e}^{\mathrm{i}\mathbf{h^{\prime}}\cdot(\mathbf{y}-\mathbf{x})}$
$\displaystyle~{}~{}\times\mathrm{e}^{\mathrm{i}\varphi(\mathbf{y})}\mathrm{e}^{-\mathrm{i}\varphi(\mathbf{x})}$
(26) $\displaystyle\times\quantity[\psi^{*}(\mathbf{y}){\sf
M}^{*}(-\mathrm{i}\overset{\leftarrow}{{\bf\it\nabla}}_{y}){\sf
M}(-\mathrm{i}\overset{\rightarrow}{{\bf\it\nabla}}_{x})\psi(\mathbf{x})]~{}.$

The integral over $\mathbf{h}^{\prime}$ produces
$\delta^{(3)}(\mathbf{x}-\mathbf{y})$ and we find

$\begin{split}\int\frac{\differential^{3}h^{\prime}}{(2\pi)^{3}}|\mathcal{M}|^{2}&=\int\differential^{3}x~{}\absolutevalue{{\sf
M}(-\mathrm{i}{\bf\it\nabla})\psi(\mathbf{x})}^{2}\\\
&=\int\frac{\differential^{3}p}{(2\pi)^{3}}~{}\absolutevalue{\tilde{\psi}(\mathbf{p})}^{2}\times\absolutevalue{{\sf
M}(\mathbf{p})}^{2}~{}.\end{split}$ (27)

This leads to a result which is identical to that obtained neglecting final
state interactions. We see that the Coulomb corrections cancel only after
integrating over final-state phase space. This result is independent of the
details of the initial state wavefunction.

##  III Multi-electron atoms

The analysis for hydrogen is greatly simplified by the presence of a single
recoiling proton in the final state. Inelastic excitations of the final state
proton are gapped-out by large hadronic energy scales $\sim 300~{}{\rm MeV}$
and can be safely neglected. By way of contrast when considering helium and
other multi-electron atoms inelastic excitations must be included. The remnant
system now contains a complex spectra of excitations with energy splittings on
the order of $\sim 1{\rm Ry}=\tfrac{1}{2}\alpha^{2}m_{e}$ and these are always
energetically accessible. Furthermore, while the coupling of a proton to
transverse photons is suppressed by $1/m_{p}$, the coupling of electrons is
only suppressed by $1/m_{e}$, which necessitates the inclusion of both
transverse and Coulomb modes.

The addition of a single electron (i.e., going from hydrogen to helium)
therefore introduces a substantial increase in theoretical complexity.
Nevertheless, as we will show below, the non-relativistic nature of the atomic
debris allows us to simplify the analysis and retain theoretical control. Many
of the properties derived above for hydrogen generalize readily to multi-
electron atoms provided they are re-formulated at an operator level; for
instance we will show that the exponentiation of Coulomb corrections persists.
In what follows we will first motivate our power counting, then analyze
Coulomb photons neglecting energy splittings in the debris, and finally
discuss sub-dominant corrections.

###  III.1 Power counting

In this section we sketch our power counting, motivating the scaling of loop
momentum, and the organizing principles behind our calculation. Following the
analysis for hydrogen we take regions where $|\mathbf{q}|\sim\Lambda\sim\alpha
m_{e}$ to dominate all loop integrals. This guarantees that the atomic debris
remains non-relativistic in both intermediate and final states. When
considering photon exchange in the non-relativistic limit it is useful to
separate different modes from one another. We will phrase our power counting
in Coulomb gauge which simplifies the analysis and work in the rest frame of
the atom.

Non-relativistic particles have an unsurpressed coupling to Coulomb photons.
The interaction strength between two particles with charges $Q_{1}$ and
$Q_{2}$ with relative velocity $\beta$ can be parameterized as
$Q_{1}Q_{2}/\beta$. For interactions among particles in the initial atom
and/or atomic debris this coupling is non-perturbative, and the Coulomb
interaction must be retained at leading order. We therefore include the intra-
Coulomb interaction in the energies of atomic states which we will sum using
completeness relations. For Coulomb exchange with the “struck” electron we
count Coulomb modes as $O(\alpha)$.

The coupling of a particle to transverse photons are velocity suppressed in
the non-relativistic limit. The atomic nucleus is sufficiently heavy that its
velocity can be neglected entirely, while atomic electrons have velocities on
the order of $v_{e}\sim\alpha$. We therefore count the exchange of transverse
photons between the struck electron and the atomic debris as $O(\alpha
v_{e})\sim O(\alpha^{2})$.

Finally, in what follows we will often expand propagators making use of non-
relativistic kinematics. For on-shell states $\ket{X}$ coupled to a photon
with momentum $\mathbf{q}$, we count energy splittings as $\Delta
E_{X}\sim\mathbf{q}^{2}/m_{e}\sim\alpha|\mathbf{q}|$. Therefore we count
powers of $\Delta E_{X}/|\mathbf{q}|$ as $O(\alpha)$.

###  III.2 Analysis of Coulomb photons

Let us first analyze the dominant final state interactions mediated by the
exchange of Coulomb modes. In a multi-electron atom the wavefunction is
replaced by the matrix element of the field operator with a state $\ket{B}$.
Let us begin from the in-out matrix element of the weak interaction
Hamiltonian density (evaluated at $x=0$),

$\displaystyle\matrixelement{\nu^{\prime}e^{\prime}B^{\prime}}{\mathcal{H}_{W}}{A\nu}=\bar{u}_{k^{\prime}}\gamma_{\mu}P_{L}u_{k}\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}\Gamma^{\mu}\psi}{A}~{}.$
(28)

Next we insert a complete set of in-states with electric charge $Q=+e$,

$\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}\Gamma^{\mu}\psi}{A}=\sumint_{B}\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}}{B}\Gamma^{\mu}\\!\matrixelement{B}{\psi}{A}$
(29)

The matrix element, $\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}}{B}$, in
the above equation involves in- and out-states and can be computed using
standard techniques for scattering amplitudes.

A diagrammatic interpretation of
$\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}}{B}$ is now clear. Drawing
the highly energetic electron with a single line, and the atomic debris with a
double line, we have

$\begin{gathered}\raisebox{10.0pt}{$\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}}{B}~{}~{}~{}=~{}~{}~{}~{}$
}\fmffile{diagram_{s}oft}\fmfstraight\fmfgraph*(20,6.5)\fmfleft{i1,i2}\fmfright{o1,o2}\fmf{double}{i1,b1,b2,o1}\fmf{plain}{i2,t1,t2,o2}\fmfv{decor.shape=square,decor.size=10,decor.filled=empty}{i2}\fmfpoly{square,tension=0,shaded}{b2,t2,t1,b1}\raisebox{10.0pt}{~{}~{}~{},}\end{gathered}$
(30)

where the white square is the insertion of $\bar{\psi}$ and the shaded box
includes all possible photon exchanges. In what follows we will expand in
$\Delta E_{X}/|\mathbf{q}|$, but resumming an arbitrary number of Coulomb
exchanges.

#### Leading order in $\Delta E_{X}/|\mathbf{q}|$

As shown in Appendix A, for Coulomb exchanges Eq. 30 can be reduced to an
operator level expression. When $\Delta E_{X}$ is neglected relative to
$|\mathbf{q}|$ the only operator that appears666If energies of intermediate
states are retained then the Hamiltonian also appears in an operator-level
identity . is the Fourier transform of the the charge density at $t=0$,
$\hat{\rho}(\mathbf{q})$. The atomic debris is non-relativistic, the energetic
electron is ballistic (i.e., eikonal), and Coulomb modes are instantaneous;
taken together this implies that only one time-ordering contributes to the
matrix element. Since $\hat{\rho}(\mathbf{q})$, commutes with itself for
different arguments, i.e.,
$[\hat{\rho}(\mathbf{q}),\hat{\rho}(\mathbf{q}^{\prime})]=0$, the sum over
orders in perturbation theory ultimately exponentiates.

The result is that the in-out matrix element in Eq. 30 can be written as

$\\!\\!\\!\matrixelement{e^{\prime}(\mathbf{p}^{\prime})B^{\prime}(\mathbf{h}^{\prime})}{\bar{\psi}}{B(\mathbf{h}^{\prime})}=\bar{u}(\mathbf{p}^{\prime})\matrixelement{B^{\prime}(\mathbf{h}^{\prime})}{\hat{W}_{\mathbf{v}^{\prime}}}{B(\mathbf{h})},$
(31)

where, up to $O(e^{4})$, and evaluating operators at $t=0$,

$\displaystyle\hat{W}_{\mathbf{v}^{\prime}}=1+e^{2}\\!\\!\int\\!\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}}\frac{\hat{\rho}(\mathbf{q}_{1})}{\mathbf{q}_{1}^{2}}$
(32)
$\displaystyle+e^{4}\\!\\!\int\\!\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\frac{\differential^{3}q_{2}}{(2\pi)^{3}}\frac{1}{\mathbf{v}^{\prime}\cdot(\mathbf{q}_{1}+\mathbf{q}_{2})}\frac{\hat{\rho}(\mathbf{q_{2}})}{\mathbf{q}_{2}^{2}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}}\frac{\hat{\rho}(\mathbf{q_{1}})}{\mathbf{q}_{1}^{2}}~{}$
$\displaystyle+~{}\ldots~{}$

The operator $\hat{W}_{\mathbf{v}^{\prime}}$ resembles a Wilson line, but is
defined in terms of the operator $\hat{\rho}(\mathbf{q})$, as opposed to the
photon field $\hat{A}_{\mu}$. Matrix elements of
$\hat{W}_{\mathbf{v}^{\prime}}$ may be thought of as resumming operator
insertions of $\hat{\rho}$,

$\begin{gathered}\raisebox{6.0pt}{$\displaystyle\langle\hat{W}_{\mathbf{v}}\rangle={{\sum_{n}}}\hskip
21.68231pt~{}$ }\fmffile{wilson-
line}\fmfstraight\fmfgraph*(30,7.5)\fmfstraight\fmfbottom{i1,b1,b2,b3,b4,b5,b6,o1}\fmftop{i2,t1,t2,t3,t4,t5,t6,o2}\fmf{phantom,tension=4}{i1,b1,b2,b3,b4,b5,b6,o1}\fmf{dbl_{d}ots,tension=0}{b1,t1}\fmf{dbl_{d}ots,tension=0}{b2,t2}\fmf{dbl_{d}ots,tension=0}{b5,t5}\fmf{dbl_{d}ots,tension=0}{b6,t6}\fmf{phantom}{t3,v,b4}\fmfv{label=$\ldots$}{v}\fmf{plain,tension=4}{i2,t1,t2,t3,t4,t5,t6,o2}\fmfv{label=$\big{\langle}\hat{\rho}(\mathbf{q}_1\\!)
\hat{\rho}(\mathbf{q}_2) ~{}~{}\ldots~{} \hat{\rho}(\mathbf{q}_{n-1}\\!)
\hat{\rho}(\mathbf{q}_n)\big{\rangle}$}{b3}\raisebox{6.0pt}{~{}~{}~{}~{}.}\end{gathered}$

Here, the dotted lines refer to Coulomb modes (with propagators
$\mathrm{i}/\mathbf{q}^{2}$), and the solid line to the eikonalized electron.
See Ref. Vaidya (2021) for a similar construction in the context of Glauber
gluons in QCD.

The second integral in Eq. 32 can be symmetrized between $\mathbf{q}_{1}$ and
$\mathbf{q}_{2}$. Then, using standard eikonal identities,

$\frac{1}{2}\quantity(\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}}+\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{2}})=\frac{1}{2}\frac{\mathbf{v}^{\prime}\cdot(\mathbf{q}_{1}+\mathbf{q}_{2})}{(\mathbf{v}^{\prime}\cdot\mathbf{q}_{1})(\mathbf{v}^{\prime}\cdot\mathbf{q}_{2})}~{},$
(33)

it is straightforward to show [in complete analogy with the eikonal
wavefunction in Eq. 24] that Eq. 32 exponentiates

$\begin{split}\hat{W}_{\mathbf{v}^{\prime}}&=\exp\quantity[e^{2}\\!\\!\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}-\mathrm{i}0}\frac{\hat{\rho}(\mathbf{q})}{\mathbf{q}^{2}+\mu_{\gamma}^{2}}]\\\
&=\exp\quantity[-\mathrm{i}e^{2}\\!\\!\int_{0}^{\infty}\differential
s~{}\hat{V}(\hat{s}\mathbf{v}^{\prime})]~{},\end{split}$ (34)

where we have made the infrared regulator $\mu_{\gamma}$ explicit [cf. Eqs. 15
and 16]. Equation 34 is the operator level generalization of Eq. 24 evaluated
at $\mathbf{x}=0$. We have introduced the Hermitian operator $\hat{V}$

$\hat{V}(\mathbf{x})=\int\frac{\differential^{3}q}{(2\pi)^{3}}\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{x}}\frac{\hat{\rho}(\mathbf{q})}{\mathbf{q}^{2}+\mu_{\gamma}^{2}}~{},$
(35)

which manifests that $\hat{W}_{\mathbf{v}^{\prime}}$ is unitary,
$\hat{W}_{\mathbf{v}^{\prime}}^{\dagger}\hat{W}_{\mathbf{v}^{\prime}}=1$. For
a point-like particle (such as the proton), the matrix element of
$\hat{W}_{\mathbf{v}^{\prime}}$ reduces to a standard Wilson line, expanded in
the limit of small energy-transfer.

Upon summing over all states $\ket{B}\bra{B}$ (which is the analog of folding
with the wavefunction for hydrogen), we obtain the compact formula

$\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}\Gamma^{\mu}\psi}{A}=\bar{u}(\mathbf{p}^{\prime})\matrixelement{B^{\prime}(\mathbf{h}^{\prime})}{\hat{W}_{\mathbf{v}^{\prime}}\hat{\psi}}{A(\mathbf{0})}~{}.$
(36)

Therefore, at the level of the cross section, we have the simple replacement
Plestid and Wise (2024),

$\begin{split}\\!\\!\matrixelement{A}{\hat{\psi}^{\dagger}\delta(\ldots)\hat{\psi}}{A}\rightarrow\\!\matrixelement{A}{\hat{\psi}^{\dagger}\hat{W}_{\mathbf{v}^{\prime}}^{\dagger}\delta(\ldots)\hat{W}_{\mathbf{v}^{\prime}}\hat{\psi}}{A}~{}.\end{split}$
(37)

where the argument of the delta function includes the Hamiltonian, which does
not commute with $\hat{W}_{\mathbf{v}^{\prime}}$. This operator level
replacement holds for arbitrary atoms. It relies only the non-relativistic
dynamics of the atomic debris and on the scale separation between the energy
of the outgoing electron, $E^{\prime}_{e}$ and the binding energies of the
atoms, $\epsilon_{A}\sim\epsilon_{B}$.

The results of this section apply to all orders in the Coulomb interaction in
the limit of negligible energy transfer i.e., neglecting corrections of order
$\Delta E_{X}/\Lambda$. Phenomenologically (e.g., for an extraction of space-
like hadronic vacuum polarization) we expect working to $O(\alpha^{2})$ to be
sufficient; this corresponds to two Coulomb modes. At this order, however,
sub-leading corrections not captured in Eq. 37 must also be considered.

#### Higher order in $\Delta E_{X}/|\mathbf{q}|$

The leading effect arrises from corrections to the struck electron’s
propagator due to finite energy-level splittings, $\Delta E_{X}$, in the
debris. The contribution from the atomic-energy $\Delta E_{X}$ appears at
$O(\alpha\Delta E_{X}/|\mathbf{q}|)\sim O(\alpha^{2})$ and must therefore also
be included. Using the expansion

${u^{\prime}_{0}\over{u^{\prime}\cdot
q_{1}+\mathrm{i}0}}={-1\over{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}-\mathrm{i}0}}+{q^{0}\over(\mathbf{v}^{\prime}\cdot{\mathbf{q}_{1}}-\mathrm{i}0)^{2}}+\ldots$
(38)

where $u_{\mu}=\gamma(1,{\bf v^{\prime}})$ is the four-velocity of the
outgoing electron.

For a single Coulomb exchange (i.e., at order $e^{2}$) the energy transfer is
given by $q_{1}^{0}=h^{{}^{\prime}0}-h^{0}$. The first correction [coming from
the second term on the right-hand side of Eq. 38] gives rise to shift in Eq.
31 that is proportional to

$\langle
B^{\prime}(\mathbf{h}^{\prime})|\left[H,\int{\differential^{3}q_{1}\over(2\pi)^{3}}{\hat{V}(\mathbf{q}_{1})\over(\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}-\mathrm{i}0)^{2}}\right]|B(\mathbf{h})\rangle~{}.$
(39)

Note that ${\bf q}_{1}$ in the integral of Eq. 39 is actually fixed by
momentum conservation. The operator in the matrix element above

${\cal O}=\left[H,\int{\differential^{3}q_{1}\over(2\pi)^{3}}{\hat{V}({\bf
q}_{1})\over(\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}-\mathrm{i}0)^{2}}\right]~{},$
(40)

is anti-Hermitian [recall that
${\hat{V}}(\mathbf{q}_{1})^{\dagger}={\hat{V}}(-\mathbf{q}_{1})$] and
therefore at linear order it’s contribution to the cross section, which is
proportional to

$\Delta\sigma\propto\int{\differential^{3}k_{2}\over(2\pi)^{3}}\int{\differential^{3}k_{1}\over(2\pi)^{3}}\langle
A|\hat{a}^{\dagger}_{\mathbf{k}_{2}}\left({\cal O}+{\cal
O}^{\dagger}\right)\hat{a}_{\mathbf{k}_{1}}|A\rangle~{},$ (41)

vanishes. This correction (at the amplitude level) was discussed in II.1 for
the case $\bf h^{\prime}=0$ when the incoming atom is hydrogen so that $B$ and
$B^{\prime}$ are a single proton.

###  III.3 Transverse photons

Having demonstrated the exponentiation of Coulomb modes, we now turn to
transverse photons which enter first at $O(\alpha^{2})$ in our power counting.
Therefore, to match the size of the binding corrections considered in Ref.
Plestid and Wise (2024), we will only study the exchange of a single
transverse photon. In Coulomb gauge a crucial difference between Coulomb and
transverse modes appears in their propagators:

$D_{00}(q)=\frac{\mathrm{i}}{\mathbf{q}^{2}}~{}~{}~{}{\rm
vs.}~{}~{}D_{ij}(q)=\frac{-\mathrm{i}}{q^{2}+\mathrm{i}0}\quantity[\delta_{ij}-\frac{q_{i}q_{j}}{\mathbf{q}^{2}}]~{}.$
(42)

Transverse modes have poles in the complex $q_{0}$ plane and are propagating
degrees of freedom. Coulomb modes are instantaneous, and non-propagating.

When considering contributions from transverse photons it is therefore not
possible to completely separate final states from initial states. We therefore
organize our discussion along separate lines for final state interactions and
interactions involving the initial state.

#### Exchange in the final state

Let us consider a contribution to Eq. 30 from the exchange of a single
transverse photon. The spatial photon propagator in Coulomb gauge we have

$D_{ij}=\frac{-\mathrm{i}}{q_{0}^{2}-\mathbf{q}^{2}+\mathrm{i}0}\quantity[\delta_{ij}-\frac{q_{i}q_{j}}{\mathbf{q}^{2}}]\rightarrow\frac{\mathrm{i}}{\mathbf{q}^{2}-\mathrm{i}0}\quantity[\delta_{ij}]~{}.$
(43)

Where we have made use of current conservation, and an expansion in
$q_{0}/|\mathbf{q}|$ in the second step. At this order (single transverse
photon exchange) $q_{0}$ is explicitly fixed to be small by on-shell
conditions from the atomic-debris side of the diagram.

It is then clear from the structure of the loop amplitude that to this order
in the expansion, transverse photons can be included by making the operator
level replacement
$\hat{\rho}\rightarrow\hat{\rho}-\mathbf{v}^{\prime}\cdot\hat{\mathbf{J}}$.
Counting non-relativistic matrix elements of the current operator
$\langle\hat{\mathbf{J}}\rangle\sim O(\alpha)$ then guarantees that transverse
photons do not spoil the exponentiation at $O(\alpha^{2})$.

#### Exchange with the initial state

There are also corrections that arise from the exchange of photons between the
outgoing struck electron and the initial atom. Hard photons, with virtuality
on the order of $m_{e}^{2}$ or larger, will factorize from bound state effects
and are already included in existing calculations of QED corrections to $\nu
e\rightarrow\nu e$ scattering with free electrons.

Corrections involving soft photons may, however, be sensitive to atomic
structure. Any transverse photon coupling from the atom to the debris is
doubly suppressed by non-relativistic velocities and enters at
$O(\alpha^{3})$. We therefore focus on diagrams (shown below) in which the
transverse photon couples to the outgoing ballistic electron. It is
instructive to compare the one-loop diagrams involving the exchange of a
Coulomb (dotted line) vs. a transverse (wavy line) photon,

$\begin{gathered}\fmffile{coulomb-
graph}\fmfstraight\fmfgraph*(16,16)\fmfstraight\fmftop{t}\fmfleft{ib,i1,i2,i3,i4,i5,it}\fmfright{ob,o1,o2,o3,o4,o5,ot}\fmfbottom{b}\fmf{dashes}{b,t}\fmf{double,tension=2}{i3,v1,v_{h}ard}\fmf{phantom}{v_{h}ard,o3}\fmffreeze\fmf{double}{v_{h}ard,o1}\fmf{plain,tension=1.1}{v_{h}ard,v2,ot}\fmf{dbl_{d}ots,tension=0,left=0.5,label=$q$}{v1,v2}\fmfv{decor.shape=square,decor.size=10,decor.filled=empty}{v_{h}ard}\fmfv{label=$A$}{i3}\fmfv{label=$B$}{o1}\fmfv{label=$e$}{ot}\hskip
86.72267pt\fmffile{transverse-
graph}\fmfstraight\fmfgraph*(16,16)\fmfstraight\fmftop{t}\fmfleft{ib,i1,i2,i3,i4,i5,it}\fmfright{ob,o1,o2,o3,o4,o5,ot}\fmfbottom{b}\fmf{dashes}{b,t}\fmf{double,tension=2}{i3,v1,v_{h}ard}\fmf{phantom}{v_{h}ard,o3}\fmffreeze\fmf{double}{v_{h}ard,o1}\fmf{plain,tension=1.1}{v_{h}ard,v2,ot}\fmf{photon,tension=0,left=0.5,label=$q$}{v1,v2}\fmfv{decor.shape=square,decor.size=10,decor.filled=empty}{v_{h}ard}\fmfv{label=$A$}{i3}\fmfv{label=$B$}{o1}\fmfv{label=$e$}{ot}\raisebox{20.0pt}{~{}~{}~{}.}\end{gathered}$

We have drawn a dashed line where we insert a complete set of states in Eq.
29. The Coulomb graph vanishes, because its propagator supplies no poles in
the complex $q_{0}$ plane, and the poles from the atom and the debris both lie
on either the upper- or lower-half of the complex plane (see Appendix B of
Plestid (2024) for a more detailed discussion). A similar argument applies to
Coulomb exchanges between the initial state atom and final state debris.
Physically these results follow from the non-relativistic nature of the bound-
state constituents, and the fact that the Coulomb interaction is
instantaneous.

The graph involving a transverse photon has additional poles from the photon
propagator that invalidate the argument given above. Performing the integral
over $q_{0}$ by closing the contour in the upper half plane picks up only the
photon’s pole. This corresponds to an intermediate state that contains a
photon [and contributes to the sum in Eq. 29]. However transverse photons
couple proportional to a three momentum (i.e.,
$\hat{\mathbf{J}}=\sum_{i}\hat{\mathbf{p}}_{e,i}/m_{e}$) and in the scattering
amplitude this is integrated against the initial state multiparticle wave
function. For rotationally invariant targets this integral gives a
contribution to the cross section that vanishes,777Strictly speaking, the
expectation value of momentum can be proportional to the spin of the atom. For
unpolarized targets this cancels upon spin averaging.  because
$\langle\hat{\mathbf{p}}_{e}\rangle\simeq 0$. Therefore, the contribution from
transverse photons to the cross section can occur first at order
$e^{2}\langle\hat{\mathbf{p}}^{2}\rangle/m_{e}^{2}\sim O(\alpha^{3})$.

##  IV Applications

We now return to the phenomenological examples discussed in the introduction.
We begin with neutrinos scattering from atomic electrons. This is convenient
since in the previous sections we have already analyzed $\nu e\rightarrow\nu
e$ scattering for atomic electrons in detail. In the absence of final-state
interactions, the cross section888The same formula applies for the
differential cross section $\differential\sigma/\differential\Pi_{e}$ where
$\Pi_{e}$ is the phase space of the struck electron Plestid and Wise (2024).
for neutrino-atom scattering can be written as Plestid and Wise (2024),

$\displaystyle\sigma$
$\displaystyle=\int\differential\epsilon\int\frac{\differential^{3}p}{(2\pi)^{3}}S_{A}(\epsilon,\mathbf{p})$
(44) $\displaystyle\hskip
65.04034pt\frac{1}{2\omega}\frac{1}{2E}\int\differential\Pi_{e}\frac{1}{2\omega^{\prime}}\delta({\cal
E}-\epsilon)\frac{1}{2}\sum_{\rm spins}|{\sf M}|^{2}~{}.$

Here ${\sf M}(\mathbf{k}^{\prime},\mathbf{p}^{\prime};\mathbf{k},\mathbf{p})$,
is the free-particle matrix element for
$\nu(\mathbf{k})e(\mathbf{p})\rightarrow\nu(\mathbf{k}^{\prime})e(\mathbf{p}^{\prime})$
scattering and $S_{A}(\epsilon,\mathbf{p})$ is the non-relativistic spectral
function of the target atom $A$,

$S_{A}=\\!\\!\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\matrixelement{A}{\hat{a}^{\dagger}_{\mathbf{q}}\delta(\epsilon_{A}+\hat{H}-\epsilon)\hat{a}_{\mathbf{p}}}{A}~{}.$
(45)

Binding corrections can be efficiently incorporated by expanding $|{\sf
M}|^{2}$ in $\epsilon$ and $\mathbf{p}$ Plestid and Wise (2024).

To include final-state Coulomb exchange one simply makes the replacement
$S_{A}\rightarrow\tilde{S}_{A}$ where

$\tilde{S}_{A}=\\!\\!\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\matrixelement{A}{\hat{a}^{\dagger}_{\mathbf{q}}\hat{W}_{\mathbf{v}^{\prime}}^{\dagger}\delta(\epsilon_{A}+\hat{H}-\epsilon)\hat{W}_{\mathbf{v}^{\prime}}\hat{a}_{\mathbf{p}}}{A}~{}.$
(46)

The same procedure (mentioned above) of expanding $|{\sf M}|$ in terms of
$\mathbf{p}$ and $\epsilon$ still applies. We note that Eq. 46 is reminiscent
of the opperatorial definition of a parton distribution function.

The unitarity of $\hat{W}_{\mathbf{v}^{\prime}}$ guarantees that corrections
disappear after integrating over $\differential\epsilon$ (which removes the
delta function). The first term for which this is not true is the energy
weighted sum $\int\differential\epsilon(-\epsilon)\times(\ldots)$, which
introduces the insertion of the Hamiltonian. The non-commutativity of
$\hat{H}$ and $\hat{W}_{\mathbf{v}^{\prime}}$ then gives

$\displaystyle\int\differential\epsilon\tilde{S}_{A}(\epsilon,\mathbf{p})$
$\displaystyle=\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\matrixelement{A}{\hat{a}^{\dagger}_{\mathbf{q}}\hat{a}_{\mathbf{p}}}{A}~{},$
(47)
$\displaystyle\int\differential\epsilon(-\epsilon)\tilde{S}_{A}(\epsilon,\mathbf{p})$
$\displaystyle=\\!\\!\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\matrixelement{A}{\hat{a}^{\dagger}_{\mathbf{q}}\hat{W}_{\mathbf{v}^{\prime}}^{\dagger}[\hat{W}_{\mathbf{v}^{\prime}}\hat{a}_{\mathbf{p}},\hat{H}]}{A}~{}.$
(48)

Equation 48 generalizes the Koltun sum rule Koltun (1972) to include final-
state Coulomb interactions in the leading non-relativistic approximation.

In our counting $\epsilon\sim O(\alpha^{2})$, and so when computing
$O(\alpha^{2})$ corrections to the cross section we can set
$\hat{W}_{\mathbf{v}^{\prime}}=1$ in Eq. 48. This demonstrates that Coulomb
corrections do not alter the predictions of the cross section until at least
$O(\alpha^{3})$.

For the same reason, when considering the sub-leading $O(\alpha^{2})$
corrections we only need to interfere these contributions with the tree-level
matrix element evaluated at $\mathbf{p}=0$ and $\epsilon=0$ (hereafter
referred to as “1”). As argued in Sections III.2 and III.3 the contributions
from final state transverse photon exchange, and corrections due to finite
energy splitting in the debris do not contribute to the cross section at
$O(\alpha^{2})$. We therefore conclude that all final-state interaction
effects can be neglected at $O(\alpha^{2})$ for observables that are inclusive
with respect to the atomic debris (i.e., that have integrated over the debris’
phase space). We note however, that corrections involving transverse photon
exchange with the initial atom must still be evaluated for a complete
$O(\alpha^{2})$ analysis.

###  IV.1 Neutrino flux normalization

One application of our results is for neutrino flux normalizations at long-
baseline experiments such as DUNE and T2HK. These experiments require the $\nu
e\rightarrow\nu e$ cross section to be known with a precision on the order of
(ideally at least a factor of ten smaller than) one percent. In reality these
experiments measure hard outgoing electrons from atomic scattering. For
example in the case of DUNE the relevant process is $\nu{\rm Ar}\rightarrow\nu
eB$ with $B$ all the soft atomic debris.

As discussed in Plestid and Wise (2024), atomic binding corrections are much
smaller than power-counting would suggest due to accidental cancellations that
come from the structure of the tree-level matrix element. Coulomb corrections
vanish upon integrating over the final state debris, and transverse photon
exchange vanishes at least until $O(\alpha^{2})$. We therefore conclude that
all possible sources of QED corrections to $\nu A\rightarrow\nu eB$ scattering
are under sufficient control (i.e., better than $1\%$) for neutrino flux
normalization.

###  IV.2 Extraction of hadronic vacuum polarization

Another process for which the results of this paper are relevant is the
$t$-channel scattering of charged leptons off atomic electrons. We have in
mind (in particular) muons scattering on electrons bound to carbon

$\mu(\mathbf{k})\\!~{}^{12}{\rm
C}(\mathbf{0})\rightarrow\mu(\mathbf{k}^{\prime})e(\mathbf{p}^{\prime})B(\mathbf{h}^{\prime})~{}.$
(49)

This is relevant for the MuonE experiment Abbiendi  _et al._ (2017); Abbiendi
(2022) which aims to extract hadronic vacuum polarization at space-like
kinematics. MuonE’s data can potentially give an independent determination of
hadronic vacuum polarization’s contribution to the anomalous magnetic moment
of the muon Carloni Calame  _et al._ (2015); Abbiendi  _et al._ (2017);
Abbiendi (2022). The theory demands of the experiment are a relative precision
of $10^{-5}$ Banerjee  _et al._ (2020), which necessitates the inclusion of
bound state effects.

In Ref. Plestid and Wise (2024) we have computed the shift in the shape of
$\differential\sigma/\differential t$ due to atomic binding effects. The
results of this paper imply that final-state photon exchange between the
struck electron and the atomic debris do not alter the conclusion of our
previous work at $O(\alpha^{2})$ (the working-order of Ref. Plestid and Wise
(2024)).

Muon electron scattering, however, involves a charged muon which may also
exchange photons with both the initial atom and the outgoing debris. The
results of this paper are therefore not sufficient for a complete account of
all of these effects, however we can comment immediately on a sub-class of
diagrams. Let us first match the hard-photon mediated $\mu-e$ scattering
vertex onto an effective four-Fermi contact operator, $\hat{\mathcal{O}}$. The
operator’s Wilson coefficient includes all short-distance radiative
corrections.

We are then interested in matrix elements of the form
$\matrixelement{B^{\prime}e^{\prime}\mu^{\prime}}{\hat{\mathcal{O}}}{A\mu}$.
If we insert a complete set of states before the operator insertion, we can
re-sum Coulomb exchange into an operator level expression involving
$\hat{W}_{\mathbf{v}_{\mu}}^{\dagger}$ where $\mathbf{v}_{\mu}$ is the
3-velocity of the initial muon. We can resum Coulomb exchange between the
final state leptons and the atomic debris as described above using the product
of $\hat{W}_{\mathbf{v}_{\mu}^{\prime}}$ and
$\hat{W}_{\mathbf{v}_{e}^{\prime}}$. These operators commute with one-another
since they only involve $\hat{\rho}(\mathbf{x})$ or equivalently
$\hat{\rho}(\mathbf{q})$. Our experience with final state Coulomb exchanges
suggests that for high energy muon atomic electron scattering, all the effects
of soft Coulomb exchanges can be captured by making the replacement

$\hat{a}_{\mathbf{p}}\rightarrow\hat{W}_{\mathbf{v}_{\mu}^{\prime}}\hat{W}_{\mathbf{v}_{e}^{\prime}}\hat{a}_{\mathbf{p}}\hat{W}_{\mathbf{v}_{\mu}}^{\dagger}~{},$
(50)

in the definition of the spectral function. The physical interpretation being
that the Wilson-line before $\hat{a}_{\mathbf{p}}$ “shakes up” the initial
state. This leads to different sum rules than in the case of neutrino electron
scattering,

$\int\differential\epsilon\tilde{S}_{A}(\epsilon,p)=\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\matrixelement{A}{\hat{W}_{\mathbf{v}_{\mu}}\hat{a}^{\dagger}_{\mathbf{q}}\hat{a}_{\mathbf{p}}\hat{W}_{\mathbf{v}_{\mu}}^{\dagger}}{A}~{}.$
(51)

We are interested in whether or not the operators $\hat{W}_{\mathbf{v}_{\mu}}$
and $\hat{W}_{\mathbf{v}_{\mu}}^{\dagger}$ modify the sum rules that are
relevant for our analysis. Corrections to the Koltun sum rule, and to the
average kinetic energy are necessarily $O(\alpha^{3})$ and can therefore be
neglected in our $O(\alpha^{2})$ analysis.

The leading-order sum rule,
$\int\differential^{3}p/(2\pi)^{3}\int\differential\epsilon\tilde{S}_{A}(\epsilon,p)=Z$,
could in principle receive an $O(\alpha)$ or $O(\alpha^{2})$ correction.
Integrating over $\differential^{3}p$, using
$\hat{\psi}(0)=\int\differential^{3}p/(2\pi)^{3}\hat{a}_{\mathbf{p}}$ and
carrying out both integration over $p$ and $q$ we find

$\int\frac{\differential^{3}p}{(2\pi)^{3}}\int\differential\epsilon\tilde{S}_{A}(\epsilon,p)=\matrixelement{A}{\hat{W}_{\mathbf{v}_{\mu}}\hat{\rho}_{e}\hat{W}_{\mathbf{v}_{\mu}}^{\dagger}}{A}~{}.$
(52)

where $\hat{\rho}_{e}=\hat{\psi}^{\dagger}\psi$ and the fields are evaluated
at $x=0$. Then, since the electron charge density $\hat{\rho}_{e}$ commutes
with $\hat{W}_{\mathbf{v}_{\mu}}$ we conclude that (provided Eq. 50 holds) the
modified spectral function satisfies the same sum rule as when Coulomb
exchanges are ignored,

$\\!\\!\\!\int\\!\frac{\differential^{3}p}{(2\pi)^{3}}\\!\\!\int\differential\epsilon\tilde{S}_{A}(\epsilon,p)=\\!\\!\int\\!\frac{\differential^{3}p}{(2\pi)^{3}}\\!\\!\int\differential\epsilon
S_{A}(\epsilon,p)=Z~{},$ (53)

where $Z$ is the atomic number of the state $A$. This is not surprising since
Coulomb exchanges do not change the number of electrons in the atom. We
therefore conclude that Coulomb exchanges in both the initial and final state
do not interfere with the leading order amplitude for $\mu\\!~{}^{12}{\rm
C}\rightarrow\mu eB^{+}$ scattering until at least $O(\alpha^{3})$.

##  V Conclusions

We have considered corrections to scattering cross sections that arise when a
target electron is bound in an atom. The focus of the present paper is on
final-state photon exchange and compliments our previous work on corrections
due to the initial bound state Plestid and Wise (2024).

We have studied which regions of loop momentum dominate virtual corrections,
and find that photons with virtuality on the order of the atomic scale
$\Lambda\sim\alpha m_{e}$ dominate the integrals. We have organized these
virtual corrections using a power counting scheme that makes use of the non-
relativistic features of the atomic bound state, $A$, and final state debris,
$B^{+}$.

Within this power-counting, Coulomb exchanges provide the dominant leading
correction (before summing over final states). We have studied these graphs in
detail and find that they can be resummed in the leading non-relativistic
approximation at an operator level. This includes arbitrary multi-particle
intermediate states (e.g., ${}^{12}{\rm C}^{3+}e^{-}e^{-}$) with charge
$Q=+1$. Our work then provides an operator level definition of Coulomb
corrections in the eikonal approximation for a system with small excitation
energies. We find that the unitary nature of the resummed Coulomb corrections
(they resemble a Wilson line) demonstrates that these corrections do not enter
until at least $O(\alpha^{3})$ for neutrino electron scattering. Our
experience suggests that they likely do not enter for $\mu e\rightarrow\mu e$
scattering either (this second result relies on Eq. 50 holding).

We have also considered sub-leading corrections in the final state, such as
transverse photon exchange (suppressed by non-relativistic velocities in the
atom) and corrections to Coulomb exchanges from finite-energy splittings among
different final states in the atomic debris. We find that both of these
corrections (already down by $\alpha^{2}$) do not interfere with the tree-
level cross section after integrating over the final states of the debris, and
therefore do not influence the cross section at $O(\alpha^{2})$. We conclude
that all effects necessary for a sub-percent prediction of the neutrino
(atomic) electron cross section are under control.

In the context of $\mu e\rightarrow\mu e$ scattering, the results of Ref.
Plestid and Wise (2024) and of the current work account for all of the leading
“$Z$-enhanced” corrections to the cross section. These correspond to the class
of diagrams that are $O(Q_{e}^{2}Q_{B}^{2})$, with $Z$-enhancements in medium-
heavy atoms; the correction to the cross section per electron scale
parametrically as $\sim Z^{4/3}\alpha^{2}$.

At $O(e^{4})$, there can be further corrections from final state interactions
that stem from “mixed” corrections involving both Coulomb and transverse
photons e.g.,

$\begin{gathered}\fmffile{diagram_{m}ixed}\fmfstraight\fmfgraph*(20,6.5)\fmfstraight\fmfleft{i1,i2}\fmfright{o1,o2}\fmf{double}{i1,b1,o1}\fmf{plain}{i2,t1,t2,t3,o2}\fmf{photon,tension=0,left=0.5}{t1,t3}\fmf{dbl_{d}ots,tension=0}{t2,b1}\fmfv{decor.shape=square,decor.size=10,decor.filled=empty}{i2}\raisebox{10.0pt}{~{}~{}~{}.}\end{gathered}$

These only receive $Z$-enhancements for Coulomb mode, but not for the
transverse photon. Furthermore, loops with a transverse photon scale like
$\alpha/(4\pi)$, whereas Coulomb modes scale like
$\alpha/|\mathbf{v}^{\prime}|\sim\alpha$. We therefore expect that these
“mixed” corrections scale as $\sim Z^{2/3}\alpha^{2}/(4\pi)$ which is smaller
than the purely kinematic binding corrections by999This estimate does not
account for any large logarithms (or accidental numerical enhancements) that
may appear in the mixed radiative corrections.  $Z^{-2/3}/(4\pi)\approx 0.02$.
We therefore defer their evaluation to future work.

In summary, we have made further progress on understanding radiative
corrections to leptonic interactions with atomic electrons. We find that
photon exchanges (in both the initial and final state) between the atomic
system and any high energy leptons cancel after integrating over the phase
space of the atomic debris. The formalism for Coulomb corrections developed in
this work is appropriate for systems with many low-lying energy levels that
should be summed over. This is different from conventional treatments in terms
of a static background potential. The results in this paper account for all
maximally $Z$-enhanced corrections at $O(\alpha^{2})$. The standard non-
relativistic power-counting applied in this work should serve as a useful
guide for future studies of atomic corrections to muon electron scattering for
the MuonE experiment.

## Acknowledgements

Some of this work was done at the Aspen Center for Physics, which is supported
by National Science Foundation grant PHY-1607611. RP is supported by the
Neutrino Theory Network under Award Number DEAC02-07CH11359. RP and MW are
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics under Award Number DE-SC0011632, and by the Walter Burke
Institute for Theoretical Physics.

##  Appendix A Exponentiation of Coulomb corrections

In this section we derive Eq. 32 by considering the perturbative series that
arises from Coulomb exchange. We show that within a non-relativistic
approximation, the Coulomb exchange leads to an expression involving a unitary
operator that is closely related to a Wilson line . The main novelty of our
derivation is the inclusion of arbitrary states of the atomic debris between
subsequent insertions of the charge-density operator. In particular, we do not
model Coulomb exchanges using a (classical) background potential, and instead
include all possible non-relativistic excited states of the debris as
intermediate states in the perturbative expansion.

We find that only one time-ordering contributes (the uncrossed ladder graphs).
After expanding the ballistic electron propagator to take advantage of the
small energy transfers (fixed by the non-relativistic atomic debris) the
expressions are independent of intermediate state energies at leading order.
This allows the intermediate states to be removed via completeness, and a
simple operator-level identity remains. We now discuss two different
derivations, on in the time-domain and one in the frequency-domain.

###  A.1 Marks Way

It is convenient to define,

$\int\differential^{4}x\mathrm{e}^{\mathrm{i}Q\cdot x}\langle
B^{\prime}(\mathbf{h}^{\prime})e(\mathbf{p}^{\prime})|\bar{\psi}(x)|B(\mathbf{h})\rangle={\bar{u}}I~{},$
(54)

where $Q$ is a residual electron four momentum and our convention is that the
final state electron has zero residual momentum. Factoring out the energy
momentum conserving delta function we write,
$I(q_{1})=(2\pi)^{4}\delta({Q}+(h^{\prime}-h))J(q_{1})$. Next, $I$ (or $J$)
can be expanded in a series $I=I^{(0)}+I^{(1)}+\ldots$ where the superscript
is the number of Coulomb exchanges. The leading order piece is
$I^{(0)}=(2\pi)^{4}\delta({Q}+(h^{\prime}-h))$, or equivalently $J^{(0)}=1$
and for the one photon exchange in Coulomb gauge $I^{(1)}$ we have, treating
the electron as ballistic and using the appropriate vertex and propagator,

$I^{(1)}=e^{2}{u^{\prime 0}\over u^{\prime}\cdot
Q}\int\differential^{4}x_{1}\mathrm{e}^{\mathrm{i}Q\cdot x_{1}}\langle
B^{\prime}(\mathbf{h}^{\prime})|J_{0}(x_{1})|B(\mathbf{h})\rangle~{}.$ (55)

Here $u^{\mu}$ is the electron four velocity ($u^{\mu}=\gamma(1,{\bf
v^{\prime}})$ in the rest frame of the atom) and we have used the fact that
the constituents of the states are non-relativistic so matrix elements of the
electromagnetic 3-vector current are suppressed. Translating the charge
density operator $J_{0}(x)\rightarrow J_{0}(x=0)$ to the origin of space-time
and performing the integration over $x_{1}$ the above is equal to

$\displaystyle I^{(1)}={u^{\prime 0}\over u^{\prime}\cdot Q}{e^{2}\over{\bf
Q}^{2}}(2\pi)^{4}$ $\displaystyle\delta(Q+(h^{\prime}-h))$ (56)
$\displaystyle\langle
B^{\prime}(\mathbf{h}^{\prime})|J_{0}(0)|B(\mathbf{h})\rangle~{}.$

Next we use

$J_{0}(0)=\int{\differential^{3}q_{1}\over(2\pi)^{3}}{\tilde{J}}_{0}({\bf
q}_{1})~{},\vspace{6pt}$ (57)

and $\langle B^{\prime}(\mathbf{h}^{\prime})|{\tilde{J}}_{0}({\bf
q})|B(\mathbf{h})\rangle\propto(2\pi)^{3}\delta({\bf Q}-{\bf q})$. Note that
${\tilde{J}}_{0}({\bf q}_{1})$ is also evaluated at $t=0$. It can be combined
with the factor $1/{\bf q}_{1}^{2}$ into the Fourier transform of the
potential energy operator ${\tilde{V}}$. Finally since non relativistic
energies are small compared to momenta we can write $u^{\prime}\cdot
Q\simeq-u^{\prime 0}{\bf v^{\prime}}\cdot{\bf Q}$ yielding

$J^{(1)}=e^{2}\langle
B^{\prime}(\mathbf{h}^{\prime})|\int{\differential^{3}q_{1}\over(2\pi)^{3}}{{\tilde{V}({\bf
q}_{1}})\over{\bf v^{\prime}}\cdot{\bf q}_{1}}|B(\mathbf{h})\rangle~{}.$ (58)

At second order we have

$\displaystyle\\!\\!\\!\\!I^{(2)}=\int{\differential^{4}q_{2}\over(2\pi)^{4}}{u^{\prime
0}\over u^{\prime}\cdot q_{2}+\mathrm{i}0}{u^{\prime 0}\over u^{\prime}\cdot
Q+\mathrm{i}0}{e^{2}\over{\bf q}_{2}^{2}}{e^{2}\over|{\bf Q}-{\bf
q}_{2}|^{2}}~{}~{}~{}~{}~{}~{}$ $\displaystyle\hskip
54.2025pt\int{\differential^{4}x}\\!\int{\differential^{4}y}~{}\mathrm{e}^{-\mathrm{i}(Q-q_{2})\cdot
x}\mathrm{e}^{-\mathrm{i}q_{2}\cdot y}$ (59) $\displaystyle\hskip
86.72267pt\langle
B^{\prime}(\mathbf{h}^{\prime})|T\\{J_{0}(y)J_{0}(x)\\}|B(\mathbf{h})\rangle~{}.$

First imagine doing the $q_{2}^{0}$ integration (see Section A.2 for an
explicit discussion of the method by contours). The ballistic electron
propagator only has a pole in the lower half $q_{2}^{0}$ complex plane.
Consequently in the time ordered product of the currents only the time
ordering where $y^{0}-x^{0}>0$ contributes and we can so we can
replace101010Since $1=\theta(x^{0}-y^{0})+\theta(y^{0}-x^{0})$ and the second
theta function gives no contribution.

$T\\{J_{0}(x)J_{0}(y)\\}\rightarrow
J_{0}(y)J_{0}(x)\theta(y^{0}-x^{0})\rightarrow J_{0}(y)J_{0}(x)~{},$

in the integral. Next insert a complete set of states
$|X_{1}(p_{X_{1}})\rangle$ between the two currents and translate the currents
to the origin of space-time. Then perform the coordinate integrations. This
gives a product of two delta functions,
$(2\pi)^{4}\delta(p_{X_{1}}-h^{\prime}+q_{2})(2\pi)^{4}\delta(h-p_{X_{1}}-q_{2}+Q)$.
They can be rewritten as a product of the overall four momentum conservation
delta function and one that performs the $q_{2}$ integration setting
$q_{2}=h^{\prime}-p_{X_{1}}$. This yields

$J^{(2)}\simeq\sumint_{X_{1}}{1\over{\bf v^{\prime}}\cdot({\bf
h^{\prime}}-{\bf h})}{1\over{\bf v^{\prime}}\cdot({\bf h^{\prime}}-{\bf
p}_{X_{1}})}{e^{2}\over|{\bf h^{\prime}}-{\bf p}_{X_{1}}|^{2}}{e^{2}\over|{\bf
h}-{\bf p}_{X_{1}}|^{2}}\langle
B^{\prime}(\mathbf{h}^{\prime})|J_{0}(0)|X_{1}(p_{X_{1}})\rangle\langle
X_{1}(p_{X_{1}})|J_{0}(0)|B(\mathbf{h})\rangle~{}.$ (60)

Next express the charge density operator at the origin and an integral over
the three dimensional Fourier transform, and use the fact that matrix elements
conserve momentum to reassemble terms into the Fourier transform of the
potential operator $\tilde{V}$. At this point the sum over intermediate states
can be done yielding

$J^{(2)}\simeq e^{4}{1\over{\bf v^{\prime}}\cdot({\bf h}^{\prime}-{\bf
h})}\langle
B^{\prime}(\mathbf{h}^{\prime})|\int{\differential^{3}q_{1}\over(2\pi)^{3}}{\tilde{V}}({\bf
q}_{2}){1\over{\bf
v^{\prime}}\cdot{\bf{q}_{2}}}\int{\differential^{3}q_{1}\over(2\pi)^{3}}\tilde{V}(\mathbf{q}_{1})|B(\mathbf{h})\rangle~{}.$
(61)

Since the matrix element under the integral is proportional to
$\delta^{(3)}([\mathbf{h}-\mathbf{h}^{\prime}]-[\mathbf{q}_{1}+\mathbf{q}_{2}])$
this can be rewritten as, cf. Eq. 32,

$\displaystyle J^{(2)}=e^{4}\int{\differential^{3}q_{1}\over(2\pi)^{3}}$
$\displaystyle\int{\differential^{3}q_{2}\over(2\pi)^{3}}{1\over{\bf
v}^{\prime}\cdot({\mathbf{q}_{1}}+\mathbf{q_{2}})}{1\over{\bf
v^{\prime}}\cdot\mathbf{q}_{2}}$ (62) $\displaystyle\langle
B^{\prime}(\mathbf{h}^{\prime})|{\tilde{V}}({\bf q}_{2})V({\bf
q}_{1})|B(\mathbf{h})\rangle~{}.$

Obviously this procedure can be repeated to deduce expressions for all the
$J^{(n)}$.

###  A.2 Ryan’s way

Let us repeat the above analysis with a different emphasis and method.
Consider the $n^{\rm th}$-order in perturbation theory. Photon exchange
between the electron line and the charged-debris $\ket{B}$ leads to a
conventional diagrammatic expansion (with fermion and photon propagators) and
a “blob” on the bottom of the diagram with an incoming ionic system $B$ and an
outgoing system $B^{\prime}$. This “blob” corresponds to a multi-current
correlator, $G(q_{1},\ldots,q_{n})$ of time-ordered electromagnetic currents
(specifically $\hat{J}_{0}=\hat{\rho}$ for Coulomb exchange),

$G^{(n)}=\\!\\!\int\\!\\!\\{\differential x\\}\mathrm{e}^{+\mathrm{i}\sum
q_{i}\cdot
x_{i}}\\!\matrixelement{B^{\prime}}{T\\{\hat{J}_{0}(x_{1})\ldots\hat{J}_{0}(x_{n})\\}}{B}.$
(63)

In analogy with the transition from $I$ to $J$ discussed above, it is
convenient to factor out an overall momentum conserving delta function and
introduce $\mathcal{G}$.

$\begin{split}G^{(n)}(q_{1},\ldots,q_{n})=&(2\pi)^{4}\delta(\Sigma q-Q)\\\
&~{}~{}\times\mathcal{G}^{(n)}(Q;~{}q_{2},\ldots,q_{N})~{}.\end{split}$ (64)

where $Q=h^{\prime}-h$ is enforced by the delta function. Notice that
$\mathcal{G}^{(n)}$ is a function of $n-1$ variables, while $Q$ is a parameter
fixed by the states $\ket{B(\mathbf{h})}$ and
$\bra{B^{\prime}(\mathbf{h}^{\prime}}$. Making use of the eikonal
approximation on the highly energetic electron leg, and working in Coulomb
gauge, we then find

$\matrixelement{e^{\prime}B^{\prime}}{\bar{\psi}}{B}=\bar{u}(\mathbf{p}^{\prime})\quantity[\innerproduct{B^{\prime}}{B}+e^{2}\frac{1}{\mathbf{q}_{1}^{2}}\frac{u^{\prime}_{0}}{u^{\prime}\cdot\mathbf{Q}}\mathcal{G}^{(1)}(Q)+e^{4}\int\frac{\differential^{4}q_{2}}{(2\pi)^{4}}\frac{1}{\mathbf{q}_{2}^{2}}\frac{u^{\prime}_{0}}{u^{\prime}\cdot
q_{2}}\frac{1}{|\mathbf{Q}-\mathbf{q}_{2}|^{2}}\frac{u^{\prime}_{0}}{u^{\prime}\cdot
Q}\mathcal{G}^{(2)}(Q;~{}q_{2})+\ldots]~{}.$ (65)

Extension to higher orders is straightforward: one appends an extra integral,
an extra fermion and photon propagator, and uses
$\mathcal{G}^{(n)}(Q;~{}q_{2},\ldots,q_{n})$. The term in square brackets in
Eq. 65 corresponds to $J$ as defined beneath Eq. 54.

It is useful to write $\hat{J}_{0}(0)$ in terms of its components which supply
definite momentum transfer $\mathbf{k}$. For example,

$\begin{split}\mathcal{G}^{(1)}(Q)=\matrixelement{B^{\prime}(\mathbf{h}^{\prime})}{\hat{J}_{0}(0)}{B(\mathbf{h})}=\int\frac{\differential^{3}\mathbf{q}_{1}}{(2\pi)^{3}}\matrixelement{B^{\prime}}{\hat{\rho}(\mathbf{q}_{1})}{B}~{},\end{split}$
(66)

where
$\hat{\rho}(\mathbf{q}_{1})=\int\differential^{3}x~{}\mathrm{e}^{-\mathrm{i}\mathbf{q}_{1}\cdot\mathbf{x}}\hat{J}_{0}(\mathbf{x})$.

We now proceed with a direct evaluation of the loop integrals by contour
integration. Let us study $\mathcal{G}^{(2)}(Q;~{}q_{2})$ as an explicit
example. Begin by explicitly time ordering the currents in
$\mathcal{G}^{(2)}$. Inserting a complete set of states, re-writing
$\hat{J}_{0}(0)$ in terms of $\hat{\rho}(\mathbf{q}_{1})$, and carrying out
the remaining integral over $\differential^{4}x$ we obtain (defining
$\omega_{2}=q_{2}^{0}$),

$\begin{split}\\!\\!\mathcal{G}^{(2)}(Q;~{}q_{2})=\\!\\!\\!\int\\!\\!\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\\!\sumint_{X}\\!\matrixelement{B^{\prime}}{\hat{\rho}(\mathbf{q}_{1})}{X}\frac{-\mathrm{i}}{\omega_{2}-\Delta
E_{X}+\mathrm{i}0}\matrixelement{X}{\hat{\rho}(\mathbf{q}_{2})}{B}\\!+\\!\matrixelement{B^{\prime}}{\hat{\rho}(\mathbf{q}_{2})}{X}\frac{-\mathrm{i}}{\omega_{2}-\Delta
E_{X}^{\prime}-\mathrm{i}0}\matrixelement{X}{\hat{\rho}(\mathbf{q}_{1})}{B},\end{split}$
(67)

where $\Delta E_{X}=E_{X}-E_{B}$ and $\Delta
E_{X}^{\prime}=E_{X}-E_{B}^{\prime}$. This result is obtained by translating
the currents to $t=0$ and integrating the exponentials. It essentially amounts
to a derivation of time-ordered perturbation theory, and the two terms in Eq.
67 can be interpreted as a crossed and uncrossed Feynman diagram.

Next notice that because $1/u^{\prime}\cdot q_{2}\equiv 1/(u^{\prime}\cdot
q_{2}+\mathrm{i}0)$ only has a pole in the lower-half plane. We may then close
our contour in the upper-half plane which selects only the second term in Eq.
67. At higher orders in perturbation theory all time orderings with poles in
the lower-half plane vanish after contour integration; these correspond to
crossed ladders. It follows that only the uncrossed ladders contribute to the
amplitude to any order in perturbation theory, in agreement with the time-
domain analysis presented in Section A.1.

Expanding the eikonal propagators and dropping the (now on-shell) energy
transfer $\Delta E_{X}$ gives,

$\quantity[\frac{u^{\prime}_{0}}{u^{\prime}\cdot
q_{2}+\mathrm{i}0}]_{\omega_{2}=\Delta
E_{X}^{\prime}}\simeq\frac{1}{-\mathbf{v}^{\prime}\cdot\mathbf{q}_{2}+\mathrm{i}0}\left(1+O\quantity(\frac{\Delta
E_{X}^{\prime}}{|\mathbf{q}|})\right)~{}.$ (68)

The propagator structure is then independent of $\Delta E_{X}$, and the sums
over intermediate states reduce to the identity by completeness. We are then
left with Eq. 32, with the $n^{\rm th}$ order contribution being given by

$e^{2n}\int\quantity[\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\ldots\frac{\differential^{3}q_{n}}{(2\pi)^{3}}]\quantity{\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}_{1}}\frac{1}{\mathbf{v}^{\prime}\cdot(\mathbf{q}_{1}+\mathbf{q}_{2})}\ldots\frac{1}{\mathbf{v}^{\prime}\cdot(\sum_{i}\mathbf{q}_{i})}}\prod_{i}\frac{\hat{\rho}_{i}(\mathbf{q}_{i})}{\mathbf{q}_{i}^{2}}=\frac{e^{2n}}{n!}\quantity[\int\\!\frac{\differential^{3}q}{(2\pi)^{3}}\frac{1}{\mathbf{v}^{\prime}\cdot\mathbf{q}-\mathrm{i}0}\frac{\hat{\rho}(\mathbf{q})}{\mathbf{q}^{2}}]^{n}~{},$
(69)

where the equality follows from symmetrizing the eikonal propagators inside
the curly braces (introducing the factor of $1/n!$), and using the soft-photon
identity Yennie  _et al._ (1961); Peskin and Schroeder (1995),

$\sum_{\rm
permutations}\frac{1}{x_{\pi(1)}}\frac{1}{x_{\pi(1)}+x_{\pi(2)}}\ldots\frac{1}{x_{\pi(1)}+x_{\pi(2)}+\ldots+x_{\pi(n)}}=\frac{1}{x_{1}}\frac{1}{x_{2}}\ldots\frac{1}{x_{n}}~{}.$
(70)

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