# Human-centric Literature on Trust for SfTI Veracity Spearhead

Kelly Blincoe  The University of Auckland, New Zealand,
<EMAIL_ADDRESS>Markus Luczak-Roesch  Victoria University of
Wellington, New Zealand<EMAIL_ADDRESS><EMAIL_ADDRESS>Tim Miller  Victoria University of Wellington, New Zealand, markus.luczak-
<EMAIL_ADDRESS><EMAIL_ADDRESS>Matthias Galster  University of
Canterbury, New Zealand<EMAIL_ADDRESS>
###### Abstract

This article summarizes the literature on trust of digital technologies from a
human-centric perspective. We summarize literature on trust in face-to-face
interactions from other fields, followed by a discussion of organizational
trust, technology-mediated trust, trust of software products, trust of AI, and
blockchain. This report was created for the Science for Technological
Innovation Veracity Spearhead supported by New Zealand’s National Science
Challenges.

##  1 Trust in General

Disclaimer: This report does not include a Māori perspective on trust. Work is
in progress to combine this Western view with a Māori perspective.

Trust has been described as facilitating cooperative behaviour [1]. Trust has
been examined extensively in the fields of experimental psychology,
philosophy, sociology, and political science [2]. In sociology, trust is
considered to be multi-faceted with distinct cognitive, emotional, and
behavioral dimensions [2]. The cognitive dimension says that trust is based on
rational decisions, while the emotional dimension, which is also referred to
as affect-based trust, says that emotional relationships between people form a
basis for trust [3, 4]. The behavioral component of trust is the action of
doing something with uncertain outcomes while assuming that all people
involved in the action will act with integrity [5]. Trust is only needed when
there is some level or risk or uncertainty involved, so trust also involves
vulnerability [6]. Research in psychology and philosophy also describe trust
as having both rational and emotional aspects [7].

Much research has considered how initial trust is formed. McKnight and
Chervany identified a set of characteristics based on a review of literature
across multiple disciplines (see Figure 1) [8]. First, a person’s own
disposition to trust, or their willingness or tendency to depend on others,
impacts how trust is formed. Second, there must be conditions in place that
could lead to success, this is called institutional-based trust. Finally,
there are three main trusting behaviours: competence, benevolence, and
integrity. Competence is defined as the belief that the other party has the
required skills, benevolence is the belief that the other party wants to do
good, and integrity relates to the belief that the other party has good values
or character.

Figure 1: Characteristics of Trust [9]

Studies have shown that people are influenced by a “truth bias”, meaning that
they are more likely to assess things as truth than lies, even when being
deceived [10, 11, 12]. People are not very good at identifying deception,
which is defined as someone purposely misleading someone else [13]. Research
in psychology has found that people assume all information as truth initially
and only later change their assessment if they find the information is false
[14]. This is in line with the Truth-Default theory, which states that people
tend to believe each other by default [15].

When trust is damaged, there are negative consequences [16, 17]. Researchers
have studied how trust is repaired after it has been damaged [18, 19]. Unlike
initial trust, significant effort is often required to rebuild trust after a
trust violation. Various factors impact trust repair, including the strength
of the initial trust [20]. A theory which is important for trust repair is
attribution theory [21]. Attribution theory considers the cause of the trust
violation. It considers three main dimensions: locus of control (internal or
external), controllability, and stability [19]. Attribution theory states that
the outcomes and reactions of trust violations will vary based on these
dimensions. The theory also suggests that the outcomes and reactions are not
permanent and that trust can be repaired following violations [22].

##  2 Organizational Trust

While most research in the field of psychology has focused on interpersonal
trust, organizational trust has been studied extensively in other domains like
management and marketing [6]. Organizational trust has been defined as “the
belief that the decision makers will produce outcomes favorable to the
person’s interests without any influence by the person” [23]. Management
researchers argue that trust can improve business performance [6, 24]. Trust
within an organization can result in improved productivity and satisfaction of
employees [25]. In the field of marketing, researchers have found that
consumers’ trust of a business is impacted by both the people within that
business that they interact with and the business’s management practices and
policies [26].

##  3 Technology-Mediated Trust

Trust has primarily been studied from a perspective of human, face-to-face
interactions [27]. When interactions occur through technology, signals of
trust are different [1]. Riegelsberger et al. proposed a framework of trust in
technology-mediated interactions (see Figure 2), which included both
contextual and intrinsic properties of trust [9]. The contextual properties
included in the framework are temporal, social, and institutional
embeddedness. Where temporal embeddedness considers likely future encounters
since repeated interactions can both encourage trustworthy behaviour and
provide signals to make decisions around trust. Social embeddedness considers
the reputation of the person or organization who is being trusted, which can
be discussed and shared across the trustors (those doing the trusting).
Institutional embeddedness refers to the institutions that govern behaviour,
such as judicial systems or organizations, since the rules imposed by these
institutions will influence trust. Yet, there is an acknowledgement that new
technology can disrupt trust formation, since new technology has the potential
to transform the way in which people interact, which can lead to uncertainty
and vulnerability until new norms are established [28].

The intrinsic properties included in Riegelsberger et al.’s framework are
ability, internalized norms, and benevolence, which are in line with the
trusting behaviours of competence, integrity, and benevolence of McKnight and
Chervany described above [8]. Here, ability refers to the capabilities and
characteristics of the person or organization who is being trusted that will
enable them to fulfill the promised outcomes. Internalized norms includes
attributes such as honesty, credibility, reliability, dependability, openness,
and good will. Benevolence represents the enjoyment obtained by person or
organization who is being trusted when good outcomes are experienced by the
person doing the trusting.

Figure 2: Characteristics of trust from Riegelsberger et al. framework for
technology-mediated interactions [9]

While this framework was created to conceptualise trust in technology-mediated
interactions, the elements of trust are still focused on the people or
organizations involved in the trust relationship. Shneiderman also recognized
this in his definition of trust: “If users rely on a computer and it fails,
they may get frustrated or vent their anger by smashing a keyboard, but there
is no relationship of trust with a computer. If users depend on a network and
it breaks, they cannot get compensation from the network. However, they can
seek compensation from people or organizations they trusted to supply a
correctly functioning computer or communication service.” [1] Based on this
definition, Shneiderman developed a set of guidelines for developers of online
services, such as e-commerce or e-services, which are underlined by two key
principles. First, the organization providing the service should ensure trust
both by providing evidence of past trustworthy performance and providing
strong assurances of trust. Second, the organization should clarify
responsibilities and obligations by providing full disclosure of terms,
guarantees, and mechanisms for disputes.

Of course, when considering technology-mediated trust, it is also important to
know that people have different perceptions of and attitudes towards
technology, and so technology-mediated trust will be subjective [29]. This is
in line with the characteristics of trust in general by McKnight and Chervany
which state that a person’s own disposition to trust impacts trust formation.

##  4 Trust of Software

In line with this, studies have shown that for software products, trust is
based on both a trust of the creators of the software product and a trust of
the software itself [30, 31]. Similarly, Siau and Wang argue that trust in
technology is determined by three main factors: human characteristics,
environment characteristics, and technology characteristics [32]. Users of
software products assess the trustworthiness of software in different ways
[31, 33, 34]. Yang et al. propose a software trust framework which considers
software correctness, security, and reliability as measures of trustworthiness
[33]. Jackson equates trust to dependability of the software product to
perform a particular task [31]. These definitions all relate to the intrinsic
ability and internalized norms of the software. On the other hand, Wang et al.
show that user feedback of software products is useful to determining levels
of trust, which relates to the reputation of the software [35]. Mercuri
considered the view of transparency as it relates to trust of software,
defining different ways that transparency can be achieved [36]. For example,
through open sourced code, certifications, and assurances. Provenance, defined
as “metadata about the origin, context or history of data”, can also promote
transparency [37].

Another perspective comes from the field of human computer interaction where
the relationship between trust and user interface design has been studied.
Interface designers argue that the visual design of the interface forms the
first impressions of trust [38]. Rendell et al. found that inclusion of nature
imagery on websites positively influenced users’ perceptions of trust [39].
Xiling found that simple and well laid out interfaces promoted trust [40].
They also found that familiarity, being able to clearly relate the “offline”
brand and experience to the online interface, was important for trust. Xiling
found that usability was important for building trust [40]. Systems that were
easy to use, consistent, and logically structured were more trusted. While
researchers have investigated the use of particular colors in an interface and
their relationship with user trust, no relationship was identified [41].

##  5 Trust of AI

With the rise of Artificial Intelligence (AI) to perform decision-making, it
is also important to consider trust as it relates to these systems in
particular. Glikson and Woolley find that the representation of an AI system
(e.g., robot, humanoid, embedded) and the system’s capabilities are important
factors in developing trust [42]. Siau and Wang present a list of factors that
are used for both building initial trust in AI systems and developing
continuous trust in those systems [32]. They find that understanding how AI
works (its transparency and explainability) and being able to trial the AI
system before adopting it (trialability) are important for initial trust
formation. In addition, the visual appearance of the AI (its representation),
reviews of the AI system written by other users, and the users’ perceptions of
AI in general based on exposure to things like media coverage or Sci-fi books
will also impact initial trust.

While many AI systems operate as black boxes (there is no way to understand
why decisions are being made), transparency, explainability, and
interpretability are still seen as important for trust of AI systems [43].
Some research defines interpretable as understandable or transparent [44].
Others define interpretable as providing explanations for decisions. In these
cases, the model may not be transparent, but some understandable reasons for
decisions are provided by the black box AI model [45]. Thus, interpretability
may be defined as both explainable or transparent. Explanations are often
proposed to improve trust in AI systems [46, 47] and recent research shows
that software users do want explanations when complex decisions are being made
[48].

For developing continuous trust in AI systems, Siau and Wang [32] find that
usability and reliability, collaboration and communication, sociability and
bonding, security and privacy protection, interpretability, concerns about job
replacement, and goal congruence are important factors. Of course, accuracy is
also important. Yin et al. found that people considered both a model’s stated
accuracy and its observed accuracy in determining their trust of the model
[49]. Siau and Wang say “trust in AI takes time to build, seconds to break,
and forever to repair once it is broken!” [32]

We have seen many examples where AI has gone wrong, and Winfield and Jirotka
argue that ethical governance is critical to building trust in AI [50].
Through a literature review of trust and AI, Lockey at al. identified five
main challenges: 1) transparency and explainability, 2) accuracy and
reliability, 3) automation resulting in job loss, 4) anthropomorphism (or
including human-like characteristics) leading to over-estimation of the AI
system, and 5) privacy concerns related to mass data extraction [51].

Another important factor related to trust in AI is fairness [52]. While one
might assume machines can make more fair decisions that are free from human
bias, it is well known that AI systems actually amplify existing bias [53,
54]. Historical bias in training data can cause AI systems to learn this bias
and make biased decisions.

Accountability is also important [52] for trust in AI. This factor considers
who will be held responsible for the decisions made by AI systems. There is
currently not a clear answer to who should be held accountable. The Law
Commission of England and Wales and the Scottish Law Commission recently
proposed that self driving car users should not be held responsible for
crashes and other driving
offenses.111https://www.forbes.com/sites/zacharysmith/2022/01/25/self-driving-
car-users-shouldnt-be-held-responsible-for-crashes-uk-report-says. However, in
the US, self driving car users are considered responsible. Research suggests
more auditing of AI is needed to reduce corporate reputation damage and assure
AI is legal, ethical, and safe [55].

Trust of AI also comes down to perceptions of how decisions are made. Machines
make decisions which are rule-based and algorithmic [56]. Machines do not
consider emotions in their decision-making nor can they learn in the same way
as humans [57]. These differences can lead to algorithmic aversion, where
people prefer human made decisions even if the decisions are inferior to those
made by a machine [58].

Jussupow et al. defined four characteristics of algorithms that influence
aversion: 1) algorithm agency which describes the level at which the algorithm
behaves autonomously; 2) algorithm performance which considers the accuracy
and failures of the algorithm; 3) perceived algorithm capabilities which
describes the algorithm’s perceived ability to perform the task; and 4) human
involvement which relates to how much humans (but not the end user) are
involved in training and using the algorithm [58].

Recent research has found that people do not want AI to make moral decisions
[59]. Through a series of studies, Bigman and Gray found that people distrust
AI to make moral decisions even when the outcome is favorable since “machines
can neither fully think nor feel” [59]. They suggest that limiting AI to
providing only advice and increasing the AI’s perceived experience and
expertise are ways to improve trust in AI for moral decisions [59].

Figure 3 summarizes the factors that influence trust for digital technologies,
including software products and AI.

Figure 3: Factors that affect trust of digital technology

##  6 Blockchain

A discussion on trust of software is not complete without a mention of
Blockchain. Blockchain has been nicknamed a “trustless” technology [60, 61].
It emerged due to a growing lack of trust in centralized systems which relied
on trust of institutions (e.g. banks) [62]. The idea of a trustless technology
goes back to the theory of Wang and Emurian that claims that trust is only
needed when there is some level or risk or uncertainty involved [6].
Blockchain is a distributed, immutable ledger. This means transactions cannot
be modified once they are written to the ledger and all participants have
access to the shared ledger. Thus, “users subject themselves to the authority
of a technological system that they are confident is immutable, rather than to
the authority of centralized institutions which are deemed untrustworthy.”
[62]

De Filippi et al. prefer to label Blockchains as “confidence machines”, since
their underlying technology creates shared expectations and confidence in the
correctness of its transactions. However, while many Blockchains remove the
need to trust a single organization, they still require “distributed trust”
since there are often a large number of actors who require a low-level of
trust [62]. There still needs to be trust that the data going into the
Blockchain can be trusted since compromised data cannot be corrected. These
actors must be trusted not to collude and cause collective harm. It should
also be noted that not all Blockchains follow this same model and some (e.g.
The Linux Foundation’s Hyperledger and Amazon’s QLDB) are maintained by
organizations, which means these organizations will also still require trust.
Using blockchain may lead to trade-offs between trust and other concerns like
energy consumption and sustainability [63]. These tradeoffs could in turn
compromise trust because benevolence and integrity can be adversely affected
if a Blockchain is perceived to be non-sustainable.

There does not appear to be literature on trust violations and repair in
relation to blockchain technology.

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The effective field theory of extended Wilson lines

Ryan Plestid

Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA, 91125 USA

We construct the effective theory of electrically charged, spatially extended,
infinitely heavy objects at leading power. The theory may be viewed as a
generalization of NRQED for particles with a finite charge distribution where
the charge radius and higher moments of the charge distribution are counted as
$O(1)$ rather than $O(1/M)$. We show this is equivalent to a Wilson line
traced by the worldline of an extended charge distribution. Our canonical use
case is atomic nuclei with large charge $Z\gg 1$. The theory allows for the
insertion of external operators and is sufficiently general to allow a
treatment of both electromagnetic and weak mediated lepton-nucleus scattering
including charged-current processes. This provides a first step towards the
factorization of Coulomb regions, including structure dependence arising from
a finite charge distribution, for scattering with nuclei.

††preprint: CALT-TH-2024-019

###### Contents

  1. I Introduction
  2. II Motivation and setup
  3. III Construction of the effective theory
    1. III.1 Construction from NRQED
    2. III.2 Wilson lines and decoupling transformations
    3. III.3 Emergence of a Coulomb field in the static limit
  4. IV Applications to $(e,e^{\prime})$ scattering
    1. IV.1 Eikonal approximation
    2. IV.2 Structure mismatch
  5. V Conclusions
  6. A Gauge invariance of background field terms

##  I Introduction

The soft limit of gauge theories is fully characterized by an effective field
theory (EFT) of Wilson lines, where particles are treated as infinitely heavy
point-like sources Caswell and Lepage (1986); Korchemsky and Radyushkin
(1992); Isgur and Wise (1989, 1990); Georgi (1990); Falk et al. (1990); Bauer
et al. (2002); Becher and Neubert (2009); Grozin (2022). Feynman rules involve
eikonal propagators, and allow one to dramatically simplify the analysis of
the soft sector of a theory. This simplification underlies our ability to
control radiative corrections for hard scattering processes, and the shuffling
of soft dynamics into Wilson lines is the basis of many factorization theorems
Bauer et al. (2003); Beneke and Feldmann (2004); Bauer et al. (2004); Becher
et al. (2007); Becher and Schwartz (2008); Stewart et al. (2010); Chiu et al.
(2012); Feige and Schwartz (2014); Rothstein and Stewart (2016).

When particles are very heavy, this is an especially powerful tool. For
example, consider the scattering of $E\sim 1~{}{\rm GeV}$ charged particles
off of a $M\sim 200~{}{\rm GeV}$ particle of charge $Z\gg 1$. The large charge
of the heavy particle, $Z\gg 1$, ensures that the matrix element is dominated
by ladder graphs and since $E\ll M$ the heavy particle is effectively static.
One can then make use of eikonal identities to reduce the leading series in
$Z\alpha$ to a potential scattering problem Brodsky (1971); Dittrich (1970);
Neghabian and Gloeckle (1983); Weinberg (2005) dramatically simplifying the
analysis.

In the example above the particles we have in mind are heavy nuclei, e.g.
lead, tungsten, gold etc. or any other heavy nucleus relevant for particle
physics experiments. Unlike elementary particles, however, heavy nuclei become
larger (i.e. more spatially extended) as their mass increases. The scaling of
length scales for a nuclei and point-like particles is given by

$R\sim M^{1/3}\mbox{\quad(nucleus)\quad},\mbox{\quad vs.\quad}~{}~{}R\sim
1/M\mbox{\quad(point-like).\quad}$ (1)

Clearly the scaling of the radii are qualitatively opposite. The consequence
of this scaling is that for heavy nuclei the inverse radius is typically
$R^{-1}\sim 25~{}{\rm MeV}$ while their mass is $M\sim 200~{}{\rm GeV}$. The
point-like limit then only applies for $Q\ll R^{-1}\ll M$, severely
restricting the phase space where a point-like treatment is appropriate as
opposed to $Q\ll M$ as in e.g., the case of a proton. In potential scattering
models, one simply replaces the Coulomb field by a static charge distribution,
solves Gauss’ law, and computes scattering in the potential. This allows one
to extrapolate the static field approximation to larger values of $Q$, however
it is not obvious how to interpret such a procedure in an EFT framework, or
how to systematically improve the calculation to account for higher order
radiative corrections.

The purpose of the present article is to construct an EFT applicable when
$qR\sim O(1)$ while $q/M\ll 1$. This is the generalization of point-like
Wilson lines to a theory of Wilson lines with extended structure as shown
schematically in Fig. 1. We focus on QED because of its simplicity and
relevance for applications to nuclear scattering and bound-state problems;
there are few, if any, composite states more ubiquitous than atomic nuclei. We
will restrict our discussion to the insertion of a single external operator.

We are motivated by experiments that require percent-level (or better)
precision for reactions involving nuclei; in particular the neutrino
oscillation program Branca et al. (2021); Tomalak et al. (2022) and super
allowed beta decay Seng et al. (2018); Czarnecki et al. (2019); Hardy and
Towner (2020). Other applications use nuclear targets and explore extremely
specialized pockets of phase space where radiative corrections may be larger
than expected and poorly understood; we have in mind Mu2e/COMET and LDMX
Åkesson et al. (2018); Bernstein (2019). In these systems, long and
intermediate wavelength photons will experience an enhancement, proportional
to $Z$, due to their coherent interaction with the nucleus. When considering
virtual QED corrections to high energy scattering with nuclei one must
integrate over all scales ranging from the coherent to the incoherent regime,
and one therefore always probes the intermediate regime $qR\sim O(1)$.

Precision measurements involving nuclei must account for the $Z$-enhanced
contributions discussed above. To do so one must account for the composite’s
structure beyond tree level, i.e. inside of loops and this necessitates a QFT
formalism that explicitly includes the scale of nuclear structure inside
higher order loop diagrams. When $1/R$ is well separated from other energy
scales in the problem, then an EFT treatment can be formalized which separates
coherently enhanced regions from high energy regions where $Z$-enhancements
are absent and conventional QED power counting holds. In what follows we study
how structure dependence from large composite particles enters the loop
expansion. While nuclei are our main focus, our results apply to any theory
with composite particles coupled to a $U(1)$ gauge theory.

The rest of the paper is organized as follows. In Section II we sketch how
coherently enhanced regions appear when considering QED loops in the “full
theory” using electron-nucleus scattering as an example. We apply the method
of regions and find that coherent enhancements occur in a particular momentum-
region. In Section III we construct an EFT which allows for composite
structure, and which reduces to NRQED in the point-like limit. We apply these
ideas to $(e,e^{\prime})$ scattering in the high energy limit in Section IV.
Finally in Section V we summarize our findings, and suggest future directions
of interest from both a formal EFT and phenomenological perspective.

Figure 1: Schematic of extended Wilson line operator for an extended charge
distribution $\rho(\mathbf{x})$ tracing out a trajectory along a worldline
parameterized by the time-like vector $v_{\mu}$. This is a generalization of a
typical point-like Wilson line. Equations 18 and 19 define the appropriate
non-local operator in general gauge. The gauge invariant Feynman rule is given
by Eq. 14.

##  II Motivation and setup

Let us consider a reaction

$A~{}+~{}X_{1}\quad\rightarrow\quad B~{}+~{}X_{2}$ (2)

where $A$ and $B$ are heavy composite objects, while $X_{1}$ and $X_{2}$
represent probe states. Working in Feynman gauge, a generic ladder
diagram111We focus on ladder diagrams here to illustrate the appearance of
coherent enhancements. In the EFT we develop there is no restriction to a
ladder topology.  in which $n$ photons are exchanged with the nucleus will be
given by (with $\differential q_{i}=\differential^{4}q/(2\pi)^{4}$)

$\begin{split}{\rm i}\mathcal{M}_{a}^{(n)}=\int&\quantity[\differential
q_{i}~{}\frac{-{\rm
i}}{q_{i}^{2}}]X^{\mu_{1}...\mu_{n}}_{a}(q_{1},...,q_{n})\\\
&\int\quantity[\differential^{4}x_{j}]{\rm e}^{-{\rm i}q_{i}\cdot
x_{i}}\matrixelement{B(\mathbf{p}^{\prime})}{T\\{J_{\mu_{1}}(x_{1})...\mathcal{J}(0)...J_{\mu_{n}}(x_{n})\\}}{A(\mathbf{p})}~{},\end{split}$
(3)

where $\mathcal{J}$ is an external current, $X^{\mu_{1}...\mu_{n}}_{a}$
represents some fixed set of photon insertions among the legs of the charged
particles that make up either $X_{1}$ or $X_{2}$ in Eq. 2. The square brackets
are a short hand for a product over indices.

The time ordering produces all possible crossed and uncrossed ladder diagrams.
For QED applications, it suffices to study the correlator (shown
diagramatically in Fig. 2)

$G_{\\{\mu_{i}\\}}(\\{q_{j}\\})=\int\quantity[\differential^{4}x_{j}~{}{\rm
e}^{-{\rm i}q_{j}\cdot
x_{j}}]\matrixelement{B(\mathbf{p}^{\prime})}{T\\{J_{\mu_{1}}(x_{1})...\mathcal{J}(0)...J_{\mu_{n}}(x_{n})\\}}{A(\mathbf{p})}~{}.$
(4)

For small photon momenta photon couplings can be enhanced by $Z_{A}$ and/or
$Z_{B}$. We analyze the problem in the limit of $q/M\rightarrow 0$, but with
$qR\sim O(1)$ held fixed. Our goal will be to identify contributions enhanced
by factors of $Z$.

Figure 2: Diagram corresponding to Eq. 4. The gray box denotes the insertion
of a single external current, and the blob includes all possible permutations
of the photon vertices.

Let us explicitly expand the time orderings by labeling the time orderings
sequentially by $\\{a,b,...\beta,\alpha\\}$ such that
$x_{a}<x_{b}<...<0<...<x_{\beta}<x_{\alpha}$. Here, Latin letters correspond
to times before the hard current $\mathcal{J}(0)$ and Greek letters to times
after the hard current. Next, insert a complete set of states between
$J_{\mu_{a}}(x_{a})$ and $J_{\mu_{b}}(x_{b})$. Integration over
$\differential^{3}x$ enforces momentum conservation via a delta function,
while the time integral gives

$~{}\int_{-\infty}^{t_{b}}\differential t_{a}~{}{\rm e}^{-{\rm i}\Delta
E_{a}t_{a}}\matrixelement{A(\mathbf{p}+\mathbf{q}_{a})}{J_{\mu_{a}}}{A(\mathbf{p})}=\frac{{\rm
e}^{-{\rm i}\Delta E_{a}t_{b}}}{-\Delta E_{a}+{\rm
i}0}\matrixelement{A(\mathbf{p}+\mathbf{q}_{a})}{J_{\mu_{a}}}{A(\mathbf{p})}~{}.$
(5)

where $\Delta
E_{a}=\epsilon_{A}(\mathbf{p}+\mathbf{q}_{a})-[\omega_{a}+\epsilon_{A}(\mathbf{p})]$
where $\epsilon_{A}(\mathbf{p})=(\mathbf{p}^{2}+M_{A}^{2})^{1/2}$ is the on-
shell energy of the heavy particle $A$. If we focus on the last photon in the
time ordering we instead find

$~{}\int^{\infty}_{t_{\beta}}\differential t_{\alpha}~{}{\rm e}^{{\rm i}\Delta
E_{\alpha}t_{\alpha}}\matrixelement{B(\mathbf{p})}{J_{\mu_{\alpha}}}{A(\mathbf{p}-\mathbf{q}_{\alpha})}=\frac{{\rm
e}^{{\rm i}\Delta E_{\alpha}t_{\beta}}}{\Delta E_{\alpha}+{\rm
i}0}\matrixelement{B(\mathbf{p})}{J_{\mu_{\alpha}}}{A(\mathbf{p}-\mathbf{q}_{\alpha})}~{}.$
(6)

where $\Delta
E_{\alpha}=\epsilon_{B}(\mathbf{p}+\mathbf{q}_{\alpha})-[\omega_{\alpha}+\epsilon_{B}(\mathbf{p})]$.
Carrying this procedure out term by term in each time ordering then yields
analogous terms with e.g. $\Delta
E_{b}=\epsilon_{A}(\mathbf{p}+\mathbf{q}_{a}+\mathbf{q}_{b})-[\omega_{a}+\omega_{b}+\epsilon_{A}(\mathbf{p})]$
and higher terms defined iteratively. After integrating over all times, the
result is that222Note that Eq. 7 can be derived directly using old-fashioned
perturbation theory.

$\begin{split}G_{\\{\mu_{i}\\}}(\\{q_{j}\\})=\sum_{\rm
perms}\prod_{a}^{N_{L}}\frac{J_{\mu_{a}}^{[A]}(\mathbf{q}_{a})}{-\Delta
E_{a}+{\rm
i}0}\prod_{\alpha}^{N_{R}}\frac{J_{\mu_{\alpha}}^{[B]}(\mathbf{q}_{\alpha})}{\Delta
E_{\alpha}+{\rm
i}0}~{}\times\mathcal{J}_{AB}&\quantity([\mathbf{p}+\Sigma\mathbf{q}_{a}]-[\mathbf{p}^{\prime}-\Sigma\mathbf{q}_{\alpha}])~{},\\\
&\hskip 43.36464pt+({\rm other~{}states})\end{split}$ (7)

where
$J^{[A]}_{\mu_{a}}(\mathbf{q}_{a})=\matrixelement{A(\mathbf{p}+\mathbf{q}_{a})}{J_{\mu_{a}}}{A(\mathbf{p})}$
and
$J^{[B]}_{\mu_{a}}(\mathbf{q}_{a})=\matrixelement{B(\mathbf{p}+\mathbf{q}_{\alpha})}{J_{\mu_{\alpha}}}{B(\mathbf{p})}$
are on-shell matrix elements of the current $J_{\mu}$, and “other states”
refers to all intermediate states other than elastic scattering.333These
states are subdominant to the elastic scattering transitions for highly
charged objects, $Z\gg 1$. The transition matrix element between $A$ and $B$
depends, in general, on the momenta transferred via virtual photons,

$\mathcal{J}_{AB}=\matrixelement{B(\mathbf{p}^{\prime}-\Sigma\mathbf{q}_{\alpha})}{\mathcal{J}}{A(\mathbf{p}+\Sigma\mathbf{q}_{a})}~{}.$
(8)

The labels $N_{L}$ and $N_{R}$ in Eq. 7 denote the number of photons inserted
prior to (left of) and after (right of) the hard current in a given time
ordering.

The most general matrix element (i.e. for arbitrary spin) of the
electromagnetic current contains a unique form factor in the
$M\rightarrow\infty$ limit Lorce (2009a, b),

$\matrixelement{A(\mathbf{p}+\mathbf{q})}{J_{\mu}}{A(\mathbf{p})}=Zv_{\mu}F(\mathbf{q}^{2})+O(q/M)~{},~{}$
(9)

where we have tacitly assumed that all higher order form factors that appear
in Lorce (2009b) are normalized with $O(1)$ coefficients such that
$(Ze)(|\mathbf{q}|^{n}/M^{n})$ provides a reasonable estimate of their
parametric size444E.g. the magnetic moment of nuclei is of order $(Ze)/M\sim
e/2m_{N}$ i.e. the nuclear magneton.  Lorce (2009a, b). Neglecting all recoil
effects such that the $\Delta E_{a}\rightarrow-\omega_{a}$ and $\Delta
E_{\alpha}\rightarrow\omega_{\alpha}$. This leads finally to

$\frac{\matrixelement{A(\mathbf{p}+\mathbf{q}_{a})}{J_{\mu_{a}}}{A(\mathbf{p})}}{-\Delta
E_{a}+{\rm
i}0}\rightarrow\frac{(Z_{A}e)v_{\mu_{a}}F_{A}(\mathbf{q}_{a}^{2})}{\Delta\omega_{a}+{\rm
i}0}=\frac{(Z_{A}e)F_{A}(\mathbf{q}_{a}^{2})v_{\mu}}{v\cdot q_{a}+{\rm
i}0}~{},$ (10)

where $v_{\mu}=(1,0,0,0)$ is the four velocity of the heavy composite in its
rest frame. This is precisely what one would obtain using a Wilson line or
heavy particle Feynman rules, but with a form factor included at every vertex.

We now have simplifying expressions for generic correlators in the limit
$R^{-1}\sim q\ll M$. This simplifying limit of the correlators naturally
appears when one considers loop-momenta regions where $q_{i}\sim R^{-1}\ll M$
Beneke and Smirnov (1998); Jantzen (2011); we will call this the soft region.
Hard regions $q\gg R^{-1}$ will factorize from soft regions. We therefore
expect the soft region, where the hadronic correlators have a simplified form
and where $Z$-enhanced contributions exist, to be described by a corresponding
EFT which we now describe.

##  III Construction of the effective theory

In the previous section we have seen that an analysis of correlation functions
of heavy particles naturally yields Feynman rules with eikonal propagators and
structure-dependent vertex functions. It is then natural to ask whether the
soft region presented above has its own EFT. We claim that it does, and that
the degrees of freedom are the Wilson lines traced out by a finite charge
density moving along an eikonalized world line.

###  III.1 Construction from NRQED

In the limit where $q\ll 1/R$ we must be able to match onto NRQED Caswell and
Lepage (1986); Manohar (1997); Hill et al. (2013); Paz (2015). For large
composite objects, the charge radius and other higher multipoles will be
“unaturally large” in the power counting of NRQED because $1/M\ll R$. Counting
powers of $R$ and $M$ independently then allows one to consistently account
for higher derivative operators that are enhanced by the large nuclear radius
while still performing an expansion in $1/M$. Beyond the leading kinetic
terms, the NRQED Lagrangian in the one-particle sector can be constructed out
of $\mathbf{E}$, $\mathbf{B}$, and the non-relativistic field $\psi_{v}$ which
carries the four-velocity label $v$.

Importantly $\mathbf{B}$ is odd with respect to parity and must be accompanied
by a non-trivial spin structure, e.g. ${\bf\it\sigma}\cdot\mathbf{B}$. By
contrast $\mathbf{E}$ is even with respect to parity and this therefore allows
operators of the form ${\bf\it\partial}\cdot\mathbf{E}$ and more generally
$({\bf\it\partial}^{2})^{n}{\bf\it\partial}\cdot\mathbf{E}$. The Wilson
coefficients of these operators may be determined by a matching calculation
that equates $\matrixelement{\mathbf{p}+\mathbf{q}}{J_{\mu}}{\mathbf{p}}$ in
both the full theory and NRQED (see e.g. Hill et al. (2013) for details). As
we have discussed in Eq. 9, if we neglect terms that are suppressed by powers
of $|\mathbf{q}|/M$ we only need the charge form factor $F_{1}(q^{2})$. In
this limit we may write the Lagrangian as

$\mathcal{L}_{\rm EFT}=\psi_{v}^{\dagger}({\rm i}v\cdot
D)\psi_{v}+\sum_{n=1}^{\infty}C_{{\bf\it\partial}\cdot\mathbf{E}}^{(n)}\psi_{v}^{\dagger}\psi_{v}~{}\quantity({\bf\it\partial}^{2})^{n-1}{\bf\it\partial}\cdot\mathbf{E}~{}+O(|\mathbf{q}|/M)~{}.$
(11)

Matching in the one particle sector then dictates

$C_{{\bf\it\partial}\cdot\mathbf{E}}^{(n)}=\frac{\overline{F}^{(n)}}{n!}~{}+O(Z|\mathbf{q}|/M)~{}.$
(12)

These are the unique set of terms that contribute to $\mathcal{L}_{\rm NRQED}$
that are i) coherently enhanced, ii) survive in the $M\rightarrow\infty$
limit.

Notice that in Eq. 11 the minimal coupling term, ${\rm i}v\cdot D$, is unique
in that it contributes to the matching of $F(q^{2})$, and contains the scalar
potential $v\cdot A$. At higher orders in $1/M$ one can always remove factors
of $v\cdot A$ with the equations of motion. We may re-sum the full set of
$R$-enhanced Wilson coefficients555See Aoude et al. (2021); Haddad and Helset
(2020) for a similar discussion but where the spin, rather than the radius, of
the object is counted as large., using the fact that
$\sum_{n=1}^{\infty}\frac{\overline{F}^{(n)}}{n!}(q^{2})^{n-1}=f(q^{2})$ where
$f(q^{2})=(F(q^{2})-1)/q^{2}$ we find

$\mathcal{L}_{\rm EFT}=\psi^{\dagger}({\rm i}v\cdot
D)\psi+\psi_{v}^{\dagger}\psi_{v}^{~{}}f({\bf\it\partial}^{2}){\bf\it\partial}\cdot\mathbf{E}+O(|\mathbf{q}|/M)~{}.$
(13)

The Feynman rule for a 1-photon vertex generated by Eq. 13 agrees order-by-
order in the derivative expansion with Eq. 11 up to corrections of
$O(|\mathbf{q}|/M)$. It is given explicitly by

$\begin{split}\raisebox{-10.0pt}{\fmffile{1coul-a} \fmfgraph*(65,35)
\fmfbottom{i1,d1,o1} \fmftop{i2,d2,o2} \fmf{double}{i1,b1,o1}
\fmf{phantom}{o2,t1,i2} \fmf{photon,tension=0}{b1,t1} \fmfblob{0.2w}{b1}
}&=~{}~{}{\rm i}(Ze)v_{\mu}+{\rm
i}(Ze)\frac{F(\mathbf{q}^{2})-1}{-\mathbf{q}^{2}}\quantity[v_{\mu}q^{2}-q_{\mu}v\cdot
q]\\\ &=~{}~{}{\rm i}(Ze)v_{\mu}+{\rm
i}(Ze)\frac{F(\mathbf{q}^{2})-1}{q_{\perp}^{2}}\quantity[v_{\mu}q_{\perp}^{2}-q_{\perp\mu}v\cdot
q]~{},\end{split}$ (14)

where $\mathbf{q}_{\perp\mu}=q_{\mu}-(v\cdot q)v_{\mu}=(0,\mathbf{q})$. Notice
that if this expression is multiplied by $\delta(v\cdot q)$ then the Feynman
rule reduces to

${\rm i}v_{\mu}(Ze)+{\rm
i}(Ze)\frac{F(\mathbf{q}^{2})-1}{q_{\perp}^{2}}\quantity[v_{\mu}q^{2}-q_{\mu}v\cdot
q]\rightarrow{\rm i}v_{\mu}(Ze)F(\mathbf{q}^{2})\mbox{\quad for\quad}v\cdot
q=0~{},$ (15)

as one would naively expect for a static background field.

Being constructed from the gauge invariant quantity
${\bf\it\partial}\cdot\mathbf{E}$, current conservation is automatically
enforced for terms proportional to $f(\mathbf{q}^{2})$. The first term in Eq.
14 comes from the minimal coupling from the kinetic term
$\psi^{\dagger}_{v}(v\cdot D)\psi_{v}$. When included in a full class of
diagrams, including any light/dynamical charged particles present in the
reaction, the resultant sum will satisfy current conservation. A similar
resummation of terms enhanced by $(qR)^{n}$ at subleading powers in $1/M$
(e.g. for the dipole density distribution) may be performed at higher power,
although we do not pursue this here. The analysis is straightforward involving
the identification of terms from a given form factor which contribute order-
by-order in the NRQED expansion.

The minimal coupling part of the Lagrangian will satisfy Ward identities when
combined with other parts of the amplitude. The structure dependent piece is
manifestly gauge invariant vertex-by-vertex. Taken together this then ensures
that the relevant Ward identities will be satisfied and that the inclusion of
form factors inside loops will produce gauge invariant amplitudes. See
Appendix A for a more detailed argument.

###  III.2 Wilson lines and decoupling transformations

Having now understood that the EFT of massive extended objects is a natural
generalization of NRQED with modified power counting, we have arrived at
Feynman rules that are very similar to those for point-like Wilson lines. In
what follows, we will construct the operator definition of a Wilson line for a
finite charge distribution that reproduces the Feynman rules derived above.
The definition is non-local and simplifies dramatically in Coulomb gauge.

Using coordinates such that $x_{\mu}=(\mathbf{x},v\cdot x)$, the QED Wilson
line, for a point-particle of charge $Z$, is given by Wilson (1974); Becher et
al. (2015); Grozin (2022)

$\begin{split}\widehat{W}_{v}(x)&=\exp\quantity[{\rm
i}Ze\int_{0}^{\infty}\differential s~{}~{}v\cdot\widehat{A}(\mathbf{x},v\cdot
x+s)]~{},\\\ &=\exp\quantity[{\rm i}Ze\int_{0}^{\infty}\differential
s~{}\int\differential^{3}y~{}\delta^{(3)}(\mathbf{x}-\mathbf{y})v\cdot\widehat{A}(\mathbf{y},v\cdot
x+s)].\end{split}$ (16)

In the second equality we have suggestively re-written the standard definition
in terms of a three dimensional point-like charge distribution. One might
guess a natural generalization for an extended charge distribution is to smear
the vector potential with a charge distribution,

$\begin{split}\widehat{W}_{v,\rho}(x)&=\exp\quantity[{\rm
i}Ze\int_{0}^{\infty}\differential
s~{}\int\differential^{3}y~{}\rho(\mathbf{y}-\mathbf{x})v\cdot\widehat{A}(\mathbf{y},v\cdot
x+s)]\mbox{\quad(naive).\quad}\end{split}$ (17)

however this turns out not to be gauge invariant. The correct expression in
general gauge is given by

$\begin{split}\widehat{W}_{v,\rho}(x)&=\widehat{W}_{v}(x){\rm e}^{{\rm
i}Ze\chi}~{}\mbox{\quad(general gauge).\quad}\end{split}$ (18)

where $\chi$ is given by,

$\begin{split}\chi&=\int\differential^{3}y~{}\quantity(\rho(\mathbf{x}-\mathbf{y})-\delta^{(3)}(\mathbf{x}-\mathbf{y}))\frac{1}{{\bf\it\partial}^{2}}[{\bf\it\partial}^{2}(v\cdot
A)-(v\cdot\partial){\bf\it\partial}\cdot{\bf\it A}](\mathbf{y},v\cdot
x+v_{\mu}s)~{},\\\
&=\int\differential^{3}z\int\differential^{3}y~{}\quantity(\rho(\mathbf{x}-\mathbf{y})-\delta^{(3)}(\mathbf{x}-\mathbf{y}))\frac{1}{|\mathbf{y}-\mathbf{z}|}\\\
&\hskip 173.44534pt[{\bf\it\partial}^{2}(v\cdot
A)-(v\cdot\partial){\bf\it\partial}\cdot{\bf\it A}](\mathbf{z},v\cdot
x+s)~{}.\end{split}$ (19)

This operator reproduces the Feynman rule Eq. 14 in arbitrary gauge. Setting
${\bf\it\partial}\cdot\mathbf{A}=0$ we arrive at the expression in Coulomb
gauge

$\begin{split}\widehat{W}_{v,\rho}(x)&=\exp\quantity[{\rm
i}Ze\int_{0}^{\infty}\differential
s~{}\int\differential^{3}y~{}\rho(\mathbf{y}-\mathbf{x})v\cdot\widehat{A}(\mathbf{y},v\cdot
x+s)]~{}\text{(Coul. gauge)~{},}\end{split}$ (20)

which agrees with the naive expectation. It is interesting to see how this
works in a diagramatic language where Coulomb gauge amounts to the following
choice for the photon propagator,

$D^{\mu\nu}=-\frac{{\rm i}}{q_{\perp}^{2}}v^{\mu}v^{\nu}-\frac{{\rm
i}}{q^{2}+{\rm
i}0}\quantity[(-g^{\mu\nu}+v^{\mu}v^{\nu})+\frac{q_{\perp}^{\mu}q_{\perp}^{\nu}}{q_{\perp}^{2}}]~{},$
(21)

such that $D^{00}={\rm i}/\mathbf{q}^{2}$ for $v_{\mu}=(1,0,0,0)$. Using the
Feynman rule for the vertex in Eq. 14 we arrive at,

$\Gamma_{\mu}D^{\mu\nu}=J_{0}D^{0\nu}-J_{i}D^{i\nu}=J_{0}D^{00}+J_{i}D^{ij}=\quantity({\rm
i}(Ze)F(\mathbf{q}^{2})\frac{{\rm i}}{\mathbf{q}^{2}},0)$ (22)

where we have used
$q_{\perp}^{i}(\delta^{ij}-q_{\perp}^{i}q_{\perp}^{j}/\mathbf{q}^{2})=0$. One
then sees that the extended Wilson line vertex, when contracted with a
Coulomb-gauge propagator reduces to the naive expectation of a static Coulomb
field with a charge form factor.

One can check that the above Wilson line reproduces the Feynman rules derived
from the heavy particle Lagrangian above. It would be interesting to develop
factorization theorems in terms of these extended Wilson lines rather than
their point-like counterparts, however we leave this for future work.

###  III.3 Emergence of a Coulomb field in the static limit

It is well known that ladder graphs resum in QED to produce the Coulomb
potential. This was first analyzed in the context of a relativistic Dirac
fermion Brodsky (1971); Dittrich (1970); Neghabian and Gloeckle (1983).
Weinberg demonstrated that this result was universal, independent of the spin
of the particle, applying for any system when momenta transfer are
sufficiently low that one is in the point-like limit Weinberg (2005). All of
Refs. Brodsky (1971); Dittrich (1970); Neghabian and Gloeckle (1983); Weinberg
(2005) restrict themselves to the following setup

  * •

Elastic scattering.

  * •

Charge conservation in the heavy sector.

  * •

The point-like limit (i.e. $q_{i}R\ll 1$ for all photon momenta in the graph).

The analysis presented in this work dispense with all of these assumptions and
demonstrates that the electrostatic field of an extended charge density
distribution emerges in the static limit independent of the above assumptions
provided the source is effectively static i.e., that $|q|\ll M$.

Let us denote the vertex function from the EFT by $\Gamma_{\mu}(q_{i})$. It
will be convenient to phrase out discussion in terms of hadronic correlators,
$G_{\\{\mu_{i}\\}}(\\{q_{j}\\})$, but now computed directly in the EFT.666This
is equivalent to including only elastic excitations of the nucleus in the full
theory. It captures the leading-in-$Z$ effects. We have,

$\begin{split}G_{\\{\mu_{i}\\}}(\\{q_{j}\\})=\sum_{\rm
perms}\prod_{a}^{N_{L}}\frac{Z\Gamma_{\mu_{a}}(q_{a})}{v\cdot q+{\rm
i}0}\prod_{\alpha}^{N_{R}}\frac{Z\Gamma_{\mu_{\alpha}}(q_{\alpha})}{-v^{\prime}\cdot
q_{\alpha}+{\rm i}0}~{}\times\mathcal{J}_{AB}~{},\end{split}$ (23)

where $\mathcal{J}_{AB}$ is an external operator we have inserted in the
theory. Notice that the sum over eikonal propagators in Eq. 23 is equivalent
to a product of delta functions which set every factor of $v\cdot q_{i}=0$
Weinberg (2005),

$G_{\\{\mu_{i}\\}}(\\{q_{j}\\})=\mathcal{J}_{AB}\quantity[\prod_{i}(2\pi{\rm
i})\delta(v\cdot
q_{i})Z_{A}\Gamma_{\mu_{i}}(q_{i})]=\mathcal{J}_{AB}\quantity[\prod_{i}(2\pi{\rm
i})\delta(v\cdot q_{i})Z_{A}v_{\mu_{i}}F(\mathbf{q}^{2}_{i})]~{}.$ (24)

where we have used $\Gamma_{\mu}(q)=v_{\mu}F(\mathbf{q}^{2})$ for $v\cdot
q=0$. Theses are the Feynman rules for a background Coulomb field sourced by a
static charge density distribution, $\rho(\mathbf{x})$, whose Fourier
transform gives $F(\mathbf{q}_{i}^{2})$.

In deriving Eq. 24 one must properly account for infinitesimal imaginary parts
in the denominator i.e. $1/(v\cdot q_{\alpha}+{\rm i}0)$ vs $1/(-v\cdot
q_{a}+{\rm i}0)$. If $\mathcal{J}_{AB}$ is a charged current then the
leading-$Z$ behavior is still Coulomb in nature, and new generalized eikonal
identities can be used to simplify sub-leading contributions Plestid (2024).

##  IV Applications to $(e,e^{\prime})$ scattering

In this section we will consider electron nucleus scattering at large momentum
transfer,

$e(\mathbf{k})~{}+~{}A(\mathbf{p})~{}\rightarrow~{}\ell^{-}(\mathbf{k}^{\prime})~{}+~{}B(\mathbf{p}^{\prime})~{},$
(25)

in the limit $R^{-1}\ll|Q|\ll M$ where $Q^{2}=(p-p^{\prime})^{2}$. In this
case the hard current is the electromagnetic current itself
$\mathcal{J}_{\nu}=J_{\nu}$. At tree level the amplitude is given by

${\rm i}\mathcal{M}^{(0)}={\rm
i}e^{2}\frac{1}{Q^{2}}\bar{u}(k)\gamma^{\nu}u(k)\matrixelement{B(\mathbf{p}+\mathbf{Q})}{\mathcal{J}_{\nu}}{A(\mathbf{p})}~{}.$
(26)

At 1-loop, ladder diagrams appear coupling the lepton to both the initial and
final state nucleus.

$\begin{split}{\rm i}\mathcal{M}^{(1)}&={\rm
i}e^{4}\int\frac{\differential^{4}q_{1}}{(2\pi)^{4}}~{}\frac{1}{q_{1}^{2}}\frac{1}{(q_{1}-Q)^{2}}\frac{\bar{u}(k^{\prime})\gamma^{\nu}(\not{k}+\not{q}_{1}+m)\gamma^{\mu_{1}}u(k)}{2k\cdot
q_{1}+q_{1}^{2}}G_{\mu_{1};\nu}(q_{1})\\\ &\hskip 43.36464pt+{\rm
i}e^{4}\int\frac{\differential^{4}q_{1}}{(2\pi)^{4}}~{}\frac{1}{q_{1}^{2}}\frac{1}{(q_{1}-Q)^{2}}\frac{\bar{u}(k^{\prime})\gamma^{\nu}(\not{k}^{\prime}-\not{q}_{1}+m)\gamma^{\mu_{1}}u(k)}{-2k^{\prime}\cdot
q_{1}+q_{1}^{2}}G_{\mu_{1};\nu}(q_{1})~{}.\end{split}$ (27)

At $n^{\rm th}$ order in perturbation theory we will encounter
$G_{\mu_{1},...\mu_{n};\nu}(q_{1}...q_{n})$.

We introduce the power counting parameter $\lambda=(QR)^{-1}$. Then, using the
method of regions, we may split the integral into a region where $q\sim 1$ and
$q\sim\lambda$ respectively. We will refer to these as the soft and hard
regions respectively. Coherent enhancements occur exclusively in the soft
region where the theory is described by the EFT outlined above. The matrix
element of this EFT, $\mathcal{M}_{S}^{(1)}$ may be obtained from Eq. 27 by
counting $q\sim\lambda$ and keeping leading order terms in $\lambda$.

###  IV.1 Eikonal approximation

In the limit $QR\gg 1$, it is well known that eikonal approximation describes
$(e,e^{\prime})$ data well Yennie et al. (1965); Aste et al. (2004, 2005);
Tjon and Wallace (2006). This corresponds to the matrix element evaluated in
the soft region. Choosing $v_{\mu}=(1,0,0,0)$, setting the electron mass to
zero, and performing the integral over $q_{0}$ via a contour integral, using
$\gamma_{0}\gamma_{i}\gamma_{0}=-\gamma_{i}$ and Dirac algebra on the external
spinors we find

$\mathcal{M}_{S}^{(1)}={\rm i}\mathcal{M}^{(0)}\times
Ze^{2}\int\frac{\differential^{3}q}{(2\pi)^{3}}~{}\frac{F(\mathbf{q}^{2})}{\mathbf{q}^{2}-{\rm
i}0}\quantity[\frac{2E(k)}{2\mathbf{k}\cdot\mathbf{q}-{\rm
i}0}+\frac{2E(k^{\prime})}{-2\mathbf{k}\cdot\mathbf{q}-{\rm i}0}]~{}.$ (28)

This can be recognized as the Feynman rule for an ultra-relativistic
eikonalized field (as appears in soft collinear effective theory) interacting
with a static charge distribution, with charge form factor
$F(\mathbf{Q}^{2})$. It reproduces the $O(Z\alpha)$ term in the standard
eikonal phase. The result relies essentially on the scale introduced by
$F(\mathbf{q}^{2})$ without which the integral would be scaleless and vanish
in dimensional regularization. Nuclear structure is therefore essential to the
eikonal approximation.

###  IV.2 Structure mismatch

For inelastic scattering $A\rightarrow B$, we may have $F_{A}(Q^{2})\neq
F_{B}(Q^{2})$. There will be “mismatch” between photon exchange with $A$ and
$B$ even for equal charges. The results presented above allows one to
calculate this effect systematically in perturbation theory. As an
illustration we work at one loop in the static limit. Expanding around the
high-energy limit we introduce a light-like reference vector
$n_{\mu}=(1,0,0,1)$ writing $k_{\mu}=E_{e}n_{\mu}$. Defining $\delta
F=F_{A}-F_{B}$ we have

$\int\frac{\differential^{4}L}{(2\pi)^{4}}~{}\frac{\delta
F(\mathbf{L}^{2})}{\mathbf{L}^{2}}\quantity[v_{\mu}L^{2}-L_{\mu}v\cdot
L]\frac{1}{v\cdot L+{\rm i}0}\frac{n^{\mu}}{n\cdot L+{\rm
i}0}\frac{1}{L^{2}+{\rm i}0}~{}.$ (29)

We perform integrals over $q_{0}$ using contour integration since
$F(\mathbf{q}^{2})$ does not depend on $q_{0}$. The result is that the
contributions at one-loop order proportional to $\delta f$ can be written as

$\int\frac{\differential^{4}L}{(2\pi)^{4}}~{}\frac{\delta
F(\mathbf{L}^{2})}{\mathbf{L}^{2}}\frac{1}{L^{2}+{\rm i}0}=\frac{{\rm
i}}{2}\int\frac{\differential^{3}L}{(2\pi)^{3}}~{}\delta
F(\mathbf{L}^{2})\quantity[\frac{1}{\mathbf{L}^{2}}]^{3/2}~{}.$ (30)

Taking
$F_{A,B}(\mathbf{L}^{2})=\Lambda_{A,B}^{2}/(\mathbf{L}^{2}+\Lambda_{A,B}^{2})$
gives for the difference of form factors,

$\delta
F(\mathbf{L}^{2})=\frac{\mathbf{L}^{2}(\Lambda_{A}^{2}-\Lambda_{B}^{2})}{(\mathbf{L}^{2}+\Lambda_{A}^{2})(\mathbf{L^{2}}+\Lambda_{B}^{2})}~{},$
(31)

which gives

$\frac{{\rm i}}{2}\int\frac{\differential^{3}L}{(2\pi)^{3}}~{}\delta
F(\mathbf{L}^{2})\frac{1}{|\mathbf{L}|^{3}}=\frac{{\rm
i}}{4\pi^{2}}\log\quantity(\frac{\Lambda_{A}}{\Lambda_{B}})~{},$ (32)

for the structure mismatch introduced above. At this order the resulting
correction only affects the imaginary part of the amplitude which then enters
at $O(\alpha^{2})$ in the cross section.

##  V Conclusions

We have constructed an EFT appropriate for the interaction of heavy, extended,
composite objects with electromagnetic probes. Our work provides a
generalization of of Wilson lines for heavy but spatially extended charge
distributions. The fact that such a generalization exists should not be
surprising since a spatial Fourier transform of a charge form factor becomes
well defined in the infinite mass limit. The theory supplies a systematic
framework to investigate structure dependent corrections for heavy particles
at higher orders in a loop expansion.

The EFT allows us to straddle the high and low momentum limits, where $pR\gg
1$ and $pR\ll 1$ respectively, within a unified framework. By including the
extended Wilson lines in the “full theory” we may identify modes using the
method of regions for different kinematic configurations. This allows us to
construct the relevant EFT for the problem at hand, and does not artificially
restrict our analysis to the point-like limit. Indeed for the high energy
limit this is essential, since the eikonal expansion for potential scattering
is ill-defined in the point-like limit. For low-momentum probes (e.g. in
nuclear beta decays) the structure dependent contribution modifies short-
distance Wilson coefficients.

An extension to sub-leading power would be interesting. We expect a similar
identification of “radius enhanced” operators in NRQED can be carried out at
sub-leading power (e.g. for the dipole density), and an extension of the same
ideas to the multi-photon sector e.g. contributions from nuclear
polarizabilities, may also be of interest. Nuclear excited states may be
straightforwardly included by allowing for additional heavy particles with an
explicit mass splitting in the theory. It would be interesting to apply the
concepts presented here to more general settings. In particular, gravity where
large coherent enhancements for heavy extended objects are ubiquitous (e.g.
for black holes and neutron stars), and structure dependent QED corrections in
heavy meson decays where the radius $R\sim 1/\Lambda_{\rm QCD}$ is
parametrically separated from the mass $M\sim m_{b}~{}{\rm or}~{}m_{c}$.

### Acknowledgments

I thank Richard J. Hill & Oleksandr Tomalak for useful discussions during
collaboration on related work, and Michele Papucci, Julio Para-Martinez, Ira
Rothstein, Florian Herren, and Andreas Helset for useful feedback. Part of
this work was completed at the Kavli Institute for Theoretical Physics during
the program “Neutrinos as a Portal to New Physics and Astrophysics” and I
thank KITP for their hospitality. KITP is supported in part by the National
Science Foundation under Grant No. NSF PHY-1748958. This work benefited from
feedback and discussions during the workshop “Power expansions on the
lightcone” at the Mainz Institute for Theoretical Physics and I would like to
thank them for their hospitality. This work is supported by the Neutrino
Theory Network under Award Number DEAC02-07CH11359, 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 Gauge invariance of background field terms

Correlators which are proportional $G_{\mu}(q_{i})\propto v_{\mu}\delta(v\cdot
q_{i})$ automatically satisfy current conservation, i.e.
$q_{i}^{\mu}G_{\mu}(q_{i})=0$. One then expects that relevant Ward identities
being satisified, the insertion of blobs involving $G_{\mu}$ inside of Feynman
diagrams will generate gauge invariant amplitudes. In a general gauge,
however, the photon propagator can be expressed as

$D_{\mu\nu}(q)={\rm
i}\quantity[-g_{\mu\nu}+q_{\mu}c_{\nu}(q)+c_{\mu}(q)q_{\nu}]/q^{2}~{}.$ (33)

with $c_{\nu}(q)$ not necessarily proportional to $q_{\nu}$.

For gauge invariance we must therefore require that both $(v\cdot
q)\delta(v\cdot q)=0$ and $(v\cdot c)\delta(v\cdot q)$ vanish. At first
glance, this seems not to hold since $c_{\nu}(q)$ is a generic function of
$q$, however closer inspection reveals that if $(c\cdot q)\delta(v\cdot q)=0$
in one gauge then it must also hold in arbitrary gauge. This condition can be
explicitly checked in Coulomb gauge, axial gauge, and arbitrary $R_{\xi}$
gauge and one finds that indeed $(c\cdot q)\rightarrow 0$ when $v\cdot
q\rightarrow 0$.

To see why this must be the case in an arbitrary gauge consider
$A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\Lambda$ (at the operator level we
take $\Lambda$ to be linear in the creation and annihilation operators of the
photon field). Consider the shift in the propagator

$D_{\mu\nu}=\int\differential^{4}x~{}{\rm e}^{-{\rm i}q\cdot
x}\matrixelement{0}{T\\{A_{\mu}(x)A_{\nu}(0)\\}}{0}~{}.$ (34)

Let us consider specifically $v^{\mu}D_{\mu\nu}$. Terms with
$v\cdot\partial\Lambda$ will be proportional to $v\cdot q$ and will vanish
when multiplied by $\delta(v\cdot q)$. We then only have to worry about the
term $\matrixelement{0}{T\\{v\cdot A(x)\partial_{\nu}\Lambda(0)\\}}{0}$. We
may reach an arbitrary gauge via a gauge transformation from any of axial
gauge, Coulomb gauge, or $R_{\xi}$ gauge, and since $v\cdot A\propto v\cdot q$
in all of these gauges, the contribution to the time-like component of the
photon propagator, $v^{\mu}D_{\mu\nu}$, will vanish when multiplied by
$\delta(v\cdot q)$. This then demonstrates that

$\delta(v\cdot q)v^{\mu}D_{\mu\nu}(q)=-{\rm i}v_{\nu}\delta(v\cdot
q)/q^{2}=-{\rm i}v_{\nu}\delta(v\cdot q)/(-\mathbf{q}^{2})~{},$ (35)

holds in an arbitrary gauge as desired.

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