# Sea-quark loop contributions to the $\bar{d}-\bar{u}$ asymmetry in the
proton

Derek B. Leinweber  Anthony W. Thomas  Centre for the Subatomic Structure of
Matter (CSSM), Department of Physics, University of Adelaide, SA, 5005,
Australia

###### Abstract

QCD interactions for equal-mass fermion flavours are flavour blind. This fact
is often used to state that sea-quark loop contributions are equal for $u$ and
$d$ quarks in the mass symmetric case and therefore these disconnected sea-
quark loop contributions cannot contribute to the well-known $\bar{d}-\bar{u}$
asymmetry in the proton. Instead, it is argued that one must look to the
connected sector of lattice QCD correlation functions to find this difference.
In this presentation, we note that these statements are true only when one
includes unphysical contributions in the sea-quark-loop sector, contributions
from baryons that do not appear in the physical spectrum. Because these
unphysical contributions must be eliminated from the sea-quark loop sector via
the connected sector, the assertion that the $\bar{d}$ and $\bar{u}$ sea-quark
loop contributions cancel in $\bar{d}-\bar{u}$ is incorrect. Upon considering
only physically observed baryons in the loop contributions, we illustrate an
important contribution from the sea-quark loop sector to $\bar{d}-\bar{u}$.

###### keywords:

Quantum Chromodynamics , lattice QCD , partially quenched chiral perturbation
theory , proton properties , anti-matter asymmetry in the nucleon sea

††journal: Physics Letters B

##  1 Introduction

Understanding the structure of the nucleon remains an exciting challenge for
modern nuclear and particle physics. One of the great surprises of the past
few decades in this area was the discovery that there is an asymmetry between
the anti-down ($\bar{d}$) and anti-up ($\bar{u}$) sea quarks in the proton.
While this was in fact predicted on the basis of chiral symmetry [1], it was
almost a decade later that a hint of such an asymmetry [2] was verified
experimentally by the New Muon Collaboration [3], as a significant discrepancy
in the Gottfried sum-rule [4, 5].

Since then, the shape of $\bar{d}-\bar{u}$ as a function of Bjorken-$x$, has
been studied in detail through the Drell-Yan process [6, 7] as well as further
measurements of $F_{2p}\,-\,F_{2n}$ [8]. There have also been extensive
studies of the capacity of chiral field theory to constrain this and other
asymmetries in the sea of the nucleon [9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20].

Apart from the intrinsic interest in nucleon structure, modern tests of the
Standard Model demand that the parton distribution functions (PDFs) be known
very accurately [21, 22, 23, 24, 25, 26, 27]. Such considerations have led to
interesting suggestions that constraints from lattice QCD might supplement
experimental data in constraining hadronic PDFs [28, 29, 30]. Here we focus on
suggestions that the difference in the sea quark contributions from so-called
connected and disconnected (or sea-quark loop) diagrams in lattice QCD could
be used to improve phenomenological PDF extractions. In particular, we examine
the suggestion that the disconnected diagrams have a physical interpretation
and do not contribute to the $\bar{d}-\bar{u}$ asymmetry [30, 31].

In the following we demonstrate that neither the connected nor the
disconnected diagrams in lattice QCD have a physical meaning on their own. We
show how the symmetry required to ensure the disconnected sea-quark loop
sector cannot contribute to the $\bar{d}-\bar{u}$ asymmetry necessarily
requires the inclusion of unphysical baryon contributions to the sea-quark-
loop sector, contributions from baryons that do not appear in the physical
spectrum.

Because these unphysical contributions must be eliminated from the sea-quark
loop sector through a cancellation with appropriate contributions in the
connected sector, the assertion that the physical $\bar{d}$ and $\bar{u}$ sea-
quark loop contributions cancel in $\bar{d}-\bar{u}$ is incorrect. Upon
considering only physically observed baryons in the loop contributions, we
illustrate an important contribution from the sea-quark loop sector to the
$\bar{d}-\bar{u}$ asymmetry.

##  2 Quark flow analysis

We begin our analysis through the consideration of quark flow diagrams
included in lattice QCD calculations of three point functions of standard
proton interpolating fields and a flavour-diagonal quark-bilinear current. In
performing the Wick contractions, one encounters two topologically distinct
diagrams, one in which the quark flow is connected through the creation
interpolating field to the annihilation interpolating field of the proton, and
a second in which the quark fields of the current are contracted to form a
loop. Such a diagram is commonly referred to as a quark-flow disconnected
diagram. However, it is connected to the two-point function through gluon
interactions.

In drawing the diagrams associated with the disconnected sea-quark loop
contribution, one can consider flavour-singlet constructions where the quark-
antiquark pair of the loop annihilates to gluons, and flavour-octet
contributions where the loop pairs with one of the quark-flow lines of the
proton interpolating fields to form flavour-octet mesons. As the flavour-
singlet mesons have an equal balance of $\bar{u}u$ and $\bar{d}d$ components
in the mass-symmetric limit neglecting electromagnetic interactions, these
contributions do not generate a $\bar{d}-\bar{u}$ asymmetry and we do not
consider them further.

Figure 1: Quark flow diagrams describing the one-loop meson dressings contributing to the $\bar{d}-\bar{u}$ asymmetry of the proton. Connected flows are listed in the left column and the corresponding flows incorporating a sea-quark-loop contribution appear in the right-hand column. Diagram (g) contains no quark labels and serves to illustrate there is no connected contribution to the $d\bar{d}$ meson component. We focus on contributions where the external current acts on the anti-quark loop.  Table 1: The contribution for each diagram of Fig. 1 from SU(3) partially-quenched chiral perturbation theory. Here we use the normalisation of Ref. [32] that sums diagrams (a) and (b) to give the physical $n\,\pi^{+}$ dressing of the proton equal to $2\,(D+F)^{2}=2\,g_{A}^{2}$. Numerical values correspond to $D=0.77$ and $F=0.50$. The right-hand column entitled Relative Contribution describes the percentage split of the strength between the connected and sea-quark-loop contributions in each row of Fig. 1. |  |  | Relative
---|---|---|---
Diagram | Contribution | Value | Contribution (%)
(a) | $2\,D^{2}/3+4\,D\,F-2\,F^{2}$ | $1.44$ | $44.5$
(b) | $(D+3\,F)^{2}/3+(D-F)^{2}$ | $1.79$ | $55.5$
(c) | $4\,(F^{2}-D^{2}/3)$ | $0.21$ | $10.5$
(d) | $(D+3\,F)^{2}/3+(D-F)^{2}$ | $1.79$ | $89.5$
(e) | $-2\,(D-F)^{2}$ | $-0.15$ | $50.0$
(f) | $2\,(D-F)^{2}$ | $0.15$ | $50.0$
(h) | $2\,(D-F)^{2}$ | $0.15$ | $100.0$

Figure 1 illustrates the quark-flow diagrams having overlap with meson
dressings of the proton contributing to the $\bar{d}-\bar{u}$ asymmetry of the
proton. The eight diagrams are associated with two choices for the light quark
propagating in the meson dressing, times two choices for the anti-quark, times
two topologically different quark flows giving rise to the meson dressing.
Connected Wick contractions are listed in the left column and the
corresponding disconnected contractions generating a sea-quark-loop
contribution appear in the right-hand column.

There are three connected flows to consider as the $d$ quark can be placed on
each of the quark flow lines. With the intermediate-state quark-flows
identified, the corresponding diagram incorporating a sea-quark loop is
illustrated. In obtaining the physical contribution, both the connected and
sea-quark-loop contributions are summed. The final row, illustrating the
$d\bar{d}$ component of the meson dressings emphasises that there is no
connected contribution to this component. One would need two $d$ quarks in the
valence sector of the proton. Thus, in this case, only the sea-quark-loop
contributes.

The symmetries manifest in lattice QCD calculations are not restricted to
specific choices for the mesons and baryons participating in the intermediate
state. Rather, the lattice correlation functions have overlap with all baryons
and mesons associated with the quantum numbers specified by the quark flow.
Thus, in this discussion of the symmetries of lattice QCD correlation
functions, we will ignore the hadron labels on the intermediate states of Fig.
1.

The symmetry of the $\bar{u}$ and $\bar{d}$ contributions is manifest in the
right-hand column of Fig. 1. When a $u$ valence quark is contributing to a
meson dressing, both the $\bar{d}$ and $\bar{u}$ sea-quarks contribute in
diagrams (b) and (d) respectively. For equal-mass $u$ and $d$ flavours, and
neglecting electromagnetic effects, these sea-quark loop contributions are
equal. While the gluonic interactions are sensitive to the mass of the quarks,
they do not differentiate flavour. Similarly, when a $d$ valence quark is
contributing to a meson dressing, both the $\bar{u}$ and $\bar{d}$ sea-quarks
contribute in diagrams (f) and (h) respectively, again maintaining the
symmetry. The naive conclusion is that this symmetry prevents a contribution
to the $\bar{d}-\bar{u}$ asymmetry. However there is a subtlety that has been
overlooked.

If one considers the connected and disconnected diagrams on their own, without
taking the sum, one must expand the list of baryons and mesons having overlap
with the quark flows to include unphysical states that do not appear in the
physical spectrum. Indeed, the Pauli exclusion principle is enforced by
summing both of the aforementioned Wick contractions leading to connected and
disconnected quark flows. On their own, a connected or disconnected graph will
receive contributions from baryons that do not satisfy the Pauli exclusion
principle. Fortunately, the symmetries highlighted in the previous paragraph
can be used to determine the properties of these unphysical states.

At this point, it is helpful to become more specific on the hadrons
contributing to the quark flows of Fig. 1. Thus we turn our attention to light
pseudoscalar meson dressings where the intermediate baryon is degenerate with
the proton. These dressings provide important contributions to the
$\bar{d}-\bar{u}$ asymmetry and the focus on baryons degenerate with the
proton, not only isolates the leading nonanalytic contribution in chiral
perturbation theory [9, 33, 34, 35], but also serves to expose the presence of
unphysical baryon contributions.

Diagrams (a) and (b) of Fig. 1 sum to provide the physical $n\,\pi^{+}$
dressing of the proton. In diagrams (c) and (d), we are introduced to the
$\eta_{u}$ meson, a meson composed of a $u\bar{u}$ pair. Similarly, the
$d\bar{d}$ meson in diagram (h) is labelled as $\eta_{d}$. The masses of these
neutral pseudoscalar mesons can be inferred from the symmetries manifest at
the level of lattice QCD correlation functions.

Because the sea-quark loop contributions from diagrams (b) and (d) are equal,
we conclude that when the $\bar{d}u$ meson of diagram (b) is a $\pi^{+}$, the
$\eta_{u}$ meson of diagram (d) has a mass degenerate with the pion.
Similarly, because the sea-quark loop contributions from diagrams (f) and (h)
are equal, when the $\bar{u}d$ meson of diagram (f) is a $\pi^{-}$, the
$\eta_{d}$ meson of diagram (h) has a mass degenerate with the pion.

The axial couplings of these mesons are provided in a meson basis in which the
diagonal entries of the nonet pseudoscalar meson matrix are $\eta_{u}$,
$\eta_{d}$, and $\eta_{s}$. In other words, the square of the axial couplings
of the $\eta_{u}$ and $\eta_{d}$ mesons is associated with a linear
combination of $\pi^{0}$, $\eta$, and $\eta^{\prime}$ mesons in the proportion
$3:1:2$ respectively for standard SU(3) symmetry. We emphasise this is simply
a change of basis. The masses of the mesons are governed by the symmetries
encountered in lattice QCD calculations.

The baryons associated with the quark flows are also labelled in Fig. 1, where
we focus on intermediate baryons degenerate with the proton. It is here that
one encounters an unphysical baryon labelled “$p^{++}$” in diagrams (e) and
(f). While the charge of $+2$ is manifest in the presence of a $\pi^{-}$ and
the necessity of charge conservation, the label of a $p$ emphasising the
propagation of a baryon degenerate with the proton requires further
explanation. Because QCD interactions are flavour blind when $m_{u}=m_{d}$,
diagrams (f) and (h) make equal contributions. It is clear that diagram (h)
has overlap with the normal proton. Indeed diagrams (c), (d) and (h) sum to
provide the physical $p\,\pi^{0}$ dressing of the proton. We can then conclude
that the mass of the $p^{++}$ is also that of the proton through the
equivalence of diagrams (f) and (h). Of course, all of the quark flow
diagrams, including diagrams (e) and (f), also have overlap with decuplet
baryons. However, those contributions are all physical and pose no issue.

Because the Pauli Exclusion Principle does not allow octet baryons with charge
$+2$, we must look to the other Wick contraction to cancel the unphysical
contribution of the sea-quark-loop in diagram (f). This is the role of diagram
(e) and thus we conclude that the contribution of diagram (e) is equal and
opposite to that of diagram (f). In other words, if we were to restrict our
diagrams to those having physical intermediate baryons, we would need to
eliminate diagrams (e) and (f) in the third row of Fig. 1. With the loss of
diagram (f), the symmetry of the sea-quark loop contributions is lost.
Flavour-blindness in QCD interactions equates the contributions of diagrams
(b) and (d), leaving the contribution from diagram (h) to break the symmetry.
Thus, we conclude that sea-quark loops do in fact make physical contributions
to the $\bar{d}-\bar{u}$ asymmetry in the proton.

##  3 Partially-quenched chiral perturbation theory

Partially quenched chiral perturbation theory for baryons [36, 37, 38, 32, 39]
provides a mechanism for understanding the relative contributions of the
connected and sea-quark-loop diagrams, which are summed to get the physical
contribution. While the label “partially-quenched” has an historical origin,
the results presented here pertain to that of physical QCD. Here we draw on
the results in Tables I and II of Ref. [32] using the diagrammatic approach
for obtaining the meson-baryon couplings in terms of the familiar
SU(3)-flavour axial couplings, $F$ and $D$. There the physical basis was used
for the axial couplings and the $\eta_{u}$ and $\eta_{d}$ couplings are
obtained by summing over the $\pi^{0}$, $\eta$, and $\eta^{\prime}$ entries.

Diagrams (a) and (b) of Fig. 1 sum to generate the full physical $\pi^{+}$
dressing of the proton, The sea-quark-loop coefficients are provided in Table
II of Ref. [32] and the coefficients of the connected diagrams are obtained by
subtracting these loop contributions from the full contributions provided in
Table I.

Working with the normalisation of Ref. [32] where the contribution of the
$n\,\pi^{+}$ dressing of the proton is proportional to
$2\,(D+F)^{2}=2\,g_{A}^{2}$, the contribution of each diagram in Fig. 1 is
summarised in Table 1. Here the numerical values correspond to $D=0.77$ and
$F=0.50$, such that $g_{A}=1.27$.

Note how the symmetries discussed in the previous section in the context of
lattice QCD quark-flow diagrams are manifest in Table 1 for pseudoscalar
dressings of the proton. The equivalence of diagrams (b) and (d) is reflected
in the equivalence of the axial couplings for these diagrams. The equivalence
of diagrams (f) and (h) is similarly reflected in equal axial couplings for
those diagrams.

With regard to the $\bar{d}-\bar{u}$ asymmetry of the nucleon, we see that
there is an enhancement of $\bar{d}$ in the sea-quark-loop sector coming from
diagram (h). Thus, the assumption of equal $\bar{d}$ and $\bar{u}$ in the
disconnected sector of the sea is wrong in principle. In the connected sector,
from the third column of Table I we see that the coupling differences between
diagrams (a) and (c) leave a net effect of $1.44-0.21=1.23$ favouring the
$\bar{d}$ contribution. Now the sea quark loop contribution also enhances the
$\bar{d}$ excess, raising it from $1.23\to 1.23+0.15=1.38$, a 12% enhancement
of the pion-nucleon contribution to $\bar{d}-\bar{u}$ in the proton arising
from the sea-quark loop sector.

##  4 Conclusions

We have carefully analysed the role of connected and disconnected quark-flow
diagrams in generating a $\bar{d}-\bar{u}$ asymmetry in the sea of the proton.
The formalisms of lattice QCD and partially-quenched chiral perturbation
theory have been considered. It has been shown that the naive assertion of the
equality of the $\bar{d}$ and $\bar{u}$ contributions from disconnected sea-
quark loops relies on contributions from an unphysical baryon. There is an
unphysical contribution to $\bar{u}$ arising from the process involving an
intermediate $\pi^{-}$ and a charge $2^{++}$ baryon degenerate with the
proton, illustrated in Fig. 1(f). In a full QCD calculation this unphysical
contribution is precisely cancelled by the connected contribution to $\bar{u}$
shown in Fig. 1(e). Once the unphysical contribution associated with Fig. 1(f)
is removed, one finds an enhancement of approximately 12% in the
$\bar{d}-\bar{u}$ asymmetry arising from pion-baryon-octet components of the
proton wave function.

## Acknowledgements

We are pleased to acknowledge helpful discussions with Anthony Williams and
Ross Young. This research was undertaken with the assistance of resources from
the National Computational Infrastructure (NCI), provided through the National
Computational Merit Allocation Scheme. This research was supported by the
Australian Research Council through ARC Discovery Project Grants DP190102215
and DP210103706 (D.B.L.).

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11institutetext: 1Indian Institute of Technology Delhi, New Delhi 110016,
INDIA
11email<EMAIL_ADDRESS>Corresponding author

# HF-Fed: Hierarchical based customized Federated Learning Framework for X-Ray
Imaging

Tajamul Ashraf† 11 0000-0002-7372-3782 Tisha Madame 11 0009-0009-7190-6591

###### Abstract

In clinical applications, X-Ray technology plays a crucial role in noninvasive
examinations like mammography, providing essential anatomical information
about patients. However, the inherent radiation risk associated with X-Ray
procedures raises significant concerns. X-Ray reconstruction is crucial in
medical imaging for creating detailed visual representations of internal
structures, and facilitating diagnosis and treatment without invasive
procedures. Recent advancements in deep learning (DL) have shown promise in
X-Ray reconstruction. Nevertheless, conventional DL methods often necessitate
the centralized aggregation of substantial large datasets for training,
following specific scanning protocols. This requirement results in notable
domain shifts and privacy issues. To address these challenges, we introduce
the Hierarchical Framework-based Federated Learning method (HF-Fed) for
customized X-Ray Imaging. HF-Fed addresses the challenges in X-Ray imaging
optimization by decomposing the problem into two components: local data
adaptation and holistic X-Ray Imaging. It employs a hospital-specific
hierarchical framework and a shared common imaging network called Network of
Networks (NoN) for these tasks. The emphasis of the NoN is on acquiring stable
features from a variety of data distributions. A hierarchical hypernetwork
extracts domain-specific hyperparameters, conditioning the NoN for customized
X-Ray reconstruction. Experimental results demonstrate HF-Fed’s competitive
performance, offering a promising solution for enhancing X-Ray imaging without
the need for data sharing. This study significantly contributes to the
evolving body of literature on the potential advantages of federated learning
in the healthcare sector. It offers valuable insights for policymakers and
healthcare providers holistically. The source code and pre-trained HF-Fed
model is available at this link

##  1 Introduction

X-Ray imaging is vital for clinical diagnosis, providing noninvasive insights
into patient anatomy, especially in breast mammography [27], [26], [9].
However, potential radiation risks related to genetic and cancer diseases have
raised concerns [12]. Common strategies to mitigate these risks, like
adjusting the X-Ray tube’s current/voltage and reducing scanning views, often
degrade imaging quality [31], negatively impacting image analysis and
diagnoses. To address these challenges, researchers have explored deep
learning (DL) for low-dose X-Ray (LD-Xray) reconstruction [13], with promising
results. However, current methods depend on centralized training (CL) with
large datasets from various hospitals, overlooking privacy concerns and domain
shifts that cause low accuracy. Anonymizing data has proven insufficient in
ensuring robust patient privacy [25, 28].

In light of privacy, legal, and ethical concerns, there is an increasing
demand for a collaborative, privacy-preserving approach in multi-hospital
training. Federated Learning (FL) is a paradigm designed to enhance data
security and privacy by decentralizing the training process, keeping
information on individual devices, and sharing only the updated model
parameters [18], [11]. FL stands out as a decentralized solution explicitly
crafted to safeguard data and its confidentiality [14]. Unlike CL methods,
which involve the transfer of private patient data, FL methods exclusively
exchange gradients, thereby minimizing privacy risks associated with data
content. FL offers a privacy-conscious alternative for training models on
X-Ray data from multiple sources, addressing the limitations of centralized
approaches and fostering a more secure and ethically sound framework for
medical imaging. A significant challenge in FL arises from the non-
independence and non-identically distributed (Non-IID) nature of data, a
particularly pronounced issue in X-Ray Imaging compared to other analysis
tasks. The diverse hardware and scanning protocols across different scanners
or hospitals exacerbate this challenge. Unfortunately, the holistic model in
FL is limited to capturing general statistical patterns and lacks the
specificity required for individual data sources, leading to inaccurate
results. To address these problems, recent efforts focus on customized FL
methods, aiming to train local models tailored to specific data distributions.
Certain methods employ hypernetworks [30] to attain personalization by
creating customized weights for the target network. Nevertheless, numerous
currently available hypernetwork-based Federated Learning approaches often
overlook the underlying physical processes and come with significant
additional training expenses. In computer-aided diagnosis, the precision of
imaging (upstream) tasks significantly impacts subsequent processes like
segmentation and detection (downstream). This paper presents an in-depth FL
framework for upstream tasks, focusing on post-processing and reconstruction.
Our hierarchical customized FL framework (HF-Fed) for X-Ray Imaging addresses
holistic optimization by personalizing feature adaptation and extracting
invariant holistic imaging features. In X-Ray Imaging, maintaining structural
similarity is crucial. Our HF-Fed framework trains a universally shared
imaging network using varied client data to capture consistent holistic
characteristics. Recognizing the impact of scanning protocols and geometry
parameters, HF-Fed introduces a hierarchical hypernetwork that dynamically
adjusts imaging features using embedded knowledge.

HF-Fed integrates a hospital-specific hierarchical framework and a Network of
Networks (NoN). The hypernetwork adapts scanning parameters to customize
features, enhancing invariant features of the imaging network. Utilizing FL,
it facilitates comprehensive gradient exchange for robust model training. The
hospital-specific hypernetwork ensures domain-specific personalization, while
NoN learns universal features across diverse domains.

The contributions of this work are as follows:

1\. Introduction of HF-Fed for customized X-Ray imaging using FL. This is the
first attempt at applying Federated Learning for customized X-Ray imaging.

2\. The proposed HF-Fed framework comprises two components: the hospital-
specific hierarchical hypernetwork and the Network of Networks (NoN). The
hypernetwork is designed for customization to address Non-IID challenges,
while NoN captures invariant and stable features across diverse data
distributions.

3\. A flexible framework that can be easily extended to various X-Ray imaging
tasks, including post-processing and reconstruction.

##  2 Related Work

X-Ray imaging technology has advanced significantly, enhancing diagnostic
capabilities [22]. Reconstruction methods include sinogram filtration,
iterative reconstruction (IR), and postprocessing. Sinogram filtration
involves filtering raw or log-transformed data using adaptive techniques such
as weighted least-squares with penalties [7]. IR methods improve objective
functions by incorporating prior knowledge like total variation and non-local
means filters [5], along with regularization terms. Postprocessing methods
offer convenience but lack flexibility without raw data. Deep Learning (DL)
has shown promise in low-dose computed tomography (LDCT) reconstruction with
models like RED-CNN for denoising [4], GAN-based approaches for super
resolution [33], and parameter-dependent frameworks (PDF) [32]. However, these
centralized models raise privacy concerns and limit clinical applicability.

###  2.1 Customized Federated Learning

Federated Learning (FL) adopts a decentralized approach that emphasizes data
privacy while allowing models to learn from diverse data sources [29]. The
traditional FL method, FedAvg, introduced by McMahan et al. [24], builds a
comprehensive model by averaging local models from various parties. FedProx
[18] improves upon FedAvg by enforcing proximity between local and global
models. Li et al. [16] introduced the model-contrastive (MOON) technique,
which minimizes contrastive distances between local and global models. FL’s
privacy-preserving features have been applied in various medical tasks [14].

In the realm of medical imaging advancements, Guo et al. [10] introduced an
intermediate latent feature alignment approach for MRI reconstruction,
requiring knowledge of target domain data, which poses practical challenges.
Dinh et al. [6] addressed non-iid challenges in multitask learning within
federated learning (FL), aiming for a universally applicable model but facing
significant hurdles. Zhang et al. [34] proposed a semi-asynchronous FL
framework for short-term solar power forecasting, showing robust performance.
Liang et al. [20] introduced LG-FedAvg, linking learning representation and
FL, primarily for high-level tasks, potentially less seamless in low-level
imaging assignments. Ditto [17], a comprehensive-regularized multitask
learning framework, integrates local and holistic models effectively, showing
promising outcomes.

Ma et al. [23] introduced an approach to assess layer significance across
diverse clients for tailored model aggregation. Li et al. [19] presented
FedBN, integrating local batch normalization (BN) to mitigate feature shift
and achieve personalization in local hospital settings. Shamsian et al. [30]
proposed pFedHN, using a hypernetwork for local model parameter generation,
particularly effective in simple models. Ashraf et al. [1] explored network
customization for transformer-based client-side models, limiting broad
applicability. Chen et al. developed cyclic knowledge distillation for
extracting semantic features from local models, focusing on classification
tasks aligning semantic features across clients. Hanzely [11] incorporated
regularization to minimize optimization gaps in federated models. Feng et al.
[8] proposed an MRI denoising model with shared encoders, addressing non-iid
challenges in X-Ray reconstruction due to protocol sensitivity. Here, we
introduce a hierarchical X-Ray reconstruction framework adaptable for diverse
imaging tasks.

##  3 Proposed Method

###  3.1 Problem Statement

In the context of X-Ray reconstruction, the general optimization problem can
be expressed as follows:

$\min_{a}\frac{1}{2}\lVert Xa-b\rVert_{2}^{2}+Y(a)$ (1)

In this context, $\lVert\cdot\rVert_{2}^{2}$ signifies the L2 norm, and $X$
symbolizes the system matrix. The variables $a$ and $b$ denote the image for
reconstruction and the measured data, respectively. The regularization term
$Y(\cdot)$ integrates prior knowledge. In customized Federated Learning (FL),
hospitals strive to attain customized models from local data, improving
performance through collaborative FL methods, such as sharing specific layers.
If there are $K$ hospitals, the learning process can be expressed as:

$\min_{\theta_{1},\ldots,\theta_{K}}\sum_{k=1}^{K}W_{k}\mathbb{E}_{(a_{l},b,a_{n})\sim
D_{k}}\left\|F_{k}(a_{l},b,\theta_{k})-a_{n}\right\|_{2}^{2}$

In this formulation, $F^{k}$ denotes the optimization model in the $k$-th
hospital, $W_{k}$ represents the weight of the $k$-th hospital in holistic
optimization, and $D_{k}$ signifies the dataset in the $k$-th hospital. The
aim of customized Federated Learning is to discover optimal local models for
different hospitals without sharing raw data.

Figure 1: The proposed HF-Fed architecture consists of a globally shared
imaging network and a hospital-specific, hierarchically-driven hypernetwork.

###  3.2 HF-Fed Architecture

Given the substantial impact of scanning protocol and geometry parameters on
the physical X-Ray reconstruction process, it is reasonable to leverage this
knowledge for guiding the network in predicting normal X-Ray images. Motivated
by this concept, we introduce the Hierarchical Framework-based Federated
Learning (HF-Fed), which consists of a hospital-specific hypernetwork and a
holistically shared imaging framework. The complete assimilation process of
the current imaging framework, denoted as $L_{\text{imag}}(a,b,w)$, involves
feeding inputs into the model $L_{\text{imag}}$ with parameters $w$, without
any constraint. While assuming a uniform distribution of inputs can reduce
complexity, meeting this assumption in real scenarios is challenging, leading
to the non-iid problem. To tackle this, the hospital-specific hypernetwork
functions as a monitor, adjusting the process. As a result, the assimilation
process of HF-Fed is redefined as $L_{\text{imag}}(a,b,w,\theta)$, where
$\theta$ denotes the result from the hypernetwork $H(\cdot)$ constrained with
$\xi$. In particular, the vector $n$, encompassing crucial scanning and
geometry parameters, is input into $H(\cdot)$ to generate the sets of scaling
factors $\gamma$ and biases $\beta$. These sets are employed to adjust the
features of $F_{\text{imag}}$, and this procedure is expressed as
$\gamma,\beta=H(n,\xi)$. Some elements in $n$ with large magnitudes, such as
the numbers of views and detector bins, and the photon number of incident
X-Ray, undergo normalization and logarithmization. The normalization is
defined as

$n_{j}=\frac{n_{j}-\min(n_{j})}{\max(n_{j})-\min(n_{j})}$ (2)

Figure 1 illustrates the architecture of our proposed HF-Fed. The structure of
the hypernetwork is closely tied to the imaging network and is designed as a
three-layer fully-connected network in this article. It generates scaling
factors $\gamma$ and biases $\beta$ based on a 7-dimensional physical
parameter vector $n$. The output sizes of the two subsequent linear layers are
256 and 512, respectively, with the final output dimensionality dependent on
the imaging network’s feature dimension, which is twice the imaging feature
dimension.

To enhance parameter efficiency, RED-CNN adopts a strategy where the encoder
and decoder layers share two groups of modulation factors. Similarly, in
LEARN, all IR blocks share the same modulation factors. The modulation
function is represented as:

$L^{\ast}_{\text{imag}}(b)=\gamma L_{\text{imag}}(b)+\beta.$

This operation is applied to feature maps from various modules in the imaging
network, resulting in the modification of equation (6):

$L^{\ast}_{\text{imag}_{i}}(b)=\gamma_{i}L_{\text{imag}_{i}}(b)+\beta_{i},$

where $L_{\text{imag}_{i}}(b)$ and $L^{\ast}_{\text{imag}_{i}}(b)$ represent
the feature map from the $i$th module and its modulated counterpart,
respectively. $\gamma_{i}$ and $\beta_{i}$ signify the regularization factor
and bias for the $i$th module.

As highlighted earlier, the imaging network in HF-Fed is adaptable for various
tasks, with imaging units varying based on different imaging methods. For
instance, convolution layers serve as imaging units for postprocessing methods
like RED-CNN [5]. Conversely, for unrolled iteration methods such as LEARN
[3], the imaging unit represents an unrolled iteration module.

###  3.3 Implementation of NoN (Network of Networks)

In this research, we propose HF-Fed, a hypernetwork-based federated learning
framework designed to tackle the non-iid challenge in X-Ray image
reconstruction. Similar to FedAvg and FedProx, HF-Fed involves local updates
for both the hypernetwork and imaging network, with only the imaging network’s
updates aggregated on the server. Unlike FedBN, which normalizes data
globally, HF-Fed adapts by performing local normalization due to varied X-Ray
data distributions from different machines, challenging FedBN’s assumptions
[21]. To address this, we introduce a hypernetwork to modulate feature maps
within the imaging network, enabling hierarchical-driven self-normalization

Privacy and security challenges in inter-hospital collaboration are addressed
by HF-Fed, ensuring that local data remains private. Each hospital refines its
local model by minimizing the loss:

$\ell_{k}=\frac{1}{2}\mathbb{E}_{(a_{i},n,b)\sim
D_{k}}\left[L_{k}(\delta_{k},b,n)-a_{i}\right]^{2}$

where $L_{k}(\cdot)$ represents the local network at the $k$-th hospital,
parameterized by $\delta_{k}$. Optimization of parameters $w_{k}$ and
$\xi_{k}$ of $L_{\text{imag}}$ and $H(\cdot)$ at the $k$-th hospital is
conducted iteratively:

$w^{w+1}_{k},\xi^{w+1}_{k}=w^{w}_{k},\xi^{w}_{k}-\lambda\nabla_{w^{w}_{k},\xi^{w}_{k}}L^{k}$

where $\lambda$ denotes the learning rate. After training epochs, gradients
from the imaging network are sent to the server for aggregation, while the
hypernetwork remains locally for modulation.

Table 1: HF-Fed’s SSIM scores for the post-processing task

SSIM Scores # Models w/o FL Fedavg FedProx FedBN Ditto pFedHN HF-Fed Hospital
#1 $94.55$$\mathrm{p}\mathrm{m}$0.15 $92.21$$\mathrm{p}\mathrm{m}$0.08
$92.94$$\mathrm{p}\mathrm{m}$0.13 $93.68$$\mathrm{p}\mathrm{m}$0.13
$93.91$$\mathrm{p}\mathrm{m}$0.17 $95.25$$\mathrm{p}\mathrm{m}$0.11
96.87$\pm$0.12 Hospital #2 $90.42$$\mathrm{p}\mathrm{m}$4.22
$94.28$$\mathrm{p}\mathrm{m}$4.23 $91.85$$\mathrm{p}\mathrm{m}$1.50
$91.23$$\mathrm{p}\mathrm{m}$1.93 $93.02$$\mathrm{p}\mathrm{m}$0.88
$90.20$$\mathrm{p}\mathrm{m}$0.95 94.64$\pm$0.22 Hospital #3 94.86$\pm$0.99
$94.01$$\mathrm{p}\mathrm{m}$0.12 $92.41$$\mathrm{p}\mathrm{m}$0.16
$95.70$$\mathrm{p}\mathrm{m}$0.14 $91.23$$\mathrm{p}\mathrm{m}$0.32
$91.72$$\mathrm{p}\mathrm{m}$0.16 $91.19$$\mathrm{p}\mathrm{m}$0.18 Hospital
#4 $92.26$$\mathrm{p}\mathrm{m}$0.61 $93.57$$\mathrm{p}\mathrm{m}$0.52
$91.45$$\mathrm{p}\mathrm{m}$0.87 $92.13$$\mathrm{p}\mathrm{m}$0.67
$93.08$$\mathrm{p}\mathrm{m}$0.72 94.94$\pm$0.91
$93.25$$\mathrm{p}\mathrm{m}$0.77 Hospital #5
$91.75$$\mathrm{p}\mathrm{m}$0.22 $94.46$$\mathrm{p}\mathrm{m}$0.11
$94.25$$\mathrm{p}\mathrm{m}$0.69 $91.13$$\mathrm{p}\mathrm{m}$0.84
$94.61$$\mathrm{p}\mathrm{m}$0.45 $95.05$$\mathrm{p}\mathrm{m}$0.47
96.63$\pm$0.98 Average $92.57$$\mathrm{p}\mathrm{m}$1.65
$93.11$$\mathrm{p}\mathrm{m}$1.65 $92.58$$\mathrm{p}\mathrm{m}$1.65
$92.70$$\mathrm{p}\mathrm{m}$1.65 $93.57$$\mathrm{p}\mathrm{m}$1.65
$93.03$$\mathrm{p}\mathrm{m}$1.65 95.12$\pm$1.65 STD $1.65$ $0.91$ $1.22$
$1.63$ $1.23$ $2.04$ $1.62$

##  4 Experiments and Results

The "RSNA Screening Mammography Breast Cancer Detection" dataset [2] includes
54.7K X-Ray mammogram images from 1,000 patients, split into 4,000, 1,000, and
1,000 images for training, validation, and testing respectively. No overlap
exists between these sets, with validation occurring every 200 rounds during
training. The dataset is divided into five groups on average, resulting in
diverse client-specific data distributions, posing a severe non-iid challenge.
Evaluation uses the correlation coefficient (CC) to assess denoised image
quality. Each hospital handles a single data type with strict transmission
restrictions. Post-processing uses RED-CNN [5], and reconstruction employs
LEARN [3] with a learning rate of $1\times 10^{-4}$. RED-CNN undergoes 3
epochs over 1000 rounds, while LEARN uses 200 rounds. HF-Fed is compared with
FedAvg [24], FedProx [18], FedBN [19], Ditto [17], pFedHN [30], and original
models without federated learning (w/o FL). FedProx applies a penalty constant
of $1\times 10^{-4}$, and FedBN adds personalized BN layers post-convolution.
Optimization uses Adam [15] with Mean Squared Error (MSE) loss. Experiments
are conducted on NVIDIA GTX 3090 and A100 GPUs using PyTorch.

###  4.1 Experiments on the Large-Scale Training Set

In this section, we compare FL-based methods and original imaging models
without FL, using a large-scale local dataset with RED-CNN as the backbone
network. Hyperparameters remain consistent, except each hospital has 200
images in the training set. Table 1 demonstrates that FL-based methods
significantly improve performance with larger datasets, mitigating the non-iid
problem. HF-Fed consistently delivers high performance across different
dataset sizes, enhancing imaging quality effectively. All methods benefit from
larger training samples, with HF-Fed remaining competitive. Our hypernetwork
uses X-Ray geometry parameters to modulate feature maps, balancing stability
and imaging performance, proving effective in achieving consistent and
competitive results

Model | PSNR | SSIM
---|---|---
W/o FL "†" | 36.58 | 0.9384
W/o FL "‡" | 38.15 | 0.9348
FedAvg "†" | 35.51 | 0.8726
FedAvg "‡" | 36.85 | 0.9158
"HF-Fed $\diamondsuit$" | 38.62 | 0.9644
"HF-Fed $\star$" | 38.95 | 0.9667
"HF-Fed $\circ$" | 39.51 | 0.9691
HF-Fed | 39.95 | 0.9654
Table 2: Results of Ablation Studies

Fig. 2: Boxplots of PSNR scores for post-processing

###  4.2 Ablation Study

In this section, we conduct ablation studies to highlight the effectiveness of
various components in HF-Fed. The experiments follow the settings outlined in
Section 3.2, with results summarized in Table 2. "†" and "‡" denote imaging
networks without and with the hypernetwork, respectively, showing the
significant role of the hypernetwork in improving imaging performance by
addressing domain gap issues. Further, we evaluate our learning strategy by
aggregating only the hypernetwork in rounds labeled "HF-Fed $\diamondsuit$,"
addressing the challenge of heterogeneous scanning parameters. Additional
experiments explore the modulation scope of the hypernetwork, where "HF-Fed
$\star$" and "HF-Fed $\circ$" scenarios focus on encoder and decoder
modulation, respectively. Figure 2 shows the boxplots of PSNR based on w/o FL,
FedAvg, FedProx, FedBN. Ditto, pFedHN, and HF-Fed for the postprocessing task.
Results suggest similar performances across all modulation scenarios,
indicating the effectiveness of modulating all layers for consistency and
generality.

##  5 Conclusion

Current X-Ray imaging networks typically use centralized learning (CL), which
overlooks privacy concerns and faces challenges with non-iid data due to
varied scanning protocols and equipment. Introducing HF-Fed, a hierarchical-
based hypernetwork for X-Ray imaging, addresses these issues by seamlessly
integrating into diverse networks (CNN-based, IR-based, transformer-based).
HF-Fed features hospital-specific hierarchical-driven hypernetworks and a
shared Network of Networks (NoN), which adaptively extract stable imaging
features across domains. Ablation experiments demonstrate that while FL-absent
methods achieve customized X-Ray reconstruction, they suffer from detail loss
with limited data. In contrast, FL-based HF-Fed maintains stability,
recovering more details across hospitals and achieving superior reconstruction
quality.

####  5.0.1 Disclosure of Interests.

The authors have no competing interests to declare relevant to this article’s
content.

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[aff1]Department of Physics and Astronomy, University of California Los
Angeles, California 90095, USA [aff2]Department of Physics, University of
Chicago, Chicago, Illinois 60637, USA [aff3]Argonne National Laboratory,
Lemont, Illinois 60439, USA<EMAIL_ADDRESS>
# Multileaf Collimator for Real-Time Beam Shaping using Emittance Exchange

Nathan Majernikcor1  Gerard Andonian  Ryan Roussel  Scott Doran  Gwanghui Ha
John Power  Eric Wisniewski  James Rosenzweig

###### Abstract

Emittance exchange beamlines employ transverse masks to create drive and
witness beams of variable longitudinal profile and bunch spacing. Recently,
this approach has been used to create advanced driver profiles and demonstrate
record-breaking plasma wakefield transformer ratios [Roussel, R., et al.,
Phys. Rev. Lett. 124, 044802 (2020)], a crucial advancement for efficient
witness acceleration. However, since the transverse masks are individually
laser cut and installed into the UHV beamline, refinement of the beam profiles
is not possible without replacing masks. Instead, this work proposes the use
of a UHV compatible multileaf collimator as a beam mask. Such a device permits
real-time adjustment of the electron distribution, permitting greater
refinement in a manner highly synergistic with machine learning. Beam dynamics
simulations have shown that a practically realizable multileaf collimator can
offer resolution that is functionally equivalent to that offered by laser cut
masks.

##  1 INTRODUCTION

In the context of beam-driven wakefield accelerators, _transformer ratio_ is
the ratio of the maximum accelerating field experienced by a witness bunch to
the maximum decelerating field experienced by the driver bunch:
$\mathcal{R}\equiv|W_{+}/W_{-}|$ Roussel (2019). Regardless of the peak
accelerating gradient achieved in the wakefield accelerator, the transformer
ratio sets an upper bound on the maximum energy gained by the witness beam
before the driver has been depleted. For temporally symmetric drive bunches,
the fundamental theorem of beam loading indicates that the transformer ratio
cannot exceed 2 Bane, Chen, and Wilson (1985). However, by using a drive bunch
with a tailored current profile it is possible to achieve values of
$\mathcal{R}$ which exceed 2 Loisch  _et al._ (2018a), thereby reducing the
required drive beam energy to accelerate a witness by a fixed, target energy.
This has been demonstrated experimentally, including by Gao  _et al._ (2018)
which measured a transformer ratio of 4.8 in a dielectric wakefield
accelerator, and by Roussel  _et al._ (2020) which measured a record-setting
transformer ratio of 7.8 in a plasma wakefield accelerator (PWFA). Both of
these examples relied on highly asymmetric drive beams to more effectively
couple energy from the drive beam to the witness beam.

A number of options have been established for creating shaped current
profiles, including laser pulse stacking Loisch  _et al._ (2018b), combining
wakefield chirping with chicanes Andonian  _et al._ (2017), the use of doglegs
and higher order multipole magnets England, Rosenzweig, and Travish (2008), as
well as emittance exchange beamlines Gao  _et al._ (2018); Roussel  _et al._
(2020). Each technique has costs and benefits but emittance exchange (EEX) is
one of the most versatile options for creating high charge bunches with a
great degree of control. EEX is an advanced phase space manipulation technique
whereby the transverse phase space of a beam is exchanged with its
longitudinal phase space; one method of achieving EEX involves placing a
transverse deflecting cavity between two doglegs Roussel (2019). By masking
the beam’s transverse profile prior to EEX, high charge current profiles can
be realized that would be difficult or impossible to achieve using other
longitudinal shaping techniques. The EEX beamline at Argonne’s AWA facility Ha
_et al._ (2017) was used to produce the record-setting transformer ratio of
Roussel  _et al._ (2020).

Presently, at AWA’s EEX beamline the transverse masking is done using laser-
cut tungsten masks. Therefore, to change the mask shape requires installing
newly cut masks in to the UHV beamline, an operation which takes days from
start to finish. This latency makes it difficult to quickly iterate and refine
the current profile to optimize a wakefield interaction. This current work
proposes to replace these laser cut masks with a multileaf collimator (MLC), a
device with dozens of independently actuated leaves which move in and out of
the beam to create a custom aperture Jordan and Williams (1994); Boyer  _et
al._ (1992); Ge  _et al._ (2014). The most common application for multileaf
collimators is their use in radiotherapy, where they can be used to shape the
beams to precisely match the shape of the tumor from any angle, reducing
damage to nearby tissue while still delivering an effective radiation dose. In
the context of an EEX beamline though, the MLC will enable real-time, nearly
arbitrary control over both the drive and witness bunch profiles. A multileaf
collimator has many free variables for tuning, making it highly synergistic
with machine learning.

##  2 BEAM DYNAMICS SIMULATIONS

Unlike a laser cut mask, a multileaf collimator is comprised of discrete
elements, resulting in a ‘pixelation’ of the masked beam. It is necessary to
establish that, for a MLC with a practical number of leaves of an achievable
width, relevant current profiles can be reproduced with sufficient fidelity to
be useful for high transformer ratio wakefield acceleration. To that end, beam
dynamics simulations using OPAL Adelmann  _et al._ (2009) have been conducted
which compare the performance of laser cut masks to their approximations by a
MLC. The simulated MLC has 32 leaves, 16 per side, with a logarithmic width
distribution with an average leaf width of 2.5 mm. To test the MLC, two masks
from Roussel  _et al._ (2020) were selected; a YAG screen shortly downstream
shows the masked beam in Fig. 1(a-b). The simulated MLC leaves were set to
approximate this aperture in Fig. 1(c-d) before the masked results were
virtually propagated through the EEX beamline to the plasma interaction point,
with the current profiles for both cases for both the MLC and laser cut masks
shown in Fig. 1(e-f). Based on the agreement between the MLC and laser cut
mask current profiles, it is evident that a practical MLC is functionally
equivalent to laser cut masks, yet provides greater flexibility for near-
arbitrary beam shaping.

Figure 1: (a-b) The left column shows the YAG screen output shortly downstream
from laser cut masks in the AWA EEX beamline from Roussel (2019); the red
lineout is the result of integrating screen brightness in the vertical
direction. (c-d) The results from the two laser cut masks are represented as
the blue regions and the pink bars represent the MLC leaf positions to
approximate this aperture. (e-f) For each row, the masked beams are virtually
propagated through the EEX beamline with the final current profiles resulting
from the target aperture (blue line) and the MLC approximation (orange line)
shown.

##  3 CONCEPTUAL MODEL

Figure 2: (a) A render of the proposed multileaf collimator with a standard 12
ounce soda can for scale. The “wye” beam pipe allows a camera to be pointed at
the leaf tips for machine vision based feedback. (b) A detailed render of the
MLC mechanism with the vacuum vessel not shown. Some of the key elements are
called out.

There are many differences between commercially available multileaf
collimators for radiotherapy and the experimental requirements for EEX masking
but the most significant is the need for UHV compatibility. In Fig. 2 a
conceptual model which addresses these experimental requirements is presented.
This design is a 40 leaf MLC, with 20 leaves per side, where each leaf is 2 mm
wide. Each leaf is independently magnetically coupled to the exterior of the
vacuum chamber and actuated via a cable driven by a stepper motor. The leaf is
mostly comprised of aluminum to minimize weight but is tipped with tungsten
for the beam interaction. Each leaf can move in and out by twelve millimeters,
enough to cover the whole beam spot size.

The magnetic coupling and cable actuation are a UHV safe and economical
solution, but it comes with a substantial potential problem: the ability to
dead reckon the positions of the leaves is lost since there is no rigid
connection to a position encoder. Determining the leaf position therefore
requires another form of feedback. To precisely read out the leaf positions
the YAG screen images, like those of Fig. 1(a-b) can be used, but this
measurement is destructive to the beam. To supplement this diagnostic, a
camera will be pointed at the MLC leaves and, using machine vision, the leaf
positions can be determined in a fully online, nondestructive fashion.

##  4 PROOF-OF-CONCEPT EXPERIMENTS

For this concept to work, the magnets on the exterior of the vacuum vessel
must stay coupled to the magnets on the interior, but there is a tradeoff
between compactness and coupling strength. The coupling strength needs to
exceed the leaf weight, magnet-to-chamber friction, other friction sources,
and have some safety margin. It is expected that the total force will be
dominated by the friction at the magnet-chamber interface since the magnetic
attraction will produce a substantial normal force.

Preliminary tests were conducted using 0.25 inch diameter, N52 grade neodymium
magnets (consistent with the 2 mm leaf design from Fig. 2) for a variety of
vacuum chamber surrogate wall thickness and magnet lengths to measure the
force required to split the inner and outer magnet with the results shown in
Fig. 3. Two different test conditions were considered: static and dynamic. In
the static cases, the wall surrogate and external magnet were held together
and moved as a unit, therefore removing the magnet-chamber friction. Those
results are representative of the breakaway strength anticipated if the inner
and outer walls of the vacuum vessel were frictionless. In the dynamic cases,
the external magnet was independently actuated and the breakaway force was
again recorded. These cases, which include the motion of unlubricated magnets
on a 2B semi-bright finish 304 stainless steel surface, breakaway at
approximately 20 to 50 percent lower force, illustrating the important
contribution of magnet-chamber friction. This performance can be recovered by
reducing the friction but the vacuum requirements of the EEX beamline impose
substantial restrictions on the options for lubrication. Overall though, the
results suggest that the coupling strength is sufficient, even without
lubrication.

Figure 3: The results of the initial magnetic breakaway tests are shown.
“Long-long” tests uses 0.25 inch diameter, 0.5 inch long neodymium magnets for
both the interior and exterior magnets while “Long-short” tests use a 0.25
inch long magnet for the interior magnet. The “static” tests represent the
upper bound for breakaway strength in the limit of zero friction while the
“dynamic” case includes the friction of magnets on an unlubricated stainless
steel surface.

##  5 SUMMARY AND FUTURE WORK

A technology to replace the current laser cut tungsten masks used on the EEX
beamline with a multileaf collimator has been presented. Beam dynamics
simulations indicate that practically realizable multileaf collimator designs
can produce current profiles that are functionally identical to laser cut
masks. Such a change will allow real-time control over driver and witness
current profiles allowing for iterative refinement that is not possible with a
fixed-mask system. The large number of new variables the MLC introduces makes
it highly synergistic with machine learning for the optimization of beam
shaping for applications in high transformer wakefield acceleration. The
conceptual design addresses the main hurdles of implementation and UHV
compatibility. A series of benchtop tests are underway to validate the
magnetic coupling and cable driven actuation concepts. A successful
demonstration of the MLC concept would find utility in other accelerator
beamlines that rely on transverse masking and require strict UHV levels, for
example at BNL’s ATF Barber (2015) or at SLAC FACET Yakimenko  _et al._
(2019).

##  6 ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under Grant No.
PHY-1549132 and DOE Grant No. DE-SC0017648.

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