# Twisted Pair Transmission Line Coil - A Flexible, Self-Decoupled and Robust
Element for 7T MRI

Jules Vliem  Ying Xiao  Daniel Wenz  Lijing Xin  Wouter Teeuwisse  Thomas
Ruytenberg  Andrew Webb  Irena Zivkovic

###### Abstract

Objective: This study evaluates the performance of a twisted pair transmission
line coil as a transceive element for 7T MRI in terms of physical flexibility,
robustness to shape deformations, and interelement decoupling.
Methods: Each coil element was created by shaping a twisted pair of wires into
a circle. One wire was interrupted at the top, while the other was interrupted
at the bottom, and connected to the matching circuit. Electromagnetic
simulations were conducted to determine the optimal number of twists per
length (in terms of B${}_{1}^{+}$ field efficiency, SAR efficiency,
sensitivity to elongation and interelement decoupling properties) and for
investigating the fundamental operational principle of the coil through fields
streamline visualization. A comparison between the twisted pair coil and a
conventional loop coil in terms of B${}_{1}^{+}$ fields, maxSAR10g, and
stability of $S_{11}$ when the coil was deformed, was performed.
Experimentally measured interelement coupling between individual elements of
multichannel arrays was also investigated.
Results: Increasing the number of twists per length resulted in a more
physically robust coil. Poynting vector streamline visualization showed that
the twisted pair coil concentrated most of the energy in the near field. The
twisted pair coil exhibited comparable B${}_{1}^{+}$ fields and improved
maxSAR10g to the conventional coil but demonstrated exceptional stability with
respect to coil deformation and a strong self-decoupling nature when placed in
an array configuration.
Discussion: The findings highlight the robustness of the twisted pair coil,
showcasing its stability under shape variations. This coil holds great
potential as a flexible RF coil for various imaging applications using
multiple-element arrays, benefiting from its inherent decoupling.

###### keywords:

Twisted Pair , Flexible Coils , 7T Arrays , Ultra-High Field MRI

††journal: *Corresponding author<EMAIL_ADDRESS>
[inst1]organization=Department of Electrical Engineering, Eindhoven University
of Technology,city=Eindhoven, country=The Netherlands

[inst2]organization=CIBM Center for Biomedical Imaging,city=Lausanne,
country=Switzerland

[inst3]organization=École Polytechnique Fédérale de Lausanne (EPFL), Animal
Imaging and Technology,city=Lausanne, country=Switzerland

[inst4]organization=C.J. Gorter MRI Centre, Department of Radiology, Leiden
University Medical Center,city=Leiden, country=The Netherlands

##  1 Introduction

Multi-channel RF coil arrays are ubiquitous in clinical and ultra-high field
MRI and can be used for both RF transmission and signal reception [1, 2, 3, 4,
5, 6]. In transmit mode, they can be advantageous for RF shimming [7],
whereas, in a receive mode, they can provide a high Signal-to-Noise Ratio
(SNR) and enable parallel imaging strategies [8, 9, 10]. C]. Common building
elements of such arrays are usually constructed using resonant copper loop
elements, segmented with capacitors, which are fixed in space. In this
approach, the array is designed such that it can fit a large population of
patients. This is often associated with a reduced filling factor of an array
leading to sub-optimal coil performance. Highly flexible RF coil arrays can be
considered a very promising solution to address this limitation, and therefore
there has been an ongoing interest in developing novel strategies for such
flexible arrays [11, 12, 13, 14, 15]. The appeal of flexible coils lies in
their ability to conform to diverse anatomies, maintaining a consistent
distance from the subject and thereby improving electromagnetic (EM)
performance through enhanced coupling to the sample [13]. Additionally,
flexible RF coil arrays improve patient comfort due to their lightweight and
form-fitting design, making them well-aligned with the goal of creating
patient-centred solutions.

Interelement coupling is considered a major challenge in any type of
multichannel array; insufficient decoupling between the elements can lead to
decreased performances of the array by transferring energy to adjacent
elements instead of the subject. For accelerated imaging, it is essential to
have as low coupling as possible. Ideally, each element should have a distinct
sensitivity map for optimal acceleration [9, 16]. When coils couple
inductively, they become sensitive to the same regions of the sample, yielding
intertwined sensitivity maps and diminished accelerated imaging efficiency
[17, 18].

Various methods are used for interelement decoupling for loop coils such as
partial overlapping [19], capacitive and inductive networks [20, 21, 22],
transformer decoupling [23] and the use of strategically placed passive
resonators [24, 25]. For receive-only arrays, preamplifier decoupling is used
[19]. The objective of these decoupling techniques is to prevent unwanted
”communication” between the coils, i.e. parasitic current induction which
could lead to a decline in coil performance. Ideally, the coil elements
utilised in an array configuration would inherently possess decoupling
properties, negating the need for any of the aforementioned decoupling
techniques. A proposed intrinsically decoupled element is the shielded-
coaxial-cable (SCC) coil, exhibiting a highly decoupled nature per se - it is
demonstrated that placing SCC elements in various array configurations keeps
elements highly decoupled which allows high-performance imaging at 7T [11,
26]. Given that a coaxial cable is a transmission line, the question arises if
other transmission line elements hold the potential for a flexible RF coil
design.

In this work an alternative transmission line transceive element is proposed -
the twisted pair transmission line coil. Prior research explored the use of
the twisted pair coil solely as receive-only elements in both 3T [27, 28] and
7T [29] MRI systems. However, these previous studies did not extensively
explore aspects such as decoupling properties, Specific Absorption Rate (SAR),
transmission efficiency, and overall coil flexibility and robustness. This is
mainly because the coil was used exclusively as a receive-only element.

In this work, we compared the twisted pair coil with a conventional coil in
terms of transmit field (B${}_{1}^{+}$) efficiency, maximum SAR10g, robustness
towards elongation and shape changes and interelement decoupling when placed
in various array configurations. The intrinsic highly decoupled nature of the
proposed coil has also been investigated and elucidated by considering the
power flow via calculation of the Poynting vector.

##  2 Theory and Methods

###  2.1 The Main Coil Concept

The twisted pair loop coil was created with two polytetrafluoroethylene (PTFE)
insulated copper wires, twisted around one another, as can be seen in Figure 1
(a). This created a flexible structure with a distributed capacitance. The
twisted pair can be thought of as a parallel plate capacitor with the PTFE
insulation of both wires as the dielectric. This distributed capacitance of
the coil contains the electric field [28]. The twisted wire was shaped into a
loop and gaps were introduced for both wires, as illustrated in Figure 1 (b).
The cut in the grey shield wire \- at the top, opposite of the feeding point,
effectively un-shields the magnetic field, making the coil sensitive to
electro-motive forces (EMF) and thus resonant. At the feeding point, a cut was
made in the complementary red signal wire for connection to the tuning and
matching network, facilitating interfacing with the MRI scanner, and enabling
the transmission and reception of signals.

When a current is excited on the red signal wire, it flows in one direction,
either clockwise or counter-clockwise. Take counter-clockwise as an example.
Due to the tightly twisted configuration, a current is induced in the
complementary grey shield wire, flowing in the opposite direction according to
Lenz’s law, in this case clockwise. In the region in between the conductors of
the wires, the magnetic fields generated by the currents point in the same
direction, resulting from the opposite current flow in each wire. Outside the
twisted wires, the magnetic fields tend to destructively interfere with each
other since they flow in opposite directions.

Within the loop, a gap is introduced in the grey shield wire, disrupting the
continuous current path at that point. This gap is an open circuit, leading to
a high voltage and a low current at the gap. A high potential difference is
present between the two grey shield wire ends, generating a strong electric
field in the gap region. The strong electric field at the gap induces a
substantial current in the red signal wire, flowing in the same direction as
the current in the grey shield wire. This induced current contributes to the
partial cancellation of the original current in the red signal wire since it
flows in the opposite direction (clockwise) to the original current
(counterclockwise). The cancellation of the current in the red signal wire
results in the grey shield wire being the dominant wire in determining the
flow of the EM field. The grey shield wire has no current close to the gap but
here the red signal wire takes over the current, resulting in a constant
current flow in the clockwise direction for the loop. For our example the
dominant current flow will thus be in the clockwise direction, creating a
downward oriented magnetic field in the middle of the loop. Since an
oscillating signal is used, the current flow and magnetic field will also
switch, following the dominant current flow of the grey shield wire. In A an
illustration is shown to visualize the workings of the twisted pair coil.

###  2.2 Simulation-Based Design of the Twisted Pair Coil

The initial step in the design process involved determining the optimal
twisting density per unit length for the wires. The twisted pair coil was
modelled using SolidWorks (Dassault Systèmes SolidWorks Corporation, Waltham,
USA) and imported in CST Studio Suite 2023 (Dassault Systèmes, Vélizy-
Villacoublay, France) for electromagnetic simulations. The twisted pair was
modelled by drawing two circles with 1 mm diameter, representing the copper
conductor, spaced 2 mm apart. These circular profiles were swept along a
predefined path, which could be circular or of arbitrary shape, to create the
coil. The number of twists per unit length was specified in SolidWorks by the
number of revelations of the profile along the path. All models have a total
length of 31.4 cm, corresponding to a loop with 10 cm diameter, and are shaped
into circular, elongated, or arbitrary forms. Cuts of 2 mm were introduced,
one at the top and one at the bottom, for the grey shield wire and red signal
wire, respectively. A discrete port was connected between the gap at the feed
point, i.e. the red signal wire. Lumped elements were connected to the port in
the RF circuit co-simulation to tune the coil to the resonance frequency
(297.2 MHz) and match the coil’s impedance to 50 $\Omega$.

To accurately mesh the complex geometry of the twisted shape, tetrahedral
meshing is preferred. Tetrahedral meshing enables rapid and accurate meshing
of intricately curved geometries compared to hexahedral meshing, which would
not accurately capture the twisted and curved structure in a reasonable number
of cells [30]. Therefore, the frequency domain solver with tetrahedral meshing
was employed, utilising adaptive mesh refinement with a minimum of 3 and a
maximum of 8 passes. The simulations were conducted by placing a coil on a
homogeneous cubic phantom ($\varepsilon_{r}=67.9$, $\sigma=0.48$ S/m,
dimensions 40$\times$40$\times$40 cm) with the coils positioned 2 cm away from
the phantom. The copper cores were modelled as annealed copper
($\sigma=5.8\cdot 10^{7}$ S/m), while the insulation of both wires was
excluded due to the increased complexity of the model. Additional simulations
were conducted with the insulation layer to confirm the negligible impact on
the simulated results. All the simulation results were normalised to 1 W of
accepted power.

To investigate the optimal number of twists per unit length, various
performance metrics on the cubic phantom were examined, including the
reflection coefficient of the coil element, transmit efficiency
(B${}_{1}^{+}$) measured at 5 cm depth in the phantom at coil center, maximum
Specific Absorption Rate at 10 grams average (maxSAR10g), SAR efficiency at 5
cm depth (B${}_{1}^{+}$ divided by the square root of the maxSAR10g), and the
coupling coefficient ($S_{21}$) between two neighbouring elements. The
mentioned parameters were analysed for coils designed with different numbers
of twists per coil length, such as 0, 10, 24, 48 and 64 twists per 31.4 cm
(the circumference of all the 10 cm diameter loop coils). See Figure 2 (a) for
an illustration of two wires with different twists per unit length.

To compare the performance of the twisted pair coil we conducted a comparison
with a conventional loop coil featuring distributed capacitors, illustrated in
Figure 3 (b), on a cubic phantom. The copper loop coil was modelled as a 10 cm
diameter loop and constructed using 1 mm diameter annealed copper wire.

To better understand the differences between the EM fields of the twisted pair
and conventional coils, we visualised the flow of these fields using
streamlines. Streamlines are a visualisation technique that can be used to
represent the flow of a vector field. It has been shown previously that using
streamlines can give a much simpler and more direct explanation compared to a
circuit approach, whose results are very close to each other (order of 0.1%)
[31]. The fields were exported from CST to MATLAB to generate the streamlines
in the slice plane. The fields were exported from CST to MATLAB to generate
the streamlines in the slice plane. This visualisation allowed us to determine
the difference in electromagnetic flow between the conventional and twisted
pair coil, as well as when two coils are placed close to each other.

To demonstrate the extraordinary flexibility of the twisted pair coil, we
further expanded its design by modelling it as a spline shape. This approach
allowed us to create diverse and intricate coil configurations by defining
arbitrary splines in SolidWorks and sweeping the circular profiles along the
spline path. The coil was tuned and matched at the beginning in a circular
shape (diameter of 10 cm) and subsequently, the shape was deformed while
keeping the same circumference and tuning and matching circuit elements.

###  2.3 Coil Construction

The twisted pair coil was constructed by tightly twisting two 18-gauge wires
(1 mm diameter) with PTFE insulation. To achieve a high number of twists per
unit length, two large sections of wire were cut, and one end was secured in a
bench vise. The other end of the wires was inserted into a drill, and by
turning the drill, the wires were twisted while maintaining tension by gently
pulling them. The maximum twists per unit length were reached when the wires
became significantly taut. At this point, the twisted pair could be removed
from the vise and drill. Twisted pair coils which we fabricated had 64 twists
per coil length (31.4 cm).

Before connecting the loop to the circuitry, a cut was made in the middle of
the grey shield wire. The size of this cut affected the coil’s tuning, where a
larger cut slightly increases the resonance frequency. The grey shield wire
cuts were approximately 2 mm in our construction. The coil’s physical
realisation and its corresponding schematic are shown in Figure 3 (a). The
coil, with a diameter of 10 cm, was tuned and matched using one parallel and
two series fixed capacitors (AVX800 E Series, Kyocera-AVX, Fountain Inn, USA).
The red signal wire was attached to both sides of the parallel capacitor,
while the grey shield wires were connected, either twisted or soldered on the
same trace of the Printed Circuit Board (PCB). The PCB was created by milling
signal traces on a copper-plated FR4 material ($\varepsilon_{r}=4.4$). The
circuitry was fed by an RG58 coaxial cable (RG58 LSZH, Multicomp PRO, London,
UK) connecting the core to the signal trace and the shield to the ground
trace.

To compare the performance of the twisted pair loop coil, a conventional
copper loop was also fabricated, as depicted in Figure 3 (b) along with its
circuitry. The conventional coil was constructed using a copper wire with a
wire diameter of 1 mm. Fixed and variable capacitors were distributed along
the 100 mm diameter loop, one parallel and two series fixed capacitors were
utilised for tuning to the resonance frequency and matching the coil’s
impedance to 50 $\Omega$.

###  2.4 Array Construction

To assess the performance of individual elements in an array configuration,
three distinct array layouts were examined. The first configuration featured
an eight-element (two-by-four) array. All elements were circular and exhibited
partial overlap, i.e. neighbouring elements within the same row and between
the two rows were overlapped. This arrangement aimed to accommodate all eight
coils within the given phantom space. Notably, no precise overlap distances or
metrics were employed, highlighting the inherent decoupling of the coils, even
when arranged in a semi-random fashion.

The second array configuration also adopted a two-by-four arrangement. Coils
within each row were slightly elongated to fit the available space without
overlapping neighbouring elements. The elements positioned between the two
rows were rotated by 180 degrees relative to each other and demonstrated
partial overlap, although no predefined overlap degree was stipulated.

Lastly, a five-element array was constituted by positioning extremely
elongated elements sequentially to fill the available phantom space.

###  2.5 Bench and MRI Measurements

####  2.5.1 Bench Measurements

The individual elements were assessed on a cubic phantom, see Figure 3 (c),
placed 2 cm away from the phantom using a foam spacer. $S$-parameters were
measured with a Vector Network Analyser (VNA) (Keysight N9914A FieldFox VNA
6.5GHz) for individual elements and when placed in an array configuration.

To investigate the coupling behaviour ($S_{21}$) of the twisted pair coil, two
elements were placed in close proximity to the phantom. The distance between
the elements was systematically adjusted, starting from 20 mm apart and
gradually reducing the separation until they overlapped by 90 mm (90% of the
coil’s diameter). The same procedure was used for two conventional coils.
Additionally, different geometry orientations were tested for the twisted pair
coils while varying the degree of overlap.

To measure the quality factors of the coil the half-power bandwidth (HPBW)
edges were found when looking at the input impedance ($Z_{in}$) of the
fabricated coils. Dividing the central frequency by the HPBW gives the Q
factor of the coil [32]. To measure the unloaded Q, the coil was placed in the
air and for the loaded Q it was placed on the phantom.

####  2.5.2 MRI Measurements

The rectangular body phantom ($\varepsilon_{r}$ = 50, $\sigma$ = 0.6 S/m, size
= $40\times 40\times 30$ cm) experiments were performed on a 7T Philips
Achieva scanner. The 2D B${}_{1}^{+}$ maps were obtained using the dual
refocusing echo acquisition mode (DREAM) [33] sequence (FOV = $350\times
350\times 54$ mm3, slices = 5, flip angle = 10°, STEAM angle = 50°, TR/TE =
4.0/0.98 ms, scan time = 120 s).

The spherical phantom ($\varepsilon_{r}=75$, $\sigma=1.09$ S/m, diameter = 17
cm) experiments were performed on a 7T MR human scanner (Magnetom, Siemens
Healthineers, Erlangen, Germany). The three-dimensional B${}_{1}^{+}$ maps
were quantitatively measured with the SA2RAGE [34] sequence (TR/TE = 2400/0.78
ms, TI1/TI2 = 45/1800 ms, $\alpha_{1}/\alpha_{2}$ = 4/10∘, FOV = 208×256 mm2,
slices = 64, resolution = 2.0×2.0×2.5 mm, BW = 1220Hz/px, scan time = 115 s).
The measurement results were normalised to the input power.

Furthermore, additional simulations were performed on the spherical phantom to
enable a comparison between measurement and simulation results. The coils
conformed to the surface of the sphere, by projecting the shape on the phantom
so that a constant distance of 5 mm to the phantom was maintained. Simulated
results were normalised to 1 W of accepted power.

##  3 Results

###  3.1 Individual Coil Design and Comparison

The simulated metrics of twisted pair coils across varying twist densities are
presented in Figure 2. It is observed that increasing the number of twists per
unit length decreases the B${}_{1}^{+}$ field efficiency, maxSAR10g, impedance
of the coil (the points move from the right to left on the Smith chart) and
coupling coefficient. A similar trend is observed for the SAR efficiency,
despite being less pronounced as it levels off at around 24 twists per unit
length. The difference in simulated B${}_{1}^{+}$ fields of coils with 24
twists and 64 twists per unit length was $\sim$15%, whereas in the
measurements the difference was approximately 25%. Additional simulated
robustness analysis results for different numbers of twists per unit length
are provided in B, and show the enhanced robustness of the 64 twists per unit
length configuration. For this research, a coil with a low coupling
coefficient and high robustness towards the shape deformation is preferred
over a high field efficiency so the coil can be efficiently used in densely
populated receive and transceive arrays. Therefore, from now on, simulated and
manufactured coils are always with 64 twists per unit length.

The fabricated twisted pair, with 64 twists per unit length, and conventional
coils with their corresponding circuitry are shown in Figure 3 together with
the cubic phantom used for $S$-parameter measurements. Additional return loss
measurements are presented in C, showcasing the robustness of the twisted pair
to elongations compared to the conventional coil. The measured unloaded
($Q_{\text{un}}$)/ loaded ($Q_{\text{lo}}$) quality factors of the twisted
pair were 55.0/49.4 whereas those of the conventional coil were 127.4/21.1,
corresponding to ratios of $Q_{\text{un}}/Q_{\text{lo}}$ of 1.1 and 6.0.

A comparison of simulated B${}_{1}^{+}$ field efficiency, maxSAR10g and SAR
efficiency of the twisted pair and conventional coil, for different degrees of
elongation is shown in Figure 4. Simulated B${}_{1}^{+}$ fields of the twisted
pair and conventional coils were very similar. Simulated maxSAR10g for a
twisted pair coil in a circular shape was 1.1 W/kg and decreased to 1.0 W/kg
in the two elongated shapes. The conventional coil had a maxSAR10g of 1.1 W/kg
in the circular shape while maxSAR10g increased to 1.4 W/kg in the maximum
elongated shape. When comparing the simulated and measured B${}_{1}^{+}$ field
patterns for different degrees of elongation and good match is observed. The
main difference is that the simulations were scaled down with 25%. The SAR
efficiency at superficial depth was higher for the elongated coils, for both
the twisted pair and conventional coil, where after approximately 2 cm all
profiles became very similar.

###  3.2 Coupling Results

The measured coupling coefficients ($S_{21}$) between two coils for varying
overlapping geometries can be seen in Figure 5. Across all the examined
scenarios, the decoupling between two twisted pair elements was below -13 dB
for the twisted pair. When comparing the performance of the twisted pair and
the conventional coil across different degrees of overlap, the measured
$S_{21}$ for the twisted pair was consistently below or equal to -15 dB,
whereas the conventional coil exhibited poorer performance with $S_{21}$
values below or equal to -4 dB. Notably, the highest interelement coupling for
the twisted pair, reaching -13 dB, was observed when the coils were rotated
180 degrees relative to each other. In the remaining two examined scenarios,
the coupling remained below or equal to -18 dB.

###  3.3 Streamline Plots

The electric ($\bm{E}$) and magnetic ($\bm{H}$) fields, together with the
power flow (Poynting vector $\bm{S}$) for conventional and twisted pair coils
without a phantom is depicted in Figure 6 in three different cross-sections.
The $\bm{E}$ field consistently maintains its pattern and remains zero in the
YZ plane. In contrast, the $\bm{H}$ field exhibits a similar pattern but
significantly greater magnitude in the inner part of the conventional coil
compared to the twisted pair coil. $\bm{S}$ lines propagate radially within
the plane of the conventional coil, while in the case of the twisted pair
coil, power lines exit from the gap opposite the port.

In Figure 7, we observe $\bm{E}$ and $\bm{H}$ fields and power flow $S$ for
conventional and twisted pair coils with a phantom, captured in three
different cross-sections. The $\bm{E}$ field maintains its shape and
magnitude, remaining zero in the YZ plane. Conversely, the $\bm{H}$ field
undergoes changes, displaying a similar magnitude and shape for both coils and
becoming non-zero in the XZ plane. Analogous to the power bubble described in
[31], we identify a magnetic field bubble here—a border surrounding the coil,
where magnetic field lines approach from one side and recede from the other.
Notably, the magnetic field bubble is considerably smaller in the twisted pair
coil compared to the conventional coil, indicating that only a small portion
of the magnetic field from the phantom couples with the twisted pair coil. In
both cases, $\bm{S}$ streamlines are directed toward the phantom.
Additionally, the conventional coil features $\bm{S}$ streamlines propagating
on both sides within its plane. Introduction of the phantom leads to the
emergence of a very strong $\bm{E}$ field and power lines between the twisted
pair coil and the phantom, driven by significant capacitive coupling.

Figure 8 illustrates $\bm{E}$ and $\bm{H}$ fields, current density $\bm{J}$,
and power flow $\bm{S}$ for pairs of conventional and twisted pair coils with
a phantom, captured in three different cross-sections. In these experiments,
both coil types were positioned at distances of 5 cm and 5 mm from each other,
with one coil actively observing the coupling to the other (non-active) coil.
When the coils were separated by 5 mm, an analysis of the current density at
the phantom’s surface revealed that the second conventional coil (the passive
one) exhibited significantly greater coupling than the twisted pair coil,
inducing a stronger current in the phantom. Magnetic field lines encircle both
the active and passive conventional coils at both separation distances due to
stronger coupling between the coils.For the twisted pair coil, the second
(passive) coil also creates a magnetic field bubble when spaced 5 mm apart but
is considerably smaller than the bubble from the conventional coil. Unlike
conventional coils, where power flows from the active coil towards the passive
one, resulting in coil coupling, the power lines in twisted pair coils do not
flow from the active towards the second passive coil.

###  3.4 Array Results

Figure 9 shows the experimentally measured interelement coupling of the three
array configurations. In all evaluated cases the return loss $S_{ii}$ of
individual elements as well as interelement decoupling, $S_{ij}$, had
excellent performance ($\leq$ -10 dB). The worst coupling results were
observed in configuration (b) – eight slightly elongated, partially overlapped
elements – for the opposing neighbours.

###  3.5 Spline Shape Evaluation

In Figure 10, two spline twisted pair coil shapes are presented. The coils
were tuned and matched in the circular shape and subsequently their shapes
were deformed without retuning the coils. Both twisted pair spline coils
stayed decoupled ($<$ -15dB) for any degree of overlap. The B${}_{1}^{+}$
field of the spline-shaped coils followed the coil shape, as seen in the
coronal view. The maxSAR10g of the first spline saddle-shaped coil slightly
increased to 1.2W/kg, compared to the circular shape, while the maximum SAR10g
of the second spline triangular-shaped coil was the same as the circular shape
(1.1 W/kg).

##  4 Discussion

In this work, we presented a novel design for an extremely flexible and self-
decoupled coil element for 7T MRI – the twisted pair transmission line coil.
The proposed coil does not require distributed lumped elements and due to its
intrinsic decoupling properties, it allows for straightforward construction of
tight-fitting multi-channel arrays. In MRI experiments, the proposed coil was
used as a transceive element and a good agreement between simulations and
measurements was achieved. To the best of our knowledge, this is the first
time that a twisted pair transmission line coil was examined as a
transmit/transceive element at 7T MRI.

The advantage of the twisted pair coil, in comparison to other highly flexible
elements such as the liquid metal coil [14]], lies in its inherent self-
decoupled nature. Unlike the liquid metal coil, which requires mutual overlap
to geometrically decouple the array, the twisted pair coil does not require
any additional decoupling methods. When comparing the twisted pair coil with
the Shielded-Coaxial-Cable (SCC) coil [11], similar decoupling properties were
observed. In [11], it is stated that the $Q_{\text{ul}}/Q_{\text{lo}}$ of the
conventional loop element was 5.7, while for the SCC coil, it was 1.7. In our
measurements, $Q_{\text{ul}}/Q_{\text{lo}}$ of the twisted pair coil was 1.1,
even lower than that of the SCC coil and over five times lower than that of
the conventional coil. If we consider that at around 300 MHz, resistive
coupling of coil-to-coil “through a sample“ is the dominant coupling mechanism
[11]], the results imply lower coupling to the sample of the twisted pair coil
vs. SCC, which further implies lower inter‐element coupling of twisted pair
coils. While Ruytenberg et al. did not report on the coupling behaviour of the
SCC when the loops have different orientations with respect to each other, in
our study the interelement decoupling properties were excellent, regardless of
the loop’s relative orientation and amount of overlap.

Considering the design parameters depicted in Figure 2, it becomes evident
that a higher number of twists per unit length, such as 64, results in an
inherently decoupled element with a lower maxSAR10g value in comparison to a
lower number of twists per unit length, for instance, 10. As the number of
twists increases, both the B${}_{1}^{+}$ efficiency at a depth of 5 cm and
maxSAR10g decrease, implying nearly constant SAR efficiency across different
twist densities. This decline in B${}_{1}^{+}$ and SAR is likely attributed to
amplified EM field cancellation stemming from the increased number of twists.
The incorporation of additional twists leads to a higher distributed
capacitance, causing a downward shift in the resonance frequency due to the
extended electrical length. This heightened capacitance simultaneously
contributes to a reduction in coil impedance, as indicated in the Smith chart
presented in Figure 2 (a). The increased distributed capacitance also retains
the electric field more, resulting in lower interelement interactions for a
higher number of twists per unit length.

The cancellation of the EM field, when currents flow in opposite directions
within the wires, depends on the configuration and relative orientation of
these wire segments. A greater number of twists facilitates more substantial
interaction between the individual wire segments’ generated fields. In
contrast, a coil with fewer twists may experience less cancellation due to
limited opportunities for EM field interactions, leading to a stronger field
that increases both B${}_{1}^{+}$ and SAR, compared to a higher twist per unit
length coil. Given the similar SAR efficiency observed across coils with
different twist densities, we opted to utilise the element with the highest
robustness and lowest interelement coupling for subsequent simulations and
experiments, specifically the configuration with 64 twists per unit length. If
an element with an increased B${}_{1}^{+}$ field efficiency and possibly
higher Q factor is desired one could use an element with lower twists per unit
length, such as 10 twists.

A valid question that might arise is why not more than 64 twists per unit
length were subjected to analysis. This limitation can be attributed to two
primary factors: Firstly, the wires employed in the fabrication of the coils
exhibit a threshold of around 64 twists per unit length, beyond which they
become susceptible to snapping. While employing alternative wires could
address this concern, such wires would need to possess a smaller wire diameter
to accommodate more than 64 twists per unit length. Secondly, simulations
involving more than 64 twists per length for a 10 cm diameter loop coil become
increasingly inaccurate. Notably, the model begins to manifest sharp edges,
resulting in imprecise mesh representation.

Streamline visualisation of electric and magnetic fields and power flow, shows
the coupling mechanism of the coils and their energy flow. The power is
concentrated around a twisted wire structure and only a small amount is
radiated away or present inside of the coil (similar results were obtained
when the SCC coil was analysed by streamline approach, not shown in this
manuscript). The origin of the power lines in the twisted pair coil was at the
gap position (opposite side from the feed point), whereas for the conventional
coil, it flows from the feed point. Very strong capacitive coupling between
the coil and phantom can be noticed and resulted in a strong decrease of the
transmit field efficiency when the twisted pair coil is moved away from the
phantom (see D). When a conventional coil was moved away from the phantom
(from a distance of 20 mm to a distance of 40 mm, for example) the transmit
field efficiency reduction was much lower (8% compared to 15% for the twisted
pair) than in the twisted pair coil because the conventional coil coupled to
the phantom not only via capacitive coupling but also via radiative field.
Close to the phantom, both twisted pair and conventional coils exhibited
similar transmit field efficiencies. By looking at the streamline fields of
two coils on a phantom setup, it can be concluded that in the case of twisted
pair coils the interelement coupling occurs dominantly through the electric
field – electric field lines originated at the first (active) coil and ended
on a second (passive coil). This trend was clearly visible at any cut. The
second (passive) twisted pair coil had little interaction with the magnetic
field originating from the first coil or from the phantom, while for the
conventional coil, the coupling through a phantom and directly through the
magnetic field produced by the active coil was present. The dominant coupling
mechanism of a twisted pair coil is through an electric field originating from
the coil itself and it does not depend on a coil distance from the phantom.
Unloaded and loaded Q factor measurements are almost identical which tells the
twisted pair coil is not dominantly influenced by the presence of the phantom.
On the other side, when the twisted pair coil is close to the phantom, there
is very strong capacitive coupling between the twisted pair coil and phantom
itself (very strong $\bm{E}$ field and concentrated power between the coil and
phantom, Figure 7), which does not affect interelement coil coupling but
B${}_{1}^{+}$ field efficiency. Due to the capacitive nature of the twisted
pair coil coupling to the phantom, there is a steeper decrease in
B${}_{1}^{+}$ field efficiency when the twisted pair coil is moved away from
the phantom than is the case with the conventional coil. As already mentioned,
twisted pair coils are less sensitive to the secondary magnetic field produced
by induced currents in a phantom (so-called resistive coupling).

When looking at the streamlines of an SSC coil (not shown in this work)
similar features are observed as for the twisted pair coil: they effectively
guide energy within the coil structure, minimizing outward radiation; have a
high capacitive coupling with the phantom; largely reduced inductive coupling
component. However, this leads to the question: What advantages does the
twisted pair coil offer over the SCC? While both operate on similar
principles, a benefit of the twisted pair coil is the additional degrees of
freedom of manufacturing (number of twists) and increased flexibility due to
the absence of the dielectric material inside the coaxial cable.

All simulated and fabricated coils were tuned and matched only once and
$S_{11}$ was stable while being elongated, deformed and placed in various
array configurations. Supplementary material 1 shows a video of different
shape deformation of the twisted pair coil element while the $S_{11}$ remains
stable.

In terms of B${}_{1}^{+}$ field efficiency, the coil has very similar
performances compared to the conventional coil of the same diameter (with
distributed capacitors) when placed close to the phantom. Insensitivity to the
coil shape deformation and stable $S_{11}$ and SAR are the biggest advantages
of the twisted pair coil compared to the conventional design. In circular
form, both twisted pair and conventional coils had the same maxSAR10g of 1.1
W/kg. In elongated forms, twisted pair coils showed decreased SAR from 1.1
W/kg to 1 W/kg, while the SAR of the conventional coils increased by 30% at
maximum elongation. Deformation of twisted pair coil to various spline shapes
produced a minimal increase of SAR (1.1 W/kg and 1.2 W/kg, for the two example
shapes).

When examined in terms of interelement decoupling properties, the twisted pair
coil showed a truly self-decoupled nature – it stayed decoupled (better than
-10 dB) in all examined array configurations without the need for additional
decoupling techniques. It is shown that even in extreme coil shape
deformations (spline coil shapes shown inFigure 10), the $S_{11}$ and SAR of
individual coils are stable, and remain well decoupled when placed at various
distances from each other. Such superior decoupling properties make twisted
pair transmission line coil a suitable element for building densely populated
tight-fit receive arrays. This leads to the idea to optimise the shape of the
coil when placed in receive array configuration in such a way as to allow high
accelerations and maximally improved g-factor.

An additional benefit of the twisted pair over the SCC coil is the absence of
a thicker, inflexible dielectric material. This particular design feature
makes the twisted pair coil well-suited for accommodating extreme shape
deformations. Its robustness in handling geometric changes positions the
twisted pair coil as an excellent element for optimised receive arrays for
highly accelerated MRI.

There are also commercially available flexible arrays such as AIR coil element
[35, 36, 37]. The AIR coil element has been proposed as a flexible and highly
decoupled element mainly in receive arrays for operation at 3T. A variety of
coil positioning in array configurations and coil shapes different than
circular, were not reported. Additionally, noise-controlling preamplifiers
have been placed at each loop coil, which could also help improve the
interelement coupling of AIR coils [35], while twisted pair transmission line
coils do not need any additional decoupling technique to be used.

##  5 Conclusion

This work introduced and investigated the twisted pair transmission line coil
as a novel, highly flexible and self-decoupled transceive element for 7T MRI.
The twisted pair transmission line coil showed comparable transmit field
efficiency (B${}_{1}^{+}$) and SAR performance to a conventional copper loop
coil and exceptional robustness to various shape deformations. The dominant
coupling mechanism is through electric fields originating from a coil itself
while the coils are insensitive to the magnetic fields (both direct field
originating from a coil and secondary magnetic field originating from a
phantom). In this paper, a streamline approach to visualise the electric and
magnetic fields and power flow has been proposed for a deeper understanding of
coil functioning and the same approach can be applied in studying any other
coil designs.

Overall, the findings of this study support the potential of the twisted pair
transmission line coil as a highly flexible and efficient solution for 7T MRI,
especially in tight-fit arrays.

## Acknowledgements

The authors would like to thank Matthijs van Osch (LUMC, Leiden, Netherlands)
for his help with the measurements.

## References

  * [1] C. Oezerdem, L. Winter, A. Graessl, K. Paul, A. Els, O. Weinberger, J. Rieger, A. Kuehne, M. Dieringer, F. Hezel, D. Voit, J. Frahm, T. Niendorf, 16-channel bow tie antenna transceiver array for cardiac MR at 7.0 tesla, Magnetic Resonance in Medicine 75 (6) (2016) 2553–2565.  doi:10.1002/mrm.25840.
  * [2] I. A. Elabyad, M. Terekhov, M. Bille, L. M. Schreiber, Design and Implementation of Two 16-Element Antisymmetric Transceiver Coil Arrays for Parallel Transmission Human Cardiac MRI at 7 T, IEEE Transactions on Microwave Theory and Techniques 69 (7) (2021) 3540–3557.  doi:10.1109/TMTT.2021.3079422.
  * [3] J. D. Clément, R. Gruetter, O. Ipek, A human cerebral and cerebellar 8-channel transceive RF dipole coil array at 7T, Magnetic Resonance in Medicine 81 (2) (2019) 1447–1458.  doi:10.1002/mrm.27476.
  * [4] S. N. Williams, S. Allwood-Spiers, P. McElhinney, G. Paterson, J. Herrler, P. Liebig, A. M. Nagel, J. E. Foster, D. A. Porter, S. Gunamony, A Nested Eight-Channel Transmit Array With Open-Face Concept for Human Brain Imaging at 7 Tesla, Frontiers in Physics 9 (July) (2021) 1–16.  doi:10.3389/fphy.2021.701330.
  * [5] J. T. Vaughan, G. Adriany, C. J. Snyder, J. Tian, T. Thiel, L. Bolinger, H. Liu, L. DelaBarre, K. Ugurbil, Efficient high-frequency body coil for high-field MRI, Magnetic Resonance in Medicine 52 (4) (2004) 851–859.  doi:10.1002/mrm.20177.
  * [6] B. R. Steensma, M. Luttje, I. J. Voogt, D. W. Klomp, P. R. Luijten, C. A. van den Berg, A. J. Raaijmakers, Comparing signal-to-noise ratio for prostate imaging at 7T and 3T, Journal of Magnetic Resonance Imaging 49 (5) (2019) 1446–1455.  doi:10.1002/jmri.26527.
  * [7] A. T. Curtis, K. M. Gilbert, L. M. Klassen, J. S. Gati, R. S. Menon, Slice-by-slice B1+ shimming at 7 T, Magnetic Resonance in Medicine 68 (4) (2012) 1109–1116.  doi:10.1002/mrm.23319.
  * [8] G. Adriany, P. F. Van De Moortele, F. Wiesinger, S. Moeller, J. P. Strupp, P. Andersen, C. Snyder, X. Zhang, W. Chen, K. P. Pruessmann, P. Boesiger, T. Vaughan, K. Uǧurbil, Transmit and receive transmission line arrays for 7 tesla parallel imaging, Magnetic Resonance in Medicine 53 (2) (2005) 434–445.  doi:10.1002/mrm.20321.
  * [9] K. P. Pruessmann, M. Weiger, M. B. Scheidegger, P. Boesiger, SENSE: Sensitivity encoding for fast MRI, Magnetic Resonance in Medicine 42 (5) (1999) 952–962.  doi:10.1002/(SICI)1522-2594(199911)42:5<952::AID-MRM16>3.0.CO;2-S.
  * [10] M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V. Jellus, J. Wang, B. Kiefer, A. Haase, Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA), Magnetic Resonance in Medicine 47 (6) (2002) 1202–1210.  doi:10.1002/mrm.10171.
  * [11] T. Ruytenberg, A. Webb, I. Zivkovic, Shielded-coaxial-cable coils as receive and transceive array elements for 7T human MRI, Magnetic Resonance in Medicine 83 (3) (2020) 1135–1146.  doi:10.1002/mrm.27964.
  * [12] C. C. van Leeuwen, B. R. Steensma, D. W. Klomp, C. A. van den Berg, A. J. Raaijmakers, The Coax Dipole: A fully flexible coaxial cable dipole antenna with flattened current distribution for body imaging at 7 Tesla, Magnetic Resonance in Medicine 87 (1) (2022) 528–540.  doi:10.1002/mrm.28983.
  * [13] L. Nohava, R. Czerny, S. Roat, M. Obermann, A. Kuehne, R. Frass-Kriegl, J. Felblinger, J. C. Ginefri, E. Laistler, Flexible Multi-Turn Multi-Gap Coaxial RF Coils: Design Concept and Implementation for Magnetic Resonance Imaging at 3 and 7 Tesla, IEEE Transactions on Medical Imaging 40 (4) (2021) 1267–1278.  doi:10.1109/TMI.2021.3051390.
  * [14] A. Port, R. Luechinger, L. Albisetti, M. Varga, J. Marjanovic, J. Reber, D. O. Brunner, K. P. Pruessmann, Detector clothes for MRI: A wearable array receiver based on liquid metal in elastic tubes, Scientific Reports 10 (1) (2020) 1–10.  doi:10.1038/s41598-020-65634-5.
URL http://dx.doi.org/10.1038/s41598-020-65634-5

  * [15] S. Hosseinnezhadian, R. Frass-Kriegl, S. Goluch-Roat, M. Pichler, J. Sieg, M. Vit, M. Poirier-Quinot, L. Darrasse, E. Moser, J. C. Ginefri, E. Laistler, A flexible 12-channel transceiver array of transmission line resonators for 7 T MRI, Journal of Magnetic Resonance 296 (2018) 47–59.  doi:10.1016/j.jmr.2018.08.013.
URL https://doi.org/10.1016/j.jmr.2018.08.013

  * [16] J. A. De Zwart, P. J. Ledden, P. Kellman, P. Van Gelderen, J. H. Duyn, Design of a SENSE-optimized high-sensitivity MRI receive coil for brain imaging, Magnetic Resonance in Medicine 47 (6) (2002) 1218–1227.  doi:10.1002/mrm.10169.
  * [17] M. Weiger, K. P. Pruessmann, C. Leussler, P. Röschmann, P. Boesiger, Specific coil design for sense: A six-element cardiac array, Magnetic Resonance in Medicine 45 (3) (2001) 495–504.  doi:10.1002/1522-2594(200103)45:3<495::AID-MRM1065>3.0.CO;2-V.
  * [18] M. A. Ohliger, D. K. Sodickson, An introduction to coil array design for parallel MRI, NMR in Biomedicine 19 (3) (2006) 300–315.  doi:10.1002/nbm.1046.
  * [19] P. B. Roemer, W. A. Edelstein, C. E. Hayes, S. P. Souza, O. M. Mueller, The NMR phased array, Magnetic Resonance in Medicine 16 (2) (1990) 192–225.  doi:10.1002/mrm.1910160203.
  * [20] X. Zhang, A. Webb, Design of a capacitively decoupled transmit/receive NMR phased array for high field microscopy at 14.1 T, Journal of Magnetic Resonance 170 (1) (2004) 149–155.  doi:10.1016/j.jmr.2004.05.004.
  * [21] M. Bilgen, Inductively-overcoupled coil design for high resolution magnetic resonance imaging, BioMedical Engineering Online 5 (2006) 1–10.  doi:10.1186/1475-925X-5-3.
  * [22] R. F. Lee, R. O. Giaquinto, C. J. Hardy, Coupling and decoupling theory and its application to the MRI phased array, Magnetic Resonance in Medicine 48 (1) (2002) 203–213.  doi:10.1002/mrm.10186.
  * [23] G. Shajan, M. Kozlov, J. Hoffmann, R. Turner, K. Scheffler, R. Pohmann, A 16-channel dual-row transmit array in combination with a 31-element receive array for human brain imaging at 9.4 T, Magnetic Resonance in Medicine 71 (2) (2014) 870–879.  doi:10.1002/mrm.24726.
  * [24] N. I. Avdievich, J. W. Pan, H. P. Hetherington, Resonant inductive decoupling (RID) for transceiver arrays to compensate for both reactive and resistive components of the mutual impedance, NMR in Biomedicine 26 (11) (2013) 1547–1554.  doi:10.1002/nbm.2989.
  * [25] X. Yan, X. Zhang, B. Feng, C. Ma, L. Wei, R. Xue, 7T transmit/receive arrays using ICE decoupling for human head MR imaging, IEEE Transactions on Medical Imaging 33 (9) (2014) 1781–1787.  doi:10.1109/TMI.2014.2313879.
  * [26] T. Ruytenberg, A. Webb, I. Zivkovic, A flexible five-channel shielded-coaxial-cable (SCC) transceive neck coil for high-resolution carotid imaging at 7T, Magnetic Resonance in Medicine 84 (3) (2020) 1672–1677.  doi:10.1002/mrm.28215.
  * [27] J. A. Maravilla, K. Gopalan, A. C. Arias, M. Lustig, Abstract 0189: Transmission Line Receiver Coils (TLCs) for MRI, Proceedings of the 30th scientific meeting, International Society for Magnetic Resonance in medicine, London, p 0189 (2022).
  * [28] J. A. Maravilla, C. T. Knizek, N. Khairallah, A. C. Arias, M. Lustig, Abstract number 3733: 15-Channel Head Cap Array using Twisted-Pair Elements for MRI, Proceedings of the 31st scientific meeting, International Society for Magnetic Resonance in medicine, Toronto, p 3733 (2023).
  * [29] R. Czerny, R. Frass-kriegl, V. Cap, E. Laistler, L. Nohava, Abstract 3730: Twisted pair coils as flexible receive elements for 7 T - SNR and active detuning efficiency, Proceedings of the 31st scientific meeting, International Society for Magnetic Resonance in medicine, Toronto, p 3730 (2023).
  * [30] J. F. Shepherd, C. R. Johnson, Hexahedral mesh generation constraints, in: Engineering with Computers, Vol. 24, 2008, pp. 195–213.  doi:10.1007/s00366-008-0091-4.
  * [31] E. Shamonina, L. Solymar, V. Kalinin, On wireless power transfer between coils in the presence of radiation, Journal of Physics D: Applied Physics 54 (40) (2021) 405502.  doi:10.1088/1361-6463/ac0c4f.
URL https://iopscience.iop.org/article/10.1088/1361-6463/ac0c4f

  * [32] D. M. Pozar, Microwave Engineering, 4th Edition, 4th Edition, Wiley, Hoboken, NJ, 2011.
  * [33] K. Nehrke, P. Börnert, DREAM-a novel approach for robust, ultrafast, multislice B1 mapping, Magnetic Resonance in Medicine 68 (5) (2012) 1517–1526.  doi:10.1002/mrm.24158.
  * [34] F. Eggenschwiler, T. Kober, A. W. Magill, R. Gruetter, J. P. Marques, SA2RAGE: A new sequence for fast B1+-mapping, Magnetic Resonance in Medicine 67 (6) (2012) 1609–1619.  doi:10.1002/mrm.23145.
  * [35] S. S. Vasanawala, R. Stormont, S. Lindsay, T. Grafendorfer, J. Y. Cheng, J. M. Pauly, G. Scott, J. X. Guzman, V. Taracila, D. Chirayath, F. Robb, Development and Clinical Implementation of Next Generation Very Light Weight and Extremely Flexible Receiver Arrays for Pediatric MRI, Pre-print (2017).
URL http://arxiv.org/abs/1705.00224

  * [36] K. P. McGee, R. S. Stormont, S. A. Lindsay, V. Taracila, D. Savitskij, F. Robb, R. J. Witte, T. J. Kaufmann, J. Huston, S. J. Riederer, E. A. Borisch, P. J. Rossman, Characterization and evaluation of a flexible MRI receive coil array for radiation therapy MR treatment planning using highly decoupled RF circuits, Physics in Medicine and Biology 63 (8) (2018).  doi:10.1088/1361-6560/aab691.
  * [37] B. D. Collick, B. Behzadnezhad, S. A. Hurley, N. K. Mathew, N. Behdad, S. A. Lindsay, F. Robb, R. S. Stormont, A. B. McMillan, Rapid development of application-specific flexible MRI receive coils, Physics in Medicine and Biology 65 (19) (2020) 0–7.  doi:10.1088/1361-6560/abaffb.

Figure 1: (a) Illustration of twisted pair cable. (b) Forming of the twisted
pair coil. A gap is introduced in the grey shield wire at the top to un-shield
the magnetic field. In the red signal wire a gap is introduced at the feeding
point (bottom). Figure 2: Simulated metrics of twisted pair coils across
varying twisting densities. (a) Denoting number of twists per coil and its
influence on the impedance of the coil, shown with reflection coefficients of
the coil on the Smith chart. (b) Magnitude of B${}_{1}^{+}$, in
$\mu\text{T}/\sqrt{\text{W}}$, measured at 5 cm depth in the phantom at coil
center. (c) Maximum SAR10g value, in W/kg, on the phantom. Both B${}_{1}^{+}$
and maxSAR10g are normalised to 1 W accepted power. d) Simulated (top) and
measured (bottom) B${}_{1}^{+}$ field maps for 24 and 64 twists per length.
(e) SAR efficiency: B${}_{1}^{+}$ magnitude at 5cm depth divided by the square
root of peak SAR, in $\mu\text{T}/\sqrt{\text{W/kg}}$. (f) Coupling
coefficient ($S_{21}$), in dB, between two coils, spaced 5 mm apart. Figure 3:
Schematics and photos of (a) twisted pair loop coil and (b) conventional loop
coil. (c) Homogeneous cubic phantom used in simulations and bench
measurements. The coil was placed 2 cm away from the phantom. Figure 4:
Comparison between the twisted pair, with 64 twists per unit length, and
conventional coil, for different degrees of elongation (100$\times$100,
80$\times$117.4, 60$\times$130.3, all in mm). All results are normalised to 1
W accepted power. (a) B${}_{1}^{+}$ field patterns, in
$\mu\text{T}/\sqrt{\text{W}}$, on the sagittal plane. The white dashed line
shows the central axis of the phantom. (b) Coronal view SAR10g maps, in W/kg,
accompanied by the corresponding maxSAR10g value. (c) Simulated (top) and
measured (bottom) B${}_{1}^{+}$ field maps on a spherical phantom for varying
degrees of elongation. The simulated results take 25% loss of the experiment
into account, while the measurements were normalised to the input power. (d)
SAR efficiency: B${}_{1}^{+}$ divided by the square root of peak SAR, in
$\mu\text{T}/\sqrt{\text{W/kg}}$, along the central axis of the phantom.
Figure 5: Measured coupling coefficients ($S_{21}$), in dB, of two coils for
varying overlapping geometries. (a) Graphical representation of the different
coupling orientations. (b) Coupling results of two loops, twisted pair and
conventional, next to each other for varying degrees of overlap $d$, in mm.
(c) Coupling results for different orientations of the twisted pair loops, as
indicated in (a). Figure 6: Streamline representation of electric ($\bm{E}$)
and magnetic ($\bm{H}$) fields and power flow (Poynting vector $\bm{S}$) of
individual conventional and twisted pair coils in air (no phantom present).
Figure 7: Streamline representation of electric ($\bm{E}$) and magnetic
($\bm{H}$) fields and power flow (Poynting vector $\bm{S}$) of individual
conventional and twisted pair coils in the presence of the phantom. Figure 8:
Streamline representation of electric ($\bm{E}$) and magnetic ($\bm{H}$)
fields, surface current density ($\bm{J}$) and power flow (Poynting vector
$\bm{S}$) of two conventional and two twisted pair coils in the presence of
phantom for distances of 5 cm and 5 mm from each other. Figure 9: Measured
$S$-matrices, in dB, for multiple twisted pair coil coupling layouts: (a)
Eight-channel circular overlap; (b) Eight-channel elongated overlap; (c) Five
channel elongated array. Figure 10: Different metrics for two realised spline
shapes, (a) the Saddle and (b) the Triangle. (c) Measured return loss
($S_{11}$), in dB, for both splines and a circular twisted pair coil. (d)
Coupling ($S_{21}$) between the splines for varying degrees of overlap. (e)
Simulated B1+ maps, in $\mu\text{T}/\sqrt{\text{W}}$, and (f) SAR maps, in
W/kg, on the coronal plane with its corresponding peak SAR value. Simulated
results were normalised to 1 W accepted power

##  Appendix A Working of the twisted pair

Figure 11: Illustration of the current flow on the twisted pair coil. A
current is excited on the red signal wire counterclockwise (CCW), resulting in
a clockwise (CW) induced current on the grey shield wire. At the gap of the
grey shield wire a high potential difference creates a strong electric field,
which induces a current on the red signal wire flowing in the opposite
direction as the original current. As a result of this induced current, there
is a partial cancellation of the current in the red signal wire. Consequently,
the grey shield wire becomes the predominant carrier of current, as
illustrated in the combined wires figure on the right (current flowing CW).

##  Appendix B Simulated robustness of the twisted pair

Figure 12: Simulated return loss ($S_{11}$) across varying numbers of twists
per unit length. The circular coil (10 cm diameter) underwent tuning and
matching, and the same circuitry was applied to the elongated coil
(dimensions: width = 6 cm, height = 13 cm) to assess coil robustness.

##  Appendix C Measured robustness of the twisted pair

Figure 13: Return loss ($S_{11}$) measurements. All coils were initially tuned
and matched in the circular geometry and subsequently subjected to deformation
using identical circuitry. (a) & (b) display $S_{11}$ for varying degrees of
elongation for the twisted pair (64 twists per unit length) and the
conventional coil, respectively. Panel (c) illustrates the $S_{11}$ of two
spline shapes, namely saddle and triangular, with the circular geometry
serving as the reference.

##  Appendix D Influence coil distance to the phantom

Figure 14: Influence on the simulated B${}_{1}^{+}$ for the distance of the
coil to the phantom. (a) The B${}_{1}^{+}$ profiles are plotted along the
central axis of the phantom for the conventional, dashed line, and the twisted
pair, full line. Both coils were placed at 20, 40 and 60 mm. (b) Simulated
magnitude values at the surface of the phantom (0 mm) and 50 mm depth.


# Constraining vector dark matter and dark photon with degenerate mass in a
hidden local $SU(2)$ model

Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Xinran Xu<EMAIL_ADDRESS>College of Physics,
Sichuan University, Chengdu 610065, China

###### Abstract

We discuss degenerate vector dark matter and dark photon that are induced from
hidden $SU(2)_{H}$ gauge sector where it is spontaneously broken by vacuum
expectation value of $SU(2)_{H}$ doublet. Kinetic mixing between $SU(2)_{H}$
and $U(1)_{Y}$ gauge fields can be generated by introducing dimension six
operator realizing dark photon interactions. In estimating relic density we
focus on the process in which dark matter annihilate into dark photons, and
search for the region of dark matter mass and gauge coupling realizing
observed relic density. We then discuss constraints from dark photon physics,
thermalization of dark sector and direct detection of dark matter. It is then
found that constraints from direct detection experiments give us the strongest
upper limits on the dark photon interactions.

##  I Introduction

Understanding of dark matter (DM) nature is one of the biggest challenge in
physics. The existence of DM definitely requires physics beyond the standard
model (SM) where we call new physics sector containing DM as dark sector. In
fact various candidates of DM are considered from new physics models. One
promising candidate of DM is a massive stable particle that weakly interacts
with SM sector as its relic density can be explained via thermal production.
Such a DM has been searched for by direct detection, indirect detection and
collider experiments. However we do not have any clear evidence of DM yet and
nature of dark sector is an open question.

One attractive scenario is to consider dark sector as hidden gauge sector as
the SM is also described by gauge symmetry. We consider hidden gauge symmetry
where SM particles are neutral but fields in dark sector including DM are
charged under it. In this kind of scenario we can explain stability of DM by a
remnant symmetry after spontaneous symmetry breaking of hidden gauge symmetry
Krauss:1988zc ; Ko:2018qxz . In particular it is interesting to consider
minimal non-Abelian group $SU(2)_{H}$ as hidden gauge symmetry. In the minimal
scenario Hambye:2008bq , $SU(2)_{H}$ is broken by vacuum expectation value
(VEV) of $SU(2)_{H}$ doublet scalar $\Phi$ and dark gauge bosons are DM
candidate where the stability of them is guaranteed by remaining custodial
symmetry 111There are various approaches applying a hidden $SU(2)$ gauge
symmetry in literatures; A remaining $Z_{2,3,4}$ symmetry with a
quadruplet(quintet) scalar in ref. Chiang:2013kqa ; Chen:2015nea ;
Chen:2015dea ; Chen:2015cqa ; Ko:2020qlt ; Nomura:2020zlm ; Chen:2017tva ,
$Z_{2}\times Z^{\prime}_{2}$ symmetry Gross:2015cwa , a custodial symmetry in
refs. Boehm:2014bia ; Hambye:2008bq ; Baouche:2021wwa , an unbroken $U(1)$
from $SU(2)$ in refs. Baek:2013dwa ; Khoze:2014woa ; Daido:2019tbm , a model
adding hidden $U(1)_{h}$ Davoudiasl:2013jma ; Choi:2019zeb , other DM
scenarios Zhang:2009dd ; Barman:2017yzr ; Barman:2018esi ; Barman:2019lvm ;
Barman:2020ifq , a model with classical scale invariance Karam:2015jta ,
Baryogengesis Hall:2019ank , connection to neutrino mass generation
Nomura:2021aep , electroweak phase transition Ghosh:2020ipy  and some general
discussion Yuan:2024tfs .. Interestingly, we can have kinetic mixing between
$SU(2)_{H}$ and $U(1)_{Y}$ gauge fields if we introduce dimension 6 operator
of $(\Phi^{\dagger}\sigma^{a}\Phi)X^{a}_{\mu\nu}B^{\mu\nu}$ where $X_{\mu\nu}$
and $B_{\mu\nu}$ are the gauge field strength of $SU(2)_{H}$ and $U(1)_{Y}$
respectively and $\sigma^{a}$ is the Pauli matrix on $SU(2)_{H}$
representation space Zhou:2022pom  222A mixing between $SU(2)_{L}$ and hidden
$U(1)$ is also discussed in ref. Arguelles:2016ney ; Dimension 5 operator is
also possible introducing $SU(2)_{H}$ triplet scalar Zhang:2009dd ;
Nomura:2020zlm ; Nomura:2021tmi ; Ko:2020qlt . With such a mixing, we obtain
both a complex vector DM $X^{\pm}$ and a dark photon $Z^{\prime}$
Holdom:1985ag ; Fabbrichesi:2020wbt  from the $SU(2)_{H}$ gauge sector where
the stability of DM is guaranteed by remnant $U(1)$ symmetry from custodial
one. In previous works Hambye:2008bq ; Zhou:2022pom  for a model with minimal
field contents, DM-scalar interactions are mainly considered in explaining DM
relic density where the new scalar boson plays a role of mediator between the
SM and dark sectors via mixing with Higgs boson, but it is also possible to
obtain observed relic density via $X^{+}X^{-}\to Z^{\prime}Z^{\prime}$
process; Although masses of DM and $Z^{\prime}$ are degenerate the process is
possible in early universe DAgnolo:2015ujb . In this case degenerate dark
gauge bosons, DM and $Z^{\prime}$, play a dominant role in dark matter physics
and we would get more stringent constraint in gauge sector including kinetic
mixing parameter. Thus it is worth exploring this scenario to constrain gauge
coupling, DM/Z’ mass and kinetic mixing parameter.

In this work, we discuss $SU(2)_{H}$ gauge sector introducing only one
$SU(2)_{H}$ doublet scalar and the effective operator providing $SU(2)_{H}$
and $U(1)_{Y}$ kinetic mixing. We then estimate relic density of DM $X^{\pm}$
focusing on $X^{+}X^{-}\to Z^{\prime}Z^{\prime}$ process and search for
parameter region explaining observation. In addition, we discuss constraints
on the kinetic mixing from dark photon physics and from DM direct detection by
calculating DM-nucleon scattering cross section.

This paper is organized as follows. In Sec. II, we introduce the model and
formulate mass eigenvalues and eigenstate in scalar/gauge secotor, and
relevant interactions. In Sec. III, we discuss relic density of DM, constraint
from dark photon physics and direct detection of DM. Finally we provide a
conclusion in Sec.IV.

##  II A model

We consider a model with hidden sector described by hidden $SU(2)_{H}$ gauge
symmetry where we introduce $SU(2)_{H}$ doublet scalar field $\Phi$ to break
the symmetry spontaneously. The new scalar field $\Phi$ will develop a vacuum
expectation value(VEV) $v_{\Phi}$ and without the loss of generality it can be
written by

$\Phi=\begin{pmatrix}G^{+}_{H}\\\
\frac{1}{\sqrt{2}}(v_{\Phi}+\phi+iG_{H})\end{pmatrix},$ (1)

where $G^{\pm}_{H}$ and $G_{H}$ correspond to massless Nambu-Goldstone(NG)
boson that are absorbed by $SU(2)_{H}$ gauge bosons. In scalar sector we also
have the SM Higgs field $H$ written by

$H=\begin{pmatrix}w^{+}\\\
\frac{1}{\sqrt{2}}(v+\tilde{h}+iG_{Z})\end{pmatrix},$ (2)

where $v$ is the VEV of $H$ and $\\{w^{\pm},G_{Z}\\}$ correspond to NG boson
absorbed by the SM massive gauge bosons $\\{W^{\pm},Z\\}$. Fermion sector is
considered to be the same as the SM and we do not focus on fermions in this
work.

The renormalizable Lagrangian in the model is

$\mathcal{L}=\mathcal{L}_{\rm
SM}+(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)-\frac{1}{4}X^{a}_{\mu\nu}X^{a\mu\nu}-\mathcal{V}$
(3)

where $X^{a}_{\mu\nu}(a=1,2,3)$ is the gauge field strength for $SU(2)_{H}$,
$\mathcal{L}_{SM}$ is the Lagrangian of the SM without scalar potential, and
$\mathcal{V}$ is the scalar potential including the SM Higgs field $H$.
Explicit form of the scalar potential is given by

$\mathcal{V}=\mu_{H}^{2}(H^{\dagger}H)+\mu_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{H}(H^{\dagger}H)^{2}+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{H\Phi}(H^{\dagger}H)(\Phi^{\dagger}\Phi).$
(4)

The covariant derivative of $\Phi$ is written by

$D_{\mu}\Phi=\left(\partial_{\mu}+ig_{H}\frac{\sigma^{a}}{2}X^{a}_{\mu}\right)\Phi$
(5)

where $g_{H}$ is gauge coupling associated with $SU(2)_{H}$ and $\sigma^{a}$
is the Pauli matrix acting on representation space of $SU(2)_{H}$. In addition
to the renormalizable Lagrangian we introduce a gauge invariant effective
interaction of

$\mathcal{L}_{\rm
eff}=\frac{1}{\Lambda^{2}}(\Phi^{\dagger}\sigma^{a}\Phi)X^{a}_{\mu\nu}B^{\mu\nu}$
(6)

where $B_{\mu\nu}$ is the gauge field strength of $U(1)_{Y}$ and $\Lambda$
represents a high energy scale 333Dimension 8 operator of
$(\Phi^{\dagger}\sigma^{a}\Phi)X^{a}_{\mu\nu}(H^{\dagger}\sigma^{b}H)W^{b}_{\mu\nu}$
is also possible that induces mixing between $SU(2)_{H}$ and $SU(2)_{L}$
Zhou:2022pom . Here we do not consider the operator considering it is more
suppressed by $1/\Lambda^{4}$ factor. After $\Phi$ developing its VEV the
effective Lagrangian provides kinetic mixing term such that

$\displaystyle\mathcal{L}_{\rm
eff}\supset-\frac{v_{\Phi}^{2}}{4\Lambda^{2}}X^{3}_{\mu\nu}B^{\mu\nu}$
$\displaystyle\equiv-\frac{\epsilon}{2}X^{3}_{\mu\nu}B^{\mu\nu}$
$\displaystyle=\frac{\epsilon}{2}(\partial_{\mu}X^{3}_{\nu}-\partial_{\nu}X^{3}_{\mu}+g_{H}(X^{1}_{\mu}X^{2}_{\nu}-X^{1}_{\nu}X^{2}_{\mu}))B^{\mu\nu},$
(7)

indicating mixing between third component of $SU(2)_{H}$ gauge field
$X^{3}_{\mu}$ and $U(1)_{Y}$ gauge field $B_{\mu}$. Kinetic terms of
$X^{3}_{\mu}$ and $B_{\mu}$ can be diagonalized by transformation

$\left(\begin{array}[]{c}X^{3}_{\mu}\\\ B_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}r&0\\\ -\epsilon r&1\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{Z}^{\prime}_{\mu}\\\
\tilde{B}_{\mu}\\\ \end{array}\right),$ (8)

where $r=\frac{1}{\sqrt{1-\epsilon^{2}}}$. We then obtain Lagrangian for gauge
sector as

$\mathcal{L}_{G}=-\frac{1}{4}X^{i}_{\mu\nu}X^{i\mu\nu}-\frac{1}{4}\tilde{Z}^{\prime\mu\nu}\tilde{Z}^{\prime}_{\mu\nu}-\frac{1}{4}W^{\alpha\mu\nu}W^{\alpha}_{\mu\nu}-\frac{1}{4}\tilde{B}^{\mu\nu}\tilde{B}_{\mu\nu},$
(9)

where $i=1,2$ and $W^{\alpha}_{\mu\nu}$ is the gauge field strength of
$SU(2)_{L}$ gauge field $W^{\alpha}_{\mu}(\alpha=1-3)$.

###  II.1 Scalar sector

Here we consider scalar sector in the model and derive mass eigenvalues with
corresponding eigenstates. The VEVs $v$ and $v_{\Phi}$ can be obtained from
the scalar potential Eq. (4) requiring stationary conditions $\partial
V/\partial v=\partial V/\partial v_{\Phi}=0$ that provide

$\displaystyle\mu_{H}^{2}+\lambda_{H}v^{2}+\frac{1}{2}\lambda_{H\Phi}v_{\Phi}^{2}=0,$
(10)
$\displaystyle\mu_{\Phi}^{2}+\lambda_{\Phi}v_{\Phi}^{2}+\frac{1}{2}\lambda_{H\Phi}v^{2}=0.$
(11)

Here we require $\mu_{H}^{2},\mu_{\Phi}^{2}<0$ to make square of VEVs positive
definite.

After spontaneous symmetry breaking, we obtain mass matrix for CP-even neutral
scalar as

$\mathcal{L}\supset\frac{1}{2}\begin{pmatrix}\tilde{h}\\\
\phi\end{pmatrix}^{T}\begin{pmatrix}2\lambda_{H}v^{2}&\lambda_{H\Phi}vv_{\varphi}\\\
\lambda_{H\Phi}vv_{\Phi}&2\lambda_{\Phi}v_{\Phi}^{2}\end{pmatrix}.\begin{pmatrix}\tilde{h}\\\
\phi\end{pmatrix}$ (12)

We obtain mass eigenvalues by diagonalizing the mass matrix such that

$\displaystyle
m_{h,H}^{2}=\lambda_{H}v^{2}+\lambda_{\Phi}v_{\Phi}^{2}\pm\sqrt{(\lambda_{H}v^{2}-\lambda_{\Phi}v_{\varphi}^{2})^{2}+\lambda_{H\Phi}^{2}v^{2}v_{\Phi}^{2}},$
(13)

where we identify $m_{h}=125$ GeV is the mass of SM Higgs boson. The mass
eigenstates and relevant mixing angle are obtained as

$\displaystyle\begin{pmatrix}\tilde{h}\\\
\phi\end{pmatrix}=\begin{pmatrix}\cos\beta&-\sin\beta\\\
\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}h\\\ H\end{pmatrix},$
$\displaystyle\tan
2\beta=\frac{vv_{\Phi}\lambda_{H\Phi}}{v^{2}\lambda_{H}-v^{2}_{\Phi}\lambda_{\Phi}},$
(14)

where $h$ is identified as the SM Higgs boson. In the analysis of this paper
we assume scalar mixing is negligibly small choosing tiny $\lambda_{H\Phi}$
and focus on the effect of gauge interactions.

###  II.2 Gauge boson masses and interactions

After spontaneous breaking of electroweak and hidden gauge symmetries, we
obtain mass terms of electrically neutral gauge fields including the SM ones
as follows

$\displaystyle\mathcal{L}_{M}=m_{X}^{2}X^{+\mu}X^{-}_{\mu}+\frac{1}{2}m^{2}_{\tilde{Z}^{\prime}}\tilde{Z}^{\prime\mu}\tilde{Z}^{\prime}_{\mu}+\frac{1}{2}m^{2}_{Z_{\rm
SM}}\tilde{Z}^{\mu}\tilde{Z}_{\mu}+\Delta
M^{2}\tilde{Z}^{\mu}\tilde{Z}^{\prime}_{\mu},$ (15)

where $X_{\mu}^{\pm}=\frac{1}{\sqrt{2}}(X^{1}_{\mu}\mp iX^{2}_{\mu})$ and
$\tilde{Z}_{\mu}=\cos\theta_{W}W^{3}_{\mu}-\sin\theta_{W}\tilde{B}_{\mu}$ with
$\theta_{W}$ being the Weinberg angle. Squared mass parameters are given by

$\displaystyle m^{2}_{Z_{\rm SM}}=\frac{g_{Z}^{2}v^{2}}{4},\quad
m_{X}^{2}=\frac{g_{H}^{2}v_{\Phi}^{2}}{4},\quad
m_{\tilde{Z}^{\prime}}^{2}=r^{2}(m_{X}^{2}+\epsilon^{2}m^{2}_{Z_{\rm
SM}}\sin^{2}\theta_{W}),$ $\displaystyle\Delta M^{2}=\epsilon
r\sin\theta_{W}m^{2}_{Z_{\rm SM}}.$ (16)

Thus $\tilde{Z}_{\mu}$ and $\tilde{Z}^{\prime}_{\mu}$ fields mix as a
consequence of kinetic mixing effect while $X^{\pm}_{\mu}$ is mass eigenstate
with mass $m_{X}$. We obtain mass eigenvalues by diagonalizing mass matrix for
$\tilde{Z}$ and $\tilde{Z}^{\prime}$ as

$m^{2}_{Z,Z^{\prime}}=\frac{1}{2}(m^{2}_{Z_{\rm
SM}}+m_{\tilde{Z}^{\prime}}^{2})\pm\frac{1}{2}\sqrt{(m^{2}_{Z_{\rm
SM}}-m_{\tilde{Z}^{\prime}}^{2})^{2}+4\Delta M^{4}}.$ (17)

The mass eigenstates are given by

$\displaystyle\begin{pmatrix}Z\\\
Z^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\chi&\sin\chi\\\
-\sin\chi&\cos\chi\end{pmatrix}\begin{pmatrix}\tilde{Z}\\\
\tilde{Z}^{\prime}\end{pmatrix},$ (18) $\displaystyle\tan
2\chi=\frac{2\epsilon\sin\theta_{W}m^{2}_{Z_{\rm SM}}}{m^{2}_{Z_{\rm
SM}}-m^{2}_{\tilde{Z}}}.$ (19)

In our analysis we consider $\chi$ to be tiny by choosing $\epsilon$
sufficiently small as the $Z$-$Z^{\prime}$ mixing is strongly constrained by
the electroweak precision tests. We thus take
$\tan\chi\simeq\sin\chi\simeq\chi$ henceforth. In addition, gauge boson masses
can be approximated as $m_{Z}\simeq m_{Z_{\rm SM}}$ and $m_{Z^{\prime}}\simeq
m_{X}$ for small $\epsilon$.

The gauge interactions among $X^{\pm}$ and $Z^{\prime}$ can be obtained from
Lagrangian Eq. (9) as

$\displaystyle\mathcal{L}_{X^{\pm}Z^{\prime}}=$ $\displaystyle\
ig_{H}\Bigl{[}(\partial_{\mu}X^{+}_{\nu}-\partial_{\nu}X^{+}_{\mu})X^{-\mu}Z^{\prime\nu}-(\partial_{\mu}X^{-}_{\nu}-\partial_{\nu}X^{-}_{\mu})X^{+\mu}Z^{\prime\nu}$
$\displaystyle\qquad+\frac{1}{2}(\partial_{\mu}Z^{\prime}_{\nu}-\partial_{\nu}Z^{\prime}_{\mu})(X^{+\mu}X^{-\nu}-X^{-\mu}X^{+\nu})\Bigr{]}$
$\displaystyle-
g_{H}^{2}[X^{+}_{\mu}X^{-\mu}Z^{\prime}_{\nu}Z^{\prime\nu}-X^{+}_{\mu}X^{-}_{\nu}Z^{\prime\mu}Z^{\prime\nu}],$
(20)

where we adopted approximation of $r\simeq 1$ and $\cos\chi\simeq 1$. In
addition $Z^{\prime}$ can interact with the SM particles through
$Z$-$Z^{\prime}$ mixing. The interactions between $Z^{\prime}$ and SM fermions
$f$ can be written by

$\mathcal{L}_{Z^{\prime}ff}=\frac{g}{\cos\theta_{W}}Z^{\prime}_{\mu}\bar{f}\gamma^{\mu}[-\sin\chi(T_{3}-Q_{f}\sin^{2}\theta_{W})-\cos\chi\epsilon
Y_{f}\sin\theta_{W}]f,$ (21)

where $T_{3}$ is diagonal generator of $SU(2)_{L}$, $Q_{f}$ is the electric
charge and $Y$ is hypercharge. In the model $Z^{\prime}$ decays into SM
particles while $X^{\pm}$ is stable and can be DM candidate. In addition, we
have DM-photon and DM-Z interactions from effective term Eq. (6) such that

$\mathcal{L}_{\rm eff}\supset-
ig_{H}\epsilon\cos\theta_{W}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})X^{+}_{\mu}X^{-}_{\nu}-ig_{H}\epsilon\sin\theta_{W}(\partial^{\mu}Z^{\nu}-\partial^{\nu}Z^{\mu})X^{+}_{\mu}X^{-}_{\nu},$
(22)

where $A_{\mu}$ is photon field and we ignore $Z$-$Z^{\prime}$ mixing effect
since it is negligibly small. We notice that coupling $g_{H}$ appears instead
of electromagnetic coupling $e$ for DM-photon interaction.

##  III Dark matter phenomenology

In this section we consider phenomenology of DM such as its relic density and
DM-nucleon scattering cross section for direct detection.

###  III.1 Relic density

Relic density of DM is estimated by solving the Boltzmann equation for number
density $n_{X}$ of $X^{\pm}$,

$\dot{n}_{X}+3Hn_{X}=\langle\sigma v\rangle(n_{X_{\rm eq}}^{2}-n_{X}^{2}),$
(23)

where $\langle\sigma\rangle$ is thermal average of DM annihilation cross
section, $H$ is Hubble parameter, and $n_{X_{eq}}$ is number density of
$X^{\pm}$ in thermal equilibrium. Here we adopt micrOMEGAs code
Belanger:2014vza  to estimate relic density by implementing relevant
interactions in the model. In estimating relic density, the relevant
parameters in the model are DM(Z’) mass $m_{X}$, new gauge coupling $g_{H}$
and scalar mass $m_{\phi}$. In this work we consider $\phi$ is heaver than
$X^{\pm}$ and $Z^{\prime}$ so that a DM annihilation mode including $\phi$ in
final states is forbidden kinematically. Thus the DM annihilation process in
our scenario is solely

$X^{+}X^{-}\to Z^{\prime}Z^{\prime}.$ (24)

Here we fix $m_{\phi}=1.5m_{X}$ for simplicity and search for parameter region
on $\\{m_{X},g_{H}\\}$ plane that can provide observed DM relic density. Here
we consider two mass range of DM (dark photon) as follows

$\displaystyle{\rm(i)}\ m_{X}=[1.0,60]\ {\rm GeV}$ (25)
$\displaystyle{\rm(ii)}\ m_{X}=[150,3000]\ {\rm GeV}.$ (26)

Here the range (i) is light WIMP like DM case where $Z^{\prime}$ is also
lighter than $Z$ and more dark photon like. The range (ii) corresponds to
heavy WIMP DM case and $Z^{\prime}$ is heavier than $Z$. We do not consider
the region $m_{X}\sim m_{Z}$ since $Z$-$Z^{\prime}$ mixing tends to be large.

In Fig. 1, we show the region providing $\Omega h^{2}<0.12$, $\Omega
h^{2}\simeq 0.12$ and $\Omega h^{2}>0.12$ on $\\{m_{X},g_{H}\\}$ plane. The
red dashed curve indicate the parameters explaining observed relic density.
The shaded region is excluded since relic density is over abundant while the
white region provides smaller relic density. We find that observed relic
density can be obtained by gauge coupling $g_{H}\in[0.01,1]$ for DM mass of 1
GeV to 3000 GeV.

Figure 1: The red dashed curve indicate parameter $\\{m_{X},g_{H}\\}$
realizing observed relic density $\Omega h^{2}\simeq 0.12$. Shaded region is
excluded since DM is over abundant.

###  III.2 Constraints on kinetic mixing

Here we summarize constraints on kinetic mixing parameter $\epsilon$ from dark
photon searches, thermalization condition of DM and mili-charged DM
interaction.
Constraints from dark photon search:
In the mass range (i) of our interest $\epsilon$ is constrained by BaBar
BaBar:2014zli ; BaBar:2017tiz  and LHCb LHCb:2019vmc  experiments that search
for dark photon. The BarBar experiment search for the process
$e^{+}e^{-}\to\gamma Z^{\prime}$ where $Z^{\prime}$ is considered to decay
into visible modes $\\{e^{+}e^{-},\mu^{+}\mu^{-},\text{light mesons}\\}$ or
invisible mode such as neutrinos and DM states. The LHCb experiment search for
the dark photon which is produced in proton-proton collision and decays into
$\mu^{+}\mu^{-}$ pair. We apply constraint from these experiments for the
allowed range of $\epsilon$ where we use the DARKCAST code to extract the
limit Ilten:2018crw .

DM thermalization condition:
We require particles in dark sector, DM and dark photon, are in thermal
equilibrium before freeze-out of DM to guarantee our relic density
calculation. In addition, dark photon lifetime should be smaller than
$\mathcal{O}(1)$s to avoid inconsistency with Big Bang Nucleosynthesis. In the
model, dark sector is thermalized in the early Universe via dark photon decays
into electrons. Then we impose the condition where the decay rate is larger
than the Hubble rate at freeze-out. This condition can be written by
Pospelov:2007mp ; delaVega:2023dmw

$|\epsilon|\gtrsim 2.0\times 10^{-8}\sqrt{\frac{10\ {\rm
GeV}}{m_{Z^{\prime}}}}\left(\frac{m_{X}}{10\ {\rm GeV}}\right).$ (27)

We adopt the condition as a lower bound of $\epsilon$ in the analysis below.

Constraint from mili-charged DM interaction:
Unlike to the Abelian hidden $U(1)$ case we have DM-photon interaction Eq.
(22) which arise from the effective interaction Eq. (6). We thus comment on
constraint from DM-Baryon scattering for milli-charged DM Xu:2018efh  through
the interaction. In this case the cross section is
$\sigma_{X}=\sigma_{X0}v^{2}$ where $v$ is DM-Baryon relative velocity. It is
suppressed by $v^{2}$ factor compared to milli-charged fermion DM case
$(\sigma_{F}=\sigma_{F0}v^{4})$ that is due to momentum factor appearing from
derivative in the interaction Eq. (6). Therefore the constraint is not severe
and DM direct detection experiment provide stronger constraint that is
estimated below.

###  III.3 Direct detection

In the model DM $X^{\pm}$ interacts with nucleon via $Z^{\prime}$ exchanging
process through interactions in Eqs. (II.2) and (21). In addition $X^{\pm}$
have interaction with the SM $Z$ boson via $Z$-$Z^{\prime}$ mixing where
relevant interaction for DM-nucleon scattering is

$\displaystyle\mathcal{L}_{ZX^{+}X^{-}}=$ $\displaystyle\
ig_{H}\sin\chi\Bigl{[}(\partial_{\mu}X^{+}_{\nu}-\partial_{\nu}X^{+}_{\mu})X^{-\mu}Z^{\nu}-(\partial_{\mu}X^{-}_{\nu}-\partial_{\nu}X^{-}_{\mu})X^{+\mu}Z^{\nu}$
$\displaystyle\qquad+\frac{1}{2}\left(1-\frac{\epsilon}{\sin\chi}\right)(\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu})(X^{+\mu}X^{-\nu}-X^{-\mu}X^{+\nu})\Bigr{]},$
(28)

where we included Eq. (22) for the third term inside the bracket. Thus we need
to take into account both $Z^{\prime}$ and $Z$ boson exchanging diagrams to
obtain DM-nucleon scattering cross section. Here vector type interaction among
$Z(Z^{\prime})$ and nucleon can be obtained as

$\displaystyle\mathcal{L}_{N}=\sum_{N=p,n}\frac{g}{\cos\theta_{W}}\bar{N}(C_{ZNN}^{V}Z_{\mu}+C_{Z^{\prime}NN}^{V}Z^{\prime}_{\mu})\gamma^{\mu}N,$
(29) $\displaystyle C_{Zpp}^{V}=\frac{1}{4}-\sin^{2}\theta_{W},\quad
C_{Znn}^{V}=-\frac{1}{4},$ (30) $\displaystyle
C_{Z^{\prime}pp}^{V}=-\sin\chi\left(\frac{1}{4}-\sin^{2}\theta_{W}\right)-\frac{1}{3}\epsilon\sin\theta_{W},\quad
C_{Z^{\prime}nn}^{V}=\frac{1}{4}\sin\chi-\frac{1}{2}\epsilon\sin\theta_{W},$
(31)

where $p$ and $n$ respectively indicate proton and neutron, and we approximate
$\cos\chi\simeq 1$. We then calculate diagrams and obtain DM-nucleon
scattering cross section in non-relativistic limit such that

$\sigma_{NX}=\frac{2g_{H}^{2}g^{2}}{\pi\cos^{2}\theta_{W}}\left(\frac{m_{X}m_{N}}{m_{X}+m_{N}}\right)^{2}\left(\frac{\sin\chi
C^{V}_{ZNN}}{m_{Z}^{2}}+\frac{\cos\chi
C^{V}_{Z^{\prime}NN}}{m_{Z^{\prime}}^{2}}\right)^{2},$ (32)

where $m_{N}$ denote nucleon mass $(N=p,n)$. Here we consider average of DM-
nucleon scattering cross section $(\sigma_{nX}+\sigma_{pX})/2$ and compare it
with experimental constraints to estimate upper limit of $\epsilon$. In
estimating the cross section we adopt $g_{H}$ value explaining relic density
$\Omega h^{2}\simeq 0.12$ in Fig. 1 for each DM (dark photon) mass.

Figure 2: The constraints on $\epsilon$. The colored region above curves are
excluded by direct detection experiments indicated by labels. In the gray
region labeled as ”Not Thermalized”, $\epsilon$ is too small so that DM and
dark photon are not thermalized before freeze out. On the left plot, region
labeled as LHCb and BarBar are excluded by dark photon searches by these
experiments. The $\epsilon$ value below blue dotted curve provides DM-nucleon
cross section in neutrino floor.

In Fig. 2, we show constraints on $\epsilon$ on $\\{m_{X},\epsilon\\}$ plane.
The colored region above curves are excluded by direct detection experiments
indicated by labels where the strongest limits are from LZ LZ:2022lsv ,
Xenon1T (2020) XENON:2019gfn  and DarkSide-50 DarkSide-50:2022qzh  for mass
range $m_{X}\gtrsim 10$ GeV, 3 GeV $\lesssim m_{X}\lesssim$ 10 GeV and 1 GeV
$\lesssim m_{X}\lesssim$ 3 GeV respectively; we also show constraints from
Xenon1T(2018) XENON:2018voc  and PandaX-4T PandaX-4T:2021bab . In the gray
region, $\epsilon$ is too small so that DM and dark photon are not thermalized
before freeze out. The $\epsilon$ value below blue dotted curve provides DM-
nucleon cross section in neutrino floor, and it is difficult to explore by
direct detection. In the left plot, we also indicate excluded region by dark
photon searches by LHCb and BarBar. We find that the direct detection cross
section gives us much stronger constraints than dark photon searches when DM
(dark photon) is relatively light. Future DM direct detection experiments can
also explore the allowed region unless it is above the region of neutrino
floor.

##  IV Summary

We have discussed DM and dark photon from a hidden $SU(2)_{H}$ model with
minimal new field content which is one $SU(2)_{H}$ doublet scalar fields,
introducing dimension 6 effective operator among $SU(2)_{H}$ and $U(1)_{Y}$
gauge fields and the doublet. The effective operator induces kinetic mixing
between these gauge fields after spontaneous gauge symmetry breaking and one
component of $SU(2)_{H}$ gauge field becomes dark photon $Z^{\prime}$ while
the other two components become vector DM $X^{\pm}$. Note that $X^{\pm}$ and
$Z^{\prime}$ have almost degenerate mass in the model.

Then we have considered the scenario in which relic density of DM is explained
by $X^{+}X^{-}\to Z^{\prime}Z^{\prime}$ process since it is determined by
gauge interaction only and more constrained due to fixed mass relation between
$X^{\pm}$ and $Z^{\prime}$. Numerically analyzing the Boltzmann equation, we
searched for parameter region that satisfy observed relic density. It has been
found that $\mathcal{O}(0.01)$ to $\mathcal{O}(1)$ gauge coupling of
$SU(2)_{H}$ can realize the relic density in the DM (dark photon) mass range
of $\sim$1 GeV to $\sim 3$ TeV. We have also discussed relevant constraints
from DM and dark photon physics to constrain the kinetic mixing parameter. In
addition, cross section of DM-nucleon scattering via dark photon exchange is
estimated to explore constraints from DM direct detection searches. As a
result, we have found the direct detection searches can provide the strongest
constraint on kinetic mixing parameter and further parameter space will be
tested in future unless the cross section is above neutrino floor.

## Acknowledgments

The work was also supported by the Fundamental Research Funds for the Central
Universities (T. N.).

## References

  * (1) P. Ko, J. Korean Phys. Soc. 73, no.4, 449-465 (2018)
  * (2) L.M. Krauss and F. Wilczek, Phys. Rev. Lett. 62, 1221 (1989).
  * (3) T. Hambye, JHEP 0901, 028 (2009) [arXiv:0811.0172 [hep-ph]].
  * (4) C. W. Chiang, T. Nomura and J. Tandean, JHEP 1401, 183 (2014) [arXiv:1306.0882 [hep-ph]].
  * (5) C. H. Chen and T. Nomura, Phys. Lett. B 746, 351 (2015) [arXiv:1501.07413 [hep-ph]].
  * (6) C. H. Chen and T. Nomura, Phys. Rev. D 93, no. 7, 074019 (2016) [arXiv:1507.00886 [hep-ph]].
  * (7) C. H. Chen, C. W. Chiang and T. Nomura, Phys. Lett. B 747 (2015), 495-499 [arXiv:1504.07848 [hep-ph]].
  * (8) C. H. Chen, C. W. Chiang and T. Nomura, Phys. Rev. D 97, no.6, 061302 (2018) [arXiv:1712.00793 [hep-ph]].
  * (9) T. Nomura, H. Okada and S. Yun, JHEP 06 (2021), 122 [arXiv:2012.11377 [hep-ph]].
  * (10) P. Ko, T. Nomura and H. Okada, Phys. Rev. D 103 (2021), 095011 [arXiv:2007.08153 [hep-ph]].
  * (11) C. Gross, O. Lebedev and Y. Mambrini, JHEP 1508, 158 (2015) [arXiv:1505.07480 [hep-ph]].
  * (12) C. Boehm, M. J. Dolan and C. McCabe, Phys. Rev. D 90, no. 2, 023531 (2014) [arXiv:1404.4977 [hep-ph]].
  * (13) N. Baouche, A. Ahriche, G. Faisel and S. Nasri, [arXiv:2105.14387 [hep-ph]].
  * (14) S. Baek, P. Ko and W. I. Park, JCAP 1410, 067 (2014) [arXiv:1311.1035 [hep-ph]].
  * (15) V. V. Khoze and G. Ro, JHEP 1410, 61 (2014) [arXiv:1406.2291 [hep-ph]].
  * (16) R. Daido, S. Y. Ho and F. Takahashi, JHEP 2001, 185 (2020) [arXiv:1909.03627 [hep-ph]].
  * (17) H. Davoudiasl and I. M. Lewis, Phys. Rev. D 89 (2014) no.5, 055026 [arXiv:1309.6640 [hep-ph]].
  * (18) S. M. Choi, H. M. Lee, Y. Mambrini and M. Pierre, JHEP 07 (2019), 049 [arXiv:1904.04109 [hep-ph]].
  * (19) H. Zhang, C. S. Li, Q. H. Cao and Z. Li, Phys. Rev. D 82 (2010), 075003 [arXiv:0910.2831 [hep-ph]].
  * (20) B. Barman, S. Bhattacharya, S. K. Patra and J. Chakrabortty, JCAP 12 (2017), 021 [arXiv:1704.04945 [hep-ph]].
  * (21) B. Barman, S. Bhattacharya and M. Zakeri, JCAP 09 (2018), 023 [arXiv:1806.01129 [hep-ph]].
  * (22) B. Barman, S. Bhattacharya and M. Zakeri, JCAP 02 (2020), 029 [arXiv:1905.07236 [hep-ph]].
  * (23) B. Barman, S. Bhattacharya and B. Grzadkowski, [arXiv:2009.07438 [hep-ph]].
  * (24) A. Karam and K. Tamvakis, Phys. Rev. D 92 (2015) no.7, 075010 [arXiv:1508.03031 [hep-ph]].
  * (25) E. Hall, T. Konstandin, R. McGehee, H. Murayama and G. Servant, JHEP 04 (2020), 042 [arXiv:1910.08068 [hep-ph]].
  * (26) T. Nomura and H. Okada, Phys. Rev. D 105 (2022) no.7, 075010 [arXiv:2106.10451 [hep-ph]].
  * (27) T. Ghosh, H. K. Guo, T. Han and H. Liu, [arXiv:2012.09758 [hep-ph]].
  * (28) T. Nomura and H. Okada, Phys. Lett. B 821 (2021), 136630 [arXiv:2104.01871 [hep-ph]].
  * (29) C. X. Yuan, Z. Zhang, C. Cai, Y. L. Tang and H. H. Zhang, [arXiv:2405.16165 [hep-ph]].
  * (30) H. Zhou, Nucl. Phys. B 1000 (2024), 116474 [arXiv:2209.08843 [hep-ph]].
  * (31) C. A. Argüelles, X. G. He, G. Ovanesyan, T. Peng and M. J. Ramsey-Musolf, Phys. Lett. B 770 (2017), 101-107 [arXiv:1604.00044 [hep-ph]].
  * (32) R. T. D’Agnolo and J. T. Ruderman, Phys. Rev. Lett. 115 (2015) no.6, 061301 [arXiv:1505.07107 [hep-ph]].
  * (33) B. Holdom, Phys. Lett. B 166 (1986), 196-198
  * (34) M. Fabbrichesi, E. Gabrielli and G. Lanfranchi, [arXiv:2005.01515 [hep-ph]].
  * (35) G. Bélanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 192 (2015), 322-329 [arXiv:1407.6129 [hep-ph]].
  * (36) J. P. Lees et al. [BaBar], Phys. Rev. Lett. 113 (2014) no.20, 201801 [arXiv:1406.2980 [hep-ex]].
  * (37) J. P. Lees et al. [BaBar], Phys. Rev. Lett. 119 (2017) no.13, 131804 [arXiv:1702.03327 [hep-ex]].
  * (38) R. Aaij et al. [LHCb], Phys. Rev. Lett. 124 (2020) no.4, 041801 [arXiv:1910.06926 [hep-ex]].
  * (39) P. Ilten, Y. Soreq, M. Williams and W. Xue, JHEP 06 (2018), 004 [arXiv:1801.04847 [hep-ph]].
  * (40) M. Pospelov, A. Ritz and M. B. Voloshin, Phys. Lett. B 662 (2008), 53-61 [arXiv:0711.4866 [hep-ph]].
  * (41) L. M. G. de la Vega, R. Ferro-Hernandez, A. García-Viltres, E. Peinado and E. Vázquez-Jáuregui, [arXiv:2311.17987 [hep-ph]].
  * (42) W. L. Xu, C. Dvorkin and A. Chael, Phys. Rev. D 97 (2018) no.10, 103530 [arXiv:1802.06788 [astro-ph.CO]].
  * (43) J. Aalbers et al. [LZ], Phys. Rev. Lett. 131 (2023) no.4, 041002 [arXiv:2207.03764 [hep-ex]].
  * (44) E. Aprile et al. [XENON], Phys. Rev. Lett. 123 (2019) no.25, 251801 [arXiv:1907.11485 [hep-ex]].
  * (45) P. Agnes et al. [DarkSide-50], Phys. Rev. D 107 (2023) no.6, 063001 [arXiv:2207.11966 [hep-ex]].
  * (46) E. Aprile et al. [XENON], Phys. Rev. Lett. 121 (2018) no.11, 111302 [arXiv:1805.12562 [astro-ph.CO]].
  * (47) Y. Meng et al. [PandaX-4T], Phys. Rev. Lett. 127 (2021) no.26, 261802 [arXiv:2107.13438 [hep-ex]].