# Single-pass STEM-EMCD on a zone axis using a patterned aperture: progress in
experimental and data treatment methods

Thomas Thersleff  Stockholm University, Department of Materials and
Environmental Chemistry, 10691 Stockholm, Sweden<EMAIL_ADDRESS>Linus Schönström  Stockholm University, Department of Materials and
Environmental Chemistry, 10691 Stockholm, Sweden  Uppsala University,
Department of Physics and Astronomy, Box 516, 75120 Uppsala, Sweden  Cheuk-Wai
Tai  Stockholm University, Department of Materials and Environmental
Chemistry, 10691 Stockholm, Sweden  Roman Adam  Forschungszentrum Jülich GmbH,
Peter Grünberg Institut, D-52425 Jülich, Germany  Daniel E. Bürgler
Forschungszentrum Jülich GmbH, Peter Grünberg Institut, D-52425 Jülich,
Germany  Claus M. Schneider  Forschungszentrum Jülich GmbH, Peter Grünberg
Institut, D-52425 Jülich, Germany  Shunsuke Muto  Nagoya University, Institute
of Materials and Systems for Sustainability, Nagoya 464-8603, Japan  Ján Rusz
Uppsala University, Department of Physics and Astronomy, Box 516, 75120
Uppsala, Sweden

###### Abstract

Measuring magnetic moments in ferromagnetic materials with atomic column
resolution is theoretically possible using the electron magnetic circular
dichroism (EMCD) technique in a (scanning) transmission electron microscope
((S)TEM). However, experimental and data processing hurdles currently hamper
the realization of this goal. Experimentally, the sample must be tilted to a
zone-axis orientation, yielding a complex distribution of magnetic scattering
intensity, and the same sample region must be scanned multiple times with sub-
atomic spatial registration necessary at each pass. Furthermore, the weak
nature of the EMCD signal requires advanced data processing techniques to
reliably detect and quantify the result. In this manuscript, we detail our
experimental and data processing progress towards achieving single-pass zone-
axis EMCD using a patterned aperture. First, we provide a comprehensive data
acquisition and analysis strategy for this and other EMCD experiments that
should scale down to atomic resolution experiments. Second, we demonstrate
that, at low spatial resolution, promising EMCD candidate signals can be
extracted, and that these are sensitive to both crystallographic orientation
and momentum transfer.

## Introduction

Rapid developments in the field of nanotechnology call for experimental
methods capable of providing information at sufficiently high spatial
resolution. In the field of nano-magnetism, there are several well-established
techniques such as x-ray magnetic circular dichroism [1] (XMCD), spin-
polarized scanning tunneling microscopy [2, 3], magnetic exchange force
microscopy [4] or electron holography [5]. However, these methods lack either
depth sensitivity or spatial resolution. An electron magnetic circular
dichroism (EMCD) technique [6], a (scanning) transmission electron microscopy
((S)TEM) analogue of XMCD, in principle offers depth-sensitivity
simultaneously with atomic spatial resolution.

EMCD has gone through a rapid development since its proposal in 2003 [7] and
the first experimental demonstration in 2006 [6]. It has been shown that the
STEM geometry can be used[8, 9] and that this can be exploited to both improve
the signal to noise ratio (SNR) of an EMCD signal [10] as well as map magnetic
moments in real space [11, 12]. In the domain of high spatial resolution, EMCD
has been detected using convergent beams of atomic size in a classical three-
beam geometry [13], utilizing phase ramps introduced by beam shift [14], in
zone axis orientation [15, 16], and using atomic size beams distorted by four-
fold astigmatism [17, 18]. Recent experiments with a weakly convergent
electron beam, using both geometric and chromatic aberration correction, led
to a detection of EMCD from individual atomic planes [19]. Despite some
impressive achievements, EMCD technique is still under development, primarily
due to the struggle with low magnetic signal strength and its sensitivity to
dynamical diffraction effects and experimental artifacts. In an effort to
overcome these difficulties, there is a need for both innovative experimental
design and data analysis methods.

Recently, it was proposed to use patterned apertures for acquisition of EMCD
signal [20]. This approach is expected to bring several advantages. First, it
enables acquisition of EMCD over the whole range of spatial resolutions, down
to atomic scale. Second, it should offer a dose-efficient approach, because it
utilizes a larger fraction of the inelastically scattered electrons than other
acquisition geometries, thereby improving the signal-to-noise ratio (SNR) of
the notoriously weak EMCD signal. Third, it offers a path to a single-pass
STEM acquisition of the spectra, if the data from the whole CCD camera can be
recorded at each scan point. Related experiments have demonstrated the
feasibility of this approach [21, 22].

Here, we report our experimental progress towards single-pass STEM-EMCD on a
cubic metallic iron sample using a patterned aperture as well as advances in
data processing that are necessary to search for and extract potential EMCD
signals. This second part proves to be particularly challenging given the
complex distribution of magnetic scattering on a zone-axis as well as the
nature of geometric distortions present in 2D electron energy-loss
spectroscopy (EELS) dispersion plane. We begin with an outline of our hardware
implementation including the design and installation of an 8-blade patterned
aperture with a mirror symmetry plane oriented parallel to the spectrometer
dispersion axis. We then detail the acquisition scheme and describe the steps
needed to reproduce the experimental method. Subsequently, we summarize our
analysis focusing on the impacts of 1) sample orientation and 2) momentum
transfer in the non-dispersive axis (denoted $q_{y}$ in this manuscript). We
reveal that the most promising candidate EMCD signals are detected at high
$q_{y}$ values extracted from regions of the sample that are oriented close to
the Fe $\left[001\right]$ zone axis. If a larger range of sample orientations
are included in the analysis, we still observe a potential EMCD signal;
however, its strength is diminished. We also observe that the range of $q_{y}$
vectors plays a crucial role in this experiment, even observing an inversion
of the signal sign on Fe $L_{3}$ that is not reciprocated on Fe $L_{2}$, which
we interpret in terms of experimental shortcomings. We conclude this
manuscript with a discussion of these observations with accompanying
theoretical considerations. The data and code required to reproduce this
analysis are provided and freely distributed[23]. A version of the script
formatted for publication detailing the analysis is provided in the
supplementary information.

## Results

The experiment presented in this manuscript was designed to test two
hypotheses that have been previously proposed. The first hypothesis is that an
8-blade patterned aperture can detect an EMCD signal on the $\left[001\right]$
zone axis of bcc iron. Given that simulations show that the EMCD signal is not
very sensitive to the convergence angle and, thus, the ultimate spatial
resolution of the probe[20], these experiments were performed with a probe
width that was much larger than what would be required for probing atomic
columns. This allows the use of a more easily modified non Cs-corrected
microscope configured with a large probe current of 2 nA to optimize the
signal-to-noise ratio of the EELS spectra.

The second hypothesis is that, given the complexity of magnetic scattering on
a zone-axis, the strength and, potentially, sign of a corresponding EMCD
signal should depend strongly on the momentum transfer vectors that are
allowed to pass to the spectrometer due to the patterned aperture. This is
expected to depend both on the rotation of the diffraction pattern with
respect to the patterned aperture as well as the tilt of the crystal. To test
this, the TEM was operated in STEM mode and STEM-diffraction was performed to
acquire convergent beam electron diffraction (CBED) patterns from the same
region that was probed for EELS. This allows for the crystallographic
orientation to be locally extracted and correlated to EMCD signal strength and
sign. We achieve this by looking at the EMCD signal as a function of both
$q_{y}$ and tilt away from the zone-axis. Methodological details are provided
in the methods section below.

### Mirrored ventilator aperture

Figure 1: (a) Design of the aperture holding arm suspending the aperture into
the beam path. (b) The aperture plate. This design includes one slit (right)
and one 8-blade mirror-symmetry patterned aperture (left). Switching between
the two was accomplished by venting the chamber and adjusting the position.
The large annular holes permit the use of the lower HAADF detector. (c)
Placement of the aperture table into the TEM camera chamber.

The pattern used for this aperture is presented on the left side of Fig. 1b.
It is an 8-blade mirror-symmetry design optimized for single-pass EMCD
acquisition on the $\left[001\right]$ zone axis of a cubic material, as
proposed by Negi et al. [20]. In this figure, the energy dispersion axis is
oriented vertically. The right side of Fig. 1b shows a slit aperture that was
not used in this experiment. The large holes surrounding the pattern allow for
electrons to pass through the aperture plate to the high-angle annular dark
field (HAADF) detector located at the spectrometer entrance. This aperture was
suspended in the beam path by an electrically grounded arm mounted on a
movable table, as shown in Fig. 1a. The table allowed for fine adjustments of
the aperture position prior to evacuation of the camera chamber. The entire
system was made vacuum compatible and placed into the negative chamber of a
JEOL 2100F TEM, as shown in Fig. 1c. Adjustments were made by opening the
chamber door, requiring venting and subsequent evacuation of the camera
chamber. The aperture position on the Ultrascan camera in the EELS
spectrometer is presented in Fig. 2a. We note that, unlike the rotation of the
diffraction pattern, the alignment between the patterned aperture and the EELS
spectrometer is fixed and, thus, critical to be correctly set prior to the
experiment.

Figure 2: (a) Experimental diffraction pattern recorded on the GIF CCD showing
the electron optical conditions. The probe was centered on the the grain of
interest (see Fig. 3). (b) 2D EELS spectrum acquired using the electron
optical conditions presented in (a). The $q_{y}$ axis represents the momentum
transfer that would be expected for a slit oriented normal to the spectrometer
dispersion plane. The oxygen edge visible at 532 eV comes from the Al2O3
capping layer and is primarily dominant in the background regions.

Following adjustment of the aperture position, a suitable iron grain was
sought out and brought into the field of view following the methodology
presented in the methods section. The goniometer was tilted to align the grain
along the $\left[001\right]$ zone axis and the diffraction pattern was rotated
using the projector system of the TEM to align the $\left[010\right]$ axis
with the spectrometer dispersion axis. Figure 2a shows the CBED pattern from
this orientation as well as its associated rotation to the patterned aperture,
allowing for exact calculation of the collection angles. The probe position
for this pattern was close to the center of the grain.

The spectrometer was operated in frame acquisition mode and a series of 2D
EELS spectra were acquired using the experimental conditions detailed in the
methods section. A summation over all of the 2D EELS images following
alignment along the energy axis is presented in Fig. 2b. The iron edges are
visible as vertical streaks at 709 and 723 eV. This $q-E$ diagram is aligned
vertically with the aperture placement on the CBED pattern in Fig. 2a,
illustrating the experimental concept.

Of note here is how the shape and orientation of the mirrored ventilator
aperture determines the regions of scattering angles collected by the
detector. The $q_{x}$ dimension is integrated out, leaving only $q_{y}$ and
$E$ dependences in the acquired dataset. Considering the shape of the mirrored
ventilator aperture, the $q_{y}$ values close to 0 thus consist of a sum over
a wide range of $q_{x}$, whereas the higher $q_{y}$ values will be more
selective in the probed $q_{x}$ range.

### 4D-STEM experiments

Figure 3: (a) Virtual HAADF of the 4D STEM datacube over the grain of
interest. The CBED patterns acquired at pixel position "A" and "B" are shown
in (b) and (c), respectively. A gamma curve of 0.3 is applied to the grayscale
mapping in these two images to enhance the visibility of the otherwise very
weak Kikuchi lines, which are indexed and highlighted in cyan. The center of
the $000$ reflection is denoted with a red dot. The virtual aperture used to
generate the VDF micrograph in figure 4c is also labeled and presented in
yellow. (d) Summation over $\Delta E$ and $q_{y}$ for the 4D EELS datacube.
The 2D EELS spectra from pixel positions "A" and "B" are presented in (e) and
(f), respectively.

The 4D datacubes from both the CBED patterns and the 2D EELS images are
summarized in Fig. 3. The STEM-DP 4D datacube reveals that the orientation of
the probed iron grain varies as a function of the probe position. This
variation may have been caused by a slight beam tilt induced while scanning
such a large area, as this TEM is not equipped with descan coils. This tilt
effect is exemplified in Fig. 3a-c. Two representative pixel positions ("A"
and "B") were chosen from this dataset, as displayed in Fig. 3a. The
corresponding CBED patterns are displayed in Fig. 3b and c, respectively. This
figure summarizes the more interactive experience of visualizing all of the
collected CBED patterns as a function of probe position, which was performed
using the PyXem plugin for Hyperspy [24, 25]. The qualitative interpretation
of this interactive experience was that the "upper-right corner" of the grain
was oriented closer to the zone axis. Moreover, the grain appears to primarily
rotate about the $\left[110\right]$ axis. This rotation can be visualized by
placing a virtual aperture away from the Bragg disks but centered on the
$\overline{1}10/1\overline{1}0$ Kikuchi line pair for the zone-axis
orientation, as shown with the yellow box in Fig. 3b,c. This is illustrated in
Fig. 4c and is discussed more quantitatively in the next section.

It should be noted that, since these datacubes were acquired using custom
written hook-up scripts with the spectrum imaging subsystem in Digital
Micrograph [26], the synchronization between the probe position and the data
acquisition is not perfect. A spatial distribution of the number of frames
acquired per pixel is provided in the supplementary information. While most
pixels contain one dataset, many contain zero or two. Nearest neighbor
interpolation was used to fill the $80\times 80$ pixel field of view.

### Orientation mapping and selection of region of interest

Figure 4: (a) Real-space micrograph created by using a virtual bright field
aperture. (b) ROI selection using a threshold for the full grain mask. (c)
Real-space micrograph created by using the virtual dark field aperture
presented in figure 3. (d) The two orientation masks, "Orient 01" and "Orient
02" were generated by thresholding this image, and both are composited here.
The exact thresholding parameters are provided in the supplementary
information.

To test the hypothesis that EMCD signal strength should vary with zone-axis
mistilt, virtual bright field (VBF) and virtual dark field (VDF) images were
generated using the STEM-DP datacube. The VBF image was formed by integrating
over a box centered in the transmitted disk, and is presented in Fig. 4a. This
effectively measures the loss of intensity to additional Bragg disks in the
CBED patterns. Since the central grain was oriented along a zone-axis,
significant intensity is lost and the grain itself appears dark. This image
was thresholded to reveal a selection mask that restricts the region of
interest to the zone-axis oriented grain, as presented in Fig. 4b and the
supplementary information.

Within this grain, a rotation about $\left[110\right]$ is observed, as
discussed above. An estimate of the degree of this rotation (corresponding to
a mistilt) can be obtained by generating a VDF image using the virtual
aperture shown in Figs. 3b and c. The resulting VDF is presented in Fig. 4c. A
significant increase in the intensity values of the pixels in this image thus
represents, to a first approximation, an orientation that is closer to the
$\left[001\right]$ zone-axis geometry.

This VDF was qualitatively thresholded using an empirically-determined value
to yield a spatial mask segmenting the grain scattering geometry close to a
zone-axis and off of a zone-axis (details are provided in the supplementary
information). The two regions of interest (ROI) are presented in Fig. 4d. We
note that this segmentation is not intended to be quantitative: it merely
serves as a way to test the hypothesis of whether there is a dependence of the
EMCD signal strength on the sample orientation. The mask labeled "Orient 01"
refers to the pixels where the CBED patterns show a closer to the zone-axis
orientation, while "Orient 02" refers to pixels with a stronger tilt away from
the zone-axis geometry. These orientation masks, as well as the "grain" mask
from figure 4b, are used in the EMCD signal extraction below.

### Candidate EMCD spectra

In this section, the two hypotheses described above are tested. The null
hypothesis is defined as "no EMCD signal is detected." As a detection
criterion, we use the methodology outlined in Thersleff et al. [13], where the
confidence in a positive EMCD signal detection is expressed as the SNR of both
the Fe $L_{3}$ and $L_{2}$ edges. Asserting the Rose criterion, a SNR of 5
(i.e. 7 dB) or more on both the Fe $L_{3}$ and $L_{2}$ edges is necessary to
confidently reject the null hypothesis (that no EMCD signal is present). EELS
difference spectra with positive SNR, but lower than the Rose criterion, will
be described here as "candidate" EMCD spectra, signifying the degraded
confidence. Since the $L_{2}$ edge is more difficult to resolve, this will be
the primary focus in the following discussion.

#### Influence of zone-axis mistilt

Figure 5: Candidate EMCD signal taken from the outer-most $q_{y}$ range and
the spatial mask restricted to the region oriented most close to the Fe [1 0
0] zone-axis.

We begin by examining the influence the zone-axis mistilt has on the presence
and sign of an EMCD signal. For this, we first use the spectra collected from
the spatial region closest to the zone-axis orientation, denoted by the
orientation mask 01 (see Fig. 4d). We also restrict ourselves to the outer
$q_{y}$ range (17.6 – 24.2 mrad), as will be justified below. The candidate
EMCD signal extracted from this region is presented in Fig. 5. Despite a
relatively weak signal strength of less than 1% on Fe $L_{2}$, the SNR is 2.2
dB using these extraction settings. While this does not meet the Rose
criterion, this gives a decent level of confidence that an EMCD signal can be
detected in the data using this combination of extraction method, spatial
sampling, and $q_{y}$ range.

Figure 6: Candidate EMCD signal taken from the outer-most $q_{y}$ range and
the mask restricted to the region oriented furthest from the Fe [1 0 0] zone
axis.

We now compare this signal to the one produced using the same $q_{y}$
integration range but collected from the spatial region exhibiting a larger
mistilt from the zone-axis orientation, denoted by the orientation mask 02
(see Fig. 4d). The EELS difference signal in this region, presented in Fig. 6,
shows a pronounced feature on Fe $L_{3}$ but no signal is visible on Fe
$L_{2}$.

#### Influence of $q_{y}$

Figure 7: Candidate EMCD signal taken from the full $q-E$ range and the mask
covering the entire zone-axis oriented grain.

We now shift the focus from the zone-axis mistilt to the influence of $q_{y}$.
We begin by looking at all mistilt angles using the "grain" mask presented in
Fig.4b and integrating over all $q_{y}$ values. This describes the scenario
most consistent with the theory and simulations from in Negi et al. [20]. The
result of the EMCD signal extraction is presented in Fig. 7. In this case, a
weak candidate EMCD signal appears to be visible with a signal strength of 2.3
dB on Fe $L_{3}$. A weak but noticable signal is visible on Fe $L_{2}$ here as
well, particularly when the cumulative sum of the EMCD signal is computed.
Given the results of zone-axis mistilt (Figs. 5 and 6), the comparatively weak
Fe $L_{2}$ signal strength may be partially explained by the inclusion of
various off-axis samples.

Figure 8: Candidate EMCD signal taken from the outer-most $q_{y}$ range and
the mask covering the entire zone-axis oriented grain.

We can explore this further by restricting the $q_{y}$ range to 17.6 – 24.2
mrad, as above. Using the same ROI mask from Fig.4b, we observe the results
presented in Fig. 8. The strength of Fe $L_{2}$ has now increased to 0.5 dB,
indicating a slightly improved confidence. This suggests that the $q_{y}$
range does influence the extracted EMCD signal, with higher $q_{y}$ values
(representing an integration over fewer $q_{x}$ values) yielding a more
convincing EMCD signal.

Figure 9: $q_{y}$ EMCD maps for the three different ROI masks.

We can study the effect of $q_{y}$ further by plotting the EMCD signal as a
function of $q_{y}$ for each of the three ROIs masks. This is presented in
Fig. 9. While the data are much noisier than with the previous figures (since
no $q_{y}$ integration was performed), a weak signal on Fe $L_{2}$ is still
visible for some $q_{y}$ values. Most significant in this visualization,
however, is the observation that the EELS difference signal on $L_{3}$ clearly
flips sign for middle $q_{y}$ values. The sign on $L_{2}$ does not appear to
change. Inspection of Fig. 2a reveals that the $q_{y}$ range where flipping
occurs has strong non-magnetic contributions from the $(110)$ Bragg
reflections, which are likely to enhance the noise and thus reduce the
magnetic SNR.

## Discussion

Overall, the results of this experiment lead us to an optimistic assessment of
the single-pass STEM-EMCD patterned aperture approach. While none of the
observed signals meet the Rose criterion, some signals of statistical
significance are nevertheless present. We also feel that this study has been
particularly instructive at highlighting a number of limitations and outlining
the challenges for this experiment, spurring the development of considerable
data processing methods. Here, we discuss the observed trends in the data as
well as the novel data processing techniques that we have developed that were
necessary for us to draw our conclusions.

The first trend we discuss is the influence of zone-axis mistilt. We do
observe a reduction in the strength of Fe $L_{2}$ for sample regions that are
tilted further away from the zone-axis orientation. Importantly, this does not
seem to come at the expense of a signal on Fe $L_{3}$. The loss of the Fe
$L_{2}$ edge is something that has been observed in classical EMCD under due
to partial mixing of the magnetic and non-magnetic signals resulting from an
asymmetrical orientation along the systematic row [27, 28]. We may be
similarly observing this mixing effect here, but in the zone-axis geometry.
The clear lesson to be learned from this is that the EMCD signal is likely to
be highly sensitive to sample orientation, requiring minimal beam tilt during
scanning operations (potentially limiting the field of view) and high
precision with the initial tilting settings.

The second observed trend is the influence of $q_{y}$, with the sign inversion
on the Fe $L_{3}$ edge for $q_{y}$ values approximately between 10 to 15 mrad
being particularly intriguing (see Fig. 9). While a sign inversion is not
observed for simulations with a perfectly oriented mirrored ventilator
aperture [20], such sign inversions would be theoretically possible for a
small rotation of the $\left[010\right]$ direction in the diffraction pattern
from the mirror symmetry axis (which is physically aligned to the spectrometer
energy dispersion axis). We observe such a rotation in this experiment, as
indicated in Fig. 2 by the yellow arrow. In addition, small mis-alignments in
the positioning of the aperture naturally lead to an asymmetry of the upper
and lower diffraction half-plane (with respect to the mirror axis of the
aperture), which, like the mistilt in the grain orientation, can also lead to
further mixing of the magnetic and non-magnetic signals [28, 27]. It is well
possible that failure to achieve a near-perfect alignment of the crystal
symmetry directions with the spectrometer is among the strongest factors
influencing the EMCD extraction. Hardware limitations of the TEM instrument
used in this work make a better alignment extremely tricky. However, more
stable microscope columns equipped with more projector lenses and deflectors
should greatly simplify this alignment step.

In addition to these experimental considerations, a serious challenge
originates from the residual aberrations in the EELS spectrometer. Inspection
of the raw data reveal that a number of spectral aberrations such as spectral
blurring, energy offset, and even dispersion variations persist in the $q_{y}$
dimension of the $q-E$ diagram. Given the complex distribution of magnetic
scattering in the zone-axis geometry, such artifacts provide a major hurdle to
the reliable extraction of an EMCD signal. While reasonable measures were
taken to ensure that the off-axis dispersion was optimal, the non-standard
approach taken in this study requires such adjustments to be made without the
use of computerized automation. Consequently, spectral focus and astigmation
were performed by hand. Although deemed to be sufficient when operating the
TEM, the extremely stringent demands of EMCD signal extraction amplify even
the smallest of alignment errors.

The approach we have adopted to meet these challenges at this stage is the
development of more robust data processing techniques. Some of the steps
necessary to extract candidate EMCD signals in this paper have not yet been
debated in the EELS literature in detail. For this reason, we devote space
here to outlining and justifying these steps, with the hope to stimulate a
wider discussion on the topic. Simultaneously, we express hopes that future
improved spectrometer hardware and automatic alignment procedures will remove
the need of most (if not all) of these post-processing steps, as this would
improve the quantitative performance over the whole range of EELS-based
studies.

The first non-standard signal processing step that appears to be necessary for
extracting an EMCD signal from these data is the addition of an energy
dependence to the post-edge normalization. This seems to be a critical step:
failure to use this approach results in a linear trend in the post-edge
regions. Consequently, when we use a simple integral normalization, we find
that the position of the post-edge window becomes overcritical, with the
potential for human bias. The energy dependence is kept as simple as possible:
a simple linear trend is regressed to the energy-dependent ratio between the
chiral plus and minus spectra in the post-edge region. This approach has been
taken previously by Schneider et al. [29] (inspired from discussions within
the XMCD community). We believe that this normalization routine is not only
less prone to such bias, but can be justified on physical grounds as
spectrometer aberrations mentioned above. In particular, we note that there is
a slight curvature in the dispersion of the EELS spectra that bends them down
along the increasing $E$ direction. This may result in a linear post-edge
trend, which we do, in fact, observe. While we are not yet certain about how
this correction affects the quantification of magnetic moments, we observe
that it is crucial in extraction of the EMCD candidate spectra presented in
the Results section. For a routine deployment of this processing step, more
detailed analysis is necessary than what can be provided in this work.
Alternatively, hardware or software correction procedures to minimize the
bending distortion need to be developed.

The second non-standard step involves the profile matching of the Fe $L_{2,3}$
edge shapes for both the chiral spectra via a peak broadening / sharpening
function. This removes the influence of spectral broadening that we observe as
a function of the $q_{y}$ dimension. This appears to be necessary due to
residual geometrical aberrations in the spectrometer causing difficulties in
achieving perfect focus along the entire $q_{y}$ dimension. Since quantitative
EMCD utilizes spectral integrals, profile matching will not modify the
quantification of an EMCD signal as long as the area under the peaks is
invariant. We achieve this by applying the standard signal processing
technique of subtracting a multiple of the smoothed second derivative of the
spectrum from the spectrum itself. The peak profiles are matched using the
optimization routine detailed in the Methods section below, and the step is
parameterized.

The final non-standard step relates to the EMCD signal extraction itself.
Rather than applying a step-by-step approach, as is typically done in the EMCD
literature, we instead opted for a more holistic procedure. To this end, we
have written an optimization routine in the Matlab programming language. This
routine optimizes several parameters in the extraction steps, by modelling the
resulting EMCD signal as two pseudo-Voigt peaks and applying a least-squares
approach. It is described at length in the methods section. We believe that
this results in a much more reliable and less bias-prone EMCD extraction and
quantification. The code for this approach is available along with the data
used for this study on Zenodo [23] and we encourage the interested reader to
explore this code on their own data and draw their own conclusions related to
its reliability.

In conclusion, we have presented our progress towards the use of a patterned
aperture designed to allow for a single-pass STEM-EMCD experiment on the zone-
axis of a magnetic crystal. The analysis necessary to process these data
spurred the development of considerable signal processing tools, which are
published alongside this work [23]. The use of these tools following the
procedures justified here lead us to observe candidate EMCD signals that
appear to be correlated to the alignment of the specimen along its zone axis
as well as having a dependence on the non-dispersion dimension of the $q-E$
diagram, which we interpret in terms of theory and the geometry of the
aperture / spectrometer coupling. Thus, we adopt an optimistic outlook for
this experimental design.

## Methods

### Sample fabrication

The sample used for this experiment was prepared in the same manner as those
used in Rusz et al. [14] and Muto et al.[30] A 10-nm-thick bcc Fe layer and a
3-nm-thick Al cap layer (to avoid oxidation of Fe) were deposited on 5-nm-
thick Si3N4 membranes by thermal evaporation in an ultra-high vacuum molecular
beam epitaxy (UHV-MBE) system. Thicknesses were controlled by calibrated
quartz microbalances. We estimate relative thickness fluctuations of about 3%,
which ensures that no significant spectral intensity variation due to film
thickness variation is expected. No ex situ or in situ preparation/cleaning
was applied to the Si3N4 membranes prior to the deposition. The membranes were
kept at room temperature during the deposition. The Fe films were post-
annealed at 750∘C for 120 min. to increase the lateral Fe grain size to about
50 nm. The deposition of the Al cap layers was carried out at room
temperature. The disordered structure of the membranes (nanocrystalline or
amorphous) led to a polycrystalline morphology of the metallic Fe/Al films.
Air exposure after the deposition oxidised the Al cap layer to a depth of 1.5
– 2 nm. Since the Al layer is 3 nm thick, a closed AlOx layer is maintained
even in the presence of surface roughness (likely for a polycrystalline film).
Some metallic Al may remain at the interface to the Fe film. As oxidation of
the Fe film could have substantial effects on the intensity ratio of the
$L_{3}$ and $L_{2}$ edges, the film was examined with EELS to probe if any
oxidisation of the Fe took place before or after the EMCD measurements; using
the fact that the oxygen K edge can be easily distinguished between aluminium
and iron oxides. Nevertheless we found no iron oxides within the detection
limit of EELS (< 1 at%). The oxygen edges visible in Fig. 2b arise from
averaging $q-E$ over all 2D EELS frames, including those from the background
region (where no iron grain was present).

### Experimental equipment

The EMCD experiments were performed on a JEOL-2100F microscope operated at 200
kV. The TEM is equipped with a Gatan Image Filter (GIF) Tridiem model using an
UltraScan 1000 CCD camera, which was used to acquire the 2D EELS data. Two
additional CCD cameras are equipped at different column heights. An Orius
camera is fitted above the viewing chamber at the height of the JEOL HAADF
detectors. This camera runs at approximately 20 frames per second and has a
large dynamic range, making it optimal for acquiring CBED patterns. This
camera was used for acquisition of the 4D STEM-Diffraction datacube as well as
inspecting the CBED patterns without the aperture shadow. The second CCD
camera is an UltraScan 1000 situated between the viewing chamber and the GIF
entrance. This was used for positioning the aperture, as it has a much wider
field of view than the UltraScan camera attached to the spectrometer.

Image contrast in STEM mode was generated with two annular detectors and a
secondary electron detector. The first annular detector is located above the
viewing chamber and was seldom used because of the strong demagnification of
the diffraction patterns necessary for the EELS experiments. The second
annular detector is located at the spectrometer entrance. This detector could
be used due to the HAADF pass-through that was built into the aperture (see
Fig. 1b). However, since different camera lengths were needed for the 4D EELS
and the STEM-diffraction experiments, yet the same area needed to be scanned,
neither of these detectors was used for spatial registration. Instead, a
secondary electron detector mounted above the sample was turned on. This
position makes the contrast invariant to changes in the projector system of
the TEM, ensuring high precision in the spatial registration of the probe
between subsequent scans with different camera lengths. While the contrast was
comparatively weak, the high probe current made it feasible to use for these
experiments.

The TEM has a chamber directly below the viewing screen that is intended for
film negatives. This was emptied and the custom aperture system described
above was inserted. The aperture was positioned by the manual adjustment
screws on the X-Y table (see Fig. 1) and its position with regards to the
dispersion axis of the spectrometer was verified by inspecting its shadow on
the CCD camera under parallel beam illumination. It was also verified by
looking at the unfocused zero-loss peak when operating the spectrometer in
spectroscopy mode. The aperture system was grounded to minimize charging and
contamination effects, which were not observed even at high probe currents.

### Data acquisition

#### STEM alignment

The condenser system in the TEM was aligned using custom settings tailored to
this experiment. The probe current was maximized by adjusting the ratio
between the CL1 and CL3 lenses, while the convergence angle was set by
adjusting the ratio between CL3 and the mini-condenser lens CM. This is
normally turned off in STEM mode for this instrument, but was manually engaged
using the free lens control. It was observed that higher convergence angles
lead to a reduction of alignment quality, so an optimal balance was found at a
convergence semiangle of 4.2 mrad and probe current of 2 nA.

Following base alignment, a survey image of the sample was taken at high
camera length using the upper HAADF detector. The probe was then moved by hand
using the mouse in the Digiscan system and placed over candidate grains.
Simultaneously, the Orius camera was used to observe the CBED patterns. The
goal was to search for grains that were close to the Fe $\left[001\right]$
zone axis yet were large enough to demonstrate the scanning capabilities of
this experiment. Thickness was estimated by observing the extinction fringes
in the CBED pattern. The grain investigated in this paper required the
goniometer to be slightly tilted to improve the orientation.

Diffraction conditions for the EELS experiments were configured using the
projector system. As precise rotation of the CBED pattern with respect to the
dispersion plane of the spectrometer is critical for this experiment, and
since this is a non-standard procedure, manual adjustment using the free lens
control was required. The rotation, focus, and demagnification of the CBED
patterns were empirically optimized through manual adjustment of the four
projector lenses in this TEM. This was performed using the Orius camera with
the probe stationary on a neighboring grain whose orientation respective to
the region of interest was known and subsequently verified by briefly placing
the probe on the region of interest (see Fig. 2a). Diffraction focus was
maintained by scanning the probe over a large area and minimizing lateral
translations (corresponding to beam tilt) with the projector lenses.

#### 4D EELS acquisition

Following optimization of the projector system, the spectrometer was switched
to spectroscopy mode. The shape of the zero-loss peak was optimized in 2D EELS
mode using both focus and stigmator lenses in the spectrometer system. Once
the zero-loss peak shape was linear over a wide $q_{y}$ range, the drift tube
was excited so that final tuning could be applied to the iron edges directly.

With the tuning completed, the sample was scanned. A survey image was acquired
using the secondary electron detector and a grid of $80\times 80$ pixels was
defined over the region of interest with a spacing of 6.3 nm. A dwell time of
0.1 s was used and Digiscan was instructed to run in EDX mode yielding a scan
speed of 8.7 pixels/s. A custom script was then executed to record the time
stamp at each pixel position using the Digital Micrograph Hook-up Script
language [26]. These time stamps were written to a Digital Micrograph
persistent tag structure. Simultaneously, the GIF CCD camera was switched to
2D EELS view mode and a second script was run that copied each frame to an
empty data container [31]. A time stamp for each frame copy was also recorded
to a persistent tag structure, allowing for the spatial position of the 2D
EELS image to be associated with the pixel position in Digiscan. The exposure
time on the camera was 0.1 s/frame which, including readout overhead, resulted
in a data acquisition rate of 9.1 frames per second. A slight undersampling of
the 2D EELS frames with respect to the probe position occurred, and this was
mollified by filling missing frames using nearest neighbor interpolation. A
spatial distribution of 2D EELS frames captured per pixel is provided in the
supplementary information.

The spectrometer was set up for a dispersion of 0.2 eV and a vertical binning
of 16. This yields a 2D EELS image for each readout cycle of $32\times 2048$
pixels, spanning an energy range of 420 – 850 eV. The camera length was
adjusted as above to yield a $q_{y}$ range from -27 to 27 mrad spanning the
vertical diameter of the aperture. During acquisition, in a background thread,
a sawtooth waveform was applied to the drift tube to continuously offset it
[32, 33]. The drift tube shifted the spectrum on the CCD camera by about 0.1
eV every second pixel. The purpose of this was to perform binned [34] and
iterative [35] gain averaging.

#### STEM Diffraction acquisition

After the 4D EELS experiment was recorded, the projector system was changed to
yield an optimal camera length for the Orius CCD camera. Critically, this very
significant change to the projector settings has no influence on the contrast
of the survey image generated with secondary electrons. Hence, the same survey
image could be used to correct for the probe starting position. The same
scripts used for recording time stamps in Digiscan [26] as well as recording
the camera in view mode [31] were used to acquire all of the CBED patterns as
the probe scanned across the region of interest. In this case, the exposure
time was lowered to 0.002 s/frame using binning 4 and the center quarter of
the CCD camera. This yielded a data acquisition rate of approximately 16
frames/s.

### Data processing

#### 4D datacube construction

Both the diffraction and 2D EELS data were stored as 3D image stacks in the
Digital Micrograph format. The recorded time stamps were used to assemble
these data into 4D datacubes. The timestamp at the beginning and end of each
row was used to determine the range of images that would be assigned to that
row. These were then resampled to 80 images using a time vector generated with
the timestamps and nearest neighbor interpolation. The resulting spatial
registration is visible in Fig. 3. It should be noted that this method does
not lead to perfect spatial registration. Some scan jitter within each
datacube is visible and the registration between datacubes is only accurate
down to approximately 10 nm. This source of error negatively impacts the
mapping capabilities of this experiment, but should have minimal influence on
the overall observation of EMCD when summing over larger areas.

#### 4D EELS pretreatment

Among the most important results of this study are the advances made in
processing the 4D EELS datacubes. All of the analysis was performed in custom-
written Matlab code that is provided as supplementary information.

First, pixel outliers (such as intense x-ray spikes) were removed by
subtracting the moving average and then looking for deviations with a sigma of
5 or more. Any spectra above this threshold were replaced by the mean of the
four nearest neighbors.

After removing strong outliers, the 2D EELS spectra were shifted to align
along the energy axis. This procedure requires two steps. In the first step,
the energy offset due to the drift tube at each probe position was estimated.
The 2D EELS image from each pixel was vertically summed (summation along
$q_{y}$), yielding an EELS spectrum image datacube $(x,y,\Delta E)$ datacube
with dimensions $N_{x}\times N_{y}\times N_{E}$. A reference spectrum was
chosen and cross-correlated with all of the summed spectra over the Fe
$L_{2,3}$ edges. Outliers were detected by looking for large deviations from
the sawtooth drift tube function and linearly interpolated over. The resulting
energy shift map was then saved and applied to a copy of the raw 4D data on a
frame-by-frame basis.

This first step does not correct for shifts of the ionization edges along the
$q_{y}$ dimension. To correct for these, an adaptation of the method employed
by Witjes et. al for Raman spectra peak shift alignment is employed [36]. The
roughly aligned 4D EELS datacube (having dimensions $N_{E}\times
N_{q_{y}}\times N_{x}\times N_{y}$) is rearranged into a 2D matrix having
dimensions $N_{E}\times(N_{q_{y}}N_{x}N_{y})$. The implicit assumption here is
that the entire dataset can be described as a series of 1D EELS spectra, which
we denote as $S_{k}(E)$ where $k$ represents the linear index of each
individual 1D spectrum. We can assume that, in the absence of chemical shifts,
the ionization edge of interest (Fe $L_{3}$ in this case) for all spectra
should be exactly centered with the mean spectrum. Since this is not the case,
we can more accurately describe the EELS datacube as:

$S_{k}(E_{j})=A_{k}f(E_{j}+\Delta E_{k})$ (1)

where $A_{k}$ represents an amplitude for the $k^{\textup{th}}$ spectrum,
$\Delta E_{k}$ represents the energy shift of the $k^{\textup{th}}$ spectrum
and $E_{j}$ represents the energy vector of the mean. Thus, $f(E_{j})$
represents the mean spectrum that would be expected if all spectra were
perfectly aligned. We can now expand $S_{k}(E_{j}+\Delta E_{k})$ in a Taylor
series about $f(E_{j})$:

$S_{k}(E_{j})=A_{k}\left\\{f(E_{j})+\Delta E_{k}\frac{\partial f}{\partial
E}\Bigr{|}_{\begin{subarray}{c}E_{j}\end{subarray}}+\frac{\Delta
E_{k}^{2}}{2}\frac{\partial^{2}f}{\partial
E^{2}}\Bigr{|}_{\begin{subarray}{c}E_{j}\end{subarray}}+\cdots\right\\}$ (2)

which can be rewritten as

$S_{k}(E_{j})=b_{k,1}f(E_{j})+b_{k,2}\frac{\partial f}{\partial
E}\Bigr{|}_{\begin{subarray}{c}E_{j}\end{subarray}}+b_{k,3}\frac{\partial^{2}f}{\partial
E^{2}}\Bigr{|}_{\begin{subarray}{c}E_{j}\end{subarray}}+\cdots.$ (3)

In this case, $b_{k,i}$ denotes the $i^{\textup{th}}$ Taylor coefficient of
the $k^{\textup{th}}$ spectrum. Critically, the ratio between the second and
first Taylor coefficients

$\frac{b_{k,2}}{b_{k,1}}=\frac{A_{k}\Delta E_{k}}{A_{k}}=\Delta E_{k}$ (4)

meaning that these coefficients can be used to estimate the energy shift for
each individual spectrum. The Taylor coefficients can thus be estimated simply
by first estimating $f(E_{j})$ to be the mean of $S_{k}(E_{j})$ and writing
this to the first column of a matrix $X$. Subsequently, this spectrum is
smoothed, numerically differentiated, and placed in the second column. Using
classical least squares regression

$B=(X^{t}X)^{-1}X^{t}S$ (5)

the Taylor coefficients and, thus, $\Delta E_{k}$ can be estimated. This
produces an energy shift correction that can be applied to each individual
spectrum using Fourier shift theory to allow for sub-channel interpolation
[37]. The procedure is iterated until $\Delta E_{k}$ becomes negligibly small.
It should be noted that the higher order Taylor coefficients can be used to
correct for additional aberrations, such as peak broadening [38, 39]. However,
this becomes increasingly difficult in the limit of strong noise corruption,
such as is present in this dataset.

The value of the energy shift $\Delta E_{k}$ for each of these steps was saved
and then added together. The final shift correction was then applied to the
raw data, resulting in the optimal energy alignment. Subsequently, the 4D EELS
datacube was truncated within the energy range 650 – 830 eV. This marks the
end of the pretreatment stage for spectral data.

#### 4D STEM DP pretreatment

The data processing workflow for the 4D STEM DP datacube is much less involved
than for the 4D EELS datacube. Following interpolation into a 4D datacube, the
data were visually explored using the PyXem software package [25], which is an
extention to Hyperspy [24]. This tool was used to produce qualitative
orientation maps using virtual dark field images generated by placing a
virtual aperture over the (0 1 1) Kikuchi line bands in the outer regions of
the diffraction patterns.

#### EMCD signal extraction

The extraction of EMCD signals was performed using the fmincon function
implemented in the optimization toolbox in the Matlab programming language. An
objective function was written that contains the following steps.

First, the desired $q$-range and orientation range was determined and single
EELS spectra for both chiral plus and chiral minus were obtained by
integrating over these regions of interest.

Second, the pre-edge background was modelled in the energy range 650–700 eV
using an inverse power law

$f_{\pm,\textup{BG}}(E)=f_{\pm}(E)-A_{\pm}E^{r_{\pm}}$ (6)

where $f_{\pm}(E)$ represents the raw extracted signal for chiral plus or
minus, $f_{\pm,BG}(E)$ is the background-subtracted signal, $E$ is the energy
vector (spanning the range 650–700 eV), and the remaining parameters are
defined in Table 1. This model was subtracted from the raw spectra as shown in
Eq. 6.

In many cases, we noticed that the peak width of the chiral plus and chiral
minus spectra was different. This yields a complex up-down-up EELS difference
signal reminiscent of what one would expect for magnetite on the Fe $L_{3}$
edge. Inspection of the spectra indicates that this was almost certainly
caused by residual spectral aberrations resulting from imperfect tuning of the
$S_{x}$ and $S_{y}$ lenses in the spectrometer. Correction of these
aberrations to the precision demanded by this EMCD experiment is likely not
possible without computer assistance. To counteract this effect, we employ a
profile matching routine at this stage of the objective function. The routine

$f_{-,s}(E)=f_{-,\textup{BG}}(E)-k_{s}\frac{\partial^{2}f_{-,BG}(E)}{\partial
E^{2}}$ (7)

takes the second derivative of the smoothed chiral minus signal, scales it by
$k_{s}$, and subtracts it from the original signal, yielding the same spectrum
with a different peak width $f_{-,s}(E)$. $k_{s}$ is treated as a parameter
that is allowed to vary during optimization. Negative values of this scalar
will result in a peak broadening, emphasizing that the intent is to ensure
that the peak shapes _match_ rather than to modify the experimental energy
resolution via peak sharpening. We also observe that the cumulative sum of the
second derivative shows a zero crossing around 718 eV, indicating that the
area underneath the modified peak is the same as before this procedure. A
figure demonstrating this is presented in the supplementary information. Once
the peak shapes are made similar to each other, a very small mismatch in the
alignment of the chiral plus and chiral minus spectra was occasionally
observed, usually on the order of 0.1 channels. We thus also shift the chiral
minus by a non-integer amount $\Delta E$ using Fourier shift theory. The shift
amount is also parameterized in the objective function and allowed to vary
when performing the optimization.

For the fourth step, a post-edge normalization was performed. The parameters
for the post-edge normalization were determined by first computing the energy-
dependent ratio between the background subtracted spectra, $D(E)$

$D(E)=\frac{f_{+,\textup{BG}}(E)}{f_{-,s}(E)}.$ (8)

A linear regression line with slope $m$ and intercept $d$ was fit to the post-
edge ratios in the energy range 730–760 eV. This line was extrapolated over
the entire background subtracted spectra and then multiplied by the chiral
minus signal to normalize it to chiral plus.

Following the post-edge normalization, the spectra were normalized to the
maximum value of either the chiral plus or chiral minus spectra and subtracted
from each other, yielding $f_{\textup{EMCD}}(E)$. The normalization here is
largely done for aesthetic purposes and simplifies the interpretation of the
EMCD signal fitting parameters. This step has no impact on quantitative values
extracted from EMCD spectra.

The sixth step entails the introduction of a model for the EMCD signal itself.
Since this experiment was performed on bcc iron, we use a simple model of two
pseudo-Voigt peaks

$\begin{split}f_{\textup{fit}}(E)=\left[a_{1}\eta\left\\{1+\left(\frac{E-c_{1}}{b_{1a}+b_{1b}(E-c_{1})}\right)^{2}\right\\}^{-1}+a_{1}(1-\eta)\exp\left\\{-\ln(2)\left(\frac{E-c_{1}}{b_{1a}+b_{1b}(E-c_{1})}\right)^{2}\right\\}\right]_{L_{3}}+\\\
\left[a_{2}\eta\left\\{1+\left(\frac{E-c_{2}}{b_{2a}+b_{2b}(E-c_{2})}\right)^{2}\right\\}^{-1}+a_{2}(1-\eta)\exp\left\\{-\ln(2)\left(\frac{E-c_{2}}{b_{2a}+b_{2b}(E-c_{2})}\right)^{2}\right\\}\right]_{L_{2}}\end{split}$
(9)

where all the parameters are defined in Table 1. Note that $a_{1}$ and $a_{2}$
are constrained to have opposite sign. This model adds nine parameters to the
optimization routine. The model is subtracted from $f_{\textup{EMCD}}(E)$,
yielding residual errors. The objective function passed to fmincon minimizes
the sum of the square of these errors.

The above steps lead to a total of 17 parameters that are allowed to vary by
the fmincon function. These parameters were constrained to ranges listed in
table 1. The errors arising from the pre-edge background range for both chiral
plus and chiral minus were passed separately, so as to allow the optimization
to favor a good fit to the input spectra rather than the subtracted result.
The code used for this approach can be downloaded from Zenodo [23] and the
full analysis and figure generation script for this manuscript is formatted
for publication in the supplementary information.

Parameter Description | Symbol | Lower bound | Upper bound
---|---|---|---
Chiral plus pre-edge amplitude | $A_{+}$ | 0 | +$\infty$
Chiral minus pre-edge amplitude | $A_{-}$ | 0 | +$\infty$
Chiral plus pre-edge slope | $r_{+}$ | -5 | 0
Chiral minus pre-edge slope | $r_{-}$ | -5 | 0
Chiral minus sharpening scalar | $k_{s}$ | -4 | 4
Chiral minus shift | $\Delta E$ | -3 | 3
Post-edge normalization slope | $m$ | -$\infty$ | +$\infty$
Post-edge normalization intercept | $d$ | -$\infty$ | +$\infty$
EMCD amplitude $L_{3}$ | $a_{1}$ | 0 | +$\infty$
EMCD broadening a $L_{3}$ [eV] | $b_{1a}$ | 0 | 5.0
EMCD broadening b $L_{3}$ [eV] | $b_{1b}$ | 0 | 5.0
EMCD center $L_{3}$ [eV] | $c_{1}$ | 700 | 712
EMCD amplitude $L_{2}$ | $a_{2}$ | -$\infty$ | 0
EMCD broadening a $L_{2}$ [eV] | $b_{2a}$ | 0 | 5.0
EMCD broadening b $L_{2}$ [eV] | $b_{2b}$ | 0 | 5.0
EMCD center $L_{2}$ [eV] | $c_{2}$ | 716 | 725
Lorentzian / Gaussian mixing parameter | $\eta$ | 0 | 1
Table 1: Constraints for the parameters passed to fmincon in the EMCD signal
extraction. Note that the EMCD amplitudes were not constrained to be positive
and negative as above; rather, they were constrained to have opposite sign
from each other.

## Data Availability

The data in the manuscript are available for download under the Creative
Commons Attribution 4.0 International license [23]. The code used for
generating the figures including the EMCD optimization routines is likewise
available under the GNU public license 3.0. This code is presented in a human-
readable format as the supplementary information to this manuscript.

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## Acknowledgements

T. T. acknowledges Wei Wan for suggesting to install the aperture device in
the negative chamber of the TEM, Sebastian Schneider and Carolin Schmitz-
Antoniak for the formative discussions on post-edge normalization routines, as
well as Stef Smeets and Bin Wang for the fruitful scientific exchanges and
inspiring discussions regarding open science publishing. L. S. acknowledges
Jakob Paulin for valuable contributions to the design and build of the
aperture holding system. T. T. and L. S. acknowledge funding from the Swedish
Research Council (project nr. 2016-05113). C.-W. T. and T. T. acknowledge
funding from the Swedish Strategic Research foundation (project nr.
ITM17-0301). J. R. acknowledges funding from the Swedish Research Council
(project nr. 2017-04026).

## Author contributions statement

T. T. proposed the single-pass mirror symmetry geometry investigated in this
manuscript, which was based off of the original idea for using a patterned
aperture by J. R.
T. T. and L. S. designed the experiment, the aperture plate and the aperture
holding system together with valuable input from S. M and C. -W. T.
L. S. authored the hook-up script allowing for 2D-EELS and STEM-DP frame
capture, and built the aperture holding system.
T. T. acquired the data and performed the analysis, including authoring all of
the relevant Matlab code (except where otherwise noted).
R. A., D. B., and C. S. grew and prepared the iron sample used in this
manuscript.
All authors contributed to the writing of the manuscript and the
interpretation of the results.

## Additional information

### Competing Interests

The authors declare no competing interests.


# A Unified Yukawa Interaction for the Standard Model of Quarks and Leptons

Ying Zhang 111E-mail<EMAIL_ADDRESS>Institute of Theoretical
Physics, School of Physics, Xi’an Jiaotong University, Xi’an, 710049, China

###### Abstract

To address fermion mass hierarchy and flavor mixings in the quark and lepton
sectors, a minimal flavor structure without any redundant parameters beyond
phenomenological observables is proposed via decomposition of the Standard
Model Yukawa mass matrix into a bi-unitary form. After reviewing the roles and
parameterization of the factorized matrix ${\bf M}_{0}^{f}$ and ${\bf
F}_{L}^{f}$ in fermion masses and mixings, we generalize the mechanism to up-
and down-type fermions to unify them into a universal quark/lepton Yukawa
interaction. In the same way, a unified form of the description of the quark
and lepton Yukawa interactions is also proposed, which shows a similar picture
as the unification of gauge interactions.

Yukawa interaction; mass hierarchy; flavor mixings;

###### pacs:

12.15.Ff, 14.60.Pq, 12.15.Hh

##  I Motivation

Although the Standard Model (SM) has been shown to be successful in
phenomenology, some puzzles remain mysterious, especially in flavor physics,
including the origin of CP violations, fermion mass hierarchy, flavor mixings,
etc. Buras2005arXiv ; Raidal2008EPJC ; ZZX2020PR . To address these
interesting issues, the SM must be updated. Some creative motivations for this
research are based on assuming mechanisms and models Feruglio2015EPJC ;
Rodejohann2003PRD ; Li2005PRD , but much are inspired by discrete symmetries
and/or patterns King2013RPP . Behind the new physics, different numbers of new
particles and/or interactions are assumed. Unfortunately, to date, experiments
have yielded null results in finding relevant information PDG2020 . This
status hints at another way to improve the SM: re-construction using only
currently found particles and interactions. Some clues can be extracted from
the existing SM. Comparing the three kinds of fundamental gauge interactions
with simple gauge couplings, Yukawa interactions include too many redundant
parameters in Yukawa couplings. In fact, to respond to fermion masses and
flavor mixings with CP violation (CPV) in both CKM and PMNS, Yukawa coupling
${\bf y}^{f}_{ij}$ is assigned a complex value. In the quark sector, for
example, ${\bf y}^{u}_{ij}$ and ${\bf y}^{d}_{ij}$ include $(9+9)\times 2$
degrees of freedom (dof). However, in phenomenology, only 10 quantities are
observable: 6 quark masses, 3 CKM mixing angles and 1 CP-violating phase.
These observables are often regard as the SM input, whereas Yukawa couplings
only appear in the building process of the Lagrangian and retain unknown
values. Another clue comes from the similarity between the quark and lepton
sectors. Considering the minimal extended SM with the Dirac neutrino mass, the
flavor characteristics in the quark sector are completely duplicated in the
lepton sector. This encourages us to seek a common mechanism that underlies
the quark and lepton flavors. As the main characteristics, both hierarchal
mass and flavor mixing with CPV point to the Yukawa term. In this paper, we
start with the decomposition of the fermion mass matrix. A close-to-flat
matrix is proposed to address hierarchal mass and guides unification of Yukawa
couplings into a universal form for all SM fermions step by step.

In the next section, we briefly review the minimal flavor structure with no
redundant parameters exceeding the phenomenological observables proposed in
Zhang2021arXiv , which has successfully reproduced not only quark masses and
CKM but also lepton masses and PMNS. After bi-unitary decomposition of the
fermion mass matrix, the real matrix ${\bf M}_{0}^{f}$ and unitary matrix
${\bf F}_{L,R}^{f}$ are responsible for the physical mass and the origin of
CPV, respectively. This result inspires to parameterize ${\bf M}_{0}^{f}$ into
a close-to-flat real symmetry matrix. The unitary matrix ${\bf F}_{L,R}^{f}$
can be absorbed into the fermion redefinition, which shows an origin of CPV
from a quantum superposition phase between gauge eigenstates and Yukawa
interaction states. In the Yukawa interaction states, family-universal
couplings can arise as a natural result. In Sec.III, we apply the same
mechanism to scalar and right-handed quarks and find a similar flat matrix.
The Yukawa couplings from up- and down-type fermions can be unified into a
universal form for quarks (lepton) when a slight breaking effect is
introduced. Continuing the same process in Sec.IV, the quark and lepton Yukawa
interactions are eventually unified into a common real coupling in appropriate
Yukawa eigenstates. As an example, we focus on the flavor structure of quarks.
All formulae can be directly generalized to the lepton sector with minimal
extended Dirac neutrinos (normal order). Finally, a summary is given.

##  II Minimal Flavor Structure in a Family

In ref. Zhang2021arXiv , a family-universal Yukawa coupling is proposed by
assuming flavor vacuum perturbations. The mechanism has successfully been used
to address not only hierarchal masses of quarks and leptons but also CKM and
PMNS mixings. Here, a short review is given, which will be used as a starting
point to realize a universal Yukawa interaction for all SM fermions.

The quark mass matrix in the SM is expressed as

$\displaystyle{\bf M}_{ij}^{q}=\frac{v_{0}}{\sqrt{2}}{\bf y}_{ij}^{q}$ (1)

with $q=u,d$ for up- and down-type quarks and Higgs vacuum expectation value
$\langle H\rangle=v_{0}/\sqrt{2}$ . As a complex matrix, ${\bf M}^{q}$ can be
diagonalized by bi-linear unitary transformations $\psi_{L,R}^{q}=({\bf
U}^{q}_{L,R})^{\dagger}\psi^{q,m}_{L,R}$ into mass eigenstates

$\displaystyle{\bf M}^{q}\rightarrow{\bf U}_{L}^{q}{{\bf M}^{q}}({\bf
U}_{R}^{q})^{\dagger}=\text{diag}(m^{q}_{1},m^{q}_{2},m^{q}_{3})$ (2)

The CKM mixing matrix can be expressed as ${\bf U}_{CKM}={\bf U}^{u}_{L}{{\bf
U}^{d}_{L}}^{\dagger}.$ Thus, the quark flavor structure is completely encoded
into a mass matrix generated from complex Yukawa couplings ${\bf y}^{q}_{ij}$.

As hinted by eq.(2), ${\bf M}^{q}$ can be decomposed into a bi-unitary form,

$\displaystyle{\bf M}^{q}={{\bf F}_{L}^{q}}^{\dagger}{\bf M}_{0}^{q}{\bf
F}^{q}_{R}$ (3)

with real matrix ${\bf M}_{0}^{q}$ and left- and right-handed unitary matrices
${\bf F}_{L,R}^{q}$, which relate to ${\bf U}_{L,R}^{q}$ by an $SO(3)$
rotation ${\bf U}_{0}^{q}$,

$\displaystyle{\bf M}_{0}^{q}={\bf
U}_{0}^{q}\text{diag}(m_{1}^{q},m_{2}^{q},m_{3}^{q})({\bf
U}_{0}^{q})^{T},~{}~{}{\bf U}_{L,R}^{q}={\bf U}_{0}^{q}{\bf F}_{L,R}^{q}$ (4)

These factorized matrices play different roles in the quark mass and mixing:

  * •

Quark physical masses are completely determined by the eigenvalues of ${\bf
M}_{0}^{q}$; neither the left- nor right-handed unitary transformation
provides any contributions.

  * •

Left-handed unitary matrix ${\bf F}^{q}_{L}$ provides complex phases required
by the non-vanishing CP violation in CKM.

$\displaystyle{\bf U}_{CKM}$ $\displaystyle=$ $\displaystyle{\bf
U}_{0}^{u}{{\bf{F}}_{L}^{u}}({\bf{F}}_{L}^{d})^{\dagger}({\bf
U}_{0}^{d})^{\dagger}$ (5)

  * •

Right-handed unitary matrix ${\bf F}^{q}_{R}$ has no effect on both the quark
mass and mixings (taken as ${\bf F}^{q}_{R}=1)$.

To address hierarchal mass eigenvalues, a close-to-flat flavor vacuum is
assumed,

$\displaystyle{\bf M}_{0}^{f}$ $\displaystyle=$
$\displaystyle\frac{m_{\Sigma}^{f}}{3}{\bf I}_{\Delta}^{f},~{}~{}~{}{\bf
I}_{\Delta}^{f}=\left(\begin{array}[]{ccc}1&1+\delta_{12}^{f}&1+\delta_{13}^{f}\\\
1+\delta_{12}^{f}&1&1+\delta_{23}^{f}\\\
1+\delta_{13}^{f}&1+\delta_{23}^{f}&1\end{array}\right)$ (9)

where $\delta^{f}_{ij}$ represents a slight perturbation. A possible origin of
non-diagonal perturbations $\delta_{ij}^{f}$ may be vacuum corrections for
different flavor Yukawa interaction. The physical masses are controlled by
$\delta_{ij}^{f}$,

$\displaystyle m^{f}_{1,2}$ $\displaystyle=$
$\displaystyle\frac{y^{f}v_{0}}{\sqrt{2}}\left(\frac{1}{3}S^{f}\mp\frac{2}{3}\sqrt{Q^{f}}\right)+\mathcal{O}(\delta^{2})$
(10) $\displaystyle m^{f}_{3}$ $\displaystyle=$
$\displaystyle\frac{y^{f}v_{0}}{\sqrt{2}}\left(3-\frac{2}{3}S^{f}\right)+\mathcal{O}(\delta^{2})$
(11)

with parameters $S^{f}$ and $Q^{f}$ given as

$\displaystyle S^{f}$ $\displaystyle\equiv$
$\displaystyle-\delta^{f}_{12}-\delta^{f}_{23}-\delta^{f}_{13}$ (12)
$\displaystyle Q^{f}$ $\displaystyle\equiv$
$\displaystyle(\delta^{f}_{12})^{2}+(\delta^{f}_{23})^{2}+(\delta^{f}_{13})^{2}-\delta^{f}_{12}\delta^{f}_{23}-\delta^{f}_{23}\delta^{f}_{13}-\delta^{f}_{13}\delta^{f}_{12}$
(13)

Differing from the quasi-democratic pattern from $S_{3}$ symmetry with more
complicated parameterization Sogami1998PTP ; Fukuura1999PRD ; Fritzsch2017CPC
. Here, $I_{\Delta}^{f}$ is inspired by hierarchal eigenvalues with only non-
diagonal real perturbations. We will see that the similar close-to-flat
structure can be generalized to unify up- and down-type fermion Yukawa
interactions and finally unify the quark and lepton into a universal form.

From eq. (5), only the combination of
${{\bf{F}}_{L}^{u}}({\bf{F}}_{L}^{d})^{\dagger}$ contributes to CKM. A general
and convenient choice is to parameterize ${\bf F}^{u}_{L}$ by diagonal phases,

$\displaystyle{\bf{F}}^{u}_{L}={\rm
diag}(1,e^{i\lambda^{u}_{1}},e^{i\lambda^{u}_{2}}),~{}~{}~{}{\bf F}^{d}_{L}=1$
(14)

This form can always be realized by choosing an appropriate quark gauge basis.

The unitary matrix ${\bf F}_{L,R}^{q}$ can be explained as a transformation
between weak gauge eigenstates and Yukawa interaction eigenstates. Defining
Yukawa eigenstates as

$\displaystyle d_{L,R}^{(Y)}={\bf
F}^{d}_{L,R}d_{L,R},~{}~{}~{}u_{L,R}^{(Y)}={\bf F}^{u}_{L,R}u_{L,R}$ (15)

the Yukawa term can be rewritten as

$\displaystyle-\mathcal{L}_{Y}^{q}={y}^{d}\bar{Q}_{L}^{(Y)}\left(\begin{array}[]{c}0\\\
\frac{v_{0}}{\sqrt{2}}{\bf
I}^{d}_{\Delta}\end{array}\right)d_{R}^{(Y)}+{y}^{u}\bar{Q}_{L}^{(Y)}\left(\begin{array}[]{c}\frac{v_{0}}{\sqrt{2}}{\bf
I}_{\Delta}^{u}\\\ 0\end{array}\right)u_{R}^{(Y)}+H.c.$ (20)

with the left-handed doublet $Q_{L}^{(Y)}\equiv(u_{L}^{(Y)},d_{L}^{(Y)})^{T}$.

The same flavor structure has been indicated to be suitable for the lepton
sector with extended normal order Dirac neutrino mass. After absorbing the
family breaking effect into the flavor independent Higgs vacuum $\langle
H\rangle=\frac{v_{0}}{\sqrt{2}}{\bf I}_{\Delta}^{f}$, the Yukawa terms can be
expressed in an elegant form with family-universal Yukawa couplings,

$\displaystyle-\mathcal{L}_{Y}=y^{u}\bar{Q}_{L}^{(Y)}\tilde{H}u_{R}^{(Y)}+y^{d}\bar{Q}_{L}^{(Y)}Hd_{R}^{(Y)}+y^{\nu}\bar{L}_{L}^{(Y)}\tilde{H}\nu_{R}^{(Y)}+y^{e}\bar{L}_{L}^{(Y)}He_{R}^{(Y)}+h.c.$
(21)

The flavor structure includes the following minimal parameterization: 3+3
perturbations for up- and down-type quark (lepton) masses, 2 phases in ${\bf
F}_{L}^{u}$ (${\bf F}_{L}^{\nu}$) provided to CKM (PMNS), and 2 family-
universal Yukawa couplings for absolute quark (lepton) masses. This
parameterization provides a general structure independent of the
characteristics from CKM or PMNS patterns. More importantly, the validity of
the minimal flavor structure has been checked by reproduction of all quark and
lepton mass (initializing the lightest neutrino mass), CKM and PMNS data
Zhang2021arXiv  (see Appendix A for details).

##  III Unified Yukawa Coupling for Up- and Down-type Fermions

In this section, we unify up- and down-type quark Yukawa interactions into a
universal form and then generalize it to the lepton sector.

Assuming two quark fields $\psi^{q1}$ and $\psi^{q2}$ in a new Yukawa
eigenstate (for the sake of convenience, the superscript (Y) has been
neglected), their left-handed components consist of an $SU(2)_{L}$ doublet
$\Psi_{L}^{q}=(\psi^{q1}_{L},~{}\psi^{q2}_{L})^{T}$, and the right-handed
$\psi^{q1}_{R}$ and $\psi^{q2}_{R}$ remain singlets. A unified quark Yukawa
interaction with an $SU(2)_{L}$ doublet scalar $\Phi$ can be written as

$\displaystyle-\mathcal{L}_{Y}=\frac{1}{2}y^{q}\bar{\Psi}_{L}^{q}(\Phi+\tilde{\Phi})(\psi^{q1}_{R}+\psi^{q2}_{R})$
(22)

or, in matrix form,

$\displaystyle-\mathcal{L}_{Y}=\frac{1}{2}y^{q}\bar{\Psi}_{L}^{q}\left(\begin{array}[]{cc}\tilde{\Phi}&\Phi\end{array}\right)\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\left(\begin{array}[]{c}\psi^{q1}_{R}\\\
\psi^{q2}_{R}\end{array}\right)$ (28)

where $y^{q}$ is a real coupling. The flat matrix in the above indicates a
non-distinguishing Yukawa interaction for one of the right-handed
$\psi^{q1}_{R}$ and $\psi^{q2}_{R}$ and the scalars $\Phi$ and $\tilde{\Phi}$,
as shown in Fig.1(a).

Figure 1: Universal Yukawa interaction: (a) for up- and down-type fermions;
(b) for quarks and leptons

Inspired by the hierarchal eigenvalues of the close-to-flat matrix, the
unified Yukawa interaction can be broken into hierarchal interactions if there
exists a perturbation in the flat matrix,

$\displaystyle\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\rightarrow\left(\begin{array}[]{cc}1&1+\delta^{q}\\\
1+\delta^{q}&1\end{array}\right)\equiv{\bf I}_{\delta}^{q}$

In appropriately chosen right-handed states, ${\bf I}_{\delta}^{q}$ can be
diagonalized into hierarchal eigenvalues $y^{u}\gg y^{d}$

$\frac{y^{q}}{2}R_{ud}{\bf I}_{\delta}^{q}R^{T}_{ud}=\text{diag}(y^{u},y^{d})$

by a real orthogonal rotation,

$\displaystyle\left(\begin{array}[]{c}\psi^{q1}_{L,R}\\\
\psi^{q2}_{L,R}\end{array}\right)\rightarrow
R_{ud}\left(\begin{array}[]{c}\psi^{q1}_{L,R}\\\
\psi^{q2}_{L,R}\end{array}\right),~{}~{}R_{ud}\equiv\left(\begin{array}[]{cc}\cos\theta_{q}&-\sin\theta_{q}\\\
\sin\theta_{q}&\cos\theta_{q}\end{array}\right)$ (36)

where the rotation angle $\theta^{q}=-\pi/4$. Rotated quarks are just the SM
up- and down-type quarks (in Yukawa eigenstates),

$\displaystyle R_{ud}\left(\begin{array}[]{c}\psi^{q1}_{L,R}\\\
\psi^{q2}_{L,R}\end{array}\right)=\left(\begin{array}[]{c}u^{(Y)}_{L,R}\\\
d^{(Y)}_{L,R}\end{array}\right).$ (41)

The ratio of $y^{u}/y^{d}$ is controlled by the perturbation $\delta_{q}$,

$\displaystyle\frac{y^{u}}{y^{d}}=\frac{-\delta^{q}}{2+\delta^{q}}$

Using $y^{u,d}=\frac{v_{0}}{3\sqrt{2}}\sum m_{u,d}^{i}$, the value of
$\delta_{q}$ is $\delta^{q}\simeq-2m^{b}/m^{t}\simeq{-0.0483}$. The rotated
scalar thus becomes the SM Higgs doublet and its charge conjugation,

$\displaystyle\left(\begin{array}[]{cc}\tilde{\Phi}&\Phi\end{array}\right)\xrightarrow{R_{ud}}{{\bf
R}_{ud}\left(\begin{array}[]{cc}\tilde{\Phi}&\Phi\end{array}\right){\bf
R}_{ud}^{T}=}\left(\begin{array}[]{cc}\tilde{H}&H\end{array}\right)$ (45)

Thus, the unified quark Yukawa term has been broken into a family-universal
form,

$\displaystyle-\mathcal{L}_{Y}=\bar{Q}^{(Y)}_{L}\left(\begin{array}[]{cc}\tilde{H}&H\end{array}\right)\left(\begin{array}[]{cc}y^{u}&\\\
&y^{d}\end{array}\right)\left(\begin{array}[]{c}u^{(Y)}_{R}\\\
d^{(Y)}_{R}\end{array}\right)$

as shown in the first two terms of eq. (21). After ${\bf R}_{ud}$ rotation,
the above Lagrangian retains its charge symmetry, which implies that the
perturbation $\delta^{q}$ may come from electromagnetic correction when
electroweak symmetry broken. The analysis must thus be performed at an energy
scale using a consistent set of input data and RGE to evolve the observables
in the future.

Generalizing these discussions to the lepton sector, we can express the quark
and lepton Yukawa terms using two universal Yukawa couplings, $y^{q}$ and
$y^{l}$, as follows:

$\displaystyle-\mathcal{L}_{Y}=\frac{1}{2}y^{q}\bar{\Psi}^{q}_{L}(\Phi+\tilde{\Phi})(\psi^{q1}_{R}+\psi^{q2}_{R})+\frac{1}{2}y^{l}\bar{\Psi}^{l}_{L}(\Phi+\tilde{\Phi})(\psi^{l1}_{R}+\psi^{l2}_{R})$
(47)

Note that the perturbation for the lepton is determined by
$\delta_{l}=-\frac{2y^{\nu}}{y^{\nu}+y^{e}}$ and the rotation angle is
$\theta_{l}=\pi/4$ to yield a lighter neutrino and a heavier charged lepton.

##  IV Unified Yukawa Coupling for Quarks and Lepton

Next, we apply a similar method to unify the quark and lepton Yukawa
interactions in eq. (47) into a universal form. Considering four fermions
$\psi^{Ai}$ and $\psi^{Bi}$ for $i=1,2$, left-handed components are arranged
into two $SU(2)_{L}$ doublets $\Psi_{L}^{A}=(\psi^{A1}_{L},\psi^{A2}_{L})^{T}$
and $\Psi_{L}^{B}=(\psi^{B1}_{L},\psi^{B2}_{L})^{T}$, while the four right-
handed components $\psi^{A1}_{R},\psi^{A2}_{R},\psi^{B1}_{R}$, and
$\psi^{B2}_{R}$ are singlets. The unified Yukawa term can be expressed as

$\displaystyle-\mathcal{L}_{Y}=\frac{y^{0}}{4}\left(\bar{\Psi}^{A}_{L}+\bar{\Psi}^{B}_{L}\right)(\Phi+\tilde{\Phi})\sum_{i=1,2}\left(\psi^{Ai}_{R}+\psi^{Bi}_{R}\right)$
(48)

or, equivalently in matrix form, as

$\displaystyle-\mathcal{L}_{Y}=\frac{y^{0}}{4}\left(\begin{array}[]{cc}\bar{\Psi}^{A}_{L},&\bar{\Psi}^{B}_{L}\end{array}\right)\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\left(\begin{array}[]{c}\sum_{i}\psi^{Ai}_{R}\\\
\sum_{i}\psi^{Bi}_{R}\end{array}\right)(\Phi+\tilde{\Phi})$ (54)

The flat matrix appears again, which indicates an undifferentiated interaction
between left-handed $\Psi_{L}^{A}$ and $\Psi_{L}^{B}$, right-handed
$\sum_{i}\psi_{R}^{Ai}$ and $\sum_{i}\psi_{R}^{Bi}$ and the scalar
$\Phi+\tilde{\Phi}$ (as shown in Fig.1(b)). Now, we assume that some of the
fermions experience a correction. A possible origin is the strong interaction
quantum effect, which only affect the fermions participating in the strong
interaction. This strong interaction splits fermions into strong and non-
strong interaction fermions, i.e., quarks and leptons.

The quantum correction as a perturbation can be labeled by $\delta_{S}$ in the
close-to-flat structure,

$\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\rightarrow\left(\begin{array}[]{cc}1&1+\delta_{S}\\\
1+\delta_{S}&1\end{array}\right)\equiv{\bf I}_{\delta}^{s}$

Taking a rotation $\bf{R}_{lq}$,

$\displaystyle\left(\begin{array}[]{c}\Psi^{A}_{L}\\\
\Psi^{B}_{L}\end{array}\right)\rightarrow{\bf
R}_{lq}\left(\begin{array}[]{c}\Psi^{A}_{L}\\\
\Psi^{B}_{L}\end{array}\right)=\left(\begin{array}[]{c}\Psi^{l}_{L}\\\
\Psi^{q}_{L}\end{array}\right)$ (61)
$\displaystyle\left(\begin{array}[]{c}\sum_{i}\psi^{Ai}_{R}\\\
\sum_{i}\psi^{Bi}_{R}\end{array}\right)\rightarrow{\bf
R}_{lq}\left(\begin{array}[]{c}\sum_{i}\psi^{Ai}_{R}\\\
\sum_{i}\psi^{Bi}_{R}\end{array}\right)=\left(\begin{array}[]{c}\sum_{i}\psi^{li}_{R}\\\
\sum_{i}\psi^{qi}_{R}\end{array}\right)$ (68)

with the mixing angle $\theta_{lq}=\pi/4$, ${\bf I}_{\delta}^{s}$ can be
diagonalized into

$\frac{y^{0}}{2}{\bf I}_{\delta}^{s}\rightarrow{\frac{y^{0}}{2}{\bf
R}_{lq}{\bf I}_{\delta}^{s}{\bf
R}^{T}_{lq}}=\left(\begin{array}[]{cc}y^{l}&\\\ &y^{q}\end{array}\right)$

The perturbation is determined by
$\delta_{S}\simeq-\frac{2m^{\tau}}{m^{t}}\simeq{-0.0205}$ to yield two
hierarchal eigenvalues $y^{l}$ and $y^{q}$ (with $y^{l}\ll y^{q}$). After
rotation ${\bf R}_{lq}$, the quark and lepton Yukawa interaction in eq.(47)
has been derived from the universal Yukawa term in eq. (48).

##  V Summary

Motivated by the redundancy of the SM Yukawa couplings and similarity of quark
and lepton mixings, we unify the SM Yukawa couplings into a universal form for
all quarks and leptons. Only one universal coupling exists when fermions are
expressed as good Yukawa eigenstates. The ambiguous flavor structure in the SM
is involved by mispairing fermions in gauge eigenstates. The entire picture
can be summarized as follows: the Yukawa interaction has a flavor-universal
structure for all flavors in eq. (48). When the energy decreases, some
fermions participate in the strong interaction, while others do not. The
Yukawa interaction for strong fermions obtains a coupling correction, which
involves the Yukawa term breaking into a strong quark and a non-strong lepton.
Quark and lepton Yukawa couplings correspond to the eigenvalues of a close-to-
flat coupling matrix. Due to the small corrections, leptons have a very small
Yukawa interaction compared to quarks. At even lower energies, electromagnetic
interactions are separated from electroweak interactions. Left- and right-
handed quarks (leptons) thus experience different Yukawa couplings. Up- and
down-type quark Yukawa couplings are also eigenvalues of a close-to-flat
matrix with non-diagonal electromagnetic correction. Thus, in each family,
generations can be split due to the presence of a close-to-flat flavor vacuum.
This scenario agrees with the unified method of gauge symmetry, as shown in
Fig.2.

Figure 2: Gauge interaction unification and Yukawa interaction unification

Flat structures are repeated throughout the entire Yukawa interaction
unification process, which exhibits a non-distinguishing flavor structure at
multiple levels when a proper Yukawa eigenstate is chosen. There is a natural
correspondence between the hierarchal masses/couplings and the flat structure.
The problem of mass hierarchy is often addressed from the pattern
$(m^{f}_{1},m^{f}_{2},m^{f}_{3})=(0,0,3)$, which is another manifestation of a
flat structure in mass eigenstates Weinberg2020PRD . Strong experimental
support of the existence of a close-to-flat flavor structure comes from the
widespread mass hierarchy:

  * •

in generations: $m^{f}_{1}\ll m_{2}^{f}\ll m_{3}^{f}$ $(f=u,d,\nu,e)$ for
$y^{f}\rightarrow y^{f}_{1},y^{f}_{2},y^{f}_{3}$;

  * •

in up- and down-type fermions:
$\sum_{i=1..3}m_{\nu}^{i}\ll\sum_{i=1..3}m_{e}^{i}$ for $y^{l}\rightarrow
y^{\nu},y^{e}$ and $\sum_{i=1..3}m_{d}^{i}\ll\sum_{i=1..3}m_{u}^{i}$ for
$y^{l}\rightarrow y^{u},y^{d}$;

  * •

in quark-lepton:
$\sum_{i=1..3}(m_{e}^{i}+m_{\nu}^{i})\ll\sum_{i=1..3}(m_{u}^{i}+m_{d}^{i})$
for $y^{0}\rightarrow y^{q},y^{l}$.

In the framework of a unified Yukawa interaction, the origin of the
perturbations at each level remains the most difficult challenge in the
future.

## Acknowledgments

I thank my collaborator Prof. Rong Li for helpful discussions and suggestions
on the subject. This work is partially supported by the Fundamental Research
Funds for the Central Universities (XJTU).

##  Appendix A Minimal Flavor Structure and Fit

Using bi-unitary decomposition of the fermion mass matrix in eq.(3),(9) and
(14), the quark mass matrix can be parameterized into the form

$\displaystyle{\bf M}^{d}=\frac{v_{0}}{3\sqrt{2}}y^{d}{\bf
I}_{\Delta}^{d},~{}~{}~{}{\bf M}^{u}=\frac{v_{0}}{3\sqrt{2}}y^{u}({\bf
F}_{L}^{u})^{\dagger}{\bf I}_{\Delta}^{u}$ (69)

The CKM has the form

$\displaystyle{\bf U}_{CKM}={\bf
U}_{0}^{u}\text{~{}diag}\left(1,e^{i\lambda^{u}_{1}},e^{i\lambda^{u}_{2}}\right)({\bf
U}_{0}^{d})^{T}$ (70)

The lepton mass matrix and PMNS are also expressed as

$\displaystyle{\bf M}^{e}=\frac{v_{0}}{3\sqrt{2}}y^{e}{\bf
I}_{\Delta}^{e},~{}~{}~{}{\bf M}^{\nu}=\frac{v_{0}}{3\sqrt{2}}y^{\nu}({\bf
F}_{L}^{\nu})^{\dagger}{\bf I}_{\Delta}^{\nu}$ (71) $\displaystyle{\bf
U}_{PMNS}={\bf
U}_{0}^{e}\text{~{}diag}\left(1,e^{-i\lambda^{\nu}_{1}},e^{-i\lambda^{\nu}_{2}}\right){{\bf
U}_{0}^{\nu}}^{\dagger}$ (72)

After matching these flavor parameters to quark/lepton masses and CKM and PMNS
mixings in PDG2020 , the following results are obtained Zhang2021arXiv :

| $\delta^{d}_{ij}$ | $\delta^{u}_{ij}$ | $\lambda^{u}_{i}$
---|---|---|---
quark input | $\begin{array}[]{c}\delta_{12}^{d}=-0.00723\\\ \delta_{23}^{d}=-0.0644\\\ \delta_{13}^{d}=-0.0377\end{array}$ | $\begin{array}[]{c}\delta_{12}^{u}=-0.0000453\\\ \delta_{23}^{u}=-0.0172\\\ \delta_{13}^{u}=-0.0165\end{array}$ | $\begin{array}[]{c}\lambda_{1}^{u}=-0.00504\\\ \lambda_{2}^{u}=0.0851\end{array}$
&$m^{d}_{i}$ | $m^{u}_{i}$ | $CKM$ |
results | $\begin{array}[]{c}m^{d}=4.750~{}MeV\\\ m^{s}=98.96~{}MeV\\\ m^{b}=4.176~{}GeV\end{array}$ | $\begin{array}[]{c}m^{u}=2.205~{}MeV\\\ m^{c}=1.303~{}GeV\\\ m^{t}=173.0~{}GeV\end{array}$ | $\begin{array}[]{c}s_{12}=0.2243\\\ s_{23}=0.04141\\\ s_{13}=0.003942\\\ \delta_{CP}=75.07^{\circ}\end{array}$
para. | $\begin{array}[]{c}\delta_{12}^{e}=-0.00443\\\ \delta_{23}^{e}=-0.146\\\ \delta_{13}^{e}=-0.105\end{array}$ | $\begin{array}[]{c}\delta_{12}^{\nu}=-0.176\\\ \delta_{23}^{\nu}=-0.435\\\ \delta_{13}^{\nu}=-0.073\end{array}$ | $\begin{array}[]{c}\lambda_{1}^{\nu}=-0.0111\\\ \lambda_{2}^{\nu}=1.60\end{array}$
| $m^{e}_{i}$ | $m^{\nu}_{i}$ | $PMNS$
results | $\begin{array}[]{c}m^{e}=0.5108~{}MeV\\\ m^{\mu}=105.55~{}MeV\\\ m^{\tau}=1.777~{}GeV\end{array}$ | $\begin{array}[]{c}m^{\nu}_{1}=0.0001~{}{eV}\\\ m^{\nu}_{2}=0.008640~{}{eV}\\\ m^{\nu}_{3}=0.04996~{}{eV}\end{array}$ | $\begin{array}[]{c}s^{2}_{12}=0.3353\\\ s^{2}_{23}=0.4393\\\ s^{2}_{13}=0.02009\\\ \delta_{CP}=1.486\pi\end{array}$
Table 1: Matched results for quarks and leptons. $v_{0}y^{f}/\sqrt{2}$ is set
as the total family mass, and the lightest neutrino mass is set as $0.0001$ eV
as an example.

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