galaxies:spiral — photometry: globular clusters

# Spin Parity of Spiral Galaxies IV - Differential Reddening of Globular
Cluster Systems of Nearby Spiral Galaxies

Masanori Iye11affiliation: National Astronomical Observatory of Japan, Osawa
2-21-1, Mitaka, Tokyo 181-8588 Japan 22affiliation: The Japan Academy,7-32
Ueno Park, Taito-ku, Tokyo110-0007, Japan<EMAIL_ADDRESS>Masafumi
Yagi11affiliationmark:

###### Abstract

The northwest side of the disk of M31 is known to be the near side because of
the differential reddening of globular clusters found from their photographic
photometry. This paper reports a simple geometric model to evaluate the
visibility of the effect and its application to published CCD photometry on
globular cluster systems of three spiral galaxies, M31, M33, and NGC253. The
color difference of globular cluster systems due to differential reddening was
confirmed for M31 and NGC253; however, the data for M33 were insufficient. The
analysis reaffirms the currently adopted interpretation that the side on the
minor axis of the galactic disk, where more conspicuous dust features and
interstellar reddening are visible, is the nearer side to us and provides an
additional basis for using spiral galaxies with identified spiral windings,
S-wise or Z-wise, to study the large-scale spin distribution of galaxies in
the universe.

##  1 Introduction

Identifying which side on the minor axis of the tilted galactic disk as
projected on the sky plane is the nearer side to us has not been a
straightforward question from an observational point of view. Indeed, back in
the 1950s, there was a big debate between de Vaucouleurs and Lindblad. The
former interpreted dust lanes on the near side of the disk as obscuring the
central bulge, whereas the latter interpreted that the dust lanes inside
spiral arms obscure bright spiral arms, and hence dust lanes are visible on
the far side of the disk. This interpretation on the near side of the tilted
disk was directly connected to the spiral winding direction, whether the
spiral arms observed are trailing spiral or leading spiral (de Vaucouleurs,
1959).

The current commonly adopted observational interpretation is that for disk
galaxies, where there is a discernible asymmetric distribution of dust lanes
or color gradient along the minor axis, the dust lane dominant side or
darker/redder side on the minor axis is the nearer side.

With this interpretation of the near side, Iye et al. (2019) made a study of
146 nearby spiral galaxies and confirmed that all the spiral structures
observed in those galaxies, with (1) identified spiral winding sense, S-wise
or Z-wise, as projected on the sky, (2) the identified approaching side on the
major axis, and (3) the darker/redder side on the minor axis, have trailing
spirals rather than leading spirals. This interpretation is also consistent
with numerical simulations of differentially rotating disk galaxies that
generate trailing spirals rather than leading spirals.

The fact that all spirals are trailing enables robust identification of the
sign of the line-of-sight component of the spin vector by identifying its
winding direction as S-wise or Z-wise. Tadaki et al. (2020) and Iye et al.
(2021) generated spin catalogs of spiral galaxies to study if there is any
parity violation in the large-scale distribution of spin vectors of galaxies.

In this work, we tried to identify the near side of two spiral galaxies in
addition to M31 by detecting the difference in the average color of globular
clusters (GCs) on both sides of the major axis as the number fraction of
reddened globular clusters should be larger on the near side rather than on
the far side.

Schlafly & Finkbeiner (2011) measured colors of Galactic stars with spectra in
the Sloan Digital Sky Survey to map galactic extinction. We use their result
to correct for the foreground Galactic extinction of extragalactic GCs to
isolate additional extinction due to the interstellar dust of the respective
host galaxies.

We describe a simple geometrical model to quantify the visibility of reddening
asymmetry in section 2. The application to M31 is given in section 3 and to
other spirals in section 4.

##  2 A Model of Halo Shadowed by a Disk

Let us assume a simple model, as shown in Figure 1, where the distribution of
globular clusters in a spherical halo of radius $R_{h}=1$ is shielded by a
thin disk with interstellar dust of radius $R_{d}=R$ tilted from the normal
plane of the line of sight by an inclination $-\pi/2<i<\pi/2$. We take the
Z-axis joining the galaxy from an observer at infinity in the $-Z$ direction.
The Y-axis is set in the plane orthogonal to the line-of-sight at the galaxy
center so that the spin vector of the disk is in the Y-Z plane. The X-axis is
orthogonal to the Y-Z plane.

Figure 1: Geometry of a spherical halo ($R_{h}=1$) containing globular
clusters and a tilted ($i$) disk ($R_{d}=R$) shielding the region behind the
disk from an observer in the -Z direction. A small circle BCD shows a cross-
section of the halo at $Y=y$.

In this configuration, the radius $r$ of the circle section BCD of the halo at
$Y=y$ is $r=\sqrt{1-y^{2}}$. The height of the disk at $Y=y$ is given by
$Z=z=y\tan{i}$. The two ends (one of which is at K) of the intersection of the
disk and the plane $Y=y$ are at $X=\pm
x=\pm\sqrt{R^{2}-y^{2}-y^{2}\tan{i}^{2}}$.

Figure 2 shows an example of the X-Z cross-section of the halo at $Y=y$. The
disk (thick horizontal line) divides this cross-section into regions behind
(Sb), in front of (Sf), and outside of (So) the disk. We denote the areas
behind, in front of, and outside the disk, $Sb$, $Sf$, and $So$, respectively,
as observed by an observer in the -Z direction.

By introducing normalized coordinates for the circle cross-section at $Y=y$

$\displaystyle
x_{0}=x/r=\frac{\sqrt{R^{2}-y^{2}-y^{2}\tan^{2}{i}}}{\sqrt{1-y^{2}}}$ (1)

and

$\displaystyle z_{0}=z/r=\frac{y\tan{i}}{\sqrt{1-y^{2}}}$ (2)

one can reduce our problem of evaluating the geometrical region affected by
disk extinction to finding the fraction $f(z_{0}(y))$ of area Sb to the total
area $\pi$ of a unit circle.

In the right panel of Figure 2, the fraction $f(z(y))$ of the shaded area of a
unit circle, cut by a line segment with endpoints $(-x,z)$ and $(x,z)$, where
$x^{2}+z^{2}=1$, is given by $f(z)=\theta-xz$, where
$\theta=\arctan{(x/z)}=\arccos{z}$.

Eliminating $x$, we have,

$\displaystyle f(z)=\cases{\arccos{z}-z\sqrt{1-z^{2}}&$|z|\leq 1$\cr
0&$|z|>1$\cr}$ (3)

Figure 2: (Left) Circle cross-section (thick circle) of a spherical halo (thin
circle) and cross-section of the disk (thick horizontal bar) at $Y=y$ plane.
(Right) The same figure is shown in coordinates normalized by $r$ to be
applied in equations. Areas Sb, Sf, and So show those behind, in front of, and
out of disk from an observer at -Z direction for a unit circle. Figure 3: Four
cases of disk shadowing the halo region.

Using equation (1), we can derive an expression for the shadowed area $Sb(y)$
for cases separately with $x_{0}^{2}+z_{0}^{2}\leq 1$ and
$x_{0}^{2}+z_{0}^{2}>1$.

(1) For a circle section with $\sqrt{x_{0}^{2}+z_{0}^{2}}\leq 1$, as shown in
the panel (1) of Figure 3 for $i<0$,

$\displaystyle
S_{b}(y)=\cases{\\{f(z_{1})+2x_{0}(z_{1}-z_{0})\\}(1-y^{2})&$0\leq|y|\leq|R\cos{i}|$\cr
0&$|y|>|R\cos{i}|$\cr}$ (4)

where $z_{1}=\sqrt{1-x_{0}^{2}}$.

For a circle section with $\sqrt{x_{0}^{2}+z_{0}^{2}}>1$, there are three
cases.

(2) If $x_{0}\geq 1$, as shown in the panel (2) of Figure 3 for $i<0$, $S(y)$
is for the entire range $-1\leq y\leq 1$

$\displaystyle S_{b}(y)=\cases{\\{\pi-f(-z_{0})\\}(1-y^{2})&for $z_{0}<0$\cr
f(z_{0})(1-y^{2})&for $z_{0}\geq 0$\cr}$ (5)

(3a) If $z_{0}<0$, as shown in the panel (3a) of Figure 3 for $i<0$,

$\displaystyle
S_{b}(y)=\cases{\\{\pi-f(-z_{0})-2f(x_{0})\\}(1-y^{2})&$0\leq|y|\leq|R\cos{i}|$\cr
0&$|y|>|R\cos{i}|$\cr}$ (6)

(3b) If $z_{0}\geq 0$, as shown in panel (3b) of Figure 3 for $i>0$,

$\displaystyle S_{b}(y)=\cases{f(z_{0})(1-y^{2})&$0\leq|y|\leq|R\cos{i}|$\cr
0&$|y|>|R\cos{i}|$\cr}$ (7)

The halo volume behind the disk, $V_{b}$, is obtained by integrating
$\pi(1-y^{2})S_{b}(y)$ over $[-R\cos{i},R\cos{i}]$.

$\displaystyle V_{b}$ $\displaystyle=$
$\displaystyle\pi\int_{-R\cos{i}}^{R\cos{i}}(1-y^{2})S_{b}(y)dy,$ (8)

except for case (2), where the integration is over the range [-1,1].

The volume in front of the disk is given similarly by

$\displaystyle V_{f}$ $\displaystyle=$
$\displaystyle\pi\int_{-R\cos{i}}^{Rcos{i}}(1-y^{2})(1-S_{b}(y))dy,$ (9)

except for case (2), where the integration is over the range [-1,1].

The asymmetry fraction $A(i,R)$ can be defined as follows:

$\displaystyle A(i,R)=|V_{b}-V_{f}|/(V_{b}+V_{f})$ (10)

Note that $V_{b}+V_{f}=4\pi/3$ only for case (2). There are unshaded regions
of the halo outside the edge of the shading disk for cases (1), (3a), and
(3b), whose volume $V_{o}$ is given by

$\displaystyle
V_{o}=2\pi\int_{R\cos{i}}^{1}(1-y^{2})dy=(\frac{4}{3}-2R\cos{i}+\frac{2R^{3}\cos{i}^{3}}{3})\pi$
(11)

for $R<1$ and $V_{o}=0$ for $R\geq 1$.

One can define a volume fraction $B(i,R)$ of the halo region where the
detection of the reddening effect of the disk is concerned,

$\displaystyle B(i,R)=(V_{b}+V_{f})/((V_{b}+V_{f}+V_{o}).$ (12)

These two fractions are both in the range (0,1) and the detection of the
interstellar reddening effect is easier for a spiral galaxy with large
$A(i,R)$ and $B(i,R)$.

The number of reddened GCs is expected to be proportional to $B(i,R)$ and the
statistical significance proportional to its square root. We can define a
geometrical asymmetry visibility parameter $G(i,R)$ for a spiral galaxy by
combining these two fractions

$\displaystyle G(i,R)=A(i,R)\sqrt{B(i,R)}.$ (13)

$G(i,R)$ satisfies $0\leq G(i,R)<1$, and a larger $G(i,R)$ means a favorable
geometry for detecting the differential reddening of the GC colors.

Figure 4 shows the geometrical asymmetry visibility $G(i,R)$ and the
parameters for the nearby tilted spirals studied in this paper. Here, the
$R_{h}$ was defined by a radius within which 95% of the sampled globular
clusters belong. $R_{d}=D25/2$ was taken from RC3(de Vaucouleurs et al.,
1991). Actual detection depends on the column density and the distribution of
interstellar dust and the quality and adopted color bands of the observational
photometry of GCs.

Figure 4: Geometrical asymmetry visibility $G(i,R)$ as a function of disk size
relative to halo size $R=R_{d}/R_{h}$ and inclination of the disk
$i=\arccos(b/a)$ from the celestial plane. The locations of the three spiral
galaxies studied are shown in this plane.

##  3 Reconfirmation for M31(NGC224)

Iye & Richter (1985) made a photometric study on the distribution of globular
clusters (GCs) color for M31 and showed that the identification of the near
side of the tilted disk could be made from differential reddening along the
minor axis of the galactic disk as there are more reddened GCs behind the
galactic plane on the near side rather than on the far side (cf. Figure 5).
The verification was made from three independent photographic photometries
available at that time for M31, all showing a consistent result that the
northwest side of the M31 disk is the near side, which is the dust lane
dominant side.

Figure 5: Illustration of differential reddening of M31 globular clusters.
(Figure 1 from Iye & Richter (1985))

Barmby et al. (2000) published a catalog of CCD photometric and spectroscopic
data on M31 GCs. Their catalog contains 268 cluster candidates with optical
photometry in four or more filters of which objects with $B-V<0.55$ are not
likely true globular clusters. Among those objects with $B-V>0.55$, their
catalog provides an estimate of metallicity [Fe/H] for 130 objects from
spectroscopic observation and the estimated interstellar reddening according
to an empirical relation

$E(B-V)=(B-V)-0.159[Fe/H]-0.92,$

using Table 7 of Barmby et al. (2000).

Figure 6: Coordinate geometry of M31.
Image from
https://subarutelescope.org/jp/gallery/pressrelease/024/04/03/3382.html

Figure 6 shows the coordinate geometry of M31. We take the +Y axis in the
direction rotated 90∘ counterclockwise from the +X axis, which is the
direction of the receding side on the major axis. We use this convention for
other spirals as well. Note that these coordinates are not identical to those
adopted in Iye & Richter (1985).

Figure 7: $E(B-V)$ distribution of globular clusters (blue dots) as a function
of $Y$, the distance from the major axis of the M31 disk in arcmin units. The
average reddening values and associated standard errors are given in red-
filled squares and red open squares over the sampled ranges,
$5^{\prime}<|Y|<20^{\prime}$, and the entire $Y$ range is shown in horizontal
red lines. The figure shows enhanced reddening for GCs behind the disk on the
northwest side at domains $-20^{\prime}<Y<-5^{\prime}$.

Figure 7 shows the distribution of $E(B-V)$ of globular clusters as a function
of position $Y$ in arcmin along the minor axis of M31. The root mean square
$e_{B-V}$ of individual measurement errors of the sample is 0.03 mag.

color | Region(′) | # | Mean | StdErr | $p$
---|---|---|---|---|---
B-V | $Y>0$ | 72 | 0.858 | 0.019 |
B-V | $Y<0$ | 58 | 0.901 | 0.027 | 0.093
E(B-V) | $Y>0$ | 72 | 0.121 | 0.014 |
E(B-V) | $Y<0$ | 58 | 0.173 | 0.024 | 0.031
B-V | $20>Y>5$ | 32 | 0.854 | 0.030 |
B-V | $-20<Y<-5$ | 20 | 1.018 | 0.047 | 0.0031
E(B-V) | $20>Y>5$ | 32 | 0.126 | 0.024 |
E(B-V) | $-20<Y<-5$ | 30 | 0.304 | 0.039 | 0.00022
Table 1: Color statistics of M31 GCs. The associated standard errors, StdErr,
are the estimated errors of the average color $B-V$ and reddening $E(B-V)$.
The Welch t-test $p$-values are given for each pair.

There is no physical reason to expect a difference in the intrinsic colors of
GCs on both sides of the major axis. However, interstellar reddening of the
M31 disk would have different effects on both sides, and the standard
deviations of the observed color of GCs could be different.

Welch t-test $p$-values for the statistical probability of finding larger mean
values of $B-V$ or $E(B-V)$ on the expected near region of the major axis
$-Y_{max}<Y<Y_{min}$ than for the far region $Y_{min}<Y<Y_{max}$ by chance
were evaluated for various choices of $Y_{max}$ and $Y_{min}$ at 5′ steps. We
observe a clear $E(B-V)$ enhancement in the region
$-20^{\prime}<Y<-5^{\prime}$. A similar enhancement is also seen in $B-V$
there.

Table 1 summarizes the results for the entire sample and the sample with the
lowest $p$-values. The GCs in $-20^{\prime}<Y<-5^{\prime}$ are redder than
those in $5^{\prime}<Y<20^{\prime}$ by 0.16 mag and more reddened by 0.18 mag.
These differences can occur to chance coincidence probability $p$-values
0.0031 or 0.00022, respectively. The $p$-value increases to 0.093 and 0.031,
respectively, if we take all the GCs on both sides of the major axis, probably
because of the increased population of outer GCs that are not suffering
interstellar reddening and inner ($|Y|<5^{\prime}$) GCs for which $S_{b}$ is
not much different from 0.5.

These $p$-values were calculated using the observed standard deviation of GC
colors, including the intrinsic color scatter and reddening effect. The actual
significance of the mean color difference should be compared using only the
standard deviation of the inherent color. Therefore, the relevant $p$-values
could be smaller than the values in Table 1.

With the inclination $i=71^{\circ}$ and the disk size normalized to the GC
halo size $R=R_{d}/R_{h}=1.1$, the geometrical asymmetry visibility defined by
equation (7) is $G=0.38$ for M31.

The presence of anisotropic reddening of the globular cluster colors of M31,
as reported in Iye & Richter (1985) and supported by Barmby et al. (2000) is
reconfirmed and provides another basis for using GCs to judge the near side of
the tilted disk of galaxies in the following section.

##  4 Application to Other Nearby Spiral Galaxies

###  4.1 NGC253

Cantiello et al. (2018) made ugri photometry of 82 bona fide GCs, 155
uncertain GC candidates, and 110 unlikely GCs (most likely background
galaxies) of NGC253. We use their $g-r$ color data, which have smaller
photometric errors than their $u-g$ color data. They applied a constant
Galactic extinction to their image frames. We applied Galactic extinction
individually for objects based on Schlafly & Finkbeiner (2011).

Figure 8: Geometry of NGC253.
Image from
%****␣Iye20240621-GC_Arxiv.tex␣Line␣300␣****https://ps1images.stsci.edu/cgi-
bin/ps1cutouts

Figure 8 shows that the dark lanes are more obvious on the northwest side of
NGC253 as cataloged in Iye et al. (2019). The axis ratio is
$R25=log(a/b)=0.61$ (de Vaucouleurs et al., 1991), where $a$ and $b$ are the
semimajor and semiminor radii of the galactic disk. The NGC253 disk is
slightly more inclined than that of M31 with $R25=0.49$.

Figure 9: $Y$ vs $g-r$ distribution of 81 bona fide GCs in NGC253. The height of the open and closed squares correspond to the mean error of the evaluated average color over the entire region and region $|Y|>2.^{\prime}5$, respectively. $objects$ | Region(′) | # | Mean | StdErr | $p$
---|---|---|---|---|---
GC* | $Y<0$ | 39 | 0.528 | 0.018 |
GC* | $Y>0$ | 42 | 0.569 | 0.019 | 0.059
GC | $Y<0$ | 39 | 0.456 | 0.017 |
GC | $Y>0$ | 42 | 0.504 | 0.019 | 0.033
GC | $Y<-2.5$ | 30 | 0.435 | 0.019 |
GC | $2.5<Y$ | 32 | 0.509 | 0.024 | 0.009
galaxies | $Y>0$ | 60 | 0.678 | 0.029 |
galaxies | $Y<0$ | 50 | 0.673 | 0.030 | 0.449
Table 2: $g-r$ color statistics for NGC253 bona fide GCs and background
galaxies. Colors for GC* are uncorrected for foreground Galactic extinction.

Figure 9 shows the distribution of $g-r$ color as a function of $Y$ for bona
fide GCs of NGC253, each corrected for the foreground galactic extinction.
Photometric errors in measuring their color depend on the GC brightness. The
root mean square $e_{g-r}$ of the quoted photometric errors for each GC of
this sample is 0.01. Table 2 summarizes the average color on both sides from
the major axis. The GCs on the northwest side are redder on average by 0.05
mag than those on the southeast side, indicating that the northwest side is
the near side of the tilted disk of NGC253. The Welch t-test $p$-value
evaluations were performed for $2.^{\prime}5$ step choices of $Y_{max}$ and
$Y_{min}$. The lowest $p$-value was obtained for $|Y_{min}|=2.^{\prime}5$ and
no limit for $Y_{max}$.

The Welch-test probability of finding by chance the observed asymmetry in the
$g-r$ color for all 81 bona fide GCs is $p=0.033$. The outer 62 GCs at
$|Y|>2.5^{\prime}$ show more conspicuous asymmetry $p=0.009$ with a color
difference of 0.07 mag. The statistical significance for asymmetry is slightly
decreased to $p=0.059$ for the entire GC sample without correction for the
foreground Galactic extinction.

Note that the halo radius $R_{h}=27^{\prime}$ of the sampled GCs in NGC253 is
significantly larger than the disk size $R_{d}=13.8^{\prime}$,
$R=R_{d}/R_{h}=0.51$, compared to $R=1.09$ for M31. This makes the geometrical
asymmetry visibility for NGC253 $G=0.06$, approximately 1/6 of $G=0.38$ for
M31, despite its large inclination $i=76^{\circ}$. Nevertheless, differential
reddening was perceptible for NGC253 because of the large number of GCs
available with high-precision photometry.

Note also that Table 2 shows that the sample of background galaxies does not
show any difference in their mean color on either side of the major axis of
NGC253. The standard deviation of the $g-r$ color for the background galaxies
is 0.22 mag, which is larger than that of the sampled GCs, 0.12 mag.

###  4.2 M33(NGC598)

Fan & de Grijs (2014) reported $UBVRI$ photometry, ages, metallicities, and
masses of 708 star clusters and cluster candidates of NGC598 (M33). We used
$B-V$ and the age of their catalog to study the color and extinction
dependence on $Y$ by isolating 105 clusters with an evaluated age older than 2
Gyr.

Figure 10: Geometry of M33.
Image from https://www.nao.ac.jp/en/contents/gallery/2017/20170307-subaru-
full.jpg

Figure 10 shows the geometry of M33. Figure 11 shows the distribution of $B-V$
color corrected for Galactic extinction as a function of the position $Y$ from
the major axis.

Figure 11: Distribution of $B-V$ corrected for Galactic extinction as a
function of $Y$ for 105 GCs older than 2Gyr in M33.

Table 3 indicates naively that the $B-V$ color of GCs at $Y>0$ is redder than
those at $Y<0$ by 0.06 mag but at a significance level of $p=0.118$. The
significance level increases for their raw color, uncorrected for the
foreground Galactic extinction, to $p=0.061$. The significance increases
slightly for a sample limited to GCs at $|Y|<10$ to $p=0.110$ and $p=0.049$,
corrected and uncorrected for individual foreground Galactic extinction,
respectively. The measured color difference is 0.07 mag for the sample at
$|Y|<10$.

The root mean square of the color measurement errors $e_{B-V}$ of M33 is 0.04
mag, larger than 0.03 mag and 0.01 mag for M31 and NGC253, respectively. This
makes the detection of differential reddening rather marginal. Although the
geometric asymmetry visibility parameter $G=0.35$ is similar to the value
$G=0.38$ for M31, differential reddening was not securely observed for M33.
The small color difference in M33 is partly due to its smaller inclination of
the disk, but it is more likely that M33 has a smaller column density of dust
along the line of sight than M31.

Objects | Region(′) | n | Mean | StdErr | $p$
---|---|---|---|---|---
GC* | $Y<0$ | 45 | 0.585 | 0.033 | -
GC* | $Y>0$ | 60 | 0.664 | 0.038 | 0.061
GC* | $-10<Y<0$ | 31 | 0.568 | 0.032 | -
GC* | $10>Y>0$ | 45 | 0.662 | 0.046 | 0.049
GC | $Y<0$ | 45 | 0.495 | 0.033 | -
GC | $Y>0$ | 60 | 0.556 | 0.039 | 0.118
GC | $-10<Y<0$ | 31 | 0.473 | 0.033 | -
GC | $10>Y>0$ | 45 | 0.544 | 0.047 | 0.110
Table 3: $B-V$ color asymmetry of M33 GC system. objects | $L$ | $B$ | $a(=R_{d})$ | $b$ | $R_{h}$ | PA+ | $i$ | $R_{d}/R_{h}$ | $G(i,R)$
---|---|---|---|---|---|---|---|---|---
|  |  |  | Regions | Color | $e_{color}$ | $n$ | $p$ | Near Side
unit | deg | deg | arcmin | arcmin | arcmin | deg | deg |  |
M31 | 121.17 | -21.57 | 95.27 | 30.83 | 87 | 35(NE) | 71 | 1.1 | 0.38
|  |  |  | $5<|Y|<20$ | $B-V$ | 0.03 | 52 | 0.0013 | NW
M33 | 133.61 | -31.33 | 35.40 | 20.84 | 36 | 203(SW) | 54 | 1.0 | 0.35
|  |  |  | $|Y|<10$ | $B-V$ | 0.04 | 76 | 0.110 | NW
NGC253 | 97.36 | -87.96 | 13.77 | 3.38 | 27 | 232(SW) | 76 | 0.5 | 0.06
|  |  |  | $2.5<|Y|$ | $g-r$ | 0.01 | 62 | 0.009 | NW
Table 4: Three spiral galaxies were studied for differential reddening of
their GC systems. PA+ is the position angle of the receding side of the major
axis measured from the north counterclockwise toward the east. $G(i,R)$ is the
geometrical asymmetry visibility defined by equation (7). The Welch t-test
probability $p$ of finding the observed color asymmetry by coincidence is
given for each pair.

##  5 Summary

We present a simple geometrical model to evaluate the effect of differential
reddening of a globular cluster system of a spiral galaxy with an inclined
disk containing interstellar dust. Table 4 summarizes geometric parameters of
the sampled regions, the observed color data for examining differential
reddening, and the $p$-values of three nearby spiral galaxies. The presence of
differential reddening of GCs due to an inclined obscuring disk of
interstellar dust is confirmed for spirals with high inclination M31
($R25=0.49$) and NGC253 ($R25=0.61$), but not in a decisive way for a low
inclination spiral M33 ($R25=0.23$). The correction of foreground Galactic
extinction for individual objects slightly increased the significance of
detection for NGC253 but not for M31 and M33.

This study was supported by a Grant-in-Aid for Scientific Research (C) JSPS
KAKENHI 22K03679.

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# Understanding the input-output relationship of neural networks in the time
series forecasting radon levels at Canfranc Underground Laboratory

Iñaki, Rodríguez-García, Miguel Cárdenas-Montes  CIEMAT, Department of
Fundamental Research.
Avda. Complutense 40. 28040. Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Underground physics experiments such as dark matter direct detection need to
keep control of the background contribution. Hosting these experiments in
underground facilities helps to minimize certain background sources such as
the cosmic rays. One of the largest remaining background sources is the radon
emanated from the rocks enclosing the research facility. The radon particles
could be deposited inside the detectors when they are opened to perform the
maintenance operations. Therefore, forecasting the radon levels is a crucial
task in an attempt to schedule the maintenance operations when radon level is
minimum. In the past, deep learning models have been implemented to forecast
the radon time series at the Canfranc Underground Laboratory (LSC), in Spain,
with satisfactory results. When forecasting time series, the past values of
the time series are taken as input variables. The present work focuses on
understanding the relative contribution of these input variables to the
predictions generated by neural networks. The results allow us to understand
how the predictions of the time series depend on the input variables. These
results may be used to build better predictors in the future.

##  1 Introduction

Nowadays, underground Physics experiments are focused on searching particles
with low energy signals such as dark matter direct detection. The searched
signals are, usually, very weak and with low expected frequency. For these
reasons, it is fundamental to control and understand as much as possible the
sources of background. For example, for minimizing the contribution of the
cosmic rays, these experiments are carried out in subterranean facilities such
as mines and caves under mountains.

These underground facilities are enclosed by rocks in the mountains. The rocks
emit radon, towards the caves, which is a potential source of background. In
the underground laboratories, the radon is produced by the decay of radium-226
present in the rocks. With an activity ranging from tens to hundreds of
$\frac{Bq}{m^{3}}$, the radon-222 decays into polonium-218 and emits an
$\alpha$ particle. Due to the high rate of emission, the $\alpha$ particles
can hide the searched signal.

The Canfranc Underground Laboratory (LSC) is a laboratory located in the
Spanish side of the Pyrenees. Over the laboratory there are about 800 meters
of rocks, that gives natural protection against cosmic rays. For this reason,
the LSC is a good host for low-energy events search experiments like ANAIS
[1], ArDM [2] and DArT [3]. For the international collaborations whose
experiments are located at LSC, it is useful to have high-quality predictions
of the radon levels in order to schedule the maintenance operations during the
low-level periods. Minimizing the exposure of the materials is therefore an
objective.

The present paper continues with the forecasting efforts of the radon levels
at LSC. In the past, classical time series algorithms such as Holt-Winters,
ARIMAs and STL Decomposition have been compared to deep models in [4]. An
ensemble of STL Decomposition joined to the Convolutional Networks, called
STL+CNN, to improve the forecasts, was presented in [5] for various datasets
and, in [6] for the radon Series. Also in [7] the authors proposed a
population-based incremental learning (PBIL) algorithm to optimize the
hyperparameters of STL+CNN models. Other novel research lines have been
tested, in [9] the authors proposed to use the weather variables from four
cities around the LSC, to improve the predictions of radon levels. The
analysis of the uncertainty in Deep Learning models and Gaussian Process was
done in [8]. The main goal of the present paper is to understand how the input
influences the predictions of the neural networks which forecast the radon-222
at levels. For this task, the algorithms of Garson [10] and Olden [11] are
applied.

The rest of the paper is organized as follows: in Section 2, the radon time
series, the neural networks and the importance algorithms are described. The
results and their posterior analysis is shown in Section 3. Finally, Section 4
presents the conclusions of this work.

##  2 Methods and Materials

###  2.1 Canfranc Underground Laboratory

The LSC is located between a road tunnel and old train tunnel that unites
Spain and France across the Pyrenees. The laboratory is composed of three
sites (LAB780, LAB2400 and LAB2500) under 850 meters of rocks which give a
cosmic rays protection of 2450 m.w.e. The main site is also divided in three
halls (A, B and C) for hosting the diverse experiments.

In the Hall A, the radon level has been recorded by an Alphaguard P30 every
ten minutes since July 2013. The raw measurements have a lot of noise as it
can be seen in Figure 1. Few missing values are in the data set. The large
gaps appear in July 2014 with 913 missing values -about a week-, in June 2015
with 1053, and in January 2016 with 585 -about four days.

Following previous work [4], the levels of radon exhibit a certain seasonality
in the medians, as Figure 2 shows, where the weekly medians are lower in
winter than in the summer. For this reason, the work-data are the weekly
medians of the raw values in order to reduce the noisy contributions. After
sampling data into weekly values, there are only two missing values. These are
filled after a Gaussian distribution over the time series.

Figure 1: Example of raw measurements of the radon-222 concentration in the
Hall A of the LSC collected in 2017.

Figure 2: Weekly box-plots of radon-222 level at Hall A of the LSC, by weekly
over range of the years (a) and by week independently the years (b).

The final time series is formed by the 383 median weekly values (orange points
in Figure 2 from July 21th, 2013 to November 8th, 2020. The last two years
(about the 30%), are reserved to the test set. This implies that the last date
of the train set is November 4th, 2018; and the test set begins the next week,
on November 11th, 2018. Both data sets are plotted in Figure 3.

Figure 3: Final dataset composed of the weekly medians of the radon-222
measured in the LSC from July of 2013 to November of 2020. The blue line
corresponds to the values of the train set, and the dashed green line to the
test set.

###  2.2 Encoding the input

In machine learning, the dimensions of the input data depend on the nature of
the problem being analyzed. When the input data is an univariate time series,
the input is a 1D vector, but, for a multivariate time series the input is a
2D vector. When the problem deals with images, the input is 2D matrix or 3D
tensors, depending if the images have one or three channels. Also the input
can have more dimensions for examples where several images are concatenated in
a sequence of images.

Time series seem as an unique vector or example with size as large as the
number of records. The most frequent way of creating the independent and
dependent variables in machine learning is to divide the series in several
frames or sub-series of a fixed size $L$. The look-back $L$ indicates how much
past values are incorporated in the input frames.

As an example, in the weekly data, if $L=4$, it indicates that the frames are
formed by the information of the previous month, but if $L=12$ the frames are
formed by the previous quarter.

To show how the time series is divided in frames, the following time series X
is used. It is composed of ten values and it will be divided into sub-series
with size of seven values.

$X=(x_{1},\ x_{2},\ x_{3},\ x_{4},\ x_{5},\ x_{6},\ x_{7},\ x_{8},\ x_{9},\
x_{10},\ x_{11})$ (1)

The sub-series are:

$\begin{array}[]{ccc}X_{1}&=&(x_{1},\ x_{2},\ x_{3},\ x_{4},\ x_{5},\ x_{6},\
x_{7})\\\ X_{2}&=&(x_{2},\ x_{3},\ x_{4},\ x_{5},\ x_{6},\ x_{7},\ x_{8})\\\
X_{3}&=&(x_{3},\ x_{4},\ x_{5},\ x_{6},\ x_{7},\ x_{8},\ x_{9})\\\
X_{4}&=&(x_{4},\ x_{5},\ x_{6},\ x_{7},\ x_{8},\ x_{9},\ x_{10})\\\
\end{array}$ (2)

And now, the sub-series $X_{1},X_{2},X_{3}$ and $X_{4}$ can be used to predict
the values $x_{8},x_{9},x_{10}$ and $x_{11}$ respectively, forming a dataset
with four events. The number of events in the dataset are the length of the
original time series and minus the input window length. Each variable of the
sub-serie is named as $lag_{i}$ where $i$ is the position in the sub-serie,
and the following value is called $actual$. In the example below,
$x_{8},x_{9},x_{10}$ and $x_{11}$ are values of the variable $actual$ for each
event. The $lag_{1}$ refers to the immediate previous value of $actual$ value,
if $actual=x_{8}$, then $lag_{1}=x_{7}$, and $lag_{L}$ is the last value of
the sub-serie; in the same example, $lag_{7}=x_{1}$.

###  2.3 Neural Networks

Artificial Neural Networks (ANN) are biological-inspired algorithms based in
the brain networks relations. The fundamental unit of ANN, called neuron, is a
nonlinear combination of input values and weights as it is shown below (Eq.
3):

$a_{n}=f_{h}(b_{n}+w^{1}_{n}x_{1}+w^{2}_{n}x_{2}+w^{3}_{n}x_{3}+...+w_{n}^{i}x_{i}+...+w_{n}^{I}x_{I})$
(3)

where $a_{n}$ is the value of the neuron $n$, $x^{i}$ is the i-input,
$w_{n}^{i}$ is the weight of the neuron $n$ respect the input $i$, $b_{n}$ is
the bias unit of the neuron $n$, and $f_{b}(\cdot)$ is the activation function
of the hidden layer, and it has to be a nonlinear function. The neurons are
grouped in layers; where the relationships between layers are: the outputs of
one layer are the inputs for the next layer. The layers are divided in three
classes; the input layer is the first layer of the neuron, the final layer is
called output layer whose values are the network results (or predictions); the
rest of layers are hidden layers. In the following only ANN with a single
hidden layer is considered.

The equation for the output neuron is similar to Eq. 3, where the input x has
to be changed by the hidden neurons (Eq. 4):

$\hat{y}=f_{o}(v_{0}+v^{1}a_{1}+v^{2}a_{2}+v^{3}a_{3}+...+v^{N}a_{N}+...)$ (4)

where $\hat{y}$ is the prediction of the neural network, $v^{n}$ the weight of
the output neuron respect the hidden neuron $n$ and $f_{o}(\cdot)$ is the
activation function of the output layer.

###  2.4 Evaluating the Input Importance

This work follows the algorithms developed by Garson [10] and Olden [12] in
order to know how the input influences the predictions of the models.

The Garson’s algorithm was presented in 1991111The original paper where the
Garson’s algorithm was presented, is not available. The details can be found
in [13, 12]., it calculates the importance of the input using the absolute
value of the products between the weights of the neurons of the hidden layer
$w_{n}^{i}$ and the weights of neurons of the output layer $v^{n}$ (Eq. 5).

$c_{i}=\sum_{j=1}^{m}\frac{|w_{j}^{i}\cdot
v_{j}|}{\sum_{k=1}^{I}|w_{j}^{k}\cdot v_{j}|}$ (5)

where $w$ are the weights of the neurons in the hidden layer, $v$ are the
weights of the neurons in the output layer, $m$ are the number of the neurons
in the hidden layers, and $I$ are the number of the dimensions of the input
variables. Variables with low importance in the prediction obtain coefficients
$c_{i}$ close to zero.

The Olden’s algorithm has a similar structure to Garson’s one but skips the
normalization. This allows coefficients positive and negative. Variables with
low relevance produce coefficients $c_{i}$ close to zero. In the comparison
[12] authors claim a higher performance of the former in comparison with the
latter one.

$c_{i}=\sum_{j=1}^{m}w_{j}^{i}\cdot v_{j}$ (6)

where $w$ are the weights of the neurons in the hidden layer, $v$ are the
weights of the neurons in the output layer, and $m$ are the number of the
neurons in the hidden layers.

###  2.5 Models and Training

Before applying the algorithms previously described, a grid search is
implemented to know which hyperparameters configuration produces the best
prediction. The architecture of the neural networks is: one input layer with
the same neurons as the window input $L$, followed by one hidden layer which
has $N$ neurons with relu as activation function. Then, an output layer with
one neuron is applied whose function is linear. The size of the input and the
number of hidden neurons are the hyperparameters being evaluated in the grid
search. The size of input ranges between 10 and 60 lags values with step of
one, and for the number of neurons from 10 to 100 with step of 5.

During the train sessions the callback EarlyStooping is used. It permits to
stop training after a number of epochs (chosen in 20) when the mean square
error does not improve. It is also able to restore the weights of the best
epoch. The initial number of epochs was chosen in 2000. The rest of the
hyperparameters are:

  * 1.

A batch with a size of 32 events.

  * 2.

The optimizer is Adam with the defaults parameters.

  * 3.

The loss function is the mean squared error (MSE), and also the root mean
squared error (RMSE) and the mean absolute error (MAE) are recorded as
metrics.

During the grid search, ten independent executions of each network are
performed, and to discriminate the best model, the RMSE is the figure of
merit.

The neural networks were implemented on a NVIDIA Pascal GPU with the use of
the library Tensorflow [14]. The libraries scikit-learn [15], pandas, numpy,
matplotlib and seaborn are also used.

##  3 Results and Discussion

###  3.1 Grid Search and Best Performance

The best one-layer architecture to forecast the radon-222 level in the LSC is
chosen after the grid search process. It searches between two hyperparameters,
the input window size —the number of lags used for forming the input— and the
number of neurons in the single hidden layer. Since the weights of neurons are
used for evaluating the relative importance of the variables, they must be as
close as possible to the optimal set of weights. The results of the grid
search are summarized in Figure 4.

Figure 4: Results from the grid search process after 10 independent runs.

The metrics for the ten best architectures from the grid search are detailed
in Table 1. The lowest error corresponds to a model with 12 past values as
input and 55 hidden neurons. It is also notable that the rest of high-quality
configurations use low inputs, in the order of a dozen of lagged values.

Input | Neurons | RMSE | MAE
---|---|---|---
12 | 55 | $10.51\pm 0.09$ | $8.28\pm 0.13$
13 | 90 | $10.58\pm 0.11$ | $8.36\pm 0.14$
14 | 50 | $10.59\pm 0.12$ | $8.36\pm 0.14$
14 | 45 | $10.61\pm 0.14$ | $8.33\pm 0.12$
13 | 95 | $10.61\pm 0.14$ | $8.39\pm 0.11$
12 | 60 | $10.61\pm 0.14$ | $8.33\pm 0.11$
13 | 80 | $10.62\pm 0.15$ | $8.37\pm 0.15$
13 | 85 | $10.63\pm 0.16$ | $8.44\pm 0.14$
14 | 85 | $10.63\pm 0.16$ | $8.35\pm 0.15$
13 | 70 | $10.63\pm 0.16$ | $8.40\pm 0.10$
Table 1: Results for the ten best models according to the Root Mean Squared
Error (RMSE) and the Mean Absolute Error on the test set for 10 independent
runs.

An intuitive initial configuration corresponds to an input 52 lags, which
corresponds to a window of one year. A priori good results are expected.
However, it is not pointed to as a high-quality configuration by this grid
search.

The radon-222 values predicted by the best model -input of 12 and 55 hidden
neurons- are shown in the 5 together with the real values.

Figure 5: Comparison between the real values (green) and the average
prediction (red) over the test set for the ten independent executions of the
best model composed by 55 hidden neurons and input of 12 past weeks. The
filled area represents the maximum and minimum values achieved for 10
independent runs.

It is notable that the forecasts seem to have shifted one value ahead. One
might think that the network only repeated using the last value of the input
window and that the rest tended to confuse the network. To compare this, the
neural network performance is compared with the model $\hat{y}_{t}=y_{t-1}$,
which essentially repeats the past value as forecast.

The RMSE and MAE achieved 11.42 and 8.40 respectively, and both are larger
than those associated with the best neural network. Then the neural network is
better than forecasting the past value. In this way, the rest of the input
variables tend to improve the prediction. Indeed in as many as 67.10% of the
networks analyzed had less error.

###  3.2 Variable Importance

Figures 6 and 7 show the coefficients of the importance of the input
variables. Garson’s algorithm gives a relative high importance to the first
five lags, being the highest for $lag_{1}$. For more delayed lags, the
importance reduces progressively.

Figure 6: Garson’s algorithm for the best model which is composed by 55 hidden
neurons and input of 12 past weeks.The filled area represents the maximum and
minimum values achieved for 10 independent runs.

In comparison with Garson’s algorithm, Olden’s one also gives a relatively
high importance to $lag_{1}$, at the same time that exhibits a more critical
reduction of the importance of the variables as far as the lags grow.

Figure 7: Olden’s Algorithm for the best model which is composed by 55 hidden
neurons and input of 12 past weeks.The filled area represents the maximum and
minimum values achieved.

##  4 Conclusions

In this paper a first attempt to understand the input importance in the deep
models which forecast the time series of radon-222 at Canfranc Underground
Laboratory is presented. A grid search was performed to find the best neural
networks with a single hidden layer to forecast the radon-222 levels. The
result is a neural network with input of 12 past values and 55 hidden neurons.

In order to know how the input influences the predictions, the algorithms of
Olden and Garson were applied to the weights of the best model previously
obtained. Both algorithms show that the $lag_{1}$ value has the greatest
importance for the prediction. For larger lags and depending on the analysis,
the relative importance goes down. Olden’s algorithm still considers relevant
the lags until $lag_{5}$. This result differs in Garson’s algorithm that gives
it a much lower importance. Furthermore, Olden’s algorithm establishes a
stronger reduction of the importance as far as the lags increase than Garson’s
one.

## Acknowledgment

Iñaki Rodríguez-García is funded through the PEJ2018-003089-P project to
promote Young Employment and Implementation of the Youth Guarantee in R+D+i
within the framework of the State Subprogram for the Incorporation of the
State Program for the Promotion of Talent and Employability in R+D+i within
the framework of the State Plan for Scientific and Technical Research and
Innovation 2017-2020. This contract is co-funded by the European Social Fund
and the Youth Employment Initiative through the Youth Employment Operational
Program. Authors wish to express their thanks for the data support and sharing
to Canfranc Underground Laboratory and, particularly to Iulian C. Bandac.

## References

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#  CP violation in the CKM mixing for degenerate quark masses

Ying<EMAIL_ADDRESS>School of Science, Xi’an Jiaotong
University, Xi’an, 710049, China

###### Abstract

CP violation in the CKM mixing is discussed for the case of quark mass
degeneracy that is approximate in the quark mass hierarchy limit. Differing
from the traditional understanding of CP vanishing for degenerate masses, we
find degenerate symmetry plays a non-trivial role in CP violation. The minimal
flavor structure model is reviewed to demonstrate the role of degenerate
symmetry in quark flavor mixing, particularly in CP violation. This relation
between mass hierarchy and CP violation helps us understand the origin of CP
violation and assists the construction of the flavor model.

CKM mixing matrix; mass hierarchy; CP violation

###### pacs:

12.15.Hh, 11.30.Hv, 11.30.Er

##  I Motivation

The standard model (the SM) has successfully described $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$ gauge interactions of the strong and electroweak
interactions in a concise mathematical form with simple guage couplings.
However, the grace of gauge interactions does not apply to Yukawa interactions
Hocker2006 ; Isidori2010 ; Zupan2019 . Yukawa couplings in the SM are a
3-order complex matrix for each kind of quarks and charged leptons, which
govern all flavor phenomenology: mass spectrum and flavor mixing. Due to these
unclear and redundant Yukawa couplings, the relation between fermion masses
and flavor mixing is still unknown.

In the quark sector, up-type/down-type quark masses have a hierarchal
structure

$\displaystyle h_{12}^{q}\equiv\frac{m_{1}^{q}}{m_{2}^{q}}\ll
1,~{}~{}~{}~{}h_{23}^{q}\equiv\frac{m_{2}^{q}}{m_{3}^{q}}\ll
1,~{}~{}~{}~{}{\rm for}~{}~{}q=u,d$

This is a good approximation to explore quark flavor structure. The CKM matrix
should keep its values approximately in the hierarchy limit, providing a key
clue to decoding quark mixing and CP violation. (A similar case is also
discussed in the lepton sector Raidal2008 .) Frequently, it is cursorily
believed that the presence of degenerate mass is a sufficient condition for CP
violation to vanish Jarlskog1985 . This point of view inevitably meets a
challenge in explaining why the CP violating phase in CKM mixing has a large
value rather than a small one as a perturbation correction from the mass
hierarchy.

In the paper, we focus on the relation between mixing matrix and CP violation
in the case of mass degeneracy. After briefly reviewing Jarlskog’s original
1986 work in Sec. II, we highlight two problems that require attention
regarding CP vanishing in mass degeneracy. In Sec. III, we illustrate how a
degenerate $SU(2)$ symmetry contributes non-trivially to CKM mixing.
Discussion on generating CP violating phase from a real mixing matrix is
presented, expressing non-vanishing CP violation in case of mass degeneracy.
We also review the minimal flavor structure proposed in recent research on
flavor structure in Sec. IV. The role of degenerate symmetry in the CP
violating phase is discussed. A summary is provided in Sec. V.

##  II Quark Mixing for Degeneracy

Considering the squared mass matrix $M_{SQ}^{q}\equiv M^{q}(M^{q})^{\dagger}$
with $q=u,d$ for up-type and down-type quarks, it can be diagonalized by left-
handed transformation $U_{L}^{q}$ as

$\displaystyle U_{L}^{q}M_{SQ}^{q}(U_{L}^{q})^{\dagger}={\rm
diag}\Big{(}(m_{1}^{q})^{2},(m_{2}^{q})^{2},(m_{3}^{q})^{2}\Big{)}$ (1)

Defining a commutator

$\displaystyle\Big{[}M_{SQ}^{u},M_{SQ}^{d}\Big{]}=iC$ (2)

the determinant of $C$ has been given in Jarlskog1985  as

$\displaystyle\det[C]$ $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle-2[(m_{3}^{u})^{2}-(m_{2}^{u})^{2}][(m_{3}^{u})^{2}-(m_{1}^{u})^{2}][(m_{2}^{u})^{2}-(m_{1}^{u})^{2}]$
(3)
$\displaystyle\times[(m_{3}^{d})^{2}-(m_{2}^{d})^{2}][(m_{3}^{d})^{2}-(m_{1}^{d})^{2}][(m_{2}^{d})^{2}-(m_{1}^{d})^{2}]J_{CP}$

with Jarlskog invariant

$\displaystyle J_{CP}=Im[V_{11}V_{22}V_{12}^{*}V_{21}^{*}]$ (4)

Here, $V_{ij}$ is an element of quark CKM mixing matrix defined by
$V=U_{L}^{u}(U_{L}^{d})^{\dagger}$. Using the invariance of $\det[C]$,
$J_{CP}$ can be expressed by the standard CKM mixing angles and CP violating
phase as

$\displaystyle J_{CP}=s_{13}c_{13}^{2}s_{23}c_{23}s_{12}c_{12}s_{\delta}$ (5)

with
$s_{ij}=\sin\theta_{ij},c_{ij}=\cos\theta_{ij},s_{\delta}=\sin\delta_{CP}$.
Here, the standard CKM matrix is expressed by

$\displaystyle V=\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\\
0&1&0\\\
-s_{13}e^{i\delta_{CP}}&0&c_{13}\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right)$ (15)

Using Eqs. (3) and (5), CP conservation, i.e. $\delta_{CP}=0$, can lead to a
vanishing $\det[C]$. However, the latter is not a sufficient condition of no
CP violation. An actual situation appears in mass degeneracy as an
approximation of the mass hierarchy limit. If quark mass degeneracy leads to
vanishing CP violation, then $\delta_{CP}$ must be a small quantity resulting
from quark mass hierarchy correction. In mathematical, $\delta_{CP}$ can
generally be expanded in terms of hierarchy $h^{q}_{23}$

$\displaystyle\delta_{CP}=c_{0}+c_{1}h^{q}_{23}+O(h^{2})$ (16)

with cofficient $c_{i}$. In a traditional perspective, the coefficient $c_{0}$
must approach zero as quark masses become degenerate, which leads to a small
value of $\delta_{CP}$ as a perturbation from mass hierarchy. However, the
current experiment value of $\delta_{CP}$ is about $65^{\circ}$ PDG2022 ,
which is too large to be regarded as a small one coming from the hierarchy
contribution.

There is additional doubt of $\delta_{CP}$ vanishing for degenerate masses,
which comes from a theoretical analysis of the CKM matrix. Quark mixing matrix
$V$ is symmetrically determined by up-type quark transformation $U_{L}^{u}$
and down-type one $U_{L}^{d}$ as

$\displaystyle V=U_{L}^{u}(U_{L}^{d})^{\dagger}$

Here, transformation $U_{L}^{q}$ transforms gauge basis to mass basis
(labelled by superscript (m)) $q_{L}=(U_{L})^{\dagger}q_{L}^{(m)}$. If
assuming a mass degeneracy only in up-type quarks rather than down-type
quarks, we can choose a basis on which $M^{u}_{SQ}$ is diagonal and the CKM
mixing matrix can be expressed by $V=(U_{L}^{d})^{\dagger}$, which is not
relative to up-type quark $U_{L}^{u}$. The information about the degeneracy of
up-type quarks is shielded, and $V$ is not affected by it. So, we have to seek
a new way to explain the relation between the current large CP violating phase
and quark mass hierarchy.

##  III Degenerate Symmetry

For degenerate eigenvalues in $M_{SQ}^{q}$, there exists a transformation
$G^{q}$ that keeps the mass eigenvalues invariant in degenerate subspace

$\displaystyle G^{q}{\rm
diag}\Big{(}(m_{1}^{q})^{2},(m_{2}^{q})^{2},(m_{3}^{q})^{2}\Big{)}{G^{f}}^{\dagger}={\rm
diag}\Big{(}(m_{1}^{q})^{2},(m_{2}^{q})^{2},(m_{3}^{q})^{2}\Big{)}$ (17)

In this section, we discuss the role of $G^{q}$ on the CKM mixing matrix and
emphasize $\delta_{CP}$.

Using Eqs. (1) and (17), $M_{SQ}^{q}$ is generally diagonalized by
transformation $G^{q}U_{L}^{q}$ as

$\Big{[}G^{q}U_{L}^{q}\Big{]}M_{SQ}^{q}\Big{[}G^{q}U_{L}^{q}\Big{]}^{\dagger}={\rm
diag}\Big{(}(m_{1}^{q})^{2},(m_{2}^{q})^{2},(m_{3}^{q})^{2}\Big{)}$

The mixing matrix can be expressed

$\displaystyle V=G^{u}U_{L}^{u}{U_{L}^{d}}^{\dagger}{G^{d}}^{\dagger}$ (18)

The formula shows the contributions of $G^{u}$ and $G^{d}$ to quark mixing.
Even in a stricter condition, i.e., the commutator $C=0$ in Eq. (2) instead of
$\det[C]=0$, the degenerate symmetry $G^{q}$ can still cause significant
mixing of quarks. Assuming $C=0$, $M^{u}_{SQ}$ and $M^{d}_{SQ}$ have common
eigenstates, labeled by column vector $v_{i}$ for $i=1,2,3$. We have

$\displaystyle\Big{(}v_{1},v_{2},v_{3}\Big{)}^{\dagger}M^{q}_{SQ}\Big{(}v_{1},v_{2},v_{3}\Big{)}=\Big{(}v_{1},v_{2},v_{3}\Big{)}^{\dagger}\left(\begin{array}[]{ccc}(m_{i}^{q})^{2}&&\\\
&(m_{i}^{q})^{2}&\\\
&&(m_{i}^{q})^{2}\end{array}\right)\Big{(}v_{1},v_{2},v_{3}\Big{)}$ (22)

So, diagonalization transformations $U_{L}^{y}$ and $U_{L}^{d}$ are determined
by

$\displaystyle U_{L}^{u}=U_{L}^{d}=\Big{(}v_{1},v_{2},v_{3}\Big{)}^{\dagger}$
(23)

and Eq. (18) becomes

$\displaystyle V=G^{u}(G^{d})^{\dagger}$ (24)

Due to the non-equality of $G^{u}$ and $G^{d}$, the quark mixing matrix $V$
does not become a unit matrix.

The role of degenerate symmetry in quark mixing gives a new understanding of
the difference between flavor basis and mass basis. The charged current weak
interaction is introduced in the flavor basis in terms of gauge fields

$\displaystyle\mathcal{L}_{cc}=\frac{g}{\sqrt{2}}\bar{u}_{L}\gamma_{\mu}d_{L}W^{+}_{\mu}+h.c.$
(25)

However, we need to diagonalize complex quark mass matrixes to obtain quark
mass eigenvalues $q=(U_{L})^{\dagger}(G^{q})^{\dagger}q^{(m)}$. The Charged
current weak interaction becomes

$\displaystyle\mathcal{L}_{cc}=\frac{g}{\sqrt{2}}\bar{u}_{L}^{(m)}\Big{[}G^{u}U_{L}^{u}(U_{L}^{d})^{\dagger}(G^{d})^{\dagger}\Big{]}\gamma_{\mu}d_{L}^{(m)}W^{+}_{\mu}+h.c.$
(26)

Notice that $G^{u}U_{L}^{u}(U_{L}^{d})^{\dagger}(G^{d})^{\dagger}$ is fixed by
the CKM measurement. So, the $G^{u,d}$ is not a free transformation, i.e. only
a special $G^{u,d}$ is chosen by experiments. It means that in the case of
mass degeneracy, the CKM matrix is built on a specially chosen mass basis, not
any mass basis. Due to this reason, the $G^{q}$ in the degeneracy case is not
treated as identity or free parameters. This is just the source of
misunderstanding on $\delta_{CP}$ vanishing for quark mass degeneracy.
Generally, $G^{q}$ is a $SU(2)$ transformation between two degenerate mass
states. If treating the $SU(2)$ symmetry as a free transformation, CP
violation in the CKM matrix can be eliminated by a suitable $SU(2)$
transformation. A detailed calculation on this procession has been given in
Appendix A.

The transformation $G^{q}$ can provide CP violation. Let us start from a real
mixing matrix $U_{L}^{u}(U_{L}^{d})^{\dagger}$ and consider $G^{q}$ role in
generating $\delta_{CP}$.

Generally, a real $U_{L}^{u}(U_{L}^{d})^{\dagger}$ is factorized by
3-dimensional orthogonal rotation

$\displaystyle
U_{L}^{u}(U_{L}^{d})^{\dagger}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\bar{c}_{23}&\bar{s}_{23}\\\
0&-s_{23}&\bar{c}_{23}\end{array}\right)\left(\begin{array}[]{ccc}\bar{c}_{13}&0&\bar{s}_{13}\\\
0&1&0\\\
-\bar{s}_{13}&0&\bar{c}_{13}\end{array}\right)\left(\begin{array}[]{ccc}\bar{c}_{12}&\bar{s}_{12}&0\\\
-\bar{s}_{12}&\bar{c}_{12}&0\\\ 0&0&1\end{array}\right)$ (36)

with $\bar{s}_{ij}=\sin\bar{\theta}_{ij},\bar{c}_{ij}=\cos\bar{\theta}_{ij}$.
Here, a bar is used just to label mixing angles in the real
$U_{L}^{u}(U_{L}^{d})^{\dagger}$ to distinguish the mixing angles in
phenomenology. When $G^{u}=G^{d}=1$, the CKM mixing matrix is completely
determined by Eq. (36), and there is no CP violation.

In the presence of a degeneracy in the first two families, $G^{q}$ can
generally be parameterized into a $SU(2)$ transformation

$\displaystyle G^{q}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}e^{-i({\phi^{\prime}}^{q}+\phi^{q})/2}\cos\frac{\theta^{q}}{2}&-e^{-i({\phi^{\prime}}^{q}-\phi^{q})/2}\sin\frac{\theta^{q}}{2}&0\\\
e^{i({\phi^{\prime}}^{q}-\phi^{q})/2}\sin\frac{\theta^{q}}{2}&e^{i({\phi^{\prime}}^{q}+\phi^{q})/2}\cos\frac{\theta^{q}}{2}&0\\\
0&0&1\end{array}\right)$ (40)

with 1 rotation angle $\theta^{q}$ and 2 phases
${\phi^{\prime}}^{q},\phi^{q}$.

In terms of Eq. (18), Jarlskog’s invariant can be written as

$\displaystyle J_{CP}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{1}\sin(\phi^{d})\sin\theta^{d}\cos\theta^{u}+\mathcal{C}_{2}\sin\phi^{u}\sin\theta^{u}\cos\theta^{d}$
$\displaystyle+\mathcal{C}_{3}\sin(\phi^{d}+\phi^{u})\sin\theta^{d}\sin\theta^{d}+\mathcal{C}_{4}\sin(\phi^{d}-\phi^{u})\sin\theta^{d}\sin\theta^{d}$

Here, coefficients $\mathcal{C}_{i}$ are

$\displaystyle\mathcal{C}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{c}_{13}^{2}\bar{s}_{13}\bar{s}_{23}\bar{c}_{23}$
$\displaystyle\mathcal{C}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{c}_{13}\bar{c}_{23}\Big{(}\bar{s}_{12}\bar{c}_{12}+\bar{s}_{12}\bar{c}_{12}\bar{c}_{13}^{2}\bar{c}_{23}^{2}-2\bar{s}_{12}\bar{c}_{12}\bar{c}_{23}^{2}-2\bar{s}_{23}\bar{c}_{23}\bar{s}_{13}\bar{c}_{12}^{2}+\bar{s}_{23}\bar{c}_{23}\bar{s}_{13}\Big{)}$
$\displaystyle\mathcal{C}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bar{c}_{13}\bar{c}_{23}\Big{(}\bar{s}_{12}^{2}\bar{c}_{23}^{2}-\bar{s}_{12}^{2}\bar{c}_{13}^{2}+\bar{c}_{12}^{2}\bar{s}_{13}^{2}\bar{s}_{23}^{2}+2\bar{s}_{13}\bar{s}_{12}\bar{c}_{12}\bar{s}_{23}\bar{c}_{23}\Big{)}$
$\displaystyle\mathcal{C}_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\bar{c}_{13}\bar{c}_{23}\Big{(}\bar{c}_{12}^{2}\bar{c}_{13}^{2}-\bar{c}_{12}^{2}\bar{c}_{23}^{2}-\bar{s}_{12}^{2}\bar{s}_{13}^{2}\bar{s}_{23}^{2}+2\bar{s}_{13}\bar{s}_{12}\bar{c}_{12}\bar{s}_{23}\bar{c}_{23}\Big{)}$

Non-vanishing $J_{CP}$ can be generated by $SU(2)$ transformation from a real
orthogonal matrix in Eq. (36). Phases ${\phi^{\prime}}^{u}$ and
${\phi^{\prime}}^{d}$ have no contribution to $J_{CP}$ because they can be
eliminated by quark field rephasing. CP violation requirement to complex
phases is provided by $\phi^{u}$ and $\phi^{d}$. Another factor affecting CP
violation is $SU(2)$ rotation angles $\theta^{u,d}$. A more accurate
requirement for CP violation is that at least one of the phases $\phi^{u,d}$
and one of the angles $\theta^{u,d}$ do not vanish, meaning that $G^{u}$ or
$G^{d}$ can serve as an independent source of CP violation.

Even if $\bar{s}_{13}=0$, there is also non-vanishing $\delta_{CP}$ generated
from $G^{u}$ or $G^{d}$. For the sake of simplicity, let us take $G^{d}=1$ and
focus on $G^{u}$. The Eq. (III) can be simplified to:

$\displaystyle
J_{CP}=\frac{1}{2}\bar{c}_{12}\bar{c}_{23}\bar{s}_{12}\bar{s}_{23}^{2}\sin\phi^{u}\sin\theta^{u}$
(41)

So, we get the condition of $\delta_{CP}\neq 0$ is

$\displaystyle\bar{\theta}_{12}\neq 0,\pi/2~{}\rm{and}~{}\bar{\theta}_{23}\neq
0,\pi/2$ (42)

More further result shows that the condition of generating CP violation from
real rotation by $G^{q}$ is that at least two of three mixing angles are not
$0$ or $\pi/2$.

Besides the CP violating phase, the contribution of $G^{q}$ to other mixing
angles can be found in Appendix B.

##  IV CP violation in The Minimal Flavor Structure

Recently, the minimal flavor structure (MFS) has been proposed to address
quark mass hierarchy and CKM mixing in terms of a flat mass pattern
ZhangJPG2023 . It successfully outputs 10 experimental values, including 6
quark masses, 3 mixing angles, and 1 CP violation, from just 10 model
parameters. In the MFS, the CKM matrix is determined by an approximate
degenerate symmetry. This example helps us understand the relationship between
flavor mixing and mass degeneracy.

As a result of the requirement for mass hierarchy Zhang2023arXiv3 , the up-
type and down-type quark mass matrices in the MFS have a common factorized
form as

$\displaystyle M^{q}$ $\displaystyle=$ $\displaystyle
m^{q}_{\Sigma}Y_{L}^{q}I^{q}(Y_{R}^{q})^{\dagger}$ (43)

with the total mass of quarks $m^{q}_{\Sigma}=m_{1}^{q}+m_{2}^{q}+m_{3}^{q}$
and diagonal Yukawa matrix $Y_{L}^{q}={\rm
diag}(e^{i\lambda_{1}^{q}},e^{i\lambda^{q}_{2}},1)$. A significant aspect of
Eq. (43) is that the left-handed $Y_{L}^{q}$ fully provides the required
complex phases for CP violation.

$I^{q}$ is responsible for generating hierarchical masses of quarks, which can
be represented by a nearly flat real matrix

$\displaystyle I^{q}$ $\displaystyle=$ $\displaystyle I_{0}^{q}+\Delta^{q}$
$\displaystyle I_{0}^{q}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\begin{array}[]{ccc}1&1&1\\\ 1&1&1\\\
1&1&1\end{array}\right)$ (47) $\displaystyle\Delta^{q}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\begin{array}[]{ccc}0&\delta^{q}_{12}&\delta^{q}_{13}\\\
\delta^{q}_{12}&0&\delta^{q}_{23}\\\
\delta^{q}_{13}&\delta^{q}_{23}&0\end{array}\right)$ (51)

In the mass hierarchy limit, quark masses can be gotten by diagonalizing the
flat matrix $I_{0}^{q}$ by transformation $S_{0}$

$\displaystyle S_{0}^{q}I_{0}^{q}(S_{0}^{q})^{T}={\rm diag}(0,0,1)$ (52)

with

$\displaystyle
S_{0}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{ccc}\sqrt{3}&0&-\sqrt{3}\\\
-1&2&-1\\\ \sqrt{2}&\sqrt{2}&\sqrt{2}\end{array}\right)$ (56)

For 2-fold degenerate eigenvalues on the right side of Equation (52), there
exists a real $SO(2)$ rotation symmetry $R_{0}^{q}$

$\displaystyle R^{q}_{0}{\rm diag}(0,0,1){R^{q}_{0}}^{T}={\rm diag}(0,0,1)$
(57)

with

$\displaystyle
R_{0}^{q}=\left(\begin{array}[]{ccc}\cos\theta^{q}&\sin\theta^{q}&0\\\
-\sin\theta^{q}&\cos\theta^{q}&0\\\ 0&0&1\end{array}\right)$ (61)

Because all complex phases have been factored into $Y_{L}^{q}$ in the MFS, the
symmetry is only $SO(2)$ rather than $SU(2)$ XuEPL2023 . So, we have

$\displaystyle\Big{(}R_{0}^{q}S_{0}^{q}\Big{)}I_{0}^{q}\Big{(}R_{0}^{q}S_{0}^{q}\Big{)}^{T}={\rm
diag}(0,0,1)$

Thus, the CKM matrix in charged current weak interaction is written in terms
of two $SO(2)$ transformations of up-type and down-type quarks as

$\displaystyle V=R_{0}^{u}S_{0}{\rm
diag}(e^{i\lambda_{1}},e^{i\lambda_{2}},1)S_{0}^{T}{R_{0}^{d}}^{T}$ (62)

The true hierarchal masses are also addressed by $I^{q}$ with correction
$\Delta^{q}$ in Eq. (51) that is responding to generate two lighter quark
masses for three real diagonal perturbations $\delta_{ij}^{q}$. Up to
$\mathcal{O}(h^{1})$, the broken $\delta_{ij}^{q}$ is set as

$\displaystyle\delta_{12}^{q}=\delta_{23}^{q}=-\frac{9}{4}h_{23}^{q},~{}~{}\delta_{13}^{q}=0$
(63)

$I^{q}$ is diagonalized by a corrected $S_{h}^{q}$

$\displaystyle S_{h}^{q}I^{q}{S_{h}^{q}}^{T}=\left(\begin{array}[]{ccc}0&&\\\
&h_{23}^{q}&\\\ &&1-h_{23}^{q}\end{array}\right)+\mathcal{O}(h^{2})$ (67)

with

$\displaystyle S_{h}^{q}$ $\displaystyle=$ $\displaystyle
S_{0}+\frac{h_{23}^{q}}{4\sqrt{3}}\left(\begin{array}[]{ccc}0&0&0\\\
\sqrt{2}&\sqrt{2}&\sqrt{2}\\\ 1&-2&1\end{array}\right)+\mathcal{O}(h^{2})$
(71)

In Zhang2023arXiv3 , it has been studied that the $SO(2)$ symmetry is still
valid in 1-order hierarchy approximation, i.e., the eigenvalues ${\rm
diag}(0,h_{23}^{q},1-h_{23}^{q})$ is invariant under an $SO(2)$ transformation
$R_{h}^{q}$

$\displaystyle R_{h}^{q}{\rm
diag}(0,h_{23}^{q},1-h_{23}^{q}){R_{h}^{q}}^{T}={\rm
diag}(0,h_{23}^{q},1-h_{23}^{q})+\mathcal{O}(h^{2})$ (72)

with the transformation

$\displaystyle
R_{h}^{q}=R_{0}^{q}+\frac{h_{23}^{q}}{\sqrt{2}}\left(\begin{array}[]{ccc}0&0&\sin\theta^{q}\\\
0&0&\cos\theta^{q}-1\\\ -\sin\theta^{q}&\cos\theta^{q}-1&0\end{array}\right)$
(76)

So, the CKM mixing matrix in the 1-order hierarchy becomes

$\displaystyle V=R_{h}^{u}S_{h}^{u}{\rm
diag}(e^{i\lambda_{1}},e^{i\lambda_{2}},1){S_{h}^{d}}^{T}{R_{h}^{d}}^{T}$ (77)

The MFS has been successfully checked by fitting hierarchal masses of up-type
and down-type quarks and the CKM mixing. A set of fit parameters is listed in
Tab. 1.

Table 1: MFS fit the CKM mixing  MFS para. |  $\theta^{u}=0.01926$, $\theta^{d}=3.389$, $\lambda_{1}=0.004102$, $\lambda_{2}=-0.04306$
---|---
Fit Results |  $s_{12}=0.2259$, $s_{23}=0.04172$, $s_{13}=0.003810$, $\delta_{CP}=1.118$
CKM exp. PDG2022  | $\begin{array}[]{ccc}&s_{12}=0.22500\pm 0.00067,&s_{23}=0.04182^{+0.00085}_{-0.00074}\\\ &s_{13}=0.00369\pm 0.00011,&\delta_{CP}=1.144\pm 0.027\end{array}$

$\theta^{u}$ and $\theta^{d}$ listed in Tab. 1 means that approximate $SO(2)$
symmetry of mass matrix is broken by charged current weak interaction. And
$SO(2)$ rotation angles $\theta^{u}$ and $\theta^{d}$ are fixed by the SM. The
$S_{h}$ transforms the quark flavor basis $q_{L}$ to a mass basis
$q_{L}^{(m)}$

$q_{L}=S_{h}^{T}q_{L}^{(m)}$

And charged current weak interaction decides which mass basis appears in the
CKM mixing. So, only the fitted $\theta^{u,d}$ is chozen

$\displaystyle u_{L}$ $\displaystyle=$ $\displaystyle
S_{h}^{T}{R_{h}^{u}}^{T}u_{L}^{(m)},$ (78) $\displaystyle d_{L}$
$\displaystyle=$ $\displaystyle S_{h}^{T}{R_{h}^{d}}^{T}d_{L}^{(m)}$ (79)

MFS also provides an understanding of the origin of CP violation. CP violation
comes from Yukawa phases in $Y_{L}^{q}$ and also depends on rotation angles
$\theta^{u},\theta^{d}$. These parameters are independent on mass hierarchy,
They do not vanish in the mass hierarchy limit, which explains why the large
CP violating phase does not stem from the mass hierarchy.

##  V Summary

The study of degenerate symmetry as an approximation of mass hierarchy has
been studied in quark CKM mixing, particularly in CP violation. Degenerate
symmetry $G^{q}$ plays a non-trivial role in CKM mixing and picks up a special
mass eigenstate as the state of the CKM mixing matrix in charged current weak
interaction. It explains why CP violation may not vanish for mass degeneracy
and also why a large CPV can exist in the hierarchy limit. As an example, the
role of degenerate symmetry is shown in the MFS. The relation between CP
violation and degenerate symmetry is also illustrated in the MFS. These
results assist in improving the understanding of CP violation and aid in
constructing a final flavor structure in the future.

## Acknowledgements

This work is supported by Shaanxi Foundation SAFS 22JSY035 of China.

##  Appendix A Elimination of $\delta_{CP}$ by a free $SU(2)$ transformation

Phenomenology, the CKM mixing matrix $V$ can be expressed by 3 mixing angles
and 1 CP violating phase by redefining quark fields. This process is known as
rephasing. For degenerate masses, there is a $SU(2)$ symmetry to keep mass
eigenvalues invariant. As we have mentioned in Sec. III, these $SU(2)$
transformations, $G^{u}$ and $G^{d}$, are decided by experiments. However, if
$G^{u,d}$ is regarded as free transformation, it can be utilized to eliminate
some d.o.f. in the CKM mixing. In the appendix, we adopt the standard CKM
matrix in Eq. (15) and demonstrate the elimination of $\delta_{CP}$ through a
free $SU(2)$ transformation. Without the loss of generality, we only consider
up-type quark degeneracy between the first two families. Degenerate symmetry
$G^{u}$ is parameterized by $\theta^{u}$, $\phi^{u}$ and $\phi^{\prime u}$ as
shown in Eq. 40.

To obtain a vanishing CP violation, a transformed CKM matrix $G^{u}V$ must
have a vanishing Jarlskog invariant. After a tedious calculation, $J_{CP}$ is
expressed by

$\displaystyle
J_{CP}=\mathcal{C}_{a}\cos\theta^{u}+\mathcal{C}_{b}\sin\phi^{u}\sin\theta^{u}+\mathcal{C}_{c}\cos\phi^{u}\sin\theta^{u}$
(80)

with coefficients

$\displaystyle\mathcal{C}_{a}$ $\displaystyle=$ $\displaystyle
c_{13}^{2}s_{13}c_{23}s_{23}c_{12}s_{12}s_{\delta}$
$\displaystyle\mathcal{C}_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}c_{13}c_{23}\Big{[}s_{12}c_{12}(c_{13}^{2}-c_{23}^{2}+s_{13}^{2}s_{23}^{2}-2s_{\delta}^{2}s_{13}^{2}s_{23}^{2})-c_{\delta}(2c_{12}^{2}-1)s_{13}s_{23}c_{23}\Big{]}$
$\displaystyle\mathcal{C}_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}c_{13}c_{23}s_{13}s_{\delta}\Big{[}2s_{12}c_{12}s_{13}s_{23}^{2}c_{\delta}-s_{23}c_{23}(2c_{12}^{2}-1)\Big{]}$

Here, mixing angles $\theta_{ij}$ and $\delta_{CP}$ are ones before $SU(2)$
transformation.

Vanishing $J_{CP}$ requires $\theta^{u}$ and $\phi^{u}$ meet

$\displaystyle\tan\theta^{u}=-\frac{\mathcal{C}_{a}}{\mathcal{C}_{b}\sin\phi^{u}+\mathcal{C}_{c}\cos\phi^{u}}$
(81)

##  Appendix B  $G^{q}$ roles to mixing angles

Eq. (III) has expressed degenerate symmetry $G^{q}$ contribution to Jarlskog
invariant initializing from a real mixing matrix. We will now examine the
influence of $G^{q}$ on the remaining three mixing angles. Three CKM mixing
angles can be calculated from mixing matrix $V$

$\displaystyle s_{13}$ $\displaystyle=$ $\displaystyle|V_{13}|$ (82)
$\displaystyle s_{12}^{2}$ $\displaystyle=$
$\displaystyle\frac{|V_{12}|^{2}}{1-|V_{13}|^{2}}$ (83) $\displaystyle
s_{23}^{2}$ $\displaystyle=$
$\displaystyle\frac{|V_{23}|^{2}}{1-|V_{13}|^{2}}$ (84)

Let us consider a real $U_{L}^{u}(U_{L}^{d})^{\dagger}$ as shown in Eq. (36).
Using Eq. (18) and Eq. (40), three mixing angles are determined by

$\displaystyle s_{13}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\bar{c}_{u})\bar{s}_{13}^{2}+\bar{c}_{13}\bar{s}_{23}\Big{[}\bar{c}_{13}\bar{s}_{23}\frac{1}{2}(1-c_{u})-\bar{s}_{13}s_{u}\cos\phi^{u}\Big{]}$
$\displaystyle s_{12}^{2}(1-s_{13}^{2})$ $\displaystyle=$
$\displaystyle\mathcal{C}_{A1}+\mathcal{C}_{A2}c_{d}+\mathcal{C}_{A3}s_{d}+\mathcal{C}_{A4}c_{u}+\mathcal{C}_{A5}c_{u}c_{d}+\mathcal{C}_{A6}c_{u}s_{d}$
$\displaystyle+\mathcal{C}_{A7}s_{u}+\mathcal{C}_{A8}s_{u}c_{d}+\mathcal{C}_{A9}s_{u}s_{d}$
$\displaystyle s_{23}^{2}(1-s_{13}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\bar{s}_{13}^{2}+\bar{c}_{13}^{2}\bar{s}_{23}^{2}+(\bar{c}_{13}^{2}\bar{s}_{23}^{2}--\bar{s}_{13}^{2})c_{u}+2\bar{s}_{13}\bar{c}_{13}\bar{s}_{23}s_{u}\cos\phi^{u}\Big{]}$

Here, coefficients $\mathcal{C}_{Ai}$ are listed in the following

$\displaystyle\mathcal{C}_{A1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(2-\bar{s}_{23}^{2}-\bar{s}_{13}^{2}\bar{c}_{23}^{2})$
$\displaystyle\mathcal{C}_{A2}$ $\displaystyle=$
$\displaystyle-\bar{c}_{12}\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}+\frac{1}{4}(1-2\bar{s}_{12}^{2})(\bar{s}_{13}^{2}-\bar{s}_{23}^{2}-\bar{s}_{13}^{2}\bar{s}_{23}^{2})$
$\displaystyle\mathcal{C}_{A3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\bar{c}_{23}(-1+2\bar{s}_{12}^{2})\bar{s}_{13}\bar{s}_{23}+\bar{c}_{12}\bar{s}_{12}(\bar{s}_{23}^{2}-\bar{s}_{13}^{2}+\bar{s}_{13}^{2}\bar{s}_{23}^{2})\Big{]}\cos\phi^{d}$
$\displaystyle\mathcal{C}_{A4}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\bar{s}_{23}^{2}-\bar{s}_{13}^{2}-\bar{s}_{13}^{2}\bar{s}_{23}^{2})$
$\displaystyle\mathcal{C}_{A5}$ $\displaystyle=$
$\displaystyle\bar{c}_{12}\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}-\frac{1}{4}(-1+2\bar{s}_{12}^{2})(-2+\bar{s}_{23}^{2}+\bar{s}_{13}^{2}+\bar{s}_{13}^{2}\bar{s}_{23}^{2})$
$\displaystyle\mathcal{C}_{A6}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\bar{c}_{23}(1-2\bar{s}_{12}^{2})\bar{s}_{13}\bar{s}_{23}+\bar{c}_{12}\bar{s}_{12}(2-\bar{s}_{23}^{2}-\bar{s}_{13}^{2}-\bar{s}_{13}^{2}\bar{s}_{23}^{2})\Big{]}\cos\phi^{d}$
$\displaystyle\mathcal{C}_{A7}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{c}_{13}\bar{s}_{13}\bar{s}_{23}\cos\phi^{u}$
$\displaystyle\mathcal{C}_{A8}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{c}_{13}\Big{[}2\bar{c}_{12}\bar{c}_{23}\bar{s}_{12}+(1-2\bar{s}_{12}^{2})\bar{s}_{13}\bar{s}_{23}\Big{]}\cos\phi^{u}$
$\displaystyle\mathcal{C}_{A9}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{c}_{13}(2\bar{c}_{23}\bar{s}_{12}^{2}-\bar{c}_{23}+2\bar{c}_{12}\bar{s}_{12}\bar{s}_{13}\bar{s}_{23})\cos\phi^{d}\cos\phi^{u}-\frac{1}{2}\bar{c}_{13}\bar{c}_{23}\sin\phi^{d}\sin\phi^{u}$

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