$|A\tilde{f}(U_{1},U_{2})|$ and obtain $\hat{\Lambda}_{\epsilon,\zeta}$ by
minimizing $B$.

In the limit of zero mixture between states
$|J=\frac{1}{2},j_{z}=\pm\frac{1}{2}\rangle$ and
$|J=\frac{3}{2},j_{z}=\pm\frac{1}{2}\rangle$ on the Fermi surfaces, the Fermi
level is contributed by $|J=\frac{1}{2},j_{z}=\pm\frac{1}{2}\rangle$ and we
obtain the interaction coefficients of each channels listed in Table.8. We can
see that the sign of the coefficients of each channel are determined by the
sign of $U_{1}$ and $U_{2}$. The results from the first principle calculation
require a small mixture angle $\theta$ so we treat the mixture as a
pertubation and discuss the following three situations,

Table 8: The interaction coefficients of each irrep in the limit of no mixture between $|J=\frac{1}{2},j_{z}=\pm\frac{1}{2}\rangle$ and $|J=\frac{3}{2},j_{z}=\pm\frac{1}{2}\rangle$ on the Fermi surfaces. Irreps of the channels | Coefficients $\tilde{f}_{\epsilon,\kappa}(U_{1},U_{2})$
---|---
$\mathrm{A}_{1u}$ |  $4U_{2}$, $2U_{1}$
$\mathrm{A}_{2u}$ | $2U_{1}$
$\mathrm{E}_{u}$ |  $4U_{2}$, $2U_{1}$, $2U_{1}$
$\mathrm{T}_{1u}$ |  $4U_{2}$, $2U_{1}$, $2U_{1}$, $0$
$\mathrm{T}_{2u}$ |  $4U_{2}$, $2U_{1}$, $2U_{1}$, $0$, $0$

#####  $U_{2}<0$, $U_{1}>0$.

According to Table.8, four channels can contribute superconductivities and
they are composed by one $\mathrm{A}_{1u}$, one $\mathrm{E}_{u}$, one
$\mathrm{T}_{1u}$ and one $\mathrm{T}_{2u}$, respectively. The difference
between the interaction coefficients $\tilde{f}_{\epsilon}(U_{1},U_{2})$ of
the four channels is treated as pertubation for they have the same coefficient
equalling to $4U_{2}$ in the limit of zero mixture. All of the channels have
only one component for $\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}$. For the one
dimensional irrep $\mathrm{A}_{1u}$, $\hat{\Lambda}_{\zeta}$ has only one
component, too. So $\hat{\Lambda}_{\mathrm{A}_{1u}}=1$. While for
$\mathrm{E}_{u}$, $\hat{\Lambda}_{\kappa}$ has two components. We write
$\hat{\Lambda}_{\kappa}=(\sin\theta,\cos\theta)^{\intercal}$. For
$\mathrm{T}_{1u}$ and $\mathrm{T}_{2u}$, $\hat{\Lambda}_{\kappa}$ has three
components. We write
$\hat{\Lambda}_{\kappa}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\intercal}$.
We sample on $\theta$, $\phi$ to obtain the minimum of $B$ and obatin
$\hat{\Lambda}_{\mathrm{E}_{u},\zeta}=(1,0)^{\intercal}$ and
$\hat{\Lambda}_{\mathrm{T}_{1u},\zeta},\hat{\Lambda}_{\mathrm{T}_{2u},\zeta}=(1,0,0)^{\intercal},(0,1,0)^{\intercal},(0,0,1)^{\intercal}$.

#####  $U_{2}>0$, $U_{1}<0$.

According to Table.8, five channels can contribute the superconductivities and
they are composed by one $A_{1u}$, one $A_{2u}$, two $E_{u}$, two $T_{1u}$ and
two $T_{2u}$, respectively. Similar to the last condition, they have the same
coefficients in the limit of zero mixture and we treat the difference between
$\tilde{f}_{\epsilon}(U_{1},U_{2})$ as pertubation. For $A_{1u}$ and $A_{2u}$,
$\hat{\Lambda}_{\epsilon}$ only has one component. However, for other
channels, they are composed by two irreps and the vector
$\hat{\Lambda}_{\kappa}$ have two components. We diagonalize the matrix
$\mathbb{A}$ selecting the eigenvector with bigger eigenvaule and obtain
$\hat{\Lambda}_{\epsilon,\kappa}=(\upsilon_{\epsilon,1},\upsilon_{\epsilon,2})^{\intercal}$.
And then we write
$\hat{\Lambda}_{E_{u},\zeta}=(\sin\theta,\cos\theta)^{\intercal}$ and
$\hat{\Lambda}_{T_{1u}(T_{2u}),\zeta}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\intercal}$.
We sample on $\theta$, $\phi$ to minimizing $B$ and obtain
$\hat{\Lambda}_{E_{u},\zeta}=(1,0)^{\intercal}$ and
$\hat{\Lambda}_{T_{1u}(T_{2u}),\zeta}=(1,0,0)^{\intercal},(0,1,0)^{\intercal},(0,0,1)^{\intercal}$.

#####  $U_{2}<0$, $U_{1}<0$.

According to Table.8, five channels can contribute the superconductivities and
they are composed by two $\mathrm{A}_{1u}$, one $\mathrm{A}_{2u}$, three
$\mathrm{E}_{u}$, three $\mathrm{T}_{1u}$ and three $\mathrm{T}_{2u}$,
respectively. Different from the two conditions above, the interaction
coefficients of different irreps within one channel are not the same and the
difference between $\tilde{f}_{\epsilon}(U_{1},U_{2})$ can not be treated as
pertubation anymore. There is only one $A_{2u}$ irrep.
$\hat{\Lambda}_{\mathrm{A}_{2u}}$ is a one component vector. While for other
channels, we consider $\frac{1}{A\tilde{f}_{\epsilon}(U_{1},U_{2})}$ as a
whole and solve the $\hat{\Lambda}_{\epsilon,\kappa}$ maximizing
$\frac{1}{A\tilde{f}_{\epsilon}(U_{1},U_{2})}$. For $\mathrm{A}_{1u}$, there
are two components in the vector $\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}$ and
we set
$\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}=(\varphi_{1},\varphi_{2})^{\intercal}$.
The effective interaction coefficient of the channel with the symmetry of
$\mathrm{A}_{1u}$ $\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})$ can be obtained
as,

$\tilde{f}_{\mathrm{A}_{1u}}=\frac{1}{\frac{\varphi_{1}^{2}}{4U_{2}}+\frac{\varphi_{2}^{2}}{2U_{1}}}.$
(64)

Under the basis of
$\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}=(1,0)^{\intercal},(0,1)^{\intercal}$,
$\mathbb{A}$ is a two by two matrix and we can obtain,

$\mathbb{A}=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right),\qquad
A=\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}^{\dagger}\mathbb{A}\hat{\Lambda}_{\mathrm{A}_{1u}}=\varphi_{1}^{2}a_{11}+\varphi_{2}^{2}a_{22}+(a_{12}+a_{21})\varphi_{1}\varphi_{2}.$
(65)

We substitute Eq.64 and Eq.65 into $A\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})$
and obtain,

$A\tilde{f}_{A_{1u}}(U_{1},U_{2})=\frac{\varphi_{1}^{2}a_{11}+\varphi_{2}^{2}a_{22}+(a_{12}+a_{21})\varphi_{1}\varphi_{2}}{\frac{\varphi_{1}^{2}}{4U_{2}}+\frac{\varphi_{2}^{2}}{2U_{1}}}=\frac{a_{11}\frac{\varphi_{1}^{2}}{\varphi_{2}^{2}}+a_{22}+(a_{12}+a_{21})\frac{\varphi_{1}}{\varphi_{2}}}{\frac{1}{4U_{2}}\frac{\varphi_{1}^{2}}{\varphi_{2}^{2}}+\frac{1}{2U_{1}}}.$
(66)

We write $x=\frac{\varphi_{1}}{\varphi_{2}}$ and calculate the derivation of
$A\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})$ about the variable $x$,

$\frac{\mathrm{d}A\tilde{f}_{\epsilon}(U_{1},U_{2})}{\mathrm{d}x}=\frac{(a_{12}+a_{21})\left(\frac{1}{2U_{1}}-\frac{1}{4U_{2}}x^{2}\right)+(-\frac{a_{22}}{2U_{2}}+\frac{a_{11}}{U_{1}})x}{\left(\frac{1}{4U_{2}}x^{2}+\frac{1}{2U_{1}}\right)^{2}}$
(67)

The matrix $\mathbb{A}$ is Hermitian,
$\mathbb{A}=\mathcal{A}^{\dagger}\mathcal{A}$. So $a_{12}=a_{21}^{*}$,
$a_{12}+a_{21}$ is a real number. In the limitation of
$x\rightarrow\pm\infty$, the denominator is in the fourth order of $x$ while
the numerator is in the order up to $2$ of $x$ depending on whether $a_{12}$
and $\frac{a_{11}}{U_{1}}-\frac{a_{22}}{2U_{2}}$ are zero or not, thus we have
$\lim_{x\rightarrow\pm\infty}\lim_{x\rightarrow\pm\infty}\mathrm{d}(A\tilde{f}_{\epsilon}(U_{1},U_{2}))/\mathrm{d}x=0$.
And in the situation of $x\rightarrow\infty$ and $x\rightarrow-\infty$, the
vector of $\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}$ is taken as
$(1,0)^{\intercal}$ and $(-1,0)^{\intercal}$ respectively, which are equal to
a global phase $-1$. So we have
$\lim_{x\rightarrow\infty}A\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})=\lim_{x\rightarrow-\infty}A\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})$.
We show the dispersion of $A\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})$ with
respect to $x$ in Fig.6 In Fig.6(a), $a_{12}<0$, the derivation of
$A\tilde{f}_{\mathrm{A}_{1u}}(U_{1},U_{2})$ in Eq.67 is positive with $x$
taken as infinity,
$\lim_{x\rightarrow\pm\infty}\mathrm{d}(A\tilde{f}_{\epsilon}(U_{1},U_{2}))/\mathrm{d}x>0$.
In Fig.6(b), $a_{12}<0$, we have
$\lim_{x\rightarrow\pm\infty}\mathrm{d}(A\tilde{f}_{\epsilon}(U_{1},U_{2}))/\mathrm{d}x<0$.
In Fig.6(c) and Fig.6(d), $a_{12}=0$ and the $\mathbb{A}$ matrix is diagonal.
The vector $\hat{\Lambda}_{\mathrm{A}_{1u},\kappa}$ is taken as
$(0,1)^{\intercal}$ for $a_{11}/U_{1}-a_{22}/2U_{2}>0$ and $(1,0)^{\intercal}$
for $a_{11}/U_{1}-a_{22}/2U_{2}<0$.

Figure 6:  Schematic plot for the dispersion of
$A\tilde{f}_{\mathrm{A}_{1u}(U_{1},U_{2})}$ with respect to $x$. In Fig.(a),
we have $a_{12}<0$. In Fig.(b) we have $a_{12}>0$. In Fig.(c) we have
$a_{12}=0$, $-\frac{a_{22}}{2U_{2}}+\frac{a_{11}}{U_{1}}>0$. In Fig.(d) we
have $a_{12}=0$, $-\frac{a_{22}}{2U_{2}}+\frac{a_{11}}{U_{1}}<0$. The red
points denote the minimum of $A\tilde{f}_{\epsilon}(U_{1},U_{2})$.

For $\mathrm{E}_{u}$, $\mathrm{T}_{1u}$ and $\mathrm{T}_{2u}$,
$\hat{\Lambda}_{\epsilon,\kappa}$ has three components and we write
$A\tilde{f}_{\epsilon,\kappa}(U_{1},U_{2})$ as,

$A\tilde{f}_{\epsilon}(U_{1},U_{2})=\frac{\hat{\Lambda}_{\epsilon,\kappa}^{\dagger}\mathbb{A}\hat{\Lambda}_{\epsilon,\kappa}}{\hat{\Lambda}_{\epsilon,\kappa}^{\dagger}\mathbb{F}\hat{\Lambda}_{\epsilon,\kappa}},$
(68)

where $\mathbb{F}$ is a three dimensional matrix,
$1/\hat{\Lambda}_{\epsilon,\kappa}^{\dagger}\mathbb{F}\hat{\Lambda}_{\epsilon,\kappa}=\tilde{f}_{\epsilon}(U_{1},U_{2})$.
Under the basis of
$\hat{\Lambda}_{\epsilon,\kappa}=(1,0,0)^{\intercal},(0,1,0)^{\intercal},(0,0,1)^{\intercal}$,
we can write matrices $\mathbb{A}$ and $\mathbb{F}$ as,

$\mathbb{A}=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{31}\\\
a_{21}&a_{22}&a_{23}\\\
a_{31}&a_{32}&a_{33}\end{array}\right),\qquad\mathbb{F}=\left(\begin{array}[]{ccc}\frac{1}{\tilde{f}_{\epsilon,1}(U_{1},U_{2})}&0&0\\\
0&\frac{1}{\tilde{f}_{\epsilon,2}(U_{1},U_{2})}&0\\\
0&0&\frac{1}{\tilde{f}_{\epsilon,3}(U_{1},U_{2})}\end{array}\right).$ (69)

We diagonalize matrix $\mathbb{A}$ and obtain three eigenvaules. Among them,
there are two double degenerated eigenvaules denoted as $a_{1}$ and the third
eigenvaule denoted as $a_{2}$. Under the basis of the eigenvectors of matrix
$\mathbb{A}$, we have,

$\mathbb{A}^{\prime}=\left(\begin{array}[]{ccc}a_{1}&0&0\\\ 0&a_{1}&0\\\
0&0&a_{2}\end{array}\right),\qquad\mathbb{F}^{\prime}=\left(\begin{array}[]{ccc}u_{1}&0&u_{13}\\\
0&u_{2}&u_{23}\\\ u_{13}&u_{23}&u_{3}\end{array}\right),$ (70)

where the $2$ by $2$ block on the left top of matrix $\mathbb{F}$ can be
diagonalized for the first two eigenvaules of matrix $\mathbb{A}$ are
degenerated. According to Table.8, in the limit of no mixture between states
$|J=\frac{1}{2},j_{z}=\pm\frac{1}{2}\rangle$ and
$|J=\frac{3}{2},j_{z}=\pm\frac{1}{2}\rangle$ on the Fermi surfaces, we have
$\tilde{f}_{\epsilon,1}(U_{1},U_{2})=\tilde{f}_{\epsilon,2}(U_{1},U_{2})=2U_{1}$,
$\tilde{f}_{\epsilon,3}(U_{1},U_{2})=4U_{2}$. The similarity transformation
matrix transforming $\mathbb{F}$ to $\mathbb{F}^{\prime}$ is composed by the
eigenvectors of $\mathbb{A}$ denoted as
$\hat{\varLambda}_{\epsilon,\kappa}=(\upsilon_{\epsilon,\kappa,1},\upsilon_{\epsilon,\kappa,2},\upsilon_{\epsilon,\kappa,3})^{\intercal}$.
So we have:

$\mathbb{F}_{12}=\frac{1}{2U_{1}}\left(\upsilon_{\epsilon,1,1}\upsilon_{\epsilon,2,1}+\upsilon_{\epsilon,1,2}\upsilon_{\epsilon,2,2}\right)+\frac{1}{4U_{2}}\upsilon_{\epsilon,1,3}\upsilon_{\epsilon,2,3}=0$
(71)

And the orthogonality relation gives
$\hat{\varLambda}_{\epsilon,\kappa_{1}}^{\dagger}\hat{\varLambda}_{\epsilon,\kappa_{2}}=\delta_{\kappa_{1},\kappa_{2}}$,

$\upsilon_{\epsilon,1,1}\upsilon_{\epsilon,2,1}+\upsilon_{\epsilon,1,2}\upsilon_{\epsilon,2,2}+\upsilon_{\epsilon,1,3}\upsilon_{\epsilon,2,3}=0.$
(72)

From Eq.71 and Eq.72 we can derivate
$\upsilon_{\epsilon,1,3}\upsilon_{\epsilon,2,3}=0$ and we set
$\upsilon_{\epsilon,1,3}=0$. Constrained by
$\hat{\varLambda}_{\epsilon,1}^{\dagger}\hat{\varLambda}_{\epsilon,3}=0$, we
can obatin,

$\mathbb{F}_{13}=u_{13}=\frac{1}{2U_{1}}\left(\upsilon_{\epsilon,1,1}\upsilon_{\epsilon,3,1}+\upsilon_{\epsilon,1,2}\upsilon_{\epsilon,3,2}\right)=0,$
(73)

In the limitation of no mixture, we have $u_{13}=0$ and we treat the mixture
as purtubation and have $u_{13}\rightarrow 0$. We set
$\hat{\Lambda}_{\epsilon,\kappa}=(\varphi_{1},\varphi_{2},\varphi_{3})^{\intercal}$
and obtain,

$A\tilde{f}_{\epsilon}(U_{1},U_{2})=\lim_{u_{13}\rightarrow
0}\frac{a_{1}\varphi_{1}^{2}+a_{1}\varphi_{2}^{2}+a_{2}\varphi_{3}^{2}}{u_{1}\varphi_{1}^{2}+2u_{13}\varphi_{1}\varphi_{3}+u_{2}\varphi_{2}^{2}+u_{3}\varphi_{3}^{2}+2u_{23}\varphi_{2}\varphi_{3}}$
(74)

For a fraction takes the form as
$\frac{a_{1}+x^{2}a_{2}}{b_{1}+x^{2}b_{2}+xb_{3}}$ with $x$ as a variable,
$a_{1,2}$ and $b_{1,2,3}$ all positive. We have the extreme values taking at
$x=\frac{2a_{1}b_{2}-2a_{2}b_{1}\pm\sqrt{(2a_{2}b_{1}-2a_{1}b_{2})^{2}+4a_{1}a_{2}b_{3}^{2}}}{2a_{2}b_{3}}$.
In the case of $b_{3}\rightarrow 0$, the extreme vaules locate at $x=0,\infty$
corresponding to
$\frac{a_{1}+x^{2}a_{2}}{b_{1}+x^{2}b_{2}+xb_{3}}=\frac{a_{1}}{b_{1}}$ and
$\frac{a_{1}+x^{2}a_{2}}{b_{1}+x^{2}b_{2}+xb_{3}}=\frac{a_{2}}{b_{2}}$,
respectively. So the maximun of $|A\tilde{f}_{\epsilon}(U_{1},U_{2})|$ is near
to $|\frac{a_{1}}{u_{1}}|$ or
$|\frac{a_{1}\varphi_{2}^{2}+a_{2}\varphi_{3}^{2}}{u_{2}\varphi_{2}^{2}+u_{3}\varphi_{3}^{2}+2u_{23}\varphi_{2}\varphi_{3}}|$.
We first seek for the maximum of
$|\frac{a_{1}\varphi_{2}^{2}+a_{2}\varphi_{3}^{2}}{u_{2}\varphi_{2}^{2}+u_{3}\varphi_{3}^{2}+2u_{23}\varphi_{2}\varphi_{3}}|$
and obtain $\frac{\varphi_{2}}{\varphi_{3}}$ by solving the equation below and
pick the $x$ corresponding to the maximum,

$\frac{\mathrm{d}}{\mathrm{d}x}\frac{a_{1}x^{2}+a_{2}}{u_{2}x^{2}+u_{3}+2u_{23}x}=0$
(75)

where $x=\frac{\varphi_{2}}{\varphi_{3}}$. And then, we solve
$\frac{\varphi_{1}}{vfi_{3}}$ in the same way. Finally we obtain
$\varphi_{1}:\varphi_{2}:\varphi_{3}$ and $\hat{\Lambda}_{\epsilon,\kappa}$ is
determined.

We also set
$\hat{\Lambda}_{\epsilon,\zeta}=(\sin\theta,\cos\theta)^{\intercal}$ for
$\mathrm{E}_{u}$ and
$\hat{\Lambda}_{\epsilon,\zeta}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\intercal}$
for $\mathrm{T}_{1u}$ and $\mathrm{T}_{2u}$ and sample on $\theta$, $\phi$ to
find the minimum of $B$. We obtain
$\hat{\Lambda}_{\mathrm{E}_{u},\zeta}=(1,0)^{\intercal}$ and
$\hat{\Lambda}_{\mathrm{T}_{1u}(\mathrm{T}_{2u}),\zeta}=(1,0,0)^{\intercal},(0,1,0)^{\intercal},(0,0,1)^{\intercal}$.

##  Appendix F Symmetries and topological properities of each channel

In the Nambu space,
$\hat{\psi}^{\dagger}_{\bm{k}}=(\hat{c}^{\dagger}_{\bm{k},\uparrow},\hat{c}^{\dagger}_{\bm{k},\downarrow},\hat{c}_{-\bm{k},\uparrow},\hat{c}_{-\bm{k},\downarrow})$,
the particle-hole symmetry operator takes the form as
$\mathcal{C}=\eta_{1}\sigma_{0}K$, where $\eta$ and $\sigma$ are Pauli
matrices with $\eta$ indicating the particle $\hat{c}^{\dagger}$ or hole
$\hat{c}$ and $\sigma$ indicating the pseudo-spin $\uparrow$ or $\downarrow$.
$K$ is the conjugation operator. The time reversal symmetry operator takes the
form as $\mathcal{T}=i\eta_{0}\sigma_{2}K$. With the particle-hole symmetry
operator and time reversal operator we can define the chiral operator which is
the product of these two,
$\mathcal{S}=\mathcal{C}\mathcal{T}=i\eta_{1}\sigma_{2}$. For the spacial
symmetry operator $R$ belonging to the $D_{3d}$ group on the $m$-th pocket, we
have,

$\begin{split}R\hat{c}_{\bm{k},d}^{m\dagger}R^{\dagger}&=\sum_{d^{\prime}}\hat{c}_{R\bm{k},d^{\prime}}^{m\dagger}\mathcal{R}_{d^{\prime}d}\\\
R\hat{c}_{\bm{k},d}^{m}R^{\dagger}&=\sum_{d^{\prime}}\hat{c}_{R\bm{k},d^{\prime}}^{m}\mathcal{R}_{d^{\prime}d}^{*},\end{split}$
(76)

where $\mathcal{R}$ is the transformation matrix with $d^{\prime}$ as the row
index and $d$ as the column index. After the mean-field approximation, pairing
part of the BdG Hamiltonian is obtained as
$\sum_{m,d_{1},d_{2}}\mathcal{H}_{d_{1}d_{2}}^{m}(\bm{k})\hat{c}_{\bm{k},d_{1}}^{m\dagger}\hat{c}_{\bm{-k},d_{2}}^{m\dagger}+\text{h.c.}$
and transformed by the operator $R$ as,

$\begin{split}R\sum_{d_{1},d_{2}}\mathcal{H}_{d_{1}d_{2}}^{m}(\bm{k})\hat{c}_{\bm{k},d_{1}}^{\dagger}\hat{c}_{\bm{-k},d_{2}}^{\dagger}R^{\dagger}&=\sum_{d_{1},d_{2},d^{\prime},d^{\prime\prime}}\mathcal{H}^{m}_{d_{1}d_{2}}(\bm{k})\hat{c}_{R\bm{k},d^{m\prime}}^{\dagger}\mathcal{R}_{d^{\prime}d_{1}}\hat{c}_{-R\bm{k},d^{\prime\prime}}^{m\dagger}\mathcal{R}_{d^{\prime\prime}d_{2}}\\\
&=\sum_{d^{\prime},d^{\prime\prime}}(\mathcal{R}\mathcal{H}^{m}(\bm{k})\mathcal{R}^{\intercal})_{d^{\prime}d^{\prime\prime}}\hat{c}_{R\bm{k},d^{\prime}}^{m\dagger}\hat{c}_{-R\bm{k},d^{\prime\prime}}^{m\dagger}.\end{split}$
(77)

In Eq.46, we write the pairing as
$\mathcal{H}^{m}(k)=i\bm{\sigma}\cdot\bm{d}(\bm{k})\sigma_{2}=i\sigma_{1}\sigma_{2}d_{1}(\bm{k})+i\sigma_{2}\sigma_{2}d_{2}(\bm{k})+i\sigma_{3}\sigma_{2}d_{3}(\bm{k})$
and $\mathcal{R}$ is the matrix representation of $R$. In the local reference
frame, we take $M_{a}$ as the mirror symmetry with the mirror plane
perpendicular to the $x$ axis and we obtain the matrix form of $M_{a}$ under
the basis
$(\hat{c}^{\dagger}_{\bm{k},\uparrow},\hat{c}^{\dagger}_{\bm{k},\downarrow})$
which is denoted as $\mathbb{M}_{a}$,
$\mathbb{M}_{a}=e^{-\frac{i}{2}\sigma_{x}\pi}=-i\sigma_{1}$. We act
$\mathbb{M}_{a}$ on the matrices $\bm{\sigma}\sigma_{2}$ and have,

$\begin{split}\mathbb{M}_{a}\sigma_{1}\sigma_{2}\mathbb{M}_{a}^{\intercal}&=\sigma_{1}\sigma_{2}\\\
\mathbb{M}_{a}\sigma_{2}\sigma_{2}\mathbb{M}_{a}^{\intercal}&=-\sigma_{2}\sigma_{2}\\\
\mathbb{M}_{a}\sigma_{3}\sigma_{2}\mathbb{M}_{a}^{\intercal}&=-\sigma_{3}\sigma_{2}.\end{split}$
(78)

Under $M_{a}$, the pairing is transformed as
$\mathbb{M}_{a}\mathcal{H}(\bm{k})\mathbb{M}_{a}^{\intercal}=\tilde{\mathcal{H}}(\bm{k})=i\sigma_{1}\sigma_{2}d_{1}(\bm{k})-i\sigma_{2}\sigma_{2}d_{2}(\bm{k})-i\sigma_{3}\sigma_{2}d_{3}(\bm{k})=\ell\mathcal{H}(M_{a}\bm{k})$,
where $\ell=\pm 1$ according to the form of $\bm{d}(\bm{k})$. In the Nambu
space, we the matrix form of $M_{a}$ as,

$\mathcal{M}_{a}=\left(\begin{array}[]{cc}\mathbb{M}_{a}&0\\\
0&\ell\mathbb{M}_{a}^{*}\end{array}\right).$ (79)

##### $\mathrm{A}_{1u}$

We start from the first pocket of the $A_{1u}$ channel. The $d$-vector is
obtained as $\bm{d}(\bm{k})=(\alpha k_{x},\alpha k_{y},\beta k_{z})$, where
$\alpha$ and $\beta$ are obtained from the mean field calculation. We write
the pairing function
$\mathcal{H}(\bm{k})=\sum_{m,n=1,2,3}ik_{m}\mathcal{A}_{mn}\sigma_{n}\sigma_{2}$
and we use $k_{1,2,3}$ to denote $k_{x,y,z}$, respectively. The determinant of
matrix $\mathcal{A}$ is nonzero, $\det{\mathcal{A}}\neq 0$. So the
$\mathrm{A}_{1u}$ channel is gapped. With the time reversal symmetry
respected, the winding number of the single band Hamiltonian equals to
$sgn(\det{\mathcal{A}})$. For the $C_{4}$ symmetry is trivial in the
$\mathrm{A}_{1u}$ channel, the winding number of each pocket is the same and
the system is characterized by the winding number $w=4sgn(\alpha^{2}\beta)$.

##### $\mathrm{A}_{2u}$

The $d$-vector of the first pocket is taken as $\bm{d}(\bm{k})=(-\alpha
k_{y},\alpha k_{x},0)$ and the determinant of matrix $\mathcal{A}$ equals to
$0$, $\det{\mathcal{A}}=0$. The $\mathrm{A}_{2u}$ state is gapless and
protected by the mirror and chiral symmetry. The gapless point is shown in
Fig.7(a). The pairing function is transformed by the mirror symmetry $M_{a}$
as,

$\mathbb{M}_{a}\mathcal{H}(\bm{k})\mathbb{M}_{a}^{\intercal}=\tilde{\mathcal{H}}(\bm{k})=-i\sigma_{1}\sigma_{2}\alpha
k_{y}-i\sigma_{2}\sigma_{2}\alpha k_{x}=\mathcal{H}(M_{a}\bm{k}).$ (80)

The mirror symmetry $M_{a}$ of the $\mathrm{A}_{2u}$ channel in the Nambu
space is even, $\ell=1$ and the chiral symmetry commutes with $M_{a}$. We
first decompose the BdG Hamiltonian in the mirror plane of $M_{a}$ ($k_{x}$ =
0) into different mirror subspaces with the eigenvaules of $M_{a}$ equalling
to $i$ and $-i$ seperately,

$H_{\text{BdG}}(\bm{k},k_{x}=0)=\left(\begin{array}[]{cccc}\frac{k_{y}^{2}}{2m}+\frac{\xi
k_{z}^{2}}{2m}-\mu&-\alpha k_{y}&0&0\\\ -\alpha
k_{y}&-\frac{k_{y}^{2}}{2m}-\frac{\xi k_{z}^{2}}{2m}+\mu&0&0\\\
0&0&\frac{k_{y}^{2}}{2m}+\frac{\xi k_{z}^{2}}{2m}-\mu&-\alpha k_{y}\\\
0&0&-\alpha k_{y}&-\frac{k_{y}^{2}}{2m}-\frac{\xi
k_{z}^{2}}{2m}+\mu\end{array}\right).$ (81)

Within each subspace, the chiral symmetry is respected. We transform the basis
of the mirror subspace to the eigenstates of chiral symmetry and the
Hamiltonin in each subspace is off diagonalized,

$H_{\text{sub-mirror}}=\left(\begin{array}[]{cc}0&i\alpha
k_{y}+\frac{k_{y}^{2}}{2m}+\frac{\xi k_{z}^{2}}{2m}-\mu\\\ -i\alpha
k_{y}+\frac{k_{y}^{2}}{2m}+\frac{\xi k_{z}^{2}}{2m}-\mu&0\end{array}\right).$
(82)

To study the edge states on the $(001)$ surface, we change Hamiltonian from
the local to the global reference frame and have
$k_{y}=\sqrt{\frac{1}{6}}(k_{X}+k_{Y})-\sqrt{\frac{2}{3}}k_{Z}$,
$k_{z}=\sqrt{\frac{1}{3}}(k_{X}+k_{Y}+k_{Z})$, $k_{X}=k_{Y}$ (constrained by
the mirror plane),

${}H_{\text{sub-
mirror}}=\left(\begin{array}[]{cc}0&i\alpha\sqrt{\frac{2}{3}}(k_{X}-k_{Z})+\frac{(k_{X}-k_{Z})^{2}}{3m}+\xi\frac{(2k_{X}+k_{Z})^{2}}{6m}-\mu\\\
-i\alpha\sqrt{\frac{2}{3}}(k_{X}-k_{Z})+\frac{(k_{X}-k_{Z})^{2}}{3m}+\xi\frac{(2k_{X}+k_{Z})^{2}}{6m}-\mu&0\end{array}\right).$
(83)

We take $k_{X}$ satisifing
$-\sqrt{\frac{2m\mu}{3\xi}}<k_{X}<\sqrt{\frac{2m\mu}{3\xi}}$ and set
$k_{X}-k_{Z}=x$. The off diagonal entry of the matrix turns to $\pm i\alpha
x+\frac{1}{3m}x^{2}+\frac{\xi}{6m}(3k_{X}-x)^{2}-\mu$. When we take $x$ from
$-\infty$ to $\infty$, equivalent for $k_{Z}$ from $\infty$ to $-\infty$, the
complex phase of the off diagonal entry changes from $0$ to $-2\pi$ shown in
Fig.7(b).

Figure 7:  (a) The Fermi surface of the first pocket of the $\mathrm{A}_{2u}$
channel. We use the transparent gray plane to denote the mirror plane. Two
black points stand for the gapless points. There exists the arc at the
intersection between the darker part of the plane and the $(001)$ surface
Brillouin zone. (b) Re and Im stand for the real and imaginary axis. $\theta$
is the complex phase of the off diagonal entry in Eq.83. When we take $x$ from
$-\infty$ to $\infty$, $\theta$ will travel on the circle along the direction
of the blue trace. We approach to the right red point on the real axis from up
and down side when we take $x\rightarrow\infty$ and $x\rightarrow-\infty$,
respectively. When we take $x=0$, the off diagonal entry is a negative real
number and plotted as the left red point on the real axis. The up and down red
points on the imaginary axis correspond to the two roots of the equation
$\frac{1}{3m}x^{2}+\frac{\xi}{6m}(3k_{X}-x)^{2}-\mu=0$ with opposite sign.

##### $\mathrm{E}_{u}$

For the $E_{u}$ channel, the unit vector $(t_{1},t_{2})$ in the order
parameter is taken as $(1,0)$. From the represenstation table in the paper we
know that $C_{2a}$ and $C_{4}$ symmetry is respected with the character of
$C_{2a}$ and $C_{4}$ symmetry equalling to $1$. The symmetry of the system
breaks from the group $O_{h}$ to the $\mathrm{A}_{2u}$ irrep of the group
$D_{4h}$. Similar with $\mathrm{A}_{1u}$, the $d$-vector of the first pocket
is taken as $(\alpha k_{x},\alpha k_{y}+\beta k_{z},\gamma k_{y})$. We have
$\det{\mathcal{A}}\neq 0$ and the system is characterized by the winding
number $w=-4sgn(\alpha\beta\gamma)$.

##### $\mathrm{T}_{2u}$

For the $\mathrm{T}_{2u}$ channel, we take the unit vector
$(t_{1},t_{2},t_{3})$ as $(0,0,1)$. $C_{2a}$ and $C_{4}$ symmetry are
respected according to the irreps table. The character of $C_{2q}$ and $C_{4}$
equal to $1$ and $-1$, respectively. The symmetry of the system breaks from
the $O_{h}$ group to the $\mathrm{B}_{2u}$ irrep of the $D_{4h}$ group. In the
first pocket we have the $d$-vector as $\bm{d}(\bm{k})=(\alpha
k_{x},\beta_{1}k_{y}+\beta_{2}k_{z},\gamma_{1}k_{y}+\gamma_{2}k_{z})$. The
winding number of the first pocket is obtained as
$w_{1}=sgn(\det\mathcal{A})=sgn(\alpha\beta_{1}\gamma_{2}-\alpha\beta_{2}\gamma_{1})$.
However, different with $\mathrm{A}_{1u}$ and $\mathrm{E}_{u}$, the $C_{4}$
symmetry in $\mathrm{T}_{2u}$ changes the sign of winding number between
different pockets and the winding numbers of the four pockets are taken as
$w_{1},-w_{1},w_{1},-w_{1}$, seperately. We can see that the total winding
number gives $0$, however, the $\mathrm{T}_{2u}$ state is topologically
nontrivial. The mirror symmetry $M_{a}$ is odd,
$\mathbb{M}_{a}\mathcal{H}(\bm{k})\mathbb{M}_{a}^{\intercal}=\tilde{\mathcal{H}}(\bm{k})=-\mathcal{H}(M_{a}\bm{k})$,
$\ell=-1$. In the mirror plane we decompose the BdG Hamiltonian into the
mirror subspace,

$\begin{split}&H_{\text{BdG}}(\bm{k},k_{x}=0)=\\\
&\begin{pmatrix}-\frac{k_{y}^{2}}{2m}-\frac{\xi
k_{z}^{2}}{2m}+\mu&-i\beta_{1}k_{y}-i\beta_{2}k_{z}-\gamma_{1}k_{y}-\gamma_{2}k_{z}&0&0\\\
i\beta_{1}k_{y}+i\beta_{2}k_{z}-\gamma_{1}k_{y}-\gamma_{2}k_{z}&\frac{k_{y}^{2}}{2m}+\frac{\xi
k_{z}^{2}}{2m}-\mu&0&0\\\ 0&0&-\frac{k_{y}^{2}}{2m}-\frac{\xi
k_{z}^{2}}{2m}+\mu&\gamma_{1}k_{y}+\gamma_{2}k_{z}-i\beta_{1}k_{y}-i\beta_{2}k_{z}\\\
0&0&\gamma_{1}k_{y}+\gamma_{2}k_{z}+i\beta_{1}k_{y}+i\beta_{2}k_{z}&\frac{k_{y}^{2}}{2m}+\frac{\xi
k_{z}^{2}}{2m}-\mu\end{pmatrix}.\end{split}$ (84)

In each mirror subspace, there exists a gapless point at $k_{x}=k_{y}=k_{z}=0$
when we take the chemical potential to zero, $\mu=0$ and when $\mu<0(>0)$, the
system is topologically trivial (nontrivial), so $\mu=0$ is a topological
phase transition point in the mirror plane. We calculate the topological
charge of the gapless point and we can obtain the mirror Chern number. In each
mirror subspace we similarly define $\mathcal{H}_{\text{mirror
sub}}(\bm{k})=\sum_{m,n=1,2}k_{m}\mathcal{A}^{\prime}_{mn}\sigma_{n}i\sigma_{2}$
with $k_{1},k_{2}$ denoting $k_{y}$ and $k_{z}$. The mirror Chern number for
the first pocket is obtained as
$sgn(\det{\mathcal{A}^{\prime}})=sgn(\beta_{1}\gamma_{2}-\beta_{2}\gamma_{1})$
and the mirror plane is shared by two pockets related by $C_{4}^{2}$ symmetry,
for expample, pocket $1$ and $3$. And $C_{4}^{2}=1$, the mirror Chern number
of these two pockets are the same, so the mirror Chern number of the mirror
plane is even, $2sgn(\beta_{1}\gamma_{2}-\beta_{2}\gamma_{1})$, which suggests
a second order topological superconductivity.

##  Appendix G Calculation of edge states

We open the boundary along $Z$ axis and write $k_{Z}$ as
$-i\frac{\partial}{\partial Z}$. And we write,

$-i\frac{\partial}{\partial
Z}|Z,k_{x},k_{y},w\rangle=-i\frac{|Z+\mathrm{d}Z,k_{x},k_{y},w\rangle-|Z-\mathrm{d}Z,k_{x},k_{y},w\rangle}{2\mathrm{d}Z},$
(85)

where we use $w$ to denote the pseudo-spin. We first write the BdG Hamiltonian
in the basis of
$(|Z_{1},k_{x},k_{y},w\rangle,|Z_{2},k_{x},k_{y},w\rangle,\cdots,|Z_{N},k_{x},k_{y},w\rangle)^{\intercal}$
and we totally have $N$ sites along $Z$ axis, $Z_{i+1}-Z_{i}=dZ\rightarrow 0$.
In the periodic boundary condition, we stick the $N$-th site with the first
site and take the first site the same as the $N+1$-th site. The operator
$\hat{k}_{Z}$ can be written as,

$\hat{k}_{Z}=\frac{1}{2\mathrm{d}Z}\left(\begin{array}[]{cccccc}0&i&0&\cdots&0&-i\\\
-i&0&i&\cdots&0&0\\\ 0&-i&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&0&i\\\
i&0&0&\cdots&-i&0\end{array}\right).$ (86)

While, for the open boundary condition, we cut off the loop and remove the
$N$-th row and column in the matrix. So for the open boundary condition, we
have the operator of $\hat{k}_{Z}$ taken as,

$\hat{k}_{Z}=\frac{1}{2\mathrm{d}Z}\left(\begin{array}[]{cccccc}0&i&0&\cdots&0&0\\\
-i&0&i&\cdots&0&0\\\ 0&-i&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&0&i\\\
0&0&0&\cdots&-i&0\end{array}\right).$ (87)

For the second order derivition of $Z$, we have,

$\hat{k}_{Z}^{2}=-\frac{\partial^{2}}{\partial
Z^{2}}|Z,k_{x},k_{y},w\rangle=-\frac{|Z+\mathrm{d}Z,k_{x},k_{y},w\rangle+|Z-\mathrm{d}Z,k_{x},k_{y},w\rangle-2|Z,k_{x},k_{y},w\rangle}{\mathrm{d}Z^{2}}.$
(88)

And in the periodic condition we have,

$\hat{k}_{Z}^{2}=-\frac{1}{\mathrm{d}Z^{2}}\left(\begin{array}[]{cccccc}-2&1&0&\cdots&0&1\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&-2&1\\\
1&0&0&\cdots&1&-2\end{array}\right).$ (89)

For the open boundary condition we have,

$\hat{k}_{Z}^{2}=-\frac{1}{\mathrm{d}Z^{2}}\left(\begin{array}[]{cccccc}-2&1&0&\cdots&0&0\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&-2&1\\\
0&0&0&\cdots&1&-2\end{array}\right).$ (90)

We substitute $\hat{k}_{Z}$ and $\hat{k}_{Z}^{2}$ under the open boundary
condition in Eq.87 and Eq.90 into the BdG Hamiltonian and diagonalize the
matrix to obtian the dispersion.


Corresponding author<EMAIL_ADDRESS>
Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
1]Fudan University, China

2]Gran Sasso Science Institute, Italy

3]Laboratori Nazionali del Gran Sasso, INFN, Italy

4]Shanghai Institute of Ceramics, Chinese Academy of Science, China

5]Department of Physics, Engineering Physics and Astronomy, Queen’s
University, 64 Bader Lane, K7L 3N6 Kingston, ON, Canada

6]INFN-TIFPA, Italy

7]Department of Physics, Trento University, Italy

8]Arthur B. McDonald Canadian Astroparticle Physics Research Institute, 64
Bader Lane, K7L 3N6, Kingston, ON, Canada

# LUCE: A milli-Kelvin calorimeter experiment to study the electron capture of
176Lu

Shihong Fu  Giovanni Benato  Carlo Bucci  Paolo Gorla  Pedro V. Guillaumon
Jiang Li  Serge Nagorny  Francesco Nozzoli  Lorenzo Pagnanini  Andrei Puiu
Matthew Stukel  [  [  [  [  [  [  [  [

###### Abstract

The LUCE (LUtetium sCintillation Experiment) project will search for the 176Lu
electron capture based on a milli-Kelvin calorimetric approach. This decay is
of special interest in the field of nuclear structure, with implications for
the s-process and for a better comprehension of the nuclear matrix elements of
neutrinoless double beta decay ($\mathrm{0\nu\beta\beta}$) and two-neutrino
double beta decay ($\mathrm{2\nu\beta\beta}$). Possible impacts also include
the development of a new class of coherent elastic neutrino-nucleus scattering
($\mathrm{CE\nu NS}$) and spin-dependent (independent) dark matter detectors.
We report on the current status and design of a novel detector cryogenic-
module for the measurement of the electron capture and detail a future
measurement plan.

###### keywords:

176Lu, electron capture, calorimeter, nuclear physics, cryogenic

##  1 Introduction

The study of rare decays is of fundamental importance in the field of nuclear
and particle astrophysics. These decays allow for a better understanding of
the nuclear structure involved in these processes by extracting the
electroweak coupling quenching to observe if any violation occurred when
comparing the predictions of the Standard Model [1, 2]. There are two main
implications in these studies: a better prediction of $2\nu\beta\beta$ and
$0\nu\beta\beta$ half-lives and of the nucleosynthesis of s-process [3, 4, 5,
6].

There are only six, known, long-lived (over a billion years) nuclides that
undergo an electron capture (EC) decay, namely 40K (1.29$\times$1012 yrs [7,
8]), 50V (2.2$\times$1017 yrs [9, 10]), 138La (1.6$\times$1011 yrs [11]),
123Te ($>$1016-1019 [12, 13]), 180Ta ($>$2.0$\times$1017 yrs [14, 15]), and
176Lu ($>$1013-1014 yrs [16]), three of them have never been detected. The
goal of the LUCE experiment (LUtetium sCintillation Experiment) is to measure
the partial half-life of 176Lu with a milli-Kelvin scintillating calorimeter
in an ultra-low-background facility. Using low-temperature calorimeters is a
new approach to this measurement and will lead to high sensitivity [17, 16].

The dominant 176Lu ground state (g.s.) decay is a $\beta^{-}$-decay
($J^{\pi}$(176Lu) = 7- $\rightarrow$ $J^{\pi}$(176Hf) = 0+) with a half-life
of $(3.5-3.8)\times 10^{10}$ years and is used in dating methods of meteorites
and rocks [17]. The precise value for this half-life has been subject to
discussion, and one of the possible explanations is the existence of an EC
decay to 176Yb. The EC decay is energetically allowed, having a Q-value of
$109$ keV to the 176Yb g.s. ($J^{\pi}$(176Lu) = 7- $\rightarrow$
$J^{\pi}$(176Yb) = 0+) and $\sim 27$ keV to the first excited state
($J^{\pi}$(176Lu) = 7- $\rightarrow$ $J^{\pi}$(176Yb) = 2+). These decays,
however, would be $\mathrm{7^{th}}$ and $\mathrm{5^{th}}$ forbidden decays and
have never been detected. Fig.1 shows the current decay scheme for 176Lu.

Figure 1: Decay scheme of 176Lu. [16]

##  2 The LUCE detector concept

LUCE is a proposed experiment for the detection of the 176Lu EC-decay based on
an innovative cryogenic calorimetric technique operated at $\sim 15$ mK. It
relies on a modular design where a LYSO [$\mathrm{(Lu,Y)_{2}SiO_{5}}$:Ce] or
LuAG [$\mathrm{Lu_{3}Al_{5}O_{12}}$:Ce] crystal scintillator containing the
176Lu source is placed in the middle of a copper box and is surrounded by two
TeO2 crystals, shown in Fig. 2. When 176Lu decays, its products can be fully
or partially absorbed in the crystal, producing a variation of temperature.
This change in temperature can be measured by an attached NTD-Ge sensor [18].
The raw signal will be saved separately for digital post-processing and anti-
coincidence analysis.

The main source of background is the 176Lu $\beta$-decay. More than 99% of
176Lu decays to the $6^{+}$ state which has an energy of 596.82 keV. This
means that the rest of the Q-value (593.88 keV) goes to a continuum
distribution of the electron. Due the small size of the crystal and depending
on the energy of the electron, the particle can deposit just a fraction of the
energy inside the crystal, giving rise to a distorted $\beta$-shape
distribution. This background can be reduced by running the TeO2 crystals in
coincidence with the 176Lu source. The higher energy gammas produced during
the $\beta$-decay will escape from the 176Lu source crystal unlike the
products of the EC. Preliminary simulations, for a 4$\times$4$\times$20 cm3
crystal, show that $\sim$50$\%$ of the beta events can be removed by the two
(50 $\times$ 50 $\times 5~{}\mathrm{mm}^{3}$) $\mathrm{TeO}_{2}$ crystals
placed 15 mm from the 176Lu source crystal. While greater than 90$\%$ of the
electron capture events will remain inside. Further simulations will be
performed to optimize the experimental geometry.

In order to avoid pile-ups due to the long decay time in the NTD, the lutetium
crystals will need to have an activity of less than $<$1 Bq, which means
crystals size of $\sim$1 mm3 and smaller sensors matching the thermal
impedance. With a crystal of this size, it is expected to have sensitivity to
a decay of $10^{14}$ years, after 1 month of measurement, see Sec. 3.3 for
further details. LuAG and LYSO crystals will be tested as cryogenic
calorimeters.

The $\mathrm{5^{th}}$ forbidden EC decay to the 2+ excited state produces a
82.1 keV de-excitation photon. Due to the rising time of the NTD, the de-
excitation photon will be visible with the Auger or x-rays produced from the
atomic re-arrangement of the 176Yb (K transition is energetically prohibited),
resulting in a peak search at 82.1 + 10.481 (L1 binding energy) = 92.6 keV.
The 7th forbidden EC g.s. decay allows for both the K and L captures of 176Yb
[17] (they are the most probable) resulting in peaks at 61.3 keV (log ft =
0.532) and 10.5 (log ft = 0.347) keV respectively [19].

Figure 2:  Proposed detector module for the LUCE experiment.

##  3 Experimental Program

The goal of LUCE phase-I is to have one module fully operational and reach or
exceed the current literature half-life sensitivity. Using the knowledge
gained during phase-I, LUCE phase-II will design an experiment that will aim
to discover or provide a world leading sensitivity for the electron capture.
In phase-I the detector response (both phonon propagation and light yield) at
low-temperature must be studied. LuAG:Ce and LYSO:Ce crystals will be studied
in order to choose the one with higher resolution in the region-of-interest
(ROI), in the heat channel. Although light yield measurements are not critical
to measure the EC of 176Lu, this will also allow to characterize this crystal
for other applications in neutrino and dark matter physics. A detailed
simulation program is being developed concurrently to support experimental
measurements. The following sections outline the current status of the
experiment and details future operations.

###  3.1 Light Yield Measurement

Light yield measurements as a function of temperature, for both crystals, will
be performed in order to have a full understanding of low-temperature detector
response. A room-temperature spectral measurement was performed with a
4$\times$4$\times$20 mm3 LYSO crystal and photomultiplier tube (PMT) to see
the intrinsic radiation and to perform a first comparison with the
simulations. The energy spectrum with simulation comparison can be seen in
Fig. 3. The data was collected over 16 hours in a surface facility with a
hardware trigger where the data points represent the pulse height. In general,
the simulated data follows the shape of experimental data. There is some
discrepancy at low and high energy but this is most likely explained by the
light collection efficiency, Birks quenching and pile-up, which is not yet
included in the simulation. The measured spectrum shows the shifting of the
$\beta^{-}$ spectrum by the different combinatorics of the gamma rays (i.e.
88.3 keV, 290.3 keV (88.3 + 202 keV), and 395.3 keV (88.3 + 307 keV)). The
spectral shape and subsequent intensities of the $\beta^{-}$ peaks will
ultimately be dependent on the final crystal size selected.

We have developed a system to measure the scintillation characteristics of
crystals, such as LYSO and LuAG, over a wide temperature range ([range-phrase
= –]1030040K) using a Xenon (Xe) arc lamp as the excitation source. This
method has been shown to determine the relationship between light yield and
temperature in previous studies on similar crystals [20]. The results are
expected to reveal a detailed temperature dependence of light yield, which is
crucial for optimizing the detector performance at cryogenic temperatures for
other applications.

Figure 3: Intrinsic energy spectrum of a LYSO crystal along with the simulated
spectrum. The different $\beta$-$\gamma$-ray/internal conversion combinations
allowed during the decay of 176Lu are also shown.

###  3.2 Calorimetric Measurements

A small sample with an attached NTD-Ge sensor, of LuAG (grown by the Shanghai
Institute Of Ceramics Chinese Academy Of Sciences) and of LYSO (grown by Saint
Gobain), will be tested in the underground facility of Laboratori Nazionali
del Gran Sasso. The primary goal is to show these crystals can work as low-
temperature calorimeters and determine their low-energy threshold and
resolution. One of the critical aspects of the LUCE project is the ability of
larger crystals to handle the high counting rate of intrinsic radioactivity
present in the sample and avoid pile-up. In that way, the evaluation of rise
and decay times of pulses is crucial.

LYSO crystals have a percentage of yttrium that varies from 5% to 70% [21].
Depending on the amount of yttrium, the arrangement of the crystal lattice
will change affecting the phonon propagation, and, by consequence, the
resolution of the crystal. Although not a perfect comparison (difference in
size and quality), garnet crystals similar to LuAG have already been tested as
cryogenic calorimeters, showing energy threshold lower than 8 keV with good
resolution pulses [22]. There is a lack of information about the heat capacity
of LuAG and LYSO crystals, but since the specific heat at room temperature of
those crystals, are comparable to other low-temperature detectors, it is
expected they will behave well as cryogenic detectors [23, 24, 25]. Also, no
high paramagnetism have been reported in LuAG and LYSO crystals. Studies on
the thermal properties of those crystals will need to be performed.

###  3.3 Projected Sensitivity

Based on the detector setup proposed in Sec. 2 the significance of a single
module was studied. This study simulated a 1$\times$1$\times$2.5 mm3 LuAG
crystal with the resolution from [26]. The signal was generated as a gaussian
at ground (61.3 keV) and excited state (92.6 keV). The significance was
evaluated at each of these energies. The number of signal events depended on
the selected half-life for the process. The only background considered was the
events from the $\beta^{-}$ decay that were not found in coincidence with the
adjacent TeO2 crystals. This would be the dominant background in an
underground run. The significance was evaluated as the signal counts over the
square-root of the background in a selected energy region, $\pm$10 keV around
the above peak means. The significance as a function of half-life and run time
can be seen in Fig. 4. Here it is shown that this small crystal can reach the
current literature sensitivities with two months of run time. Improvement to
the detectable half-life or significance are easily achievable by increasing
the size of the crystal or the number of detector modules in the experiment.
Additionally, the resolution of the calorimeter will effect the sensitivity.
It is expected that the initial cryogenic tests will show that the actual
resolution will be smaller than the one used in this study. It has been
reported a FWHM of 7 keV at 3 MeV with NTD-Ge macrocalorimeters [27] and of
3.1 eV at 6keV with NTD-Ge microcalorimeters [28], which is better than other,
typical, experimental setup.

Figure 4: Significance of the LUCE experiment. The significance is defined as
the signal (S) over the square root of the background (B). Current
experimental lower limits are at 1013-1014 years [16]. (Left) Ground state
electron capture significance (61.3 keV). (Right) Excited state electron
capture significance (92.6 keV).

##  4 Conclusion and Perspectives

The LUCE project is the first experiment to search for the EC-decay of 176Lu
by using a lutetium based cryogenic calorimeter. The modular and flexible
detector concept is designed to handle the high intrinsic activity and can be
repurposed for other experiments. LUCE phase-I will be the proof-of-concept
for the design and will have the sensitivity to reach or exceed current
literature values of the EC-decay half-life ($\sim 10^{14}$ years) in a two
month data taking campaign. Phase-II is expected to have multiple, optimized
detector modules to pursue discovery of the branching ratio or achieve world
leading limits. The LUCE experiment is a unique application of underground
particle physics techniques applied to nuclear physics problems and could open
a new frontier of precision nuclear measurements.

## References

  *   * Leach et al. [2017] Leach, K.G., Dillmann, I., Klawitter, R., Leistenschneider, E., Lennarz, A., Brunner, T., Frekers, D., Andreoiu, C., Kwiatkowski, A.A., Dilling, J.: Electroweak decay studies of highly charged radioactive ions with titan at triumf. Atoms 5(1) (2017) https://doi.org/10.3390/atoms5010014
  * Severijns et al. [2006] Severijns, N., Beck, M., Naviliat-Cuncic, O.: Tests of the standard electroweak model in nuclear beta decay. Rev. Mod. Phys. 78, 991–1040 (2006) https://doi.org/10.1103/RevModPhys.78.991
  * Walter et al. [2007] Walter, S., Stedile, F., Carroll, J.J., Fransen, C., Friessner, G., Hollmann, N., Garrel, H.v., Jolie, J., Karg, O., Käppeler, F., Kneissl, U., Kohstall, C., Neumann-Cosel, P.v., Linnemann, A., Mücher, D., Pietralla, N., Pitz, H.H., Rusev, G., Scheck, M., Scholl, C., Schwengner, R., Werner, V., Wisshak, K.: Photon scattering experiments on the quasistable, odd-odd mass nucleus ${}^{176}\mathrm{Lu}$. Phys. Rev. C 75, 034301 (2007) https://doi.org/10.1103/PhysRevC.75.034301
  * Käppeler et al. [2011] Käppeler, F., Gallino, R., Bisterzo, S., Aoki, W.: The $s$ process: Nuclear physics, stellar models, and observations. Rev. Mod. Phys. 83, 157–193 (2011) https://doi.org/10.1103/RevModPhys.83.157
  * Hayakawa et al. [2005] Hayakawa, T., Iwamoto, N., Shizuma, T., Kajino, T., Umeda, H., Nomoto, K.: Scaling relations for the supernova $\gamma$-process and the 176lu nuclear cosmochronometer. Nuclear Physics A 758, 525–528 (2005) https://doi.org/%****␣sn-article.bbl␣Line␣150␣****10.1016/j.nuclphysa.2005.05.095
  * Heil et al. [2008] Heil, M., Winckler, N., Dababneh, S., Käppeler, F., Wisshak, K., Bisterzo, S., Gallino, R., Davis, A.M., Rauscher, T.: 176lu/176hf: A sensitive test of s-process temperature and neutron density in agb stars. The Astrophysical Journal 673(1), 434 (2008) https://doi.org/10.1086/523892
  * Stukel et al. [2022] Stukel, M., Hariasz, L., Di Stefano, P.C.F., et al.: Rare 40k decay with implications for fundamental physics and geochronology (2022) https://doi.org/10.48550/ARXIV.2211.10319
  * Hariasz et al. [2022] Hariasz, L., Stukel, M., Di Stefano, P., et al.: First observation of the ground-state electron-capture of k-40. arXiv preprint arXiv:2211.10343 (2022)
  * Danevich et al. [2020] Danevich, F.A., Hult, M., Kasperovych, D.V., et al.: Decay scheme of ${}^{50}\mathrm{V}$. Phys. Rev. C 102, 024319 (2020) https://doi.org/10.1103/PhysRevC.102.024319
  * Laubenstein et al. [2019] Laubenstein, M., Lehnert, B., Nagorny, S., Nisi, S., Zuber, K.: New investigation of half-lives for the decay modes of v 50. Physical Review C 99(4), 045501 (2019)
  * Firestone et al. [1999] Firestone, R.B., Ekstrom, L.P., Chu, S.Y.F.: WWW Table of Radioactive Isotopes. In: APS Division of Nuclear Physics Meeting Abstracts. APS Meeting Abstracts, p. 13 (1999)
  * Münstermann and Zuber [2003] Münstermann, D., Zuber, K.: An alternative search for the electron capture of 123te. Journal of Physics G: Nuclear and Particle Physics 29(2), 1 (2003) https://doi.org/10.1088/0954-3899/29/2/701
  * Alessandria et al. [2013] Alessandria, F., Ardito, R., Artusa, D.R., et al.: The low energy spectrum of teo2 bolometers: results and dark matter perspectives for the cuore-0 and cuore experiments. Journal of Cosmology and Astroparticle Physics 2013(01), 038 (2013) https://doi.org/10.1088/1475-7516/2013/01/038
  * Lehnert et al. [2017] Lehnert, B., Hult, M., Lutter, G., Zuber, K.: Search for the decay of nature’s rarest isotope ${}^{180m}\mathrm{Ta}$. Phys. Rev. C 95, 044306 (2017) https://doi.org/10.1103/PhysRevC.95.044306
  * Cerroni et al. [2023] Cerroni, R., Dell’Oro, S., Formicola, A., Ghislandi, S., Ioannucci, L., Laubenstein, M., Lehnert, B., Nagorny, S., Nisi, S., Pagnanini, L.: Deep-underground search for the decay of 180m-ta with an ultra-low-background hpge detector. arXiv preprint arXiv:2305.17238 (2023)
  * Ghezzer et al. [2023] Ghezzer, L.E., Nozzoli, F., Nicolaidis, R., Iuppa, R., Zuccon, P., De Santis, C.: Search for electron capture in ${}^{176}\mathrm{Lu}$ with a lutetium yttrium oxyorthosilicate scintillator. Phys. Rev. C 107, 045504 (2023) https://doi.org/10.1103/PhysRevC.107.045504
  * Norman et al. [2004] Norman, E.B., Browne, E., Goldman, I.D., Renne, P.R.: Improved limit on the electron capture decay branch of 176lu. Applied Radiation and Isotopes 60(5), 767–770 (2004) https://doi.org/10.1016/j.apradiso.2003.12.001
  * Haller et al. [1984] Haller, E., Palaio, N., Rodder, M., Hansen, W., Kreysa, E.: Ntd germanium: a novel material for low temperature bolometers. In: Neutron Transmutation Doping of Semiconductor Materials, pp. 21–36. Springer, ??? (1984)
  * Quarati et al. [2022] Quarati, F.G.A., Bollen, G., Dorenbos, P., et al.: Measurements and computational analysis on the natural decay of 176Lu. arXiv (2022). https://doi.org/10.48550/ARXIV.2207.14195 . https://arxiv.org/abs/2207.14195
  * Kim et al. [2021] Kim, C., Kim, D., Lee, Y., Park, C., Ullah, M.N., Kim, D., Kwon, I., Hur, S., Yeom, J.-Y.: Radiation resistance and temperature dependence of ce:gps scintillation crystal. Radiation Physics and Chemistry 183, 109396 (2021) https://doi.org/10.1016/j.radphyschem.2021.109396
  * Knoll [2010] Knoll, G.F.: Radiation Detection and Measurement, 4th Ed. Wiley, New York, NY (2010). https://cds.cern.ch/record/1300754
  * Bertoldo et al. [2020] Bertoldo, E., Derbin, A.V., Drachnev, I.S., Laubenstein, M., Lis, D.A., Mancuso, M., Muratova, V.N., Nagorny, S., Nisi, S., Petricca, F., Ryabchenkov, V.V., Sarkisov, S.E., Semenov, D.A., Subbotin, K.A., Unzhakov, E.V., Zharikov, E.V.: A test of bolometric properties of tm-containing crystals as a perspective detector for a solar axion search. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 949, 162924 (2020) https://doi.org/10.1016/j.nima.2019.162924
  * Kuwano et al. [2004] Kuwano, Y., Suda, K., Ishizawa, N., Yamada, T.: Crystal growth and properties of (lu, y)3al5o12. Journal of Crystal Growth 260(1-2), 159–165 (2004) https://doi.org/10.1016/j.jcrysgro.2003.08.060
  * Cong et al. [2009] Cong, H., Zhang, H., Wang, J., Yu, W., Fan, J., Cheng, X., Sun, S., Zhang, J., Lu, Q., Jiang, C., et al.: Structural and thermal properties of the monoclinic lu2sio5 single crystal: evaluation as a new laser matrix. Journal of Applied Crystallography 42(2), 284–294 (2009)
  * Musikhin et al. [2015] Musikhin, A.E., Naumov, V.N., Bespyatov, M.A., Ivannikova, N.V.: The heat capacity of li2moo4 in the temperature range 6–310 k. Journal of Alloys and Compounds 639, 145–148 (2015) https://doi.org/10.1016/j.jallcom.2015.03.159
  * Ghezzer et al. [2022] Ghezzer, L.E., Nozzoli, F., De Santis, C., Iuppa, R., Zuccon, P.: Search for Electron Capture in 176Lu with LYSO scintillator. arXiv (2022). https://doi.org/10.48550/ARXIV.2211.15203 . https://arxiv.org/abs/2211.15203
  * and D. Q. Adams et al. [2022] D. Q. Adams, Alduino, C., Alfonso, K., Avignone, F.T., Azzolini, O., Bari, G., Bellini, F., Benato, G., Beretta, M., Biassoni, M., Branca, A., Brofferio, C., Bucci, C., Camilleri, J., Caminata, A., Campani, A., Canonica, L., Cao, X.G., Capelli, S., Cappelli, L., Cardani, L., Carniti, P., Casali, N., Celi, E., Chiesa, D., Clemenza, M., Copello, S., Cremonesi, O., Creswick, R.J., D’Addabbo, A., Dafinei, I., Dell’Oro, S., Domizio, S.D., Dompè, V., Fang, D.Q., Fantini, G., Faverzani, M., Ferri, E., Ferroni, F., Fiorini, E., Franceschi, M.A., Freedman, S.J., Fu, S.H., Fujikawa, B.K., Giachero, A., Gironi, L., Giuliani, A., Gorla, P., Gotti, C., Gutierrez, T.D., Han, K., Hansen, E.V., Heeger, K.M., Huang, R.G., Huang, H.Z., Johnston, J., Keppel, G., Kolomensky, Y.G., Ligi, C., Liu, R., Ma, L., Ma, Y.G., Marini, L., Maruyama, R.H., Mayer, D., Mei, Y., Moggi, N., Morganti, S., Napolitano, T., Nastasi, M., Nikkel, J., Nones, C., Norman, E.B., Nucciotti, A., Nutini, I., O’Donnell, T., Ouellet, J.L., Pagan, S., Pagliarone, C.E., Pagnanini, L., Pallavicini, M., Pattavina, L., Pavan, M., Pessina, G., Pettinacci, V., Pira, C., Pirro, S., Pozzi, S., Previtali, E., Puiu, A., Rosenfeld, C., Rusconi, C., Sakai, M., Sangiorgio, S., Schmidt, B., Scielzo, N.D., Sharma, V., Singh, V., Sisti, M., Speller, D., Surukuchi, P.T., Taffarello, L., Terranova, F., Tomei, C., Vetter, K.J., Vignati, M., Wagaarachchi, S.L., Wang, B.S., Welliver, B., Wilson, J., Wilson, K., Winslow, L.A., Zimmermann, S., Zucchelli, S.: Search for majorana neutrinos exploiting millikelvin cryogenics with CUORE. Nature 604(7904), 53–58 (2022) https://doi.org/10.1038/s41586-022-04497-4
  * Silver et al. [2005] Silver, E., Austin, G., Beeman, J., Goulding, F., Haller, E., Landis, D., Madden, N.: An ntd germanium-based microcalorimeter with 3.1 ev energy resolution at 6 kev. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 545(3), 683–689 (2005)