Efficient nuclear norm approximation via the randomized UTV algorithm

N. Heavner111Department of Applied Mathematics, University of Colorado at
Boulder, 526 UCB, Boulder, CO 80309-0526, USA, P.G. Martinsson111Department of
Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO
80309-0526, USA

Abstract: The recently introduced algorithm randUTV provides a highly
efficient technique for computing accurate approximations to all the singular
values of a given matrix $\bm{\mathsf{A}}$. The original version of randUTV
was designed to compute a full factorization of the matrix in the form
$\bm{\mathsf{A}}=\bm{\mathsf{U}}\bm{\mathsf{T}}\bm{\mathsf{V}}^{*}$ where
$\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ are orthogonal matrices, and
$\bm{\mathsf{T}}$ is upper triangular. The estimates to the singular values of
$\bm{\mathsf{A}}$ appear along the diagonal of $\bm{\mathsf{T}}$. This
manuscript describes how the randUTV algorithm can be modified when the only
quantity of interest being sought is the vector of approximate singular
values. The resulting method is particularly effective for computing the
nuclear norm of $\bm{\mathsf{A}}$, or more generally, other Schatten-$p$
norms. The report also describes how to compute an estimate of the errors
incurred, at essentially negligible cost.

##  1\. Overview

This note describes an efficient algorithm for computing an accurate estimate
for the nuclear norm $\|\cdot\|_{*}$ of a given matrix
$\bm{\mathsf{A}}\in\mathbb{R}^{m\times n}$. The nuclear norm has recently
found uses in numerical optimization, beginning with the introduction by Fazel
et al. in [3, 4] of its use as an effective heuristic for solving the rank
minimization problem

minimize $\displaystyle\phantom{=}\text{rank }\bm{\mathsf{X}}$ subject to
$\displaystyle\phantom{=}\bm{\mathsf{X}}\in\mathcal{C}$

where $\bm{\mathsf{X}}\in\mathbb{R}^{m\times n}$ is the decision variable and
$\mathcal{C}$ is some given convex constraint set. Recht et al. later proved
in [11] that in certain cases, minimizing the nuclear norm also yields the
theoretical solution to the corresponding rank minimization problem,
solidifying the validity of the heuristic. The algorithm discussed in this
note, randNN, may be used in a line search to choose the step size for nuclear
norm minimization algorithms such as projected subgradient methods [11] or
mirror descent [10, 2].

The recently proposed algorithm randUTV [8] is designed to compute, given a
matrix $\bm{\mathsf{A}}\in\mathbb{R}^{m\times n}$, a factorization of the form

(1)
$\begin{array}[]{ccccccccccc}\bm{\mathsf{A}}&=&\bm{\mathsf{U}}&\bm{\mathsf{T}}&\bm{\mathsf{V}}^{*}.\\\
m\times n&&m\times m&m\times n&n\times n\end{array}$

In (1), the matrices $\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ are orthogonal,
$\bm{\mathsf{T}}$ is upper triangular, and the diagonal entries of
$\bm{\mathsf{T}}$ are good approximations to the singular values of
$\bm{\mathsf{A}}$. This note describes a modified version of randUTV which is
of particular use in the sub-problem of conducting a line search during
nuclear norm minimization. Specifically, we re-derive the randUTV algorithm
from the perspective of nuclear norm estimation. In this case, since only the
singular value estimates for $\bm{\mathsf{A}}$, rather than the entire matrix
factorization, are desired, several steps of randUTV may be omitted or
modified to yield randNN, which sees modest acceleration over randUTV and
major acceleration over a full (SVD) computation of the singular values. We
also mention a rough upper bound on the accuracy of the computed singular
values that may computed as a part of randNN at very little extra cost from
the middle matrix $\bm{\mathsf{T}}$.

The structure of this note is as follows. In Section 2, we review the notation
used throughout the remainder. We derive the algorithm randNN in 3, and in 4
we discuss an error bound for the resulting approximations to the singular
values. Finally, Section 5 contains numerical experiments exploring the
accuracy of the singular values estimated by randNN and the performance of the
algorithm compared to the industry standard.

##  2\. Preliminaries

In this note, we use the notation $\bm{\mathsf{A}}\in\mathbb{R}^{m\times n}$
to denote a matrix $\bm{\mathsf{A}}$ of dimension $m\times n$ with real
entries. $A_{i,j}$ denotes the $i$,$j$th entry of $\bm{\mathsf{A}}$. We use
the notation of Golub and Van Loan [5] to specify submatrices: If
$\bm{\mathsf{A}}$ is an $m\times n$ matrix, and
$I=[i_{1},\,i_{2},\,\dots,\,i_{k}]$ and $J=[j_{1},\,j_{2},\,\dots,\,j_{\ell}]$
are index vectors, then $\bm{\mathsf{A}}(I,J)$ denotes the corresponding
$k\times\ell$ submatrix. We let $\bm{\mathsf{A}}(I,:)$ denote the matrix
$\bm{\mathsf{A}}(I,[1,\,2,\,\dots,\,n])$, and define $\bm{\mathsf{A}}(:,J)$
analogously. The transpose of a matrix $\bm{\mathsf{A}}$ is denoted
$\bm{\mathsf{A}}^{*}$, and we use $\sigma_{i}(\bm{\mathsf{A}})$ to reference
the $i$th leading singular value of $\bm{\mathsf{A}}$.

We measure matrices with either the spectral norm $\|\cdot\|,$ nuclear norm
$\|\cdot\|_{*}$, or Frobenius norm $\|\cdot\|_{F}$, with definitions given by

$\|\bm{\mathsf{A}}\|=\sup_{\|\bm{\mathsf{x}}\|=1}\|\bm{\mathsf{A}}\bm{\mathsf{x}}\|_{2}=\sigma_{1}(\bm{\mathsf{A}}),\quad\|\bm{\mathsf{A}}\|_{*}=\sum_{i=1}^{\min(m,n)}\sigma_{i}(\bm{\mathsf{A}}),\quad\|\bm{\mathsf{A}}\|_{F}=\left(\sum_{i=1}^{m}\sum_{j=1}^{n}A_{i,j}^{2}\right)^{1/2}$

(where $\|\cdot\|_{2}$ is the Euclidean 2-norm for vectors in
$\mathbb{R}^{n}$).

##  3\. Description of the algorithm

###  3.1. Approach

Consider a matrix $\bm{\mathsf{A}}\in\mathbb{R}^{m\times n}$. Now partition
$\bm{\mathsf{A}}$ as

$\begin{tabular}[]{ccccccc}&&$b$&\ldelim[{2}{3mm}&$\bm{\mathsf{A}}_{11}$&$\bm{\mathsf{A}}_{12}$&\rdelim]{2}{3mm}\\\
$\bm{\mathsf{A}}$&=&$m-b$&&$\bm{\mathsf{A}}_{21}$&$\bm{\mathsf{A}}_{22}$&\\\
&&&&$b$&$n-b$&\end{tabular}.$

The construction of the algorithm will be guided by several fundamental
observations:

  * •

If $\bm{\mathsf{A}}_{12}=\bm{\mathsf{0}}$ and
$\bm{\mathsf{A}}_{21}=\bm{\mathsf{0}}$, then
$\|\bm{\mathsf{A}}\|_{*}=\|\bm{\mathsf{A}}_{11}\|_{*}+\|\bm{\mathsf{A}}_{22}\|_{*}$.

  * •

Multiplying $\bm{\mathsf{A}}$ by an orthogonal matrix does not change the
nuclear norm.

  * •

Consider the matrix
$\tilde{\bm{\mathsf{A}}}=\bm{\mathsf{U}}^{*}\bm{\mathsf{A}}\bm{\mathsf{V}}$,
where $\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ are orthogonal matrices whose
first $b$ columns span the same space as the leading $b$ left and right
singular vectors of $\bm{\mathsf{A}}$, respectively. Then giving
$\tilde{\bm{\mathsf{A}}}$ the same partition as $\bm{\mathsf{A}}$, we have
$\tilde{\bm{\mathsf{A}}}_{12}=\bm{\mathsf{0}}$ and
$\tilde{\bm{\mathsf{A}}}_{21}=\bm{\mathsf{0}}$. For this reason, we will call
such a $\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ the “optimal” rotation matrices
for the problem.

These remarks lay out a path for estimating $\|\bm{\mathsf{A}}\|_{*}$. We will
choose $b$ to be a relatively small block size, say 50 or 100, and will
compute an orthonormal matrices $\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ whose
first $b$ columns approximately span the same space as the respective leading
$b$ left and right singular vectors of $\bm{\mathsf{A}}$. We will then form
$\bm{\mathsf{A}}^{(1)}=\bm{\mathsf{U}}^{*}\bm{\mathsf{A}}\bm{\mathsf{V}}$,
after which we will have $\bm{\mathsf{A}}^{(1)}_{21}=\bm{\mathsf{0}}$ and
$\|\bm{\mathsf{A}}^{(1)}_{12}\|$ is small. At this point, we may estimate the
norm by finding the norms of the small $\bm{\mathsf{A}}^{(1)}_{11}$ block and
the large $\bm{\mathsf{A}}^{(1)}_{22}$ block separately.
$\|\bm{\mathsf{A}}^{(1)}_{11}\|_{*}$ may be computed efficiently with an SVD
computation, and we may recursively apply our strategy for $\bm{\mathsf{A}}$
to $\bm{\mathsf{A}}^{(1)}_{22}$ to estimate its norm.

###  3.2. Computing $\bm{\mathsf{V}}$

At this point in the derivation, the most pressing question is how to find
$\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ that satisfy the aforementioned
conditions. We first consider the construction of $\bm{\mathsf{V}}$, at which
point $\bm{\mathsf{U}}$ will easily follow. Recent work in randomized subspace
iteration, discussed in papers including [7, 12, 14, 6], enables the efficient
computation of the desired $\bm{\mathsf{V}}$.

In particular, consider a sampling matrix
$\bm{\mathsf{Y}}=\bm{\mathsf{A}}^{*}\bm{\mathsf{G}}$, where $\bm{\mathsf{G}}$
is an $m\times b$ Gaussian random matrix. Then with high probability, the
column space of $\bm{\mathsf{Y}}$ often aligns closely to the subspace spanned
by the leading $b$ right singular vectors of $\bm{\mathsf{A}}$. As discussed
in [12, 7], in certain situations when the decay of singular values of
$\bm{\mathsf{A}}$ makes the alignment suboptimal, then using
$\bm{\mathsf{Y}}=(\bm{\mathsf{A}}^{*}\bm{\mathsf{A}})^{q}\bm{\mathsf{A}}^{*}\bm{\mathsf{G}}$
instead provides considerable correction, where $q$ is some small nonnegative
integer. Typically, $q=0,1,2$ suffices to obtain a close-to-optimal
approximation to the desired subspace.

Thus, the following steps compute a matrix $\bm{\mathsf{V}}$ whose columns
span approximately the same subspace as the leading $b$ right singular vectors
of $\bm{\mathsf{A}}$:

  * •

Draw an $n\times b$ random Gaussian matrix $\bm{\mathsf{G}}$.

  * •

Compute
$\bm{\mathsf{Y}}=(\bm{\mathsf{A}}^{*}\bm{\mathsf{A}})^{q}\bm{\mathsf{A}}^{*}\bm{\mathsf{G}}$
for some small nonnegative integer $q$.

  * •

Compute an orthonormal basis for the column space of $\bm{\mathsf{Y}}$ with a
QR factorization using Householder reflectors to obtain
$\bm{\mathsf{Y}}=\bm{\mathsf{V}}\bm{\mathsf{R}}$.

###  3.3. Computing $\bm{\mathsf{U}}$

To compute $\bm{\mathsf{U}}$, we first note that if our computed
$\bm{\mathsf{V}}$ were optimal, then the first $b$ columns of
$\bm{\mathsf{A}}\bm{\mathsf{V}}$ would be composed exclusively of linear
combinations of the leading $b$ left singular vectors of $\bm{\mathsf{A}}$.
Thus, to form $\bm{\mathsf{U}}$, we may simply compute an orthonormal basis
for the first $b$ columns of $\bm{\mathsf{A}}\bm{\mathsf{V}}$ via Householder
reflectors to obtain
$\bm{\mathsf{A}}\bm{\mathsf{V}}(:,1:b)=\bm{\mathsf{U}}\bm{\mathsf{R}}$.

######  Remark 1.

In applications, the input matrix $\bm{\mathsf{A}}$ is often rank-deficient,
with numerical rank $k$ determined by the $k$ for which
$\sigma_{i}(\bm{\mathsf{A}})$ is less than some specified tolerance for
$k<i\leq n$. In the case where $k\ll n$, randNN may be sped up substantially
by terminating the algorithm once the estimated singular values drop below
some user-defined threshold.

    function ss = randNN(A,b,q)
      m = size(A,1); n = size(A,2);
      ss = zeros(n,1);
      T = A;
      for i=1:ceil(n/b)
        I1 = (b*(i-1)+1):min((b*i),n);
        I2 = (b*i+1):m;
        J1 = (b*(i-1)+1):min((b*i),n);
        J2 = (b*i+1):n;
        if isempty(J2) == 0
          T_work = T([I1 I2],[J1 J2]);
          [TT,ss_part] = step_nn(T_work,b,q);
          ss(I1) = ss_part;
          T([I1 I2],[J1 J2]) = TT;
        else
          ss_part = svd(T(I1,J1));
          ss(I1) = ss_part;
        end
      end
    return

|

    function [T,ss_part] = step_nn(A,b,q)
      G = randn(size(A,1),b);
      Y = A’*G;
      for i=1:q
        Y = A’*(A*Y);
      end
      [V,~] = qr(Y);
      T = A*V;
      [U,R] = qr(T(:,1:b));
      T(:,1:b) = R;
      T(:,(b+1):end) = U’*T(:,(b+1):end);
      ss_part = svd(R);
    return

---|---
Figure 1. Matlab code for the algorithm randNN that, given an $m\times n$
matrix $\bm{\mathsf{A}}$, computes estimates for each of its singular values.
The input parameters $b$ and $q$ reflect the block size and the number of
steps of power iteration, respectively. This code is simplistic in that it
does not store or apply the transformation matrices $\bm{\mathsf{U}}$ and
$\bm{\mathsf{V}}$ efficiently as products of Householder vectors. It also does
not apply a stopping criterion to terminate the algorithm if the estimated
singular values become small, nor does it compute the error bound discussed in
Section 4.

##  4\. Error bounds

In this section, we make note of an error bound for the accuracy of the
estimated singular values resulting from the randNN algorithm.

First, note that after applying $\lceil n/b\rceil$ left and right rotation
matrices to $\bm{\mathsf{A}}$ as described in 3, we are left with an upper
triangular matrix which we shall call $\bm{\mathsf{T}}$ whose mass is
concentrated in blocks of size $b\times b$ along its diagonal. Therefore, let
$\bm{\mathsf{T}}_{d}$ be the block diagonal matrix consisting of the $b\times
b$ main diagonal blocks of $\bm{\mathsf{T}}$, and let $\bm{\mathsf{T}}_{u}$ be
the upper triangular matrix defined by the relation

$\bm{\mathsf{T}}=\bm{\mathsf{T}}_{d}+\bm{\mathsf{T}}_{u}.$

Observe that the estimated singular values of $\bm{\mathsf{A}}$ in our
algorithm consist precisely of the computed singular values of
$\bm{\mathsf{T}}_{d}$. Next, since $\bm{\mathsf{A}}$ and $\bm{\mathsf{T}}$
only differ by orthogonal transforms, we have that
$\|\bm{\mathsf{A}}\|_{*}=\|\bm{\mathsf{T}}\|_{*}$, and more specifically,
$\sigma_{i}(\bm{\mathsf{A}})=\sigma_{i}(\bm{\mathsf{T}})$ for
$i=1,2,\ldots,\min(m,n)$. Thus an error bound given in [13] and [9] states
that

(2)
$\sqrt{\sum_{i=1}^{\min(m,n)}\bigl{(}\sigma_{i}(\bm{\mathsf{A}})-\sigma_{i}(\bm{\mathsf{T}}_{d})\bigr{)}^{2}}\leq\|\bm{\mathsf{T}}_{u}\|_{F}.$

Since the calculation of $\|\bm{\mathsf{T}}_{u}\|_{F}$ is
$\mathcal{O}(n^{2})$, this bound may be calculated at negligible relative cost
as part of randNN, serving as an assurance of the estimation’s validity.

##  5\. Numerical experiments

###  5.1. Computational speed

In this section, we investigate the speed of the proposed algorithm randNN and
compare it to the speed of an exact computation of the singular values using
LAPACK’s highly optimized dgesvd.

All experiments reported in this note were performed on an Intel Core i7-6700K
processor (4.0 GHz) with 4 cores. In order to be able to show scalability
results, the clock speed was throttled at 4.0 GHz, turning off so-called turbo
boost. Other details of interest include that the OS used was Ubuntu (Version
16.04.2), and the code was compiled with Intel’s icc (Version 17.0.4.196).
Main routines for computing the singular values (dgesvd) were taken from
Intel’s MKL library (Version 2017.0.3). Standard BLAS routines used in our
implementation of randNN were also taken from the Intel MKL library.

Each of the three algorithms we tested (randNN,randUTV,SVD) was applied to
double-precision real matrices of size $n\times n$. We report the following
times:

  * •

$T_{\text{svd}}$. The time in seconds for the LAPACK function dgesvd from
Intel’s MKL, where the orthogonal matrices $\bm{\mathsf{U}}$ and
$\bm{\mathsf{V}}$ in the SVD were not built.

  * •

$T_{\text{randUTV}}$. The time in seconds for the function randUTV described
in [8]. The authors’ original implementation was used, and the orthogonal
matrices $\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ in the UTV decomposition were
not built.

  * •

$T_{\text{randNN}}$. The time in seconds for our implementation of randNN.

In all cases, we used a block size of $b=64$. While likely not optimal for all
problem sizes, this block size yields near best performance and allows us to
easily compare and contrast the performance of the different implementations.

Figure 2. Computational speed of randNN compared to the speeds of the LAPACK
routine dgesvd and the recently introduced randUTV [8]. The algorithms were
applied to double-precision real matrices of size $n\times n$.

As expected, Figure 2 shows that both randUTV and randNN are markedly faster
than LAPACK’s dgesvd, even in the single case. When more cores are added to
the computation, the gain in speed is even more dramatic, hi-lighting the fact
that the SVD algorithm is not designed to make efficient use of parallel
computing architectures. Finally the increase in speed from randUTV to randNN
is modest, but in a line search setting where the nuclear norm must be
computed many times, this saving can add up over the course of the solution of
the rank minimization problem.

###  5.2. Errors

In this section, we report the results of numerical experiments that were
conducted to test the accuracy of the approximation to the singular values
provided by randNN. Specifically, we compare the estimated singular values to
the true singular values of two different test matrices:

  * •

Matrix 1 (S shaped decay): This is an $n\times n$ matrix of the form
$\bm{\mathsf{A}}=\bm{\mathsf{U}}\bm{\mathsf{D}}\bm{\mathsf{V}}^{*}$ where
$\bm{\mathsf{U}}$ and $\bm{\mathsf{V}}$ are randomly drawn matrices with
orthonormal columns (obtained by performing QR on a random Gaussian matrix),
and where the diagonal entries of $\bm{\mathsf{D}}$ are chosen to first hover
around 1, then decay rapidly, and then level out at $10^{-6}$, as shown in
Figure 3 (black line) on the left.

  * •

Matrix 2 (Single Layer BIE): This matrix is the result of discretizing a
Boundary Integral Equation (BIE) defined on a smooth closed curve in the
plane. To be precise, we discretized the so called “single layer” operator
associated with the Laplace equation using a $6^{\rm th}$ order quadrature
rule designed by Alpert [1]. This is a well-known ill-conditioned problem for
which column pivoting is essential in order to stably solve the corresponding
linear system.

In each case, a matrix size of $m=n=5000$ was used with a “power iteration
parameter” (see Section 3.2) of $q=2$ and a block size of $b=64$.

For each test matrix, we plot both the estimated and true singular values
themselves and the relative error of the estimates

$\frac{|\sigma_{i}(\bm{\mathsf{A}})-\sigma_{i}(\bm{\mathsf{T}}_{d})|}{|\sigma_{i}(\bm{\mathsf{A}})|},\quad
i=1,2,\ldots,\min(m,n).$

Figure 3. Errors for the test matrices described in Section 5.2. Left: “Matrix
1,” the matrix artificially to have the singular value decay pattern shown
here. Right: “Matrix 2,” a matrix resulting from the discretization of of
Boundary Integral Equation.

##  6\. Availability of code

An implementation of the discussed algorithm is available under 3-clause
(modified) BSD license from:

https://github.com/nheavner/nn_code

## References

  * [1] Bradley K Alpert, _Hybrid gauss-trapezoidal quadrature rules_ , SIAM Journal on Scientific Computing 20 (1999), no. 5, 1551–1584.
  * [2] Amir Beck and Marc Teboulle, _Mirror descent and nonlinear projected subgradient methods for convex optimization_ , Operations Research Letters 31 (2003), no. 3, 167–175.
  * [3] Maryam Fazel, _Matrix rank minimization with applications_ , Ph.D. thesis, PhD thesis, Stanford University, 2002.
  * [4] Maryam Fazel, Haitham Hindi, and Stephen P Boyd, _A rank minimization heuristic with application to minimum order system approximation_ , American Control Conference, 2001. Proceedings of the 2001, vol. 6, IEEE, 2001, pp. 4734–4739.
  * [5] Gene H Golub and Charles F Van Loan, _Matrix computations_ , vol. 3, JHU Press, 2012.
  * [6] Nathan Halko, Per-Gunnar Martinsson, Yoel Shkolnisky, and Mark Tygert, _An algorithm for the principal component analysis of large data sets_ , SIAM Journal on Scientific computing 33 (2011), no. 5, 2580–2594.
  * [7] Nathan Halko, Per-Gunnar Martinsson, and Joel A Tropp, _Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions_ , SIAM review 53 (2011), no. 2, 217–288.
  * [8] Per-Gunnar Martinsson, Gregorio Quintana-Orti, and Nathan Heavner, _randutv: A blocked randomized algorithm for computing a rank-revealing utv factorization_ , arXiv preprint arXiv:1703.00998 (2017).
  * [9] Leon Mirsky, _Symmetric gauge functions and unitarily invariant norms_ , The quarterly journal of mathematics 11 (1960), 50–59.
  * [10] Arkadii Nemirovskii, David Borisovich Yudin, and Edgar Ronald Dawson, _Problem complexity and method efficiency in optimization_ , (1983).
  * [11] Benjamin Recht, Maryam Fazel, and Pablo A Parrilo, _Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization_ , SIAM review 52 (2010), no. 3, 471–501.
  * [12] Vladimir Rokhlin, Arthur Szlam, and Mark Tygert, _A randomized algorithm for principal component analysis_ , SIAM Journal on Matrix Analysis and Applications 31 (2009), no. 3, 1100–1124.
  * [13] Gilbert W Stewart, _Perturbation theory for the singular value decomposition_ , Tech. report, 1998.
  * [14] Franco Woolfe, Edo Liberty, Vladimir Rokhlin, and Mark Tygert, _A fast randomized algorithm for the approximation of matrices_ , Applied and Computational Harmonic Analysis 25 (2008), no. 3, 335–366.


University of Bergen
Department of informatics
Latency in Mesh Networks
_Author:_ Joakim Algrøy
_Supervisor:_ Albin Severinson

December 15, 2020

###### Abstract

In this work, we consider latency in wireless networks, motivated by the need
for reliable and low-latency wireless communication for applications such as
vehicle-to-vehicle communication and industrial automation. The environment
that these applications operate in is often highly dynamic, and, as a result,
the wireless network technology used needs to be able to adapt to changing
conditions, which makes mesh networking a good fit. Hence, in this work, we
consider Thread and Bluetooth Mesh, two prominent mesh networking
technologies. We measure one-way point-to-point latency using these
technologies and show the cumulative probability distribution of the latency
for varying payload size. We carry out the measurements under static
conditions in an open-air environment. Our results show that Thread achieves
lower latency than Bluetooth Mesh, especially for larger payloads.
Furthermore, we show that Thread meets some of the latency targets set out for
5G. However, since Thread networks are slow to adapt to network changes,
Thread may still not be suitable for application requiring low latency.
Bluetooth mesh, on the other hand, does not meet any of the latency targets,
although it is close when the payload is small (less than $11$ bytes).
Overall, there needs to be further optimization before these technologies can
be used for reliable and low-latency communication in dynamic environments.

##  1 Introduction

Reliable and low-latency wireless communication is a critical for many
important applications. For example, a vehicle-to-vehicle communication system
designed to prevent accidents is dependent on low latency in order to react to
dangerous situations in time. Other examples include factory automation, where
the latency targets can be less than 1 millisecond (ms) for, e.g., precise
robotic motion control, or 10 ms to 50 ms (depending on the application) for
cooperative driving [1]. However, achieving high reliability and low latency
for these applications is challenging. For example, the network topology
(including which devices are part of the network) and radio environment may
change significantly as vehicles travel or machines are shut on or off.

A potential solution is provided by mesh networks. In these networks, end
devices communicate directly in a peer-to-peer fashion if they are within
range, or via intermediary relay nodes (often any node in the network can act
as a relay node) otherwise. This is in contrast to so-called _star topology
networks_ (e.g., WiFi and cellular networks for mobile phones), where
communication between end devices has to go through a router, or a hierarchy
of routers, whose only role is to relay traffic. We depict the difference in
fig. 1. Mesh networks are appealing since they require little to no existing
infrastructure, which can reduce the cost of deployment, do not depend on
routers, which may fail, and are able to adapt to changes by constructing new
routes through the network when the environment changes. Furthermore, the
direct nature of the communication can also reduce latency.

figures/mesh-vs-star

Figure 1: Mesh (left) and star (right) network topology. End devices shown in
blue and routers in red. Mesh networks do not rely on special devices for
routing network traffic. Instead, network nodes collaborate to construct paths
through the network.

Hence, in this report we benchmark existing mesh network technologies to
assess how suitable they are for latency-critical applications. In particular,
we measure one-way latency between pairs of devices over Thread and Bluetooth
mesh networks, which are two of the most prominent mesh network technologies.
We find that for 1 byte messages, the median latency is similar for both
technologies, however for the 90th and 99th percentile Thread has lower
latency than Bluetooth Mesh. For 20 and 100 byte messages, we find that
Bluetooth Mesh is significantly impacted by it’s fragmentation and re-
transmission schemes, and latency is not compatible with latency targets for
the discussed applications. For these message sizes Thread is above the 50ms
latency target in the 99.9th percentile, but remains within 30ms in the 99th
percentile. None if the technologies come close to the 1ms target for precise
robotic motion control.

###  1.1 Previous work

Latency in mesh networks has been measured in several previous reports. In
particular, in two reports, published by Silicon Labs and the OpenThread
project, respectively.

Silicon Labs published a performance comparison [7], where the authors carry
out open air tests of Bluetooth Mesh, Thread, and Zigbee in an actively used
office building. Note that these technologies use the 2.4GHz frequency band,
which is part of the unlicensed spectrum. As a result, a wide variety of
different technologies use this spectrum, which will impact the test results.
Their test networks ranged in size from 24 to 192 devices, and measured round
trip time (as opposed to one-way latency). In the report, the authors conclude
that performance is similar across the mesh systems considered for small
payloads (e.g. 8 bytes), however as the payload size is increased (e.g., to 50
bytes), Thread has a lower average and less spread in latency than Bluetooth
Mesh. The authors attribute this to there being more fragmentation in
Bluetooth mesh due to the smaller packet size. Furthermore, the flooding
approach of Bluetooth Mesh may cause a higher number of total transmissions in
the network, causing delays due to busy channels.

The OpenThread project continually measure latency, loss rate, and throughput
to evaluate the performance impact of new software versions [9]. The testing
method is described in [8]. In particular, the method is based on connecting
the devices using coaxial cables, such that the wireless signals are contained
within the cables and is isolated from interfering signals from other nearby
devices. Testing is done using the nRF52840 Development Kit from Nordic
Semiconductors [12], the same device used in this work. As opposed to the
report by Silicon Labs, OpenThread measures one-way latency. This is achieved
by using a wired connection between devices to synchronize time. They use a
network with 12 devices for measuring loss rate and throughput, but perform
latency tests with just 1, 2 and 3 hops (2, 3, and 4 devices, respectively).
OpenThread does not run similar tests of other mesh network technologies, so
we cannot use their results for direct comparison with Bluetooh Mesh. However,
we can compare our OpenThread results with theirs to assess the impact of the
differences in testing methods.

##  2 An overview of Thread and Bluetooth Mesh

In this work, we consider two prominent wireless mesh network systems; Thread
and Bluetooth Mesh. Specifically we consider Google’s implementation of
Thread, OpenThread [2], which is, for example, used in the Nest line of home
automation products, and the implementation of Bluetooth Mesh found in the the
Zephyr real-time operating system [10]. Thread and Bluetooth Mesh differ
significantly in their design, both at the physical and protocol layers, which
makes comparing their respective performance especially interesting.

Thread’s [3] physical and media access control (MAC) layers are based on the
IEEE 802.15.4 wireless standard [5]. The physical layer uses the 2.4GHz
wireless band with a data rate of 250 kbit/s and a frame size of up to 127
bytes. Routing in a Thread network is handled by nodes assigned the routing
role. These nodes maintain routing tables and forward frames based on the
routing information protocol (RIP) algorithm. Every device capable of routing
can be assigned the router role when the network deems it necessary, and this
happens automatically without user intervention. If a router is no longer
needed, it will automatically be demoted to a regular end device. Only keeping
the necessary number of routers allows the entire network to remain connected
without overcrowding it with relayed traffic. Even though the network reacts
to changes dynamically, there can be significant delay before the network
reacts, meaning that, e.g., a node leaving the network, can result in long
outages.

Bluetooth Mesh [13] makes use of the widespread Bluetooth Low Energy (LE)
protocol for what is referred to in the specification as the bearer layer,
which essentially corresponds to the link and physical layers of Thread. Thus,
any device designed for Bluetooth LE can also run Bluetooth Mesh with only
software modifications. Bluetooth Mesh makes use of the Bluetooth LE
advertising channels for communication, which can be set to 1Mbit/s,
500Kbit/s, or 125Kbit/s mode depending on the configuration. The advertising
packets have 11 bytes available for the application layer, however 1, 2, or 3
bytes must be used for an opcode, leaving 8 to 10 bytes for application data.
Routing is performed by relay nodes in the network, who do not maintain
routing tables, but re-transmits any messages that are not intended for them
and that meet certain requirements. This approach to routing is referred to as
network flooding. Not all nodes need to have relay capability, and among those
that do have the relay capability it can be enabled or disabled. As opposed to
Thread, enabling and disabling relay capability does not happen automatically
in Bluetooth Mesh. Instead this must be manually configured when deploying the
network. As shown in [6], latency in large Bluetooth Mesh networks can be
significantly improved by optimizing the number of relay nodes, meaning manual
configuration might be necessary to achieve the best possible latency.

Bluetooth Mesh has several configurable parameters that will have a big impact
on latency. To achieve high reliability, Bluetooth Mesh will send each packet
multiple times, both when transmitting at the source and when relaying a
received packet. Two parameters, network transmit count ($N_{\mathsf{TC}}$)
and Network transmit interval ($N_{\mathsf{TI}}$) determine how many times a
packet is transmitted and the minimum time between the start of each
transmission, respectively. In addition to $N_{\mathsf{TI}}$, a random delay
between 0 and 10 ms is added between each packet. While Bluetooth LE v5
supports interleaving packets through _advertising sets_ , Bluetooth Mesh
remains compatible with Bluetooth LE v4.2 and does not make use of this
feature [4]. Thus, packets are sent sequentially, leading to a delay of at
least $N_{\mathsf{TC}}\times N_{\mathsf{TI}}$ between each packet. Given a
message requiring M packets, a minimum latency when sending directly between
two devices (1 hop), can be given by eq. 1, not including processing delays
[4]:

$t=T_{trans,0}+\sum_{i=1}^{M-1}\sum_{j=1}^{N_{\mathsf{TC}}}{(N_{\mathsf{TI}}+T_{\mathsf{trans},i}+T_{\mathsf{rand},i,j})}$
(1)

Where $T_{\mathsf{trans},i}$ is the time it takes to transmit packet $i$, and
$T_{\mathsf{rand},i,j}$ is a random value between 0 and 10 ms.

The different approaches to routing will have an impact on latency. In
general, we expect routing to be more efficient in Thread compared to
Bluetooth Mesh, leading to improved latency. However, Thread may be slow to
adapt to changes (e.g. when a router leaves the network) as it needs to
reconfigure the routing tables to reflect the new network state. Bluetooth
Mesh will not see any such delay, as the nodes do not keep track of routes.
Differences in packet sizes will also affect latency. Bluetooth Mesh packets
have an application payload of at most 10 bytes, while Thread has a payload of
up to 63 bytes. Thus, Bluetooth Mesh will have increased fragmentation
compared to Thread, which can lead to higher latency.

figures/testbed

Figure 2: Testing setup with two mesh nodes. A host computer connected via
serial is used for test management and to record results.

##  3 Latency measurements

We use the nRF52840 development kit [12], which is based on the nRF52840
system on a chip, for our testing. The nRF52840 supports the physical layer of
both Bluetooth Mesh and Thread. There are also implementations of the Thread
and Bluetooth Mesh protocol layers available by projects that support this
device, namely OpenThread and the Zephyr RTOS, respectively.

Our setup for measuring one-way latency between two devices consists of two
nRF52840 devices in close proximity, connected via a GPIO line. The system is
depicted in fig. 2. The following procedure is used for each individual
latency measurement:

  1. 1.

The transmitting device sends a signal, via the wired GPIO line to the
receiving device. Immediately afterwards the transmitting device sends a mesh
network message.

  2. 2.

The GPIO signal triggers a hardware interrupt on the receiving device, which
immediately records a timestamp indicating the time of transmission for the
current packet.

  3. 3.

The receiving device records a new timestamp once it has received the wireless
mesh message. The difference between these two timestamps is recorded as the
one-way latency of that message.

  4. 4.

We wait 500 ms, and then return to step 1.

This testing process is performed for both mesh technologies, using messages
of 1, 20, and 100 bytes.

figures/1_and_20B_cdf

Figure 3: Results of latency tests with 1 and 20 byte messages. 100 000
packets with 250ms interval between receive and next transmission.

figures/100B_cdf_comparison

Figure 4: Results of latency tests with 100 byte messages. 10 000 packets with
500ms interval. Note the different scale on the time axis.

###  3.1 Latency distribution

As seen in fig. 3, for 1 byte messages, median latency is similar for
Bluetooth Mesh and Thread. However, Bluetooth Mesh has a worse tail latency
than Thread. For both technologies, we see that the latency target of 1 ms for
precise robotic motion control is not in reach, even for this minimal message
size. However, OpenThread remains within 10ms with $>99.9\%$ probability, and
Bluetooth Mesh remains within 50ms with close to $99.9\%$ probability, which
was mentioned in the introduction as the lower and upper end of a range of
latency targets for cooperative driving.

Once we increase the message size to 20 bytes for Bluetooth Mesh, we see the
results of the transmission delays between packets discussed in the
introduction. A 20 byte message with a 3-byte opcode (3 bytes needed for
vendor-specific opcodes, see section 2), will need 3 packets to be transmitted
with Bluetooth Mesh. We have set parameters $N_{\mathsf{TC}}=3$ and
$N_{\mathsf{TI}}=20$, giving an absolute minimum latency of 120 ms by eq. 1,
assuming all random delays are zero. However, looking at the source code of
the Zephyr Project’s implementation of Bluetooth Mesh, we see that the actual
time used to send a packet is $30+(N_{\mathsf{TI}}+10)\cdot N_{\mathsf{TC}}$
[11], indicating that 240 ms is the minimum expected latency, which is also
clear from the 20 byte plot in fig. 3. OpenThread maintains a low latency, and
is still within 10ms with $>99.9\%$ probability for 20 byte messages.

###  3.2 Latency time series

In fig. 5 we plot latency of 1 byte messages in the measured order for
OpenThread and Bluetooth Mesh. In both cases, we see a clear range of high
density where the most common latency results lie. We also see a number of
measurements above the dense range, which seem to be more uniformly
distributed in Bluetooth Mesh, whereas in OpenThread they are more sparse and
appear in bursts. These bursts may correspond to changes in interference from
the environment.

(a) OpenThread

(b) Bluetooth Mesh

Figure 5: Latency of 1 byte packets in order.

##  4 Conclusions

Low-latency wireless mesh networks can be an important technology for enabling
applications such as cooperative autonomous driving and factory automation
since the network is able to automatically adapt to changing conditions.
However, meeting the latency targets set out for these applications is
challenging. In our results, Thread maintains a one-way single-hop latency
below 10ms for messages with a payload of up to 20 bytes with $>99.9\%$
probability, which is within the the 10-50ms latency targets for cooperative
driving suggested in [1]. However, given that Thread is slow in adapting to
changes in network topology (which occur, e.g., when a vehicle moves out of
range), it may be unfeasible to use for applications such as such as
cooperative driving. Bluetooth Mesh does not have this limitation, however we
show that latency is significantly higher than for Thread. For example, for
messages with a 1 byte payload, the $99.9$-th percentile latency is about
50ms. For messages with a payload larger than 11 bytes, the latency for
Bluetooth mesh is more than an order of magnitude larger than that of Thread
due to latency caused by fragmentation and re-transmission. For example, the
minimum latency observed for Bluetooth mesh and messages with a 20 and 100
byte payload is 240ms and 950ms, respectively.

Our results indicate that Bluetooth Mesh is only suitable for applications
requiring very small message sizes (less than 11 bytes), or without stringent
latency requirements. Thread, on the other hand, is suitable for applications
requiring a $99.9$-th percentile latency of up 10 ms or more when the payload
size is at most 20 bytes, or a $99.9$-th percentile latency of about 50 ms or
more when the payload is at most 100 bytes. However, Thread is only suitable
for applications operating in a static environment, due to the fact that it is
slow to adapt to changing conditions. Hence, neither technology is able to
fully meet the requirements set out for, e.g., cooperative autonomous driving.

## References

  * [1] 3GPP  “Study on Communication for Automation in Vertical domains”, 2020
  * [2] Google  “OpenThread”  URL: https://openthread.io/
  * [3] Thread Group  “Thread Stack Fundamentals”  URL: https://www.silabs.com/documents/public/white-papers/Thread-Stack-Fundamentals.pdf
  * [4] Hernández-Solana et al.  “Bluetooth Mesh Analysis, Issues, and Challenges”  In _IEEE Access_ 8, 2020, pp. 53784–53800  DOI: 10.1109/ACCESS.2020.2980795
  * [5] “IEEE Std 802.15.4-2006 (Revision of IEEE Std 802.15.4-2003): IEEE Standard for Information technology– Local and metropolitan area networks– Specific requirements– Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications”  IEEE, 2006
  * [6] Silicon Labs  “AN1137: Bluetooth® Mesh Network Performance”  URL: https://www.silabs.com/documents/public/application-notes/an1137-bluetooth-mesh-network-performance.pdf
  * [7] Silicon Labs  “AN1142: Mesh Network Performance Comparison”  URL: https://www.silabs.com/documents/public/application-notes/an1142-mesh-network-performance-comparison.pdf
  * [8] OpenThread  “OpenThread Quality Dashboard”  URL: https://openthread.io/testing/OpenThread_Quality_Dashboard.pdf
  * [9] OpenThread  “Quality Dashboard”  URL: https://openthread.io/testing/quality-dashboard
  * [10] Zephyr Project  “Bluetooth Mesh”  URL: https://docs.zephyrproject.org/latest/samples/bluetooth/mesh/README.html
  * [11] Zephyr Project  “Zephy RTOS”  In _GitHub repository_ GitHub, https://github.com/zephyrproject-rtos/zephyr, 2020
  * [12] Nordic Semiconductors  “nRF52840 DK”  URL: https://www.nordicsemi.com/Software-and-Tools/Development-Kits/nRF52840-DK
  * [13] Bluetooth SIG  “Mesh Networking Specifications”  URL: https://www.bluetooth.com/specifications/mesh-specifications/


# BINGO innovative assembly for background reduction in bolometric
$0\nu\beta\beta$ experiments

A. Armatol  C. Augier  I.C. Bandac  D. Baudin  G. Benato  V. Berest  L. Bergé
J. Billard  J.M. Calvo-Mozota  P. Carniti  M. Chapellier  F.A. Danevich  M. De
Jesus  T. Dixon  L. Dumoulin  F. Ferri  J. Gascon  A. Giuliani  H. Gomez  C.
Gotti  Ph. Gras  M. Gros  A. Juillard  H. Khalife<EMAIL_ADDRESS>V.V.
Kobychev  M. Lefevre  P. Loaiza  P. de Marcillac  S. Marnieros  C.A. Marrache-
Kikuchi  M. Martinez  Ph. Mas  E. Mazzucato  J.F. Millot  C. Nones  E.
Olivieri  A. Ortiz de Solórzano  G. Pessina  D.V. Poda  A. Rojas  J.A.
Scarpaci  B. Schmidt  O. Tellier  V.I. Tretyak  G. Warot  Th. Zampieri  M.M.
Zarytskyy  A. Zolotarova  IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-
Yvette, France  Institut de Physique des 2 Infinis, Villeurbanne, France
Laboratorio Subterráneo de Canfranc, 22880 Canfranc-Estación, Spain  INFN
Laboratori Nazionali del Gran Sasso, I-67100 Assergi (AQ), Italy  Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Escuela Superior de
Ingeniería, Ciencia y Tecnología, Universidad Internacional de Valencia – VIU,
46002 Valencia, Spain  Escuela Superior de Ingeniería y Tecnología,
Universidad Internacional de La Rioja, 26006 Logroño, Spain  INFN Sezione di
Milano-Bicocca, I-20126 Milano, Italy  Institute for Nuclear Research of NASU,
03028 Kyiv, Ukraine  INFN Sezione di Roma Tor Vergata, Rome, Italy  Centro de
Astropartículas y Física de Altas Energías, Universidad de Zaragoza, 50009
Zaragoza, Spain  ARAID Fundación Agencia Aragonesa para la Investigación y el
Desarrollo, 50018 Zaragoza, Spain  Univ. Grenoble Alpes, CNRS, Grenoble INP,
LPSC/LSM-IN2P3, 73500 Modane, France

###### Abstract

BINGO is a project aiming to set the grounds for large-scale bolometric
neutrinoless double-beta-decay experiments capable of investigating the
effective Majorana neutrino mass at a few meV level. It focuses on developing
innovative technologies to achieve a very low background index, of the order
of $10^{-5}$ counts/(keV kg yr) in the region of interest. The BINGO
demonstrator, called MINI-BINGO, will be composed of Li2MoO4 and TeO2 crystals
coupled to bolometric light detectors designed to investigate the promising
double-beta-decay isotopes 100Mo and 130Te. This will allow us to reject a
significant background in bolometers caused by surface contamination from
$\alpha$-active radionuclides by means of light yield selection. In addition,
BINGO introduces new methods to mitigate other sources of background, such as
surface radioactive contamination, external $\gamma$ radioactivity, and pile-
up due to random coincidence of background events. This paper focuses on the
description of an innovative assembly designed to reduce the passive materials
in line of sight of the detectors, which is expected to be a dominant source
of background in next-generation bolometric experiments. We present the
performance of two prototype modules — housing four Li2MoO4 crystals in total
— operated in the Canfranc underground laboratory in Spain within a facility
developed for the CROSS double-beta-decay experiment.

###### keywords:

Neutrino Physics , Low temperature detectors , Scintillating bolometers ,
Cryogenics

††journal: Nuclear Instruments and Methods in Physics Research Section A

##  1 Introduction

Neutrinoless double-beta decay ($0\nu\beta\beta$) is a Standard-Model-
forbidden rare nuclear transition that can occur only if neutrinos are
Majorana particles ($\nu=\bar{\nu}$) [1]. It has been proposed by W. Furry in
1939 [2], four years after M. Goeppert-Mayer suggested the existence of
ordinary two-$\nu$-double-beta decay ($2\nu\beta\beta$) [3]. In
$2\nu\beta\beta$, a nucleus ($A,Z$) spontaneously decays into an isobar
($A,Z+2$) with the emission of two electrons and two anti-neutrinos, whereas
in $0\nu\beta\beta$ we expect no emission of neutrinos and the signature would
be a peak in the summed energy spectrum of the final state electrons at the
$Q$-value of the decay ($Q_{\beta\beta}$). The $0\nu\beta\beta$ process would
violate lepton number and requires physics beyond the Standard Model [4, 5].
Moreover, the observation of this decay would result in the measurement of the
effective Majorana neutrino mass, if the decay is mediated by light Majorana
neutrinos, as foreseen by the most economic extension of the Standard Model.

There are several tens of nuclei capable of undergoing $2\nu\beta\beta$ and
only 11 have been observed experimentally [6]. Potentially, all these nuclei
can undergo $0\nu\beta\beta$ too. However, only a few are considered as good
candidates to search for $0\nu\beta\beta$ experimentally. This is related to
the availability of a suitable detector technology, a low background at the
$Q_{\beta\beta}$, a high isotopic abundance of the isotope of interest and/or
the possibility of isotopic enrichment at large scale. The sensitivity of the
experiment can be maximised by working on four factors: an ultra low
background, an excellent energy resolution, a large $\beta\beta$ isotopic
mass, and a long live-time. BINGO (Bi-Isotope $0\nu\beta\beta$ Next Generation
Observatory) [7, 8] is a project that focuses on developing techniques to
achieve an ultra low background for $0\nu\beta\beta$ bolometric experiments
[9, 10, 11, 12], setting the grounds for future searches [13].

BINGO will study the two promising $0\nu\beta\beta$ isotopes 100Mo
($Q_{\beta\beta}=3034$ keV) and 130Te ($Q_{\beta\beta}=2527$ keV) embedded in
the well known bolometric compounds Li2MoO4 and TeO2 respectively. TeO2
crystals have been established as a good bolometric material in CUORE [14, 15]
and its predecessors [16, 17, 18]. CUORE has demonstrated the efficient
scaling of bolometric experiments employing 750 kg of TeO2 in 988 detectors,
but the region of interest (ROI) in CUORE is dominated by surface $\alpha$
radioactivity which contributes to a background level of about 10-2
counts/(keV kg yr). CUPID-Mo [19, 20, 21] and CUPID-0 [22, 23], demonstrators
of CUPID (CUORE Upgrade with Particle IDentification) based on Li2100MoO4 [24,
25] and Zn82Se scintillators respectively, have established and validated the
performance of a new detector technique employing a heat-light readout to
reject the surface $\alpha$ background. The heat-light readout is achieved
thanks to an auxiliary light detector (LD) facing the main crystal and
operated as a bolometer as well. For the same deposited energy, the light
emitted after an $\alpha$ interaction in the main absorber is different
compared to $\beta/\gamma$ interactions (typically, by a factor 5–10 lower for
Li2MoO4 [26, 12]). This allows for a full $\alpha$ rejection in case of an
efficient scintillator. In the TeO2 case, we rely on the detection of the
faint Cherenkov radiation [27, 28]. The energy threshold of electrons and
$\alpha$’s to produce Cherenkov light is $\sim$50 keV and $\sim$400 MeV
respectively [27]. So we will have zero Cherenkov radiation emitted from
$\alpha$ particles that will allow us to distinguish them from $\beta/\gamma$.
However, this sets very stringent requirements on the LD energy resolution
($\sim$20 eV RMS in optimal light collection conditions) and threshold as the
collected Cherenkov signal from $\beta/\gamma$ interaction is of the order of
only $\sim$ 100 eV per 2.6 MeV energy deposited in a large volume crystal as
in CUORE ($50\times 50\times 50$ mm3) [12].

One of the most viable technologies of high performance bolometric LDs relies
on the amplification of the light signal in the LD exploiting the Neganov-
Trofimov-Luke (NTL) effect [29, 30]. This technology — that is implemented in
BINGO — is also essential to suppress the most critical source of background
in 100Mo, induced by the random coincidences of events from the relatively
fast $2\nu\beta\beta$ ($T_{1/2}\approx 7\times 10^{18}$ yr) [31, 32] of this
isotope. The NTL effect will improve substantially the signal-to-noise ratio
of the Ge LDs currently used in bolometric experiments like CUPID-Mo and
CUPID. This will allow us to fully exploit the faster rise-times of the Ge LD
($\sim 1$ ms or even less at high bias currents) compared to the heat channel
($\sim 10$ ms) to reject effectively the random coincidences using the light
signal [33]. In BINGO, the two types of crystals and the Ge LDs will be
assembled in an innovative holder and will be surrounded by an active shield
for additional background reduction from $\gamma$ radioactivity [7].

The first demonstrator of BINGO — MINI-BINGO — will consist of two towers: a
Li2MoO4 tower (crystal size: $45\times 45\times 45$ mm3) and a TeO2 tower
(crystal size: $50\times 50\times 50$ mm3). Each tower will comprise 6
modules, each hosting 2 crystals. All crystals will be coupled to an NTL Ge LD
with dimensions of $45\times 45\times 0.3$ mm3 for Li2MoO4 and $50\times
50\times 0.3$ mm3 for TeO2. The thickness of the LD Ge wafers is not important
for the detector’s performance and has been set based on the characteristics
of commercial wafers that meet our purity requirements. The towers will be
surrounded by an active shield consisting of BGO scintillators that are
coupled to NTL Ge LDs as well. MINI-BINGO will be hosted in the Modane
underground laboratory (LSM, France) in a new pulse-tube cryostat in 2025.

In this article we will focus on the description of the design of the new
detector assembly and on the results from a cryogenic test of two BINGO
modules with four Li2MoO4 crystals.

##  2 Detector assembly and operation

The BINGO assembly is designed to fulfill three requirements: less passive
materials facing the crystals; open structure (no reflective foil around the
crystals); and easiness of assembly. The remaining passive materials in the
BINGO assembly are a copper support, a nylon wire 111Here and the following we
will use the expression “nylon wire”, even though the material used in the
detectors here described is a commercial fishing line made of polyamide with a
fluorocarbon coating (Caperlan RESIST®, Decathlon). Pure nylon can be used in
the future if required, as this material can be obtained with extremely low
concentration of radioactive impurities [34]. and PTFE
(polytetrafluoroethylene) pieces, as shown in Fig. 1. The crystals are
shielded from the copper support by the LDs, which are positioned vertically,
effectively reducing backgrounds from residual surface radioactive impurities.
These can produce electrons through a high $Q$-value beta decay (typically,
214Bi in the 238U chain) that can populate the $0\nu\beta\beta$ ROI in the
energy spectrum. The remaining nylon wire and PTFE elements cover about
$\sim$1$\%$ of the crystal surfaces. They will be carefully selected and
subjected to a strict cleaning and handling protocol to minimize their level
of surface radioactivity. In an arrangement consisting of several BINGO
towers, the five sides of the crystal not shielded by the LD will face
directly another crystal rather than a passive element.

The BINGO assembly design leads to a substantial reduction of passive surfaces
— by at least an order of magnitude — compared to classical structures that
are usually composed of frames and columns to host the crystals (see for
example Ref. [17] for CUORE-0 and CUORE, and Ref. [35] for CUPID).
Furthermore, the resulting open structure allows us to efficiently identify
crystal surface contamination through crystal-crystal coincidences and reject
it through anti-coincidence cuts. The easiness of the assembly procedure,
which will be described in details below, minimizes the likelihood of mistakes
and failures and reduces handling, which in turn helps to control the
radiopurity and mitigates the risk of exposure during assembly.

Figure 1: A scheme showing the details of the BINGO nylon assembly module. It
consists of a single copper piece (in orange), three PTFE pieces per side to
keep the light-detector Ge wafer and the crystal. A nylon wire is used to fix
the crystal against the PTFE pieces, which clamp the wafer. One of the two
crystals is removed for a better visualisation of the assembly.

Figure 2: (top): Single module placed in the mounting jig. The photo shows a
nylon wire around the left Li2MoO4 crystal to be fixed with the required
tension using two torque screwdrivers for each wire end. (bottom): The two
wire ends are pulled by passing them inside a hole made in the shaft of a
screw that is subsequently rotated (violin string approach).

The core of the detector assembly (Fig. 1) comprises a copper piece serving as
the primary support for the crystals and the wafers, providing both mechanical
and thermal connections to the cryostat. As in CUPID-0 and in CUPID-Mo, both
the crystals and the Ge wafers are equipped with NTD (Neutron Transmutation
Doped) Ge thermistors [36] for registering the thermal signal after particle
or light interaction in the absorber. The copper piece works as a heat bath at
temperatures around O(10 mK) and serves as the anchoring point for the bonding
pads, to which the thermistor readout wires are attached. We have developed
and implemented a mounting jig (Fig. 2) to streamline the assembly process,
including applying the necessary tension to the nylon wire. This ensures
efficiency, with a full two-crystal module assembly taking less than 5
minutes, and precision, as all elements are positioned with sub-millimeter
accuracy. Initially, the aforementioned copper piece is secured to the
mounting jig (Fig. 2, top). Three PTFE pieces are affixed to the copper piece
on each side, each featuring a 0.3 mm slot to securely hold the LD Ge wafer
(Fig. 1). The crystal is positioned against these PTFE pieces and temporarily
secured using a plastic screw, part of the mounting jig. A nylon wire with a
diameter of 0.45 mm, capable of withstanding a maximum tension of 16 kg, is
wrapped around each crystal. A slot in the plastic screw allows us to position
the nylon wire around the crystal at mid-height and guide it under a copper
washer, inserted in a corresponding copper screw. Each end of the nylon wire
passes through a small hole in the shaft of a winding screw, which can adjust
the tension (Fig. 2, bottom) by rotation, akin to tightening a violin or
guitar string. Two torque-calibrated screwdrivers rotate the perforated
screws, enabling us to apply a known tension (chosen as 4 kg on both sides) to
pull the wire ends. Consequently, the crystal is pressed against the PTFE
pieces, effectively fastening the Ge wafer. The 4 kg tension value was
empirically determined as sufficient to prevent the Ge LDs from sliding
between the PTFE pieces. The low thermal conductivity of PTFE and nylon
ensures thermal decoupling of both the Ge wafer and crystal from the heat
bath. Most of the thermal conductance to copper occurs through the NTD Ge
thermistors and the gold bonding wires for signal readout. The perforated
screws are then secured from rotation using a nut. After fixing the second
crystal in the same manner, the nylon wires are anchored in place using the
aforementioned copper screw and corresponding washer, compressing them against
the copper holder. Subsequently, the wires are trimmed to their minimum
necessary length, and the module is detached from the jig. Each two-crystal
module is assembled individually. The assembled modules are vertically stacked
and held in place by a rod, similar to a skewer. This offers another advantage
of the BINGO approach over CUORE and CUPID: the modules are fully independent
and can be tested individually before tower construction.

Cooling down detectors to operation temperature will change the tension of the
wires, primarily because the wire will contract much more than the crystal
(typically, we expect an integral $\Delta L/L\sim-10^{-2}$ from room
temperature to millikelvin range for polymers, to be compared with
$\sim-10^{-3}$ for crystalline material) [37]. In addition, nylon will change
its physical properties. At very low temperatures, polyamide materials tend to
experience an increase in brittleness and a decrease in ductility. This can
result in reduced elongation at break. Tensile strength and modulus of
elasticity may also be affected, typically decreasing as the temperature
decreases. It is important to note that these trends are generalizations, and
the specific mechanical properties of our assembly is impossible to predict in
detail on the basis of the existing data. Since many physical properties
change significantly already at the temperature of liquid nitrogen, we
immersed modules made according to the previous design in this cryogenic
fluid, subjecting them to repeated thermal cycles. We did not observe any
detectable changes in the module structure or wire tension after these tests.
This opened the way to test the bolometric performance of the BINGO detector
modules in dilution refrigerators at millikelvin temperatures.

Figure 3: Two BINGO modules, each hosting two Li2MoO4 crystals and two Ge
light detectors.

The first proof of concept of the assembly was performed on two $20\times
20\times 20$ mm3 Li2MoO4 crystals tested above-ground in a dry pulse-tube
cryostat at IJCLab (Orsay, France) [7]. This test showed a good bolometric
performance that cleared the way for testing the final BINGO size crystals.

Four $45\times 45\times 45$ mm3 Li2MoO4 crystals and four $45\times 45\times
0.3$ mm3 LDs (operated without the NTL mode) were assembled and measured
underground in the Canfranc laboratory (LSC) in Spain (Fig. 3). In this
assembly, one module had PTFE pieces and the other 3D printed PLA (polylactic
acid) pieces to compare their impact on the bolometric performance. The
advantage of 3D printing is that the shape and the size of the pieces can be
changed quickly if required, but their radiopurity needs to be validated at
least at the PTFE level [38]. The NTD Ge thermistors are glued at the
crystal/wafer by means of Araldite® Rapid two-component epoxy glue. A heavily
doped Si chip [39] is also glued on each crystal, functioning as a heater to
simulate a particle interaction. It is used for the monitoring and offline
correction of thermal gain drifts of the bolometer response by delivering
periodically a constant amount of energy [40]. For the readout, NTDs and
heaters are bonded using gold wires to pads on a Kapton® foil glued at the
copper supporting piece.

In the same run, six more Li2MoO4 scintillating bolometers were assembled and
tested, using holders designed for the CROSS experiment and consisting of
copper and polymeric elements to fasten the crystals and the Ge wafers, as in
CUORE, CUPID-Mo and CUPID-0. The comparison with these CROSS detectors, which
present the same crystal and wafer shape and size as BINGO, is useful to
assess the performance of the innovative BINGO holder.

All the detectors were installed in the pulse-tube cryostat developed for the
CROSS experiment and located at LSC in Spain [41, 42]. The rock-overburden of
2450 m water equivalent, a 25-cm-thick external lead shield, an internal lead
shield and a muon veto provide an excellent low radioactivity environment
suitable for rare-event search campaigns with large bolometers. The facility
furthermore employs an advanced magnetically damped vibrational decoupling
system, which has proven particularly beneficial for the performance of
bolometric LDs [42].

The detector readout was provided by a room-temperature front-end electronics
consisting of low-noise DC-coupled voltage sensitive amplifiers, based on a Si
JFET as input component [43]. The acquisition system consists of two
12-channel boards with integrated 24-bit ADC [44, 45]. The data were acquired
with a sampling rate of 2 kS/s and low-pass Bessel with 300 Hz cut-off
frequency.

The facility is provided with room-temperature LED that allows us to deliver
light to the experimental space by optic fibers that shine into the 10 mK
cavity surrounding the detector. For the detector optimization, we deliver
pulses through the heaters and the LEDs to the Li2MoO4 bolometer and the Ge LD
respectively, allowing us to find the optimum working point in terms of best
signal-to-noise ratio.

During the experiment we regulated the detector plate temperature at 16 mK and
performed a set of $\gamma$ calibrations with a 232Th source, and a series of
background measurements. The 232Th source was inserted between the external
lead shielding and the outer vacuum chamber of the cryostat. Depending on the
position of the source with respect to our detectors, the intensity can vary
because of the internal lead shield placed on the top of our experimental
volume. Calibration data were acquired with high (close) and low (far)
intensity 232Th source. The high intensity calibration allows us to increase
the $\gamma$ interaction rate in the Li2MoO4 crystals which leads to an
emission of X-ray fluorescence of Mo at 17.5 keV that is used to calibrate the
Ge LDs [19]. However, large detectors can suffer from an increased pile-up
rate degrading detector performance estimates.

The detectors were cooled down for over 3 months in stable conditions. A
possible variation of the crystal and wafer fastening strength could lead to
an increase of the microphonic noise, given the high sensitivity of bolometers
to vibrations. On the contrary, we observed a remarkable stability of the
bolometric performance and of the noise structure and level. This was
confirmed by a careful room-temperature inspection of the assembly after the
conclusion of the run, which showed no change in the detector arrangement.

A set of 251 h of 232Th calibration data, corresponding to the ones acquired
at the optimized working point, was selected to study the bolometric
performance of our detectors. Typical bias currents in the NTD thermistors
were chosen of the order of 1 nA with operation resistances in the range 5–50
M$\Omega$. For the radiopurity measurement of the crystals (background
measurements), around 609 h of data were selected and studied. The data
processing is done offline using a MATLAB-based analysis tool [46] exploiting
the Gatti-Manfredi optimum filter [47]. The purpose of optimum filter is to
maximise the signal-to-noise ratio in the signal bandwidth, providing the best
estimation of the signal amplitude under a given template. The template is
based on two inputs: an average mean pulse built using tens of high energy
signals and an average noise power spectral density constructed from
$\sim$10000 waveforms with no signals. The data are then triggered based on
having an energy threshold above 5-RMS baseline-noise fluctuations. An n-tuple
file is produced after data processing that contains a set of parameters
characterizing our data such as the uncalibrated filtered amplitude.

##  3 Detector performance

The bolometric performances are summarized in Table 1 in terms of sensitivity
and baseline resolution FWHM222In the tables and in the text, the detectors
are designated with the nomenclature used within our research group. We remark
— as this could generate some confusion — that one light detector in the BINGO
assembly is dubbed LD-CROSS according to a name assigned in a previous run..
The sensitivity provides the conversion between an energy deposition in the
absorber and the corresponding sensor voltage variation usually given in
nV/keV or $\mu$V/keV. The sensitivities of the BINGO detectors are typical and
similar to those achieved by the devices in the CROSS holders. The spread in
sensitivity is expected to be related to non-uniformity in the coupling
between the NTD and the crystal through the glue. Concerning the baseline
resolution of Li2MoO4, three crystals showed a good performance with an
average of 2.4 keV FWHM baseline width, which is a good indicator of the
intrinsic energy resolution. This value is in line with those observed in the
best CROSS-holder detectors. The remaining crystal (LMO26 in Table 1) was
affected by a higher noise that led to a baseline width about a factor two
larger. As similar outliers in terms of baseline resolution have been observed
with traditional assemblies (like CUPID-Mo) and we don’t have any indication
for a non-uniformity in the assembly of the module, we presently assume that
this excess noise is likely explained by noise pick-up along the readout
chain. Since the energy resolution of Li2MoO4 crystals at the $0\nu\beta\beta$
ROI (3034 MeV) is typically dominated by an energy dependent term [20], this
excess noise has a minimal impact on the $0\nu\beta\beta$ sensitivity, so we
can quote an average resolution of all 4 detectors of 6.3 keV FWHM at 2615 keV
(208Tl calibration line). This resolution is slightly better than the harmonic
mean resolution of 6.6 keV obtained for the full calibration data acquired in
the CUPID-Mo $0\nu\beta\beta$ search [20] and is close to the CUPID goals
[25].

In terms of time structure of the pulses, we observe no difference with
respect to the detectors housed in the CROSS holders within the typical spread
of the pulse time constants. In particular, we have rise/decay times in the
range 15–30 ms/50–120 ms for the heat channels (Li2MoO4) and 4–8 ms/13–23 ms
for the light detectors. We remark that much faster rise-times can be obtained
by raising the base temperature form 16 mK to $\sim 20$ mK and by increasing
the bias current in the NTD Ge thermistors, with an acceptable loss in terms
of energy resolution.

Thanks to the good energy resolution one can clearly see the thorium chain
peaks used for the energy calibration of our detectors in Fig. 4: the plot
shows the combined high intensity source data which provides the best
statistics in the low energy peaks, due to its unshielded position. In
contrast the low-intensity source configuration partially behind the lead
shield data allowed us to assess pile-up effects in the 2615 keV peak. It
provided compatible resolutions, ruling out any significant peak broadening
and we quote combined detector performance values in Table 1.

As for the LDs, all of them work well in terms of sensitivity and baseline
resolution, so one can expect a good particle discrimination. They have a
baseline width in the range 70–110 eV RMS (Table 1). Considering that a gain
of a factor at least 10 in the signal-to-noise ratio is expected when the NTL
mode is applied [30], this performance can achieve not only full $\alpha$
rejection, but also suppression of pile-up events due to $2\nu\beta\beta$
compatible with a background index below 10-4 counts/(keV kg yr) [33], close
to the level of what requested in next-generation bolometric $0\nu\beta\beta$
experiments. The energy spectrum of a single LD is shown in Fig. 5, with two
apparent peaks of Mo X-rays at 17.5 keV and LED induced events.

We note also that there is no observable difference between PTFE (LMO18 and
LMO21-1) and PLA (LMO26 and LMO21-2) assemblies as for their overall
performances, which is an important result as PLA pieces are easier and
cheaper to produce.

Table 1: Performance of Li2MoO4 bolometric detectors (rows 1–4) and Ge light detectors (rows 5–8) in the BINGO assembly obtained during a calibration with a 232Th source at 16 mK. The Li2MoO4 detector in row $i$ is coupled to the light detector in raw $i+4$. The energy resolution of the Li2MoO4 bolometers at the 2615 keV $\gamma$ peak is quoted for a combined data with a high (88 h) and low (163 h) intensity of the source. Detector | Bias | Resistance | Sensitivity | FWHMbsl | FWHM2615
---|---|---|---|---|---
| (nA) | (M$\Omega$) | (nV/keV) | (keV) | (keV)
LMO18 | 1.0 | 11 | 31 | 2.5 | 7.1 $\pm$ 0.4
LMO21-2 | 0.8 | 13 | 85 | 1.5 | 5.6 $\pm$ 0.2
LMO26 | 1.0 | 8.6 | 57 | 4.6 | 6.0 $\pm$ 0.4
LMO21-1 | 1.0 | 9.3 | 44 | 2.6 | 6.6 $\pm$ 0.4
LD-CROSS | 1.0 | 5.8 | 985 | 0.24 | –
LD21-2 | 1.0 | 8.8 | 1660 | 0.16 | –
LD56b | 1.0 | 8.7 | 1790 | 0.21 | –
LD21-1 | 1.0 | 9.2 | 1320 | 0.26 | –
Figure 4: Energy spectrum obtained after 88 h of calibration with 232Th source
(high intensity source mode) at 16 mK for all four Li2MoO4 bolometers. Figure
5: Energy spectrum obtained after 88 hours of calibration with 232Th source at
16 mK for a single Ge light detector.

Since LDs are very sensitive to noise induced by vibrations, it is interesting
to compare the noise power spectra of the LDs in the BINGO holder with those
of the two best-performing detectors in the CROSS holder (Fig. 6). The figure
shows that no additional specific noise structure is induced by the nylon wire
or any other BINGO assembly components. In more quantitative terms, we can
compare the RMS voltage noise across the NTD Ge thermistors over a 5–50 Hz
interval, which corresponds approximately to the useful bandwidth used by the
optimum filter to provide the best signal-amplitude estimation. The integrated
noise is 0.27 $\mu$V RMS, averaged over the four BINGO LDs, to be compared
with an average 0.56 $\mu$V RMS for LDs in the CROSS structure. The small
number of channels and the usual large spread in noise figures do not allow us
to claim that the BINGO fastening method improves noise induced by vibrations,
but we can surely conclude that it adds no relevant component.

Figure 6: Comparison of the noise power spectra of the light detectors in the
BINGO assembly (solid lines) with those in the CROSS assembly (dashed lines)
at 16 mK base temperature. Figure 7: Light yield plot for LMO21-2/LD21-2 at 16
mK (Thorium calibration). The inset shows the Gaussian fit of the $\alpha$ and
$\beta/\gamma$ distributions, where events were selected having an energy
$>2.5$ MeV. The estimated discrimination power is $\sim 5$.

For particle identification we use the light yield ($LY$) parameter, defined
as the energy of the light signal detected by the Ge LD (measured in keV after
Ge LD energy calibration) for 1 MeV heat signal in Li2MoO4 (Fig. 7). We point
out that this parameter is not the absolute scintillation yield, as it depends
also on the light collection efficiency. Table 2 summarizes the $LY$ of
$\beta/\gamma$ events and the quenching (QF) of the $\alpha$-particle
scintillation light with respect to $\beta/\gamma$ for all detectors, where QF
= $LY_{\alpha}$ / $LY_{\beta/\gamma}$. We remark that LDs are not operated in
the NTL mode. The discrimination capability between $\alpha$ and
$\beta/\gamma$ can be quantified in terms of the discrimination power (DP):

$DP=\frac{|\mu_{\alpha}-\mu_{\beta/\gamma}|}{\sqrt{\sigma^{2}_{\alpha}+\sigma^{2}_{\beta/\gamma}}},$
(1)

where $\mu$ and $\sigma$ are the mean and the standard deviation respectively
extracted from the Gaussian fit applied to the $LY$ distributions of $\alpha$
and $\beta/\gamma$ events. The discrimination power was estimated from the
$LY$ distribution corresponding to events higher than 2.5 MeV, close to our
ROI. To achieve an $\alpha$ rejection higher than 99.9$\%$ (requested for a
background level of $\sim 10^{-5}$ counts/(keV kg y), a discrimination power
higher than 3.2 is essential [28, 12]. In our case, two detectors over four
showed a discrimination power 4–5, while the other two detectors around 3.
This is a result of the worse baseline resolution in the corresponding LDs.
The light yield achieved in the BINGO assembly is, as expected, lower compared
to what can be achieved in a closed structure with a reflective foil around
the crystals, as in CUPID-Mo ($LY_{\beta/\gamma}\sim$ 0.7 keV/MeV) [19], and
similar to the open detector structure prototypes of CROSS and CUPID
($LY_{\beta/\gamma}\sim$ 0.2–0.3 keV/MeV) [48, 35, 49]. This drawback,
however, will be exceedingly compensated by the use of the NTL effect which
will result in an increase by an order of magnitude in the Ge LD signal-to-
noise ratio, leading to a much higher discrimination power.

Table 2: Parameters of Li2MoO4 scintillation detection. We report on the relative light yield value for $\beta/\gamma$ events and the quenching factor of $\alpha$-particle induced scintillation light for events with energy $>$2.5 MeV. Uncertainties of light yield values are given (conservatively) as RMS values of the light yield distributions, while the uncertainties of mean values extracted from the fit are used to compute the uncertainties of the quenching factor. Detector | Light yield (keV/MeV) | Quenching factor ($\%$)
---|---|---
| $\beta/\gamma$ | $\alpha$
LMO18 | 0.22(3) | 15(3)
LMO21-2 | 0.25(3) | 12(2)
LMO26 | 0.24(4) | 17(1)
LMO21-1 | 0.22(4) | 17(2)

To estimate the contamination levels in our crystals, we use the background
measurements. This evaluation is important, as the four Li2MoO4 crystals
belong to the production batch that will be used to assemble MINI-BINGO. Using
the $LY$ plot, $\alpha$ events were selected corresponding to a $LY$ below
0.15 keV/MeV and a heat energy above 3 MeV. The energy spectra of the $\alpha$
events detected by the Li2MoO4 bolometers are shown in Fig. 8. The spectra
were calibrated using the $\alpha$ peaks to account for their thermal
quenching [50]. Three peaks are apparent. The 4.7 MeV line corresponds to the
neutron capture on 6Li resulting in an emission of an $\alpha$ and a triton:
6Li(n,t)$\alpha$. The two higher energy peaks correspond to the Q-value and
alpha-peak from 210Po decay to 206Pb. The 5.4 MeV line ($Q$-value) is for bulk
210Po contamination where all the emitted $\alpha$ and recoil nucleus energies
are absorbed in the crystal. The lower energy $\alpha$’s at around 5.3 MeV
correspond to surface 210Po contamination of the detector components, where
the nuclear recoil energy is not absorbed in the Li2MoO4 thermal detectors.
The average 210Po internal activity was estimated to be 124(8) $\mu$Bq/kg,
consistent with what was found previously for Li2MoO4 crystals [50, 21].
Taking into account that we do not find a clear evidence of 226Ra and 228Th,
we set upper limits on their activities on the level of 10 $\mu$Bq/kg, and we
expect that the contamination will likely be more than an order of magnitude
below these values as found in CUPID-Mo [21]. The results of the study of
$\alpha$ spectra are summarized in Table 3.

Figure 8: Fragments of the energy spectra in the $\alpha$ region of the following Li2MoO4 thermal detectors: (a) LMO18; (b) LMO21-2; (c) LMO26; (d) LMO21-1. Data of all detectors were acquired over 609 h, except the LMO18 bolometer (497 h), which was grounded over a single dataset. The common prominent signatures seen in the spectra are labelled as follows: 1 — products of 6Li(n,t)$\alpha$ reaction ($\alpha$ \+ triton) in the detector medium; 2 — $\alpha$ particles emitted in 210Po decays occurred at surfaces of materials surrounding the detector; 3 — products of 210Po decays ($\alpha$ \+ nuclear recoil) in the detector. Table 3: Results on analysis of alpha events detected by Li2MoO4 bolometers. We report the rate of $\alpha$+t events (from neutron capture on 6Li) as well as estimates of activities of radionuclides from U/Th chain; uncertainties are given at 68% C.L., while limits are set at 90% C.L. Detector | Rate (counts/d) | Activity ($\mu$Bq/kg)
---|---|---
|  $\alpha$+t |  210Po |  226Ra |  228Th
LMO18 | 2.5(4) | 137 $\pm$ 17 |  $\leq$ 5 |  $\leq$ 9
LMO21-2 | 2.2(3) | 129 $\pm$ 15 |  $\leq$ 8 |  $\leq$ 8
LMO26 | 2.1(3) | 117 $\pm$ 14 |  $\leq$ 10 |  $\leq$ 10
LMO21-1 | 2.4(3) | 113 $\pm$ 14 |  $\leq$ 13 |  $\leq$ 10

##  4 Conclusions

In this paper, we demonstrate the effective bolometric performance of the
BINGO nylon wire assembly tested with two modules containing 4 Li2MoO4
crystals ($45\times 45\times 45$ mm3 each) and 4 Ge light detectors ($45\times
45\times 0.3$ mm3). The BINGO detector structure offers stable mechanical
coupling, and the detectors exhibit noise levels comparable to those in more
conventional mounting structures. This finding holds significant implications
for future $0\nu\beta\beta$ bolometric experiments, as it introduces an
innovative assembly structure that drastically reduces the surface of passive
materials facing the crystals by at least an order of magnitude. Consequently,
this will lead to a substantial reduction in the radioactive background within
the region of interest originating from surrounding materials. Prior to
finalizing the MINI-BINGO towers, we will conduct additional prototype tests
using $50\times 50\times 50$ mm3 TeO2 crystals, which are heavier, to validate
the performance of the tellurium section of BINGO. Simultaneously, ongoing
simulations aim to quantify the potential background reduction achievable
through this novel assembly method.

## Acknowledgements

The BINGO project is funded by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (No. 865844, ERC
Consolidator Grant). We thank the CROSS collaboration for providing the
underground cryogenic facility and the detector readout system. CROSS is also
supported by the ERC in the framework of H2020 (No. 742345, ERC Advanced
Grant). The INR NASU group was supported in part by the National Research
Foundation of Ukraine Grant No. 2020.02/0011 and by the National Academy of
Sciences of Ukraine in the framework of the project “Development of bolometric
experiments for the search for double beta decay”.

## References

  * [1] J. Schechter, J. W. F. Valle, Neutrinoless double-$\beta$ decay in SU(2)×U(1) theories, Phys. Rev. D 25 (1982) 2951.
  * [2] W. H. Furry, On Transition Probabilities in Double Beta-Disintegration, Phys. Rev. 56 (1939) 1184.
  * [3] M. Goeppert-Mayer, Double Beta-Disintegration, Phys. Rev. 48 (1935) 512.
  * [4] M. Agostini, et al., Toward the discovery of matter creation with neutrinoless double-beta decay, Rev. Mod. Phys. 95 (2023) 025002.  arXiv:2202.01787.
  * [5] D. Q. Adams, et al., Neutrinoless Double Beta Decay, arXiv:2212.11099.
  * [6] B. Pritychenko, Systematic analysis of double-beta decay half lives, Nucl. Phys. A 1033 (2023) 122628.
  * [7] A. Armatol, et al., First cryogenic tests on BINGO innovations, arXiv:2204.14161.
  * [8] http://www.bingo-neutrino.eu/.
  * [9] E. Fiorini, T. O. Niinikoski, Low Temperature Calorimetry for Rare Decays, Nucl. Instrum. Methods Phys. Res. A 224 (1984) 83.
  * [10] D. Poda, A. Giuliani, Low background techniques in bolometers for double-beta decay search, Int. J. Mod. Phys. A 32 (2017) 1743012.
  * [11] S. Pirro, P. Mauskopf, Advances in Bolometer Technology for Fundamental Physics, Annu. Rev. Nucl. Part. Sci. 67 (2017) 161.
  * [12] D. Poda, Scintillation in Low-Temperature Particle Detectors, Physics 3 (2021) 473.
  * [13] A. Armatol, et al., Toward CUPID-1T, arXiv:2203.08386.
  * [14] D. Q. Adams, et al., Improved Limit on Neutrinoless Double-Beta Decay in 130Te with CUORE, Phys. Rev. Lett. 124 (2020) 122501.
  * [15] D. Q. Adams, et al., Search for Majorana neutrinos exploiting millikelvin cryogenics with CUORE, Nature 604 (2022) 53.
  * [16] E. Andreotti, et al., 130Te neutrinoless double-beta decay with CUORICINO, Astropart. Phys. 34 (2011) 822.
  * [17] C. Alduino, et al., CUORE-0 detector: design, construction and operation, JINST 11 (2016) P07009.
  * [18] C. Brofferio, S. Dell’Oro, The saga of neutrinoless double beta decay search with TeO2 thermal detectors, Rev. Sci. Instrum. 89 (2018) 121502.
  * [19] E. Armengaud, et al., The CUPID-Mo experiment for neutrinoless double-beta decay: performance and prospects, Eur. Phys. J. C 80 (2020) 44.
  * [20] C. Augier, et al., Final results on the $0\nu\beta\beta$ decay half-life limit of 100Mo from the CUPID-Mo experiment, Eur. Phys. J. C 82 (2022) 1033.
  * [21] C. Augier, et al., The background model of the CUPID-Mo $0\nu\beta\beta$ experiment, Eur. Phys. J. C 83 (2023) 675.
  * [22] O. Azzolini, et al., CUPID-0: the first array of enriched scintillating bolometers for $0\nu\beta\beta$ decay investigations, Eur. Phys. J. C 78 (2018) 428.
  * [23] O. Azzolini, et al., Final Result on the Neutrinoless Double Beta Decay of ${}^{82}\mathrm{Se}$ with CUPID-0, Phys. Rev. Lett. 129 (2022) 111801.
  * [24] G. Wang, et al., CUPID: CUORE (Cryogenic Underground Observatory for Rare Events) Upgrade with Particle IDentification, arXiv:1504.03599.
  * [25] W. R. Armstrong, et al., CUPID pre-CDR, arXiv:1907.09376.
  * [26] V. I. Tretyak, Semi-empirical calculation of quenching factors for ions in scintillators, Astropart. Phys. 33 (2010) 40.
  * [27] T. Tabarelli de Fatis, Cerenkov emission as a positive tag of double beta decays in bolometric experiments, Eur. Phys. J. C 65 (2010) 359.
  * [28] L. Bergé, et al., Complete event-by-event $\alpha/\gamma(\beta)$ separation in a full-size ${\mathrm{TeO}}_{2}$ CUORE bolometer by Neganov-Luke-magnified light detection, Phys. Rev. C 97 (2018) 032501.
  * [29] V. Novati, et al., An innovative bolometric Cherenkov-light detector for a double beta decay search, Nucl. Instrum. Methods Phys. Res. A 912 (2018) 82.
  * [30] V. Novati, et al., Charge-to-heat transducers exploiting the Neganov-Trofimov-Luke effect for light detection in rare-event searches, Nucl. Instrum. Methods Phys. Res. A 940 (2019) 320.
  * [31] D. M. Chernyak, et al., Random coincidence of 2$\nu$2$\beta$ decay events as a background source in bolometric 0$\nu$2$\beta$ decay experiments, Eur. Phys. J. C 72 (2012) 1989.
  * [32] C. Augier, et al., Measurement of the $2\nu\beta\beta$ decay rate and spectral shape of ${}^{100}\mathrm{Mo}$ from the cupid-mo experiment, Phys. Rev. Lett. 131 (2023) 162501.
  * [33] A. Ahmine, et al., Enhanced light signal for the suppression of pile-up events in Mo-based bolometers for the $0\nu\beta\beta$ decay search, Eur. Phys. J. C 83 (2023) 373.
  * [34] Y. Gando, et al., The nylon balloon for xenon loaded liquid scintillator in KamLAND-Zen 800 neutrinoless double-beta decay search experiment, JINST 16 (2021) P08023.
  * [35] K. Alfonso, et al., Optimization of the first CUPID detector module, Eur. Phys. J. C 82 (2022) 810.
  * [36] E. E. Haller, Advanced far-infrared detectors, Infrared Phys. Techn. 35 (1994) 127.
  * [37] F. Pobell, Matter and Methods at Low Temperatures, Springer-Verlag Berlin, Heidelberg, 2007.
  * [38] C. Alduino, et al., The projected background for the CUORE experiment, Eur. Phys. J. C 77 (2017) 543.
  * [39] E. Andreotti, et al., Production, characterization and selection of the heating elements for the response stabilization of the CUORE bolometers, Nucl. Instrum. Methods Phys. Res. A 664 (2012) 161.
  * [40] A. Alessandrello, et al., Methods for response stabilization in bolometers for rare decays, J. Cryst. Growth 412 (1998) 454.
  * [41] E. Olivieri, et al., The new CROSS Cryogenic Underground (C2U) facility: An overview. In Proceedings of the XXIX International (online) Conference on Neutrino Physics and Astrophysics (Neutrino 2020), Chicago, IL, USA, 22 June–2 July 2020., https://indico.fnal.gov/event/19348/contributions/186315/.
  * [42] A. Ahmine, et al., Test of 116CdWO4 and Li2MoO4 scintillating bolometers in the CROSS underground facility with upgraded detector suspension, JINST 18 (2023) P12004.
  * [43] C. Arnaboldi, et al., The programmable front-end system for CUORICINO, an array of large-mass bolometers, IEEE Trans. Nucl. Sci. 49 (2002) 2440.
  * [44] P. Carniti, C. Gotti, G. Pessina, High-Resolution Digitization System for the CROSS Experiment, J. Low Temp. Phys. 199 (2020) 833.
  * [45] P. Carniti, C. Gotti, G. Pessina, High resolution filtering and digitization system for cryogenic bolometric detectors, Nucl. Instrum. Methods Phys. Res. A 1045 (2023) 167658.
  * [46] M. Mancuso, PhD Thesis: Development and optimization of scintillating bolometers and innovative light detectors for the search for neutrinoless double beta decay, Université Paris-Sud, Orsay, France, https://tel.archives-ouvertes.fr/tel-01315155 (February 2016).
  * [47] E. Gatti, P. Manfredi, Processing the signals from solid-state detectors in elementary-particle physics, Riv. Nuovo Cim. 9 (1986) 1.
  * [48] A. Armatol, et al., Characterization of cubic Li2100MoO4 crystals for the CUPID experiment, Eur. Phys. J. C 81 (2021) 104.
  * [49] K. Alfonso, et al., Twelve-crystal prototype of Li2MoO4 scintillating bolometers for CUPID and CROSS experiments, JINST 18 (2023) P06018.
  * [50] E. Armengaud, et al., Development of 100Mo-containing scintillating bolometers for a high-sensitivity neutrinoless double-beta decay search, Eur. Phys. J. C 77 (2017) 785.