# Dynamic RF Beam Codebook Reduction for
Cost-Efficient mmWave Full-Duplex Systems

Gee Yong Suk1, Jongwoo Kwak1, Jaeyoung Choi2, and Chan-Byoung Chae1 1School of
Integrated Technology, Yonsei University, Korea
2Samsung Electronics Co., Ltd., Korea
Email:1{gysuk, kjw8216<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The recent attempts to realize full-duplex (FD) communications in millimeter
wave (mmWave) systems have garnered a significant amount of interest for its
potential. In this paper, we present a cost-efficient design of mmWave FD
systems, where the system dynamically reduces the RF beam codebook in a
computationally efficient manner, so that it is comprised of the RF beams that
will prevent the Rx receive chain from saturating due to the self-interference
(SI). The analog beamformer will suppress the SI to the level that the
residual SI can be completely removed with digital SI cancellation, allowing
the digital beamformer to concentrate on the desired channel, free of the SI.
To reduce the computation required for the proposed method, we propose two
sufficient conditions that prevent the Rx side saturations, which are
practically tight. Through performance evaluations conducted in realistically
modeled mmWave FD scenarios, we demonstrate that the proposed design achieves
comparable performance with the ideal FD and other benchmarks with
significantly lower costs.

##  I Introduction

The large spectrum available at millimeter-wave (mmWave) frequencies offers a
potential boost in transmission speed. Hybrid beamforming has been widely
adopted as a typical architecture for the mmWave systems [1], where the hybrid
of analog and digital beamforming balances the trade-off of multiplexing gain
and hardware (H/W) cost. It was shown that hybrid beamforming can achieve a
performance near that of full-digital beamforming with reasonable hardware
cost by exploiting the spatially sparse channel characteristic of the mmWave
system [2].

Nevertheless, these studies have been confined by the half-duplex (HD)
framework, whereas the ever-increasing demand for a high data rate motivates
us to consider doubling the system capacity with full-duplex (FD). The FD is
one of the promising technologies for future wireless communications, which
maximizes the spectral efficiency by allowing transmission and reception to
occur simultaneously at the same frequency band. Furthermore, aside from
doubling the spectral efficiency, the FD also has the potential to increase
the flexibility in the design of wireless protocols [3].

The critical technical challenge regarding the FD is the mitigation of the
self-interference (SI), an undesired interfering signal at the FD-node,
transmitted by itself. Due to the proximity of the interference source
compared to that of the desired signal’s source, the SI is a powerful
interference signal that can cause intolerable performance loss. Conventional
FD radios have extensively investigated the self-interference cancellation
(SIC) methods of which a transceiver, with the knowledge of its own transmit
signal, reconstructs the SI and subtracts it at the receiver. The SIC may be
done at the passband via analog circuit (active analog SIC) and/or at the
baseband (digital SIC) [4, 5, 6, 7], aiming to suppress the SI down to the
noise floor.

Realizing the FD communications in the mmWave system introduced new approaches
due to the unique characteristics of the mmWave systems [8, 9]. Previous works
have focused on the beamforming-based FD designs, which exploit numerous
antennas herein [10, 11, 12]. These works have investigated the design of
analog/digital beamformers that suppresses the SI at the propagation domain,
under the assumption that the full channel state information (CSI) of the
desired channel is given. Furthermore, designs in [12, 11] took the advantage
of an increased number of RF chains in order to provide the sufficient
dimensionality of the SI channel’s null space. However, these solutions are
rather infeasible because practical systems generally rely on beam training
with finite-resolution RF beam codebooks, and the number of RF chains would
typically match the number of transmitting and receiving data streams.

The joint design of the SIC and beamforming for the mmWave FD systems appears
favorable, as the responsibility of mitigating the SI is shared across the SIC
modules and hybrid beamformers [8]. By offloading the burden of mitigating the
SI, beamformers can better concentrate on the desired links. The authors in
[13] proposed the joint design, where the hybrid beamformer was acquired by
the repitition of solving a convex optimization problem under the constraints
that prevent the receiver side saturations. However, the design in [13] is
costly, raising concerns for the feasibility of real-time implementations.
Besides, considering that the typical orthogonal frequency division
multiplexing (OFDM) will require the computational complexity proportional to
the number of subcarriers, it is imperative to devise a cost-efficient design
for the mmWave FD systems.

Figure 1: A block diagram of the mmWave FD OFDM system with two HD-nodes and
one FD-node, where the FD-node employs a joint design of analog/digital
beamforming (BF) and the digital SIC.

In this paper, we propose a cost-efficient design of the mmWave FD systems
with the dyanmic RF beam codebook reduction to lower the required number of
beam measurements with the minimum computations. Specifically, a reduced-size
RF beam codebook is acquired from a set of RF beam combinations that solely
prevent the Rx side saturations. Then, the joint operation of the digital SIC
allows the digital beamformer design to focus on the desired channel, free of
the SI. We propose two sufficient conditions for preventing the Rx side
saturations with analog beamformers that involve significantly reduced
computational overhead, which are yet practically tight. We evaluate the
system performance in practical mmWave FD scenarios and demonstrate that the
proposed design offers comparable performance with the ideal FD and other
leading benchmarks with significantly lower cost.

##  II System Model

This work considers a mmWave FD system that consists of FD-node $i$ and HD-
nodes $j$ and $k$, as illustrated in Fig. 1. Node $i$ simultaneously transmits
data streams to node $j$, while receiving data streams from node $k$ in the
same frequency band. Node $i$ performs the SI mitigation via the joint
cooperation of beamforming and SIC. The nodes considered in the system deploy
hybrid beamforming in a fully-connected manner, where each antenna element is
connected to each RF chain through a network of phase shifters. Also, we
assume that node $i$ uses separate arrays for its transmission and reception,
separated by distance $r^{\left(i\right)}$, which is one of the factors that
governs the amount of passive SI suppression.

To cover the wide spectrum of the mmWave channel, we consider a wideband
system with the OFDM, along with $U$ subcarriers. For the node index
$(m,n)\in\\{(i,j),(k,i)\\}$, let $N_{t}^{(m)},N_{r}^{(n)}$,
$L_{t}^{(m)},L_{r}^{(n)}$, and $N_{s}^{\left(mn\right)}$ denote the number of
transmit and receive antennas, the number of the Tx and Rx RF chains, and the
number of symbol streams that are transmitted by $m$ and received by $n$,
respectively. The symbol streams transmitted by the $u$-th subcarrier of node
$m$ are denoted by
$\textbf{s}^{(m)}[u]\in\mathbb{C}^{N_{s}^{\left(mn\right)}\times 1}$, and the
symbol streams received by the $u$-th subcarrier of $n$ are denoted by
$\textbf{y}^{(n)}[u]\in\mathbb{C}^{N_{s}^{\left(mn\right)}\times 1}$,
respectively, where $u\in\\{1,2,\dots,U\\}$. Accordingly, we denote the
digital and analog precoding matrices by
$F_{\text{BB}}^{(m)}[u]\in\mathbb{C}^{L_{t}^{(m)}\times N_{s}^{(m)}}$ and
$F_{\text{RF}}^{(m)}\in\mathbb{C}^{N_{t}^{(m)}\times L_{t}^{(m)}}$, the
digital and analog combining matrices by
$W_{\text{BB}}^{(n)}[u]\in\mathbb{C}^{L_{r}^{(n)}\times N_{s}^{(n)}}$ and
$W_{\text{RF}}^{(n)}\in\mathbb{C}^{N_{r}^{(n)}\times L_{r}^{(n)}}$,
respectively. Unlike the digital precoder/combiners, the analog
precoder/combiners are frequency-flat across the entire $U$ subcarriers. The
symbol vector,
$\textbf{s}^{(m)}[u]\sim\mathcal{N}_{\mathbb{C}}(\textbf{0},\frac{1}{N_{s}^{\left(mn\right)}}\cdot\mathbf{I}{})$,
assumes Gaussian signaling and equal power distribution among the subcarriers.
Let
$\textbf{n}^{(n)}[u]\sim\mathcal{N}_{\mathbb{C}}(\textbf{0},\sigma_{n}^{2}\cdot\mathbf{I}{})$
be the additive noise vector at the array of receiver $n$ per subcarrier with
noise variance $\sigma_{n}$. Let
$\text{SNR}_{mn}=P_{\text{tx}}^{\left(m\right)}\cdot
G_{mn}^{2}/\sigma_{n}^{2}$ be the signal-to-noise ratio (SNR) without any
beamforming gains from node $m$ to $n$, where $P_{\text{tx}}^{\left(m\right)}$
is the transmit power at $m$ and $G_{mn}^{2}$ is the large-scale power gain of
the propagation from $m$ to $n$.

For practical considerations, we select the unit-norm columns of
$\mathbf{F}^{\left(m\right)}_{\text{RF}}$ and
$\mathbf{W}^{\left(n\right)}_{\text{RF}}$ from the predefined RF beam
codebooks that account for the H/W constraints such as phase shifter
resolution and lack of amplitude control. We define

$\big{[}\mathbf{F}^{\left(m\right)}_{\text{RF}}\big{]}_{:,\ell}\in\mathbf{\mathcal{F}}^{\left(m\right)}_{\text{RF}},\quad\ell=1,2,\dots,L_{t}^{\left(m\right)},$
(1)
$\big{[}\mathbf{W}^{\left(n\right)}_{\text{RF}}\big{]}_{:,\ell}\in\mathbf{\mathcal{W}}^{\left(n\right)}_{\text{RF}},\quad\ell=1,2,\dots,L_{r}^{\left(n\right)},$
(2)

where $\mathbf{\mathcal{F}}^{\left(m\right)}_{\text{RF}}$ and
$\mathbf{\mathcal{W}}^{\left(n\right)}_{\text{RF}}$ are the analog precoding
and combining codebooks, respectively. Also, we define
$C_{t}^{\left(m\right)}$ and $C_{r}^{\left(n\right)}$ as the number of
candidate beams in $\mathbf{\mathcal{F}}^{\left(m\right)}_{\text{RF}}$ and
$\mathbf{\mathcal{W}}^{\left(n\right)}_{\text{RF}}$, respectively. Then, the
transmitter’s total power constraint is enforced by

$\left\lVert\mathbf{F}^{\left(m\right)}_{\text{BB}}[u]\right\rVert^{2}\leq
N_{s}^{\left(mn\right)}.$ (3)

For the desired channels $\mathbf{H}_{{ij}}$ and $\mathbf{H}_{{ki}}$, we use
the far-field channel model based on the Saleh-Valenzuela model [2], where the
multiple-input multiple-output (MIMO) channel matrix between the transmit
array and receive array at tap $d$ is

$\mathbf{H}^{\text{FF}}_{{d}}[d]=\sum_{\ell=1}^{N_{\text{ray}}}\alpha_{\ell}\delta(dT_{s}-\tau_{\ell})\mathbf{a}_{{\text{r}}}(\theta^{\text{AoA}}_{\ell},\phi^{\text{AoA}}_{\ell})\mathbf{a}^{*}_{{\text{t}}}(\theta^{\text{AoD}}_{\ell},\phi^{\text{AoD}}_{\ell}),$
(4)

where $\alpha_{\ell}$ is the complex gain, $\tau_{\ell}$ is the time delay,
$\mathbf{a}_{{\text{r}}}(\cdot)$ and $\mathbf{a}_{{\text{t}}}(\cdot)$ are the
receive and transmit array responses of the elevation and azimuth angle-of-
arrival (AoA), $\theta^{\text{AoA}}_{\ell},\phi^{\text{AoA}}_{\ell}$, and the
elevation and azimuth angle-of-departure (AoD),
$\theta^{\text{AoD}}_{\ell},\phi^{\text{AoD}}_{\ell}$, for the $\ell$-th path,
respectively. The sampling rate of the channel is $1/T_{s}$ samples per
second. For the SI channel $\mathbf{H}_{{ii}}$, we consider the following
expression with the Rician factor $\kappa$ as

$\mathbf{H}_{{d,ii}}[d]=\sqrt{\frac{\kappa}{\kappa+1}}\mathbf{H}^{\text{NF}}_{{d,ii}}+\sqrt{\frac{1}{\kappa+1}}\mathbf{H}^{\text{FF}}_{{d,ii}}[d].$
(5)

where the ($p,q$)-th entry of the near-field channel
$\mathbf{H}^{\text{NF}}_{{d,ii}}$ is modeled as

$[\mathbf{H}^{\text{NF}}_{{d,ii}}]_{p,q}=\frac{\rho}{r_{p,q}}e^{-j2\pi\frac{r_{p,q}}{\lambda_{c}}},$
(6)

where $\rho$ is the normalization constant, $r_{p,q}$ is the distance between
the $p$-th element of the transmit array to the $q$-th element of the receive
array, $\lambda_{c}$ is the carrier wavelength. Since we consider an OFDM
system, we define a $U$-point discrete Fourier transform (DFT) of the channel
matrices across time to get the MIMO channels across the subcarriers as

$\mathbf{H}[u]=\sum_{d=1}^{D}\mathbf{H}_{{\text{d}}}[d]e^{-j\frac{2\pi
ud}{U}}.$ (7)

##  III Proposed mmWave FD Design

###  III-A Sufficient Conditions for Preventing Rx Saturation

Receive chains at the FD node are especially vulnerable to saturation, due to
the overwhelming SI [13]. The SI can introduce nonlinearities at the low noise
amplifiers (LNA) by imposing the input power level beyond its capacity, and/or
degrade the quality of the desired receive signal by increasing the
quantization noise at the analog-to-digital converters (ADC). Thus, it is
necessary to mitigate the portion of the SI before it reaches the receive
chain. Once the system sufficiently mitigates the SI to the levels that
prevent the receive saturation, and thus secure the linearities of the receive
chain, the digital SIC can effectively cancel out the residual SI at the
baseband [14].

The authors in [13] have well established the conditions to prevent the Rx
saturation in a narrowband system. In this section, we expand these conditions
to cover the wideband OFDM system and further formulate sufficient conditions
through Theorem 1, where its usage and advantage will be later discussed. Note
from Fig. 1 that the LNAs are located per-antenna and the ADCs per-RF chain.
To prevent saturation at the LNAs and ADCs, the SI entering each antenna and
RF chain should be suppressed below certain power thresholds. For the LNAs,
achieving this in expectation over $\mathbf{s}^{\left(i\right)}_{\text{}}[u]$
can be expressed as

$\begin{split}\sum_{u=1}^{U}\text{diag}\Big{(}\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\mathbf{F}^{\left(i\right)*}_{\text{BB}}[u]\mathbf{F}^{\left(i\right)*}_{\text{RF}}\mathbf{H}^{*}_{{ii}}[u]\Big{)}\\\
\leq\frac{U~{}N_{\text{s}}^{\left(ij\right)}P^{max}_{\text{LNA}}}{P_{\text{Tx}}^{\left(i\right)}G_{ii}^{2}}\cdot\mathbf{1}_{{N_{\text{r}}^{\left(i\right)}}},\end{split}$
(8)

where $P^{max}_{\text{LNA}}$ is the maximum power level to prevent the LNA
saturation. For the ADCs,

$\begin{split}\sum_{u=1}^{U}\text{diag}\Big{(}\mathbf{W}^{\left(i\right)*}_{\text{RF}}\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\mathbf{F}^{\left(i\right)*}_{\text{BB}}[u]\mathbf{F}^{\left(i\right)*}_{\text{RF}}\mathbf{H}^{*}_{{ii}}[u]\mathbf{W}^{\left(i\right)}_{\text{RF}}\Big{)}\\\
\leq\frac{U~{}N_{\text{s}}^{\left(ij\right)}P^{max}_{\text{ADC}}}{P_{\text{Tx}}^{\left(i\right)}G_{ii}^{2}}\cdot\mathbf{1}_{{L_{\text{r}}^{\left(i\right)}}},\end{split}$
(9)

where $P^{max}_{\text{ADC}}$ is the maximum power level to prevent the ADC
saturation. Exploiting the majorization property of a Hermitian matrix, the
respective sufficient condtions for (8) and (9) are given as

$\sum_{u=1}^{U}\sigma^{2}_{\text{max}}\left(\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\right)\leq
N_{\text{s}}^{\left(ij\right)}\cdot\eta_{\text{LNA}}$ (10)

and

$\sum_{u=1}^{U}\sigma^{2}_{\text{max}}\left(\mathbf{W}^{\left(i\right)*}_{\text{RF}}\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\right)\leq
N_{\text{s}}^{\left(ij\right)}\cdot\eta_{\text{ADC}},$ (11)

where
$\eta_{\text{LNA}}=\frac{U~{}P^{max}_{\text{LNA}}}{P_{\text{Tx}}^{\left(i\right)}G_{ii}^{2}}$
and
$\eta_{\text{ADC}}=\frac{U~{}P^{max}_{\text{ADC}}}{P_{\text{Tx}}^{\left(i\right)}G_{ii}^{2}}$.
Also, $\sigma_{\text{max}}^{2}(\cdot)$ denotes the maximum singular value of a
given input matrix. From
$\sigma_{\text{max}}^{2}\left(\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\right)\leq{\left\lVert\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\right\rVert}^{2}_{\text{F}}\leq
N_{\text{s}}^{\left(ij\right)}$, the respective sufficient condtions for (10)
and (11) can be similarly derived from [13] as

$\sum_{u=1}^{U}\sigma^{2}_{\text{max}}\left(\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\right)\leq\eta_{\text{LNA}}$
(12)

and

$\sum_{u=1}^{U}\sigma^{2}_{\text{max}}\left(\mathbf{W}^{\left(i\right)*}_{\text{RF}}\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\right)\leq\eta_{\text{ADC}}.$
(13)

TABLE I: Comparative analysis of required beam measurement number and computational cost for various system designs. Design methods |  Required beam measurements ($@$Transmit link) | Major computation-inducing functions |  CPU run time
---|---|---|---
Conventional [13] | $64\times 64~{}(C_{r}^{\left(j\right)}\times C_{t}^{\left(i\right)})$ | Solving convex optimization problem (by CVX SeDuMi [15]) | 76.778 sec
Power reduction | $64\times 64~{}(C_{r}^{\left(j\right)}\times C_{t}^{\left(i\right)})$ | Testing saturation (w/ (8),(9)) | 0.058 sec
Proposed (w/ (12), (13)) | $64\times 40.16~{}(C_{r}^{\left(j\right)}\times C_{\text{AL},t}^{\left(i\right)})$ | Constructing allowlist (w/ (12), (13)) | 26.471 sec
Proposed (w/ (14), (15)) |  $64\times 39.33$$~{}(C_{r}^{\left(j\right)}\times C_{\textbf{AL},t}^{\left(i\right)})$ | Constructing allowlist (w/ (14), (15)) | 0.223 sec

Finally, we define the sufficient conditions for (12) and (13) to reduce the
related computational complexity that will later be discussed in detail.

######  Theorem 1.

The following conditions, (14) and (15), are the sufficient conditions for the
per-LNA and per-ADC constraints, i.e., (12), (13), respectively.

$\sum_{\ell=1}^{L_{t}^{(i)}}\sum_{u=1}^{U}\left\lVert\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},\ell}\right\rVert^{2}\leq\eta_{\text{LNA}},$
(14)
$\sum_{\ell=1}^{L_{t}^{(i)}}\sum_{u=1}^{U}\left\lVert\mathbf{W}^{\left(i\right)*}_{\text{RF}}\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},\ell}\right\rVert^{2}\leq\eta_{\text{ADC}},$
(15)

where $\mathbf{f}^{\left(i\right)}_{\text{RF},\ell}$ is the $\ell$-th column
vector of $\mathbf{F}^{\left(i\right)}_{\text{RF}}$, i.e.,
$\mathbf{f}^{\left(i\right)}_{\text{RF},\ell}=\big{[}\mathbf{F}^{\left(i\right)}_{\text{RF}}\big{]}_{:,\ell}$.

###### Proof.

Note that for a matrix $\mathbf{A}\in\mathbb{C}^{N_{\text{row}}\times
N_{\text{col}}}$,

$\sigma_{\text{max}}^{2}(\mathbf{A})\leq\sum_{n=1}^{N_{\text{col}}}\sigma_{\text{max}}^{2}\Big{(}\big{[}\mathbf{A}\big{]}_{:,n}\Big{)}.$
(16)

Then,

$\displaystyle\eta_{\text{LNA}}$
$\displaystyle\geq\sum_{\ell=1}^{L_{t}^{(i)}}\sum_{u=1}^{U}\left\lVert\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},\ell}\right\rVert^{2}=\sum_{u=1}^{U}\sum_{\ell=1}^{L_{t}^{(i)}}\sigma_{\text{max}}^{2}(\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},\ell})$
(17)
$\displaystyle\geq\sum_{u=1}^{U}\sigma_{\text{max}}^{2}\left(\Big{[}\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},1}\>\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},2}\>\dots\>\mathbf{H}_{{ii}}[u]\mathbf{f}^{\left(i\right)}_{\text{RF},L_{t}^{(i)}}\Big{]}\right)$
$\displaystyle=\sum_{u=1}^{U}\sigma_{\text{max}}^{2}\left(\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}\right).$

Thus, (14) is a sufficient condition for (12). By replacing
$\mathbf{H}_{{ii}}[u]$ in (17) with
$\mathbf{W}^{\left(i\right)*}_{\text{RF}}\mathbf{H}_{{ii}}[u]$, one can show
that (15) is a sufficient condition for (13). ∎

###  III-B Design and Operation of RF Beam Allowlists

Our system design is motivated by (12) and (13). These are the sufficient
conditions for preventing the Rx saturation, which can be met without
considering the digital precoder
$\Big{\\{}\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\Big{\\}}_{u=1:U}$, given
(3). This implies that there exist rooms of design to prevent the Rx
saturation with only analog beamforming. Thus, our design requires the FD-node
to form the analog beamformers
$\mathbf{F}^{\left(i\right)}_{\text{RF}},\mathbf{W}^{\left(i\right)}_{\text{RF}}$
that solely prevent the Rx saturations, and at the same time, maintain the
strongest available beam alignments at its respective transmit/receive link.
The Rx saturation-preventing conditions may either be (12) and (13), or (14)
and (15). Then, the SI can be completely removed with the additional support
from the digital SIC, while the digital beamformer concentrates on the desired
channel, free of the SI.

Specifically, the FD-node should periodically update its feasible beam
combination set, which is a set of beam combinations that form an analog
beamformer satisfying certain Rx saturation-preventing conditions. Then, an RF
beam codebook with reduced size can be formed by the set of the unique beams
contained in the feasible beam combination set, which we call the allowlist.
As a toy example, consider an original RF beam codebook set comprised of 5
beams
$\mathbf{\mathcal{C}}=\Big{\\{}\mathbf{c}_{{1}},\mathbf{c}_{{2}},\cdots,\mathbf{c}_{{5}}\Big{\\}}$.
Suppose that the feasible beam combination set satisfying the given Rx
saturation-preventing conditions is
$\mathbf{\mathcal{P}}=\Big{\\{}\left(\mathbf{c}_{{2}},\mathbf{c}_{{3}},\mathbf{c}_{{4}}\right),\left(\mathbf{c}_{{3}},\mathbf{c}_{{4}},\mathbf{c}_{{5}}\right)\Big{\\}}$,
where the number of the RF chains is 3. Then, the allowlist is the reduced RF
beam codebook
$\mathcal{C}_{\text{AL}}=\Big{\\{}\mathbf{c}_{{2}},\mathbf{c}_{{3}},\mathbf{c}_{{4}},\mathbf{c}_{{5}}\Big{\\}}$,
necessary for the HD fashion beam sweeping measurements. Thus, with the
reduced number of beam measurements based on $\mathcal{C}_{\text{AL}}$, beam
combination that maximizes the beam alignment of the desired link will be
selected from $\mathbf{\mathcal{P}}$. The resultant RF beam combination will
inherently prevent the SI from saturating the Rx components. Fig. 3
illustrates an example of differently realized allowlists.

In the actual design, $\mathbf{\mathcal{C}}$ may either be
$\mathbf{\mathcal{F}}^{\left(i\right)}_{\text{RF}}$ or
$\mathbf{\mathcal{W}}^{\left(i\right)}_{\text{RF}}$. The allowlist may be
deployed at the transmit link alone or at both the transmit and receive link.
For the former case, the Rx saturation-preventing conditions ((12) and (13),
or (14) and (15)) will restrain only the transmit link, while the latter will
first restrain the transmit link with the LNA saturation-preventing condition
((12) or (14)), then the receive link with the ADC saturation-preventing
condition ((13) or (15)) for the predetermined analog precoder
$\mathbf{F}^{\left(i\right)}_{\text{RF}}$. Although the latter case is
expected to be more practical considering the typical load balance between the
downlink and uplink in a base station scenario, we focus our analysis on the
former case for brevity.

Figure 2: The array factors of the original $64$ beam DFT codebook and the two
different realizations of allowlist with a $16\times 4$ uniform planar array.

###  III-C Costs of System Design

In this subsection, we discuss the major cost factors for implementing the
proposed beamforming designs; the number of required beam measurements and the
computational complexity111Computational loads for digital SIC, overheads for
the estimation of the SI channel are also the cost factors that should be
considered in practice. However, We omit these factors in our comparison since
they are common and nearly identical for all the considered methods.. As a
benchmark, we consider (i) the expanded version of the method proposed by [13]
for OFDM, which acquires the digital precoder per subcarrier as the solution
of a convex optimization problem maximizing the transmit spectral efficiency
subjected to Rx preventing conditions, (10) and (11)222The maximization
problem formulated in [13] consists of two sequential optimization problems,
where solving the inner convex problem repeats for the number of predefined
candidate beam pairs. Due to its time complexity, we only consider the case
with single candidate beam pair that maximizes the beam alignment at the beam
sweeping phase.; (ii) Power reduction method, which simply reduces the
transmit power to ensure that (8) and (9) are not violated, given the
preselected analog beamformer that maximizes the beam alignment.

For the proposed methods, dynamic construction of allowlist to reduce the
required number of beam measurements becomes the major compuational burden.
Applying (12) and (13) constraints leads to
${}_{C_{t}^{\left(i\right)}}C_{L_{t}^{\left(i\right)}}$ singular value
decomposition (SVD) operations per subcarrier $\sim
O$($U(L_{t}^{\left(i\right)})^{2}N_{t}^{\left(i\right)}{}_{C_{t}^{\left(i\right)}}C_{L_{t}^{\left(i\right)}}$),
which is copmutationally heavy. However, we could significantly reduce the
required computations by applying (14) and (15), leveraging their structure.
Vector norm operation, which is by itself computationally lighter than the
SVD, can be called for only $C_{t}^{\left(i\right)}$ times per subcarrier,
while the condition testing is replaced by the summation of scalar norm
values333Furthermore, (14) and (15) also allow us to avoid the naive approach
of exhaustively computing all possible combinations using some tricks,
although it is not applied in our analysis for the sake of fairness. By
exploiting the reverse-lexicograhical order of the evaluated
$C_{t}^{\left(i\right)}$ norm values, each combination can be added
successively until the given constraint value is met. After this point, a
chunk of combinations can be skipped, since they will also meet the given
constraint value [16]. $\sim$${O}$($U\cdot C_{t}^{\left(i\right)}\cdot
N_{t}^{\left(i\right)}+{}_{C_{t}^{\left(i\right)}}C_{L_{t}^{\left(i\right)}}$).
Table I summarizes the overall comparison of the cost factors for various
system designs, where the specific values are the averaged results from
multiple simulations with the detailed parameter settings analogous to those
described in the next section. The CPU run time is measured via Intel i7-6700K
processor.

Figure 3: Visual example of 3D modeling and ray-tracing algorithm for
generating the SI channels of FD IAB scenarios.

###  III-D Digital SIC and Baseband Beamforming Design

With the designated analog precoder/combiners at the transmit/receive links
$\mathbf{F}^{\left(i\right)}_{\text{RF}}$,
$\mathbf{W}^{\left(j\right)}_{\text{RF}}$,
$\mathbf{F}^{\left(k\right)}_{\text{RF}}$, and
$\mathbf{W}^{\left(i\right)}_{\text{RF}}$, the remaining design are that of
the digital SIC, the digital beamformers
$\\{\mathbf{F}^{\left(i\right)}_{\text{BB}}[u]\\}_{u=1:U}$,
$\\{\mathbf{W}^{\left(j\right)}_{\text{BB}}[u]\\}_{u=1:U}$,
$\\{\mathbf{F}^{\left(k\right)}_{\text{BB}}[u]\\}_{u=1:U}$, and
$\\{\mathbf{W}^{\left(i\right)}_{\text{BB}}[u]\\}_{u=1:U}$. First, let us
define the effective SI channel as

$\mathbf{\tilde{H}}_{{ii}}[u]=\mathbf{W}^{\left(i\right)*}_{\text{RF}}\mathbf{H}_{{ii}}[u]\mathbf{F}^{\left(i\right)}_{\text{RF}}.$
(18)

Having preserved the linearities of the receive chain, the digital SIC can
effectively cancel out the residual SI at the baseband by reconstructing the
residual SI with the knowledge of (18) [14].

Finally, owing to the analog beamformers and the digital SIC module, the
design of the digital beamforming can completely focus on the effective
desired channel, from node $m$ to $n$, where $(m,n)\in\\{(i,j),(k,i)\\}$,
defined as

$\mathbf{\tilde{H}}_{{mn}}[u]=\mathbf{W}^{\left(n\right)*}_{\text{RF}}\mathbf{H}_{{mn}}[u]\mathbf{F}^{\left(m\right)}_{\text{RF}}.$
(19)

For evaluation, we considered eigenbeamforming for the digital beamformer
designs, i. e.,
$\mathbf{U}_{{mn}}[u]{\bm{\Sigma}}_{mn}[u]\mathbf{V}_{{mn}}[u]=\text{SVD}\left(\mathbf{\tilde{H}}_{{mn}}[u]\right)$
and
$\mathbf{W}^{\left(n\right)}_{\text{BB}}[u]=[\mathbf{U}_{{mn}}[u]]_{1:L_{r}^{\left(n\right)}}$
and
$\mathbf{F}^{\left(m\right)}_{\text{BB}}[u]=[\mathbf{V}_{{mn}}[u]]_{1:L_{t}^{\left(m\right)}}$

##  IV Performance Evaluation

In this section, we evaluate the performance of the proposed system in a
realistic integrated access and backhaul (IAB) scenario, which we envision as
one of the promising applications for the mmWave FD [17]. The IAB is a new
network framework being investigated by the 3GPP, where only a few of the BSs
are connected to the traditional wired infrastructures while the other BSs
relay the backhaul traffic wirelessly. To precisely model the SI channel under
the given network scenario, we adopt ray-tracing algorithms. Shown in Fig. 3
is a 3D model of an urban area as a visual example. As such, we randomly pick
a total of 269 points of potential FD base station positions covering urban,
suburban, and rural spots in the 3D model of Texas, in the U.S. The ray-
tracing software outputs, the attenuation, time delay, elevation/azimuth
AoA/AoDs of maximum 15 paths, which are used to generate the MIMO SI channel
using (4) and (5). The apparant characteristic of the considered SI channels
is the dominance of the line-of-sight (LoS) component and the first none-line-
of-sight (NLoS) componet.

Figure 4: Sum spectral efficiency as a function of SNR
($\text{SNR}_{ij}=\text{SNR}_{ki}$) for various beamforming methods.

For numerical evaluations, we consider $N_{t}^{(m)}=N_{r}^{(n)}=16\times 4=64$
(uniform planar array), $L_{t}^{(m)}=L_{r}^{(n)}=2$, and
$N_{s}^{\left(mn\right)}=2$. We use DFT codebooks for
$\mathbf{\mathcal{F}}^{\left(m\right)}_{\text{RF}}$and
$\mathbf{\mathcal{W}}^{\left(n\right)}_{\text{RF}}$, where the node index
$(m,n)\in\\{(i,j),(k,i)\\}$. The total number of $U=128$ subcarriers were
considered. For the LNAs and the ADCs, we assume $P^{max}_{\text{LNA}}=-10$
dBm, $P^{max}_{\text{ADC}}=-25.98$ dBm. The input power threshold of the $B$
bit ADC was determined by its dynamic range $6.021\cdot B+1.763$ dB, noise
floor NF $-90\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$, and an extra
$10\text{\,}\mathrm{d}\mathrm{B}$ power margin to account for the power-to-
average-ratio (PAPR) of the OFDM system,
$P^{max}_{\text{ADC}}=\text{NF}+6.021\cdot B+1.763-10$ dBm. The transmit power
of the FD node is set to
$P_{\text{tx}}^{\left(i\right)}=$40\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$
and the vertical separation of T/Rx arrays to
$r^{\left(i\right)}=$0.1\text{\,}\mathrm{m}$$

Figure 5: Sum spectral efficiency of proposed methods at
$10\text{\,}\mathrm{d}\mathrm{B}$ SNR ($\text{SNR}_{ij}=\text{SNR}_{ki}$)
under various configurations of Rx components.

Fig. 4 illustrates the sum spectral efficiency of transmit/receive link
according to the varying mmWave FD design methods. We observe that our
proposed designs shows a performance comparable to the ideal FD and other
benchmarks, while requiring less than 63% of the transmit link beam
measurements and 82% of the total beam measurements compared to those of the
benchmarks. Also, the proposed method with (14) and (15) constraints used only
0.29% amount of the CPU run time when compared to the method proposed by [13]
to achieve a comparable performance.

Fig. 5 illustrates the effect of varying $P^{max}_{\text{LNA}}$ and
$P^{max}_{\text{ADC}}$. It could be intuitively understood that strict LNA and
ADC constraints result in allowlists with smaller sizes. Thus, although we
also benefit from the reduced number of required beam measurements, sacrifices
are being made at the transmit link by excluding the beam candidates that
potentially could have provided a strong link to the desired receiver (node
$j$). Furthermore, the narrow performance gap between the proposed method with
(12) and (13) constraints and the method with (14) and (15) constraints
implies that the sufficient conditions derived in Theorem 1 are practically
very tight. Since Theorem 1 allows us to significantly reduce the
computational burden imposed by (12) and (13), the proposed method with (14)
and (15) constraints would be the preferred candidate for cost-efficient
mmWave FD system designs.

## Acknowledgment

This research was in part supported by Networks Business, Samsung Electronics
and Institute of Information & communications Technology Promotion (IITP)
grant funded by the Korea government (MSIT) (No. 2021-0-00486, 2021-0-02208).

##  V Conclusion

In this paper, we have presented a cost-efficient design of the mmWave FD
systems. The key design is to dynamically reduce the RF beam codebook in a
computationally efficient manner, so that it is comprised of the RF beams that
collectively prevent the Rx saturation from the SI. Then, the residual SI can
be completely removed with the digital SIC, allowing the digital beamformer to
concentrate on its desired channel, free of the SI. To reduce the computation
required for the proposed method, we introduced two sufficient conditions that
prevent the Rx side saturations, which are sufficiently tight. Through
performance evaluations conducted in practical mmWave FD scenarios,
realistically modeled via 3D ray-tracing, we demonstrated that the proposed
design achieves comparable performance with the ideal FD and other benchmarks
with significantly reduced costs.

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# Thermally activated switching in the presence of non-Gaussian noise

Lora Billings(1), Mark I. Dykman(2)∗, and Ira B. Schwartz(3<EMAIL_ADDRESS>(1)Department of Mathematical Sciences, Montclair State University, Montclair,
NJ 07043
(2) Department of Physics and Astronomy, Michigan State University, East
Lansing, MI 48824
(3) US Naval Research Laboratory, Code 6792, Nonlinear System Dynamics
Section, Plasma Physics Division, Washington, DC 20375

###### Abstract

We study the effect of a non-Gaussian noise on interstate switching activated
primarily by Gaussian noise. Even weak non-Gaussian noise can strongly change
the switching rate. The effect is determined by all moments of the noise
distribution. The explicit analytical results are compared with the results of
simulations for an overdamped system driven by white Gaussian noise and a
Poisson noise. Switching induced by a purely Poisson noise is also discussed.

###### pacs:

05.40.-a, 72.70.+m, 05.70.Ln, 05.40.Ca

Much progress has been made recently in the studies of switching between
coexisting stable states, primarily because switching can be now investigated
for a large variety of well-controlled micro- and mesoscopic systems ranging
from trapped electrons and atoms to Josephson junctions and to nano- and
micro-mechanical oscillators Lapidus et al. (1999); Siddiqi et al. (2005);
Aldridge and Cleland (2005); Kim et al. (2005); Gommers et al. (2005);
Stambaugh and Chan (2006); Abdo et al. (2007); Lupaşcu et al. (2007); Katz et
al. (2007); Serban and Wilhelm (2007). Fluctuations in these systems are
usually due to thermal or externally applied Gaussian noise. However, as the
systems become smaller, an increasingly important role may be played also by
non-Gaussian noise. It may come, for example, from one or a few two-state
fluctuators hopping at random between the states, in which case the noise may
be often described as a telegraph noise.

The switching probability is sensitive to a non-Gaussian noise. This
sensitivity attracted much attention after it was proposed Tobiska and Nazarov
(2004) to use switching in Josephson junctions to measure the full counting
statistics in electronic circuits Levitov and Lesovik (1992); Nazarov (2007).
Several theoretical Pekola (2004); Ankerhold (2007); Sukhorukov and Jordan
(2007); Grabert (2008) and experimental Timofeev et al. (2007); Huard et al.
(2007) papers on measuring the 3rd moment of the current distribution from the
switching rates were published recently, and different theoretical approaches
were compared in Refs. Novotný, 2008; Sukhorukov and Jordan, 2008.

In this paper we study switching induced by Gaussian noise in the presence of
an additional non-Gaussian noise. Even where the latter has a smaller
intensity than the Gaussian noise, its effect on the switching rate may be
exponentially strong. We show that it can be described in a simple form in
terms of the noise characteristic functional, thus accounting for all moments
of the noise distribution. The analytical results are compared with
simulations for an overdamped system driven by white Gaussian noise and a
Poisson noise. We also consider switching induced by a Poisson noise alone;
here, the result for the rate may be qualitatively different from that in the
weakly non-Gaussian noise approximation.

The potentially strong effect of an extra modulation, whether random or
regular, on the rate of Gaussian noise induced switching can be understood
from the well-established picture of the switching dynamics. Switching events
result from large rare noise outbursts. For Gaussian noise, the switching rate
is $W\propto\exp(-R/D)$, where $R$ is the activation energy and $D$ is the
noise intensity Freidlin and Wentzell (1998). Even though switching happens at
random, the system trajectories followed in switching form a narrow tube in
the space of dynamical variables ${\bf q}=(q_{1},q_{2},\ldots)$ centered at
the most probable (optimal) switching path ${\bf q}_{\rm opt}(t)$.

One can think of the effect of an additional modulation in terms of a
generalized work done by the modulation on the system moving along ${\bf
q}_{\rm opt}(t)$ Smelyanskiy et al. (1997); Dykman et al. (1997). This work
changes the activation barrier. The change $\delta R$ is proportional to the
modulation amplitude. Therefore the overall change of the switching rate
$\propto\exp(-\delta R/D)$ depends on the modulation amplitude exponentially.
The switching rate gives the probability current from the occupied state
Kramers (1940), making it an observable quantity. Thus, if the modulation is
random, one has to simply average the factor $\exp(-\delta R/D)$ over
realizations of the modulation.

Since $\delta R$ is linear in the characteristic amplitude of random
modulation, the ratio $\delta R/D$ does not have to be small even where the
modulation intensity, which is quadratic in the amplitude, is smaller than
$D$. However, the distribution of non-Gaussian modulation may decay slower
than Gaussian on the tail. To determine whether the effect of a non-Gaussian
noise on switching may be regarded as a perturbation one has to compare the
probabilities of appropriate large fluctuations induced by the Gaussian and
non-Gaussian noises, taking into account all moments of the distribution.

We study switching for a system described by the Langevin equation

$\dot{\bf q}={\bf K}({\bf q})+{\bf f}(t)+{\bm{\xi}}(t).$ (1)

We assume that, in the absence of noise, the system has a stable stationary
state ${\bf q}_{A}$ and a saddle point ${\bf q}_{\cal S}$ on the boundary of
the basin of attraction to ${\bf q}_{A}$, with ${\bf K}({\bf q}_{A})={\bf
K}({\bf q}_{\cal S})=0$. Switching from the stable state is due to the forces
${\bf f}(t)$ and ${\bm{\xi}}(t)$, which are the Gaussian and non-Gaussian
noises, respectively. We separate them, since physically they often come from
different sources. It is convenient to characterize ${\bf f}(t)$ by its
probability density functional ${\cal P}_{{\bf f}}[{\bf
f}(t)]=\exp\left(-{\cal R}_{{\bf f}}/D\right)$,

${\cal R}_{{\bf f}}[{\bf f}(t)]=\frac{1}{4}\int dt\,dt^{\prime}\,{\bf
f}(t)\hat{{\cal F}}(t-t^{\prime}){\bf f}(t^{\prime}),$ (2)

where $\hat{{\cal F}}(t-t^{\prime})/2D$ is the inverse of the pair correlator
of ${\bf f}(t)$. The characteristic noise intensity $D$ is small, so that the
switching rate $W\ll t_{\rm r}^{-1},t_{\rm c}^{-1}$, where $t_{\rm r}$ is the
relaxation time of the system and $t_{\rm c}$ is the noise correlation time.
The non-Gaussian noise is more conveniently described for our purpose by the
characteristic functional

$\tilde{\cal P}_{{\bm{\xi}}}[{\bf k}]=\left\langle\exp\left[i\int dt{\bf
k}(t){\bm{\xi}}(t)\right]\right\rangle_{{\bm{\xi}}},$ (3)

where $\langle\ldots\rangle_{{\bm{\xi}}}$ means averaging over
${\bm{\xi}}(t)$.

We first consider the case where the intensity of the non-Gaussian noise
${\bm{\xi}}$ is smaller than $D$. We will disregard corrections proportional
to this intensity, but the ratio of the characteristic amplitude $g_{0}$ of
${\bm{\xi}}$ to $D$ will not be assumed small. The switching rate can be
written as

$\displaystyle W=C\left\langle\exp[-{\cal
R}[{\bm{\xi}}]/D]\right\rangle_{{\bm{\xi}}},$ (4) $\displaystyle{\cal
R}[{\bm{\xi}}]=\min\left\\{{\cal R}_{{\bf f}}+i\int dt\,{\bf
k}(t)\left[\dot{\bf q}-{\bf K}-{\bf f}(t)-{\bm{\xi}}(t)\right]\right\\},$

where $C$ is the prefactor that weakly depends on the noise intensity. The
minimum is taken over trajectories ${\bf f}(t),{\bf q}(t),{\bf k}(t)$ that
satisfy boundary conditions ${\bf f}(t),{\bf k}(t)\to 0$ for $t\to\pm\infty$,
${\bf q}_{t\to-\infty}\to{\bf q}_{A},{\bf q}_{t\to\infty}\to{\bf q}_{\cal S}$.
This formulation was proposed in the weak-noise limit Dykman (1990); Dykman
and Smelyanskiy (1998) for a time-periodic ${\bm{\xi}}(t)$, in which case
${\bf q}_{A,{\cal S}}$ are also periodic and there is no averaging over
${\bm{\xi}}$. The variational problem (4) describes coupled optimal
trajectories ${\bf f}_{\rm opt}(t),{\bf q}_{\rm opt}(t),{\bf k}_{\rm opt}(t)$,
with ${\bf f}_{\rm opt}(t)$ being the most probable noise realization that
brings the system to the saddle on the basin boundary of the initially
occupied state.

It is known from variational calculus that, to first order in ${\bm{\xi}}$,
the effect of ${\bm{\xi}}(t)$ on ${\cal R}$ can be calculated along the
optimal trajectory unperturbed by ${\bm{\xi}}(t)$. Such a trajectory is an
instanton. Its typical duration is $\sim\max(t_{\rm r},t_{\rm c})$. It is
translation-invariant with respect to time and can be centered at any time
$t_{0}$. If ${\bm{\xi}}(t)$ is periodic, it lifts time-translation symmetry
and fixes $t_{0}$ (modulo the period) so as to minimize ${\cal
R}[{\bm{\xi}}]$.

If ${\bm{\xi}}(t)$ is a stationary noise, the switching rate $W$ is
independent of time. In this case one can think not of the adjustment of the
instanton center $t_{0}$ to ${\bm{\xi}}(t)$, but, equivalently, of the
adjustment of ${\bm{\xi}}(t)$ to $t_{0}$ so as to maximize the overall
probability of switching. This adjustment provides the major contribution to
the value of $W$ when the averaging over realizations of ${\bm{\xi}}(t)$ is
performed in Eq. (4) using a solution with a given $t_{0}$.

From Eqs. (3), (4) one obtains a simple expression for the switching rate

$\displaystyle W=W^{(0)}A_{\rm sw},\qquad A_{\rm sw}=\tilde{\cal
P}_{{\bm{\xi}}}[i{\bm{\chi}}/D],$ (5)

where $W^{(0)}$ is the switching rate in the absence of non-Gaussian noise.
The factor $A_{\rm sw}$ describes the effect of non-Gaussian noise. It is
expressed in a closed form in terms of the noise characteristic functional
calculated for function ${\bm{\chi}}(t)=-i{\bf k}_{\rm opt}^{(0)}(t)$, where
${\bf k}_{\rm opt}^{(0)}(t)$ is the solution of the variational problem (4)
for ${\bm{\xi}}={\bf 0}$. The real function ${\bm{\chi}}(t)$ is the
logarithmic susceptibility which describes the linear response of $\log W$ to
a perturbation Smelyanskiy et al. (1997); Dykman et al. (1997); Assaf et al.
(2008). The structure of Eq. (5) resembles that of the expression for a large
fluctuation probability in a birth-death system with non-Gaussian modulation
of reaction rates Dykman et al. (2008).

From Eq. (5), the effect of a non-Gaussian noise on the switching rate is
determined by the ratio of the noise amplitude to the Gaussian noise intensity
$D$. Equation (5) applies to both underdamped and overdamped systems. Examples
of calculating ${\bf k}_{\rm opt}^{(0)}$ can be found in Refs. Smelyanskiy et
al., 1997; Dykman et al., 1997; Assaf et al., 2008 and papers cited therein.

As an illustration we will consider the case of a one-component
$\delta$-correlated Poisson noise $\xi(t)$ with pulse area $g$ and mean
frequency $\nu$. Using the explicit form of the noise characteristic
functional Feynman and Hibbs (1965), we obtain

$\displaystyle A_{\rm sw}=\exp\left\\{-\nu\int
dt\,\left[1-\exp\left(-\chi(t)g/D\right)\right]\right\\},$ (6)

where $\chi(t)$ is the corresponding component of the logarithmic
susceptibility. If $g/D\ll 1$, $\log A_{\rm sw}$ is a series in $g/D$. The
coefficients in this series describe the effects of the moments of the Poisson
noise on the switching rate. In the opposite case, $g/D\gg 1$ (but the Poisson
noise intensity $\nu g^{2}\ll D$), if $\chi(t)$ becomes negative, then $\log
A_{\rm sw}\approx\nu\left[2\pi
D/g\ddot{\chi}(t_{m})\right]^{1/2}\exp[-\chi(t_{m})g/D]$ where $t_{m}$ is the
instant where $-\chi(t)$ is maximal. If $\chi(t)\geq 0$ for all $t$ and
$g/D\gg 1$, the major contribution to $A_{\rm sw}$ comes from the region of
small $\chi(t)$. If $\chi(t)$ is small only for $|t|\to\infty$, where it
decays exponentially with $|t|$, then $\log A_{\rm sw}\propto\nu t_{\rm
r}\log(g/D)$, to leading order in $g/D$.

The Poisson noise distribution does not fall off as steeply as Gaussian. This
imposes a limitation on the range of $g/D$ where Poisson noise may be treated
as a perturbation and the above theory applies. To see the far-tail effect we
consider switching due to a purely Poisson noise, where ${\bf f}={\bf 0}$ in
equation of motion (1). We will use the method of optimal fluctuation, as for
some other types of non-Gaussian noise McKane (1989).

The switching rate is determined, to logarithmic accuracy, by the integral
over trajectories ${\bf k}(t),{\bf q}(t)$ of the functional
$\left\langle\exp\left\\{-i\int dt{\bf k}(t)[\dot{\bf q}-{\bf
K}-{\bm{\xi}}(t)]\right\\}\right\rangle_{{\bm{\xi}}}$ Luciani and Verga
(1987). We assume for brevity that different components of the Poisson noise
are independent short pulses with areas ${\bf g}=(g_{1},g_{2},\ldots)$ and
with average frequency $\nu$. It is convenient to consider a zero-mean noise,
${\bm{\xi}}(t)\to{\bm{\xi}}(t)-\nu{\bf g},\;{\bf K}\to{\bf K}+\nu{\bf g}$; we
assume that ${\bf q}_{A},{\bf q}_{\cal S}$ are also appropriately shifted. Of
interest for switching are trajectories that approach the saddle point Dykman
(1990). In the spirit of the method of optimal fluctuation, for small $|{\bf
g}|$ and for $\nu\lesssim t_{\rm r}^{-1}$ the integral over trajectories ${\bf
q}(t),{\bf k}(t)$ can be calculated by steepest descent. This gives

$\displaystyle W=C^{\prime}\exp[-R_{P}],\quad R_{P}=\min\int dt\left(i{\bf
k}\dot{\bf q}-H\right),$ (7) $\displaystyle H\equiv H({\bf q},i{\bf
k})=-\nu\left(1+i{\bf g}{\bf k}-e^{i{\bf g}{\bf k}}\right)+i{\bf k}{\bf
K}({\bf q}).$

The variational problem (7) determines the optimal switching trajectory ${\bf
q}_{\rm opt}(t),{\bf k}_{\rm opt}(t)$. It starts at $t\to-\infty$ at ${\bf
q}\to{\bf q}_{A},{\bf k}\to{\bf 0}$ and goes to ${\bf q}\to{\bf q}_{\cal
S},{\bf k}\to{\bf 0}$ for $t\to\infty$. On this trajectory $H=0$. As in
systems driven by white Gaussian noise Freidlin and Wentzell (1998), the
optimal trajectory can be described as a Hamiltonian trajectory of an
auxiliary system with coordinate ${\bf q}$, momentum $i{\bf k}$, and
Hamiltonian $H$. A similar approach was proposed in Ref. Sukhorukov and
Jordan, 2007. However, the characteristic functional was not specified and the
explicit analysis took into account only the 3rd moment of the noise
distribution, which was considered a perturbation; therefore the results do
not describe switching due to large Poisson fluctuations.

The switching exponent in Eq. (7) is $R_{P}\gg 1$ for small $|{\bf g}|$.
However, in contrast to the case of Gaussian noise, $R_{P}$ is not
proportional to the reciprocal noise intensity $\nu{\bf g}^{2}$. Nor does it
scale like reciprocal noise amplitude $|{\bf g}|^{-1}$, although $R_{P}|{\bf
g}|$ appears to slowly vary with $|{\bf g}|$.

An explicit dependence of the switching rate on the intensity $g$ of Poisson-
distributed pulses can be found for a one-variable overdamped system with
equation of motion

$\dot{q}=-U^{\prime}(q)+f(t)+\xi(t).$ (8)

Here, $U(q)$ is the effective potential. The stationary states $q_{A}$ and
$q_{\cal S}$ correspond to the minimum and the barrier top of $U(q)$.

If $f(t)$ is white Gaussian noise, $\langle
f(t)f(t^{\prime})\rangle=2D\delta(t-t^{\prime})$, and the Poisson noise is
weak, the Poisson-noise induced factor in the switching rate $A_{\rm sw}$ is
described by Eq. (6) with $\chi(t)=-f_{\rm opt}^{(0)}(t)/2=-\dot{q}_{\rm
opt}^{(0)}(t)$ and with $\dot{q}_{\rm opt}^{(0)}=U^{\prime}\left(q_{\rm
opt}^{(0)}\right)$.

In the opposite case where switching is due to purely Poisson noise, i.e.,
$f=0$ in Eq. (8), from Eq. (7)

$\displaystyle R_{P}=\frac{1}{g}\int\nolimits_{\tilde{q}_{A}}^{\tilde{q}_{\cal
S}}dq\kappa(q),\quad\kappa=\log\left\\{1+\left[\kappa
U^{\prime}(q)/g\nu\right]\right\\}.$ (9)

Here, $\tilde{q}_{A}$ and $\tilde{q}_{\cal S}$ are the shifted extrema of the
potential given by equation $U^{\prime}(q)=g\nu$. From Eq. (9), $R_{P}\sim
r_{P}\log(r_{P}/\nu t_{\rm r})$, with $r_{P}=(\tilde{q}_{\cal
S}-\tilde{q}_{A})/g$.

A qualitative feature of unipolar (pulses of one sign) Poisson noise is that,
for an overdamped system, it causes switching only provided the noise pulses
push the system from the stable state towards the saddle. In this case
$r_{P}>0$. There is no switching for pulses of the opposite sign. The “one-
sidedness” of fluctuations in overdamped systems has other manifestations,
which includes the work fluctuation distribution Cohen . On the other hand, we
expect that an underdamped system should be able to switch for Poisson pulses
of any sign, in which case there should be a critical value of damping for
which switching from the state is possible for a given sign of $g$. Equations
(7), (9) apply if $r_{P},r_{P}/\nu t_{\rm r}\gg 1$.

As a cause of switching, a Poisson noise is effectively weaker than a Gaussian
noise if the switching exponent $R_{P}$ is larger than the switching exponent
for the Gaussian noise. For white Gaussian noise of intensity $D$ in Eq. (8),
the switching exponent is $\Delta U/D$ with $\Delta U=U(q_{\cal S})-U(q_{A})$
Kramers (1940). The condition $R_{P}>\Delta U/D$ effectively limits the range
of applicability of Eq. (6). We saw that $\log A_{\rm sw}$ becomes large
provided $-\chi(t_{m})g/D\gg 1$. An order of magnitude estimate shows that
$-\chi(t_{m})g/D\sim(\Delta U/D)R_{P}^{-1}\log(r_{P}/\nu t_{\rm r})$, and
therefore from Eq. (9) the large $\log A_{\rm sw}$ asymptotics applies only
provided $\log(r_{P}/\nu t_{\rm r})\gg 1$. This condition is compatible with
$-\chi(t_{m})g/D\gg 1$ only for very small $D$.

We now apply the above results to an overdamped system (8) with a double-well
potential

$U(q)=-q^{2}/2+q^{4}/4,$ (10)

which has been extensively studied in the context of white-noise driven
systems. In the absence of Poisson noise, the escape rate in this case is
$W^{(0)}=(\sqrt{2}/\pi)\exp(-1/4D)$, and the logarithmic susceptibility for
escape from the negative-$q$ well ($q_{A}=-1$) is $\chi(t)=-\exp(t/2)(2\cosh
t)^{-3/2}$ ($\chi(t)$ has opposite sign for switching from $q_{A}=1$).

Figure 1: Poisson noise induced change of the switching exponent for an
overdamped Brownian particle in potential (10). The dimensionless Poisson
noise frequency is $\nu=0.5$. The data of numerical simulations are shown with
circles and squares for the cases where in escape the particle moves along and
opposite to Poisson pulses, respectively. The solid curves show the weak-
Poisson noise theory (6) for these cases, and the dashed curves show the
approximation where only the three moments of Poisson noise are taken into
account.

In Fig. 1 we present results of Monte Carlo simulations of switching of an
overdamped Brownian particle described by Eqs. (8), (10). They are compared
with the weak Poisson noise prediction, Eq. (6), and with the approximation
where only terms up to $g^{3}$ are kept in Eq. (6). The Poisson noise
intensity $\nu g^{2}<D$ in the whole range of studied $g/D$. For $g/D\lesssim
1$ the effect of Poisson noise is small and comes, primarily, to the change of
the activation barrier, $\Delta U\to\Delta U\pm\nu g$, and the effective noise
intensity, $D\to D+\nu g^{2}$. For larger $g/D\gtrsim 3$ the switching
exponent changes significantly, as expected.

For switching in the direction opposite to Poisson noise pulses, where $\log
W/W^{(0)}<0$, the numerics agrees well with Eq. (6). This is to be expected,
since the far tail of Poisson noise distribution is immaterial here; for
$g/D\gtrsim 5$ the results differ noticeably from the three-moments
approximation. For switching along the pulses, because of the far-tail effect,
with increasing $g$ Poisson noise quickly becomes as important as white noise
for chosen $D$. Therefore the perturbation theory fails and the dependence of
the switching exponent on $g$ is much weaker than the exponential dependence
expected from Eq. (6).

Figure 2: Switching exponent $R_{P}=-\ln W$ for an overdamped particle in a
potential (10) driven by a Poisson noise with dimensionless mean frequency
$\nu=0.5$ and pulse intensity $g$. Squares show the results of Monte-Carlo
simulations, the solid line is the asymptotic theory (9).

Numerical simulations of the switching rate for purely Poisson noise are shown
in Fig. 2. There is good agreement between the data and the asymptotic theory
(9) for small $g$. In this range $R_{P}g$ slowly varies with $g$.

In conclusion, we have considered switching in systems simultaneously driven
by a Gaussian and a non-Gaussian noise. Even where the non-Gaussian noise has
intensity smaller than that of the Gaussian, it may strongly change the
switching rate. The effect is determined by the ratio of the non-Gaussian
noise amplitude to the Gaussian noise intensity. It is described by the
characteristic functional of the non-Gaussian noise calculated for a function
determined by the system dynamics without this noise. A non-Gaussian tail of
the noise distribution may strongly modify the switching rate even for a small
noise intensity. We demonstrate this effect using Poisson noise as an example.
Analytical results are compared with Monte Carlo simulations.

MID acknowledges valuable discussions with M. Büttiker and A. Korotkov. LB is
supported by ARO grant No. W911NF-06-1-0320. MID is supported by ARO grant No.
W911NF-06-1-0324 and NSF grant DMR-0305746. IBS is supported by the Office of
Naval Research.

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11institutetext: Institut für Experimentelle Teilchenphysik, Karlsruher
Institut für Technologie (KIT), 76131 Karlsruhe, Germany22institutetext:
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

#  Belle II sensitivity to long–lived dark photons

Torben Ferber 2 Camilo Garcia-Cely 2 and Kai Schmidt-Hoberg
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this letter we point out that the Belle II experiment has a unique
sensitivity to visibly decaying long-lived dark photons. Concentrating on the
signatures with a single high energy photon in association with a displaced
pair of charged particles, we find that Belle II will be able to probe large
regions of parameter space that cannot be covered by any other running or
proposed experimental facility. While the signature with charged muons or
pions in the final state is expected to be background-free after all
selections are applied, the case of final state electrons is more involved and
requires an in-depth study. We discuss possible ways to further suppress
backgrounds and the corresponding experimental prospects.

††preprint: DESY-22-023

##  1 Introduction

Additional $U(1)$ gauge groups naturally appear in many scenarios beyond the
Standard Model and are often part of a so-called ‘hidden sector’ Fayet:1990wx
; Patt:2006fw ; Batell:2009di , which may contain the dark matter particle as
well as other weakly coupled states. Even though Standard Model (SM) particles
do not necessarily possess direct charges under this U(1) factor, effective
couplings between the associated $A^{\prime}$ gauge boson and SM states can
naturally be induced through kinetic mixing with the SM hypercharge field
strength Holdom:1985ag . This mixing therefore provides a renormalisable
portal through which $A^{\prime}$ gauge bosons can be produced via the
collision of SM particles, and corresponding signatures have been proposed and
searched for extensively in recent years covering a very large dark photon
mass range Ahlers:2007rd ; Ahlers:2008qc ; Batell:2009yf ; Andreas:2012mt ;
Essig:2013vha ; Izaguirre:2013uxa ; Morrissey:2014yma ; Curtin:2014cca ;
Frandsen:2012rk ; Fairbairn:2016iuf ; Ilten:2016tkc ; Battaglieri:2017aum ;
Alekhin:2015byh ; Beacham:2019nyx ; Batell:2014mga ; BaBar:2017tiz ;
Banerjee:2019pds ; ATLAS:2014pcp ; LHCb:2019vmc ; BaBar:2014zli ;
NA482:2015wmo . In particular, light dark sectors with masses in the MeV to
GeV range have triggered a lot of attention Dolan:2017osp ; Bondarenko:2019vrb
; Filimonova:2019tuy ; Berger:2016vxi ; Biswas:2015sha ; Knapen:2017xzo ;
Baek:2020owl ; Dolan:2014ska ; Krnjaic:2015mbs ; Jodlowski:2019ycu  and the
Belle II experiment is known to have excellent sensitivity to many of these
scenarios.

Depending on the mass ordering and coupling structure within the dark sector,
the $A^{\prime}$ decays invisibly into dark sector final states Batell:2014mga
; BaBar:2017tiz ; Banerjee:2019pds , visibly into SM final states
ATLAS:2014pcp ; LHCb:2019vmc ; BaBar:2014zli ; NA482:2015wmo  or into a
combination of both, leading to semi-visible final states Izaguirre:2015zva ;
Izaguirre:2017bqb ; Berlin:2018jbm ; Duerr:2019dmv ; Duerr:2020muu . In the
following we concentrate on the case in which the $A^{\prime}$ decays visibly.
For small mixing angles and small $A^{\prime}$ masses the effective couplings
to the SM are photon-like, so that the $A^{\prime}$ decays into electrically
charged SM particles.

In this letter we point out that the Belle II experiment has a unique
sensitivity to a weakly coupled visibly decaying dark photon $A^{\prime}$
whose decays have a displaced vertex. Specifically, the signature we
concentrate on here is a single high energy photon in association with a
displaced pair of charged particles. We find that the regions in parameter
space that can be probed by these searches are not covered by any other
running or proposed experiment and significantly extend the overall
experimental reach to kinetically mixed dark photons. While non-electron final
states will lead to signatures which can be made essentially background-free
with a number of simple selections, the case of $A^{\prime}\to e^{+}e^{-}$
proves much more difficult. We discuss the problematic backgrounds in detail
and sketch what needs to be done on the experimental side to reduce them to an
acceptable level.

In the next section we briefly introduce our notation. In section 3 we then
describe the steps required to evaluate the sensitivity of Belle II to this
scenario. Our results are presented in section 4 and we conclude with a short
discussion in section 5.

##  2 A brief recap of dark photons

Many theories beyond the SM exhibit additional $U(1)$ gauge factors. It is
well known that the cancellation of gauge anomalies constrain the SM charges
under this $U(1)_{X}$ to only a few possibilities, with the simplest scenario
corresponding to vanishing charges of SM fermions. However, even in this case
an effective interaction between the mass eigenstate dark photon
$A_{\mu}^{\prime}$ and the SM particles can arise from a simple kinetic mixing
in the Lagrangian, which is allowed by all symmetries. The Lagrangian
including such a kinetic mixing term between the dark $U(1)_{X}$ and
$U(1)_{Y}$ is given by

$\displaystyle{\mathcal{L}}=\mathcal{L}_{\text{SM}}-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}-\frac{\epsilon}{2c_{\text{W}}}\hat{X}_{\mu\nu}\hat{B}^{\mu\nu}\,,$
(1)

together with additional terms which give rise to a mass for the dark photon -
either a Stückelberg mass or a mass arising from a dark Higgs mechanism. In
this letter we are agnostic about the mass generation mechanism as our results
apply to both cases.

The required gauge-boson diagonalisation has been thoroughly discussed in the
literature, see e.g. Babu:1997st ; Frandsen:2011cg . After the fields have
been diagonalised and canonically normalised one obtains the physical
$Z$-boson, the SM photon $A_{\mu}$, and the dark photon $A^{\prime}_{\mu}$.
For $m_{A^{\prime}}\ll m_{Z}$ as studied in this article, the field
$A^{\prime}_{\mu}$ inherits the coupling structure of the photon to the SM
fermions up to a common factor $\epsilon$.

A comprehensive discussion of constraints and future prospects for dark
photons in the mass range relevant to us has been presented in Ilten:2018crw ;
Bauer:2018onh ; Fabbrichesi:2020wbt ; Graham:2021ggy  and we refer to those
works for further details. For our plots we adapt the limits from
Bauer:2018onh  and add limits which were subsequently published from LHCb
LHCb:2019vmc  and NA64 NA64:2019auh , as well as projections from FASER2
FASER:2018eoc .111We thank Patrick Foldenauer for providing the corresponding
data points.

##  3 Dark photons at Belle II

In this letter we concentrate on direct production of a dark photon
$A^{\prime}$ in association with the SM photon $\gamma$, with the former
subsequently decaying into a pair of charged SM states. This results in a
single high energy photon with a displaced pair of charged particles. We
concentrate on the case where the pair of charged particles has a significant
displacement to reduce backgrounds from prompt SM processes.

Light dark photons with smaller couplings could also be targeted at a long-
lived particle detector near the Belle II interaction point Dreyer:2021aqd .
However, the corresponding region of parameter space is already largely
constrained by beam dump experiments and hence of less interest.

###  3.1 The Belle II experiment

The Belle II experiment at the SuperKEKB accelerator is a next generation
B-factory Abe:2010gxa  that started physics data taking in 2019. SuperKEKB is
a circular asymmetric $e^{+}e^{-}$ collider with a nominal collision energy of
$\sqrt{s}=10.58\,\mathrm{GeV}$ and a design instantaneous luminosity of
$8\times 10^{35}$ cm-2 s-1. The detector is described in detail in Abe:2010gxa
.

We study the Belle II sensitivity for a dataset corresponding to an integrated
luminosity of $0.5\,\mathrm{ab^{-1}}$, $2\,\mathrm{ab^{-1}}$ and
$50\,\mathrm{ab^{-1}}$, spanning present and future prospects.

###  3.2 Event Generation

To generate events for the process $e^{+}e^{-}\to\gamma A^{\prime}$ with
subsequent decays of $A^{\prime}$ we employ MadGraph5_aMC@NLO v2.7.2
Alwall:2014hca . For final states induced by the decays of $A^{\prime}$ we
consider $A^{\prime}\to l^{+}l^{-}/h^{+}h^{-}$ ($l^{\pm}=e^{\pm},\mu^{\pm}$
and $h^{\pm}=\pi^{\pm},K^{\pm}$). Given that pions and kaons behave very
similarly to muons in the Belle II detector, see e.g. the discussion in
Duerr:2019dmv , we do not simulate these particles explicitly. Instead, we
rescale the simulated events with final state muons taking into account the
relevant branching ratios, which we take from Ilten:2018crw . As in
Duerr:2019dmv  events are generated in the centre-of-mass frame with
$\sqrt{s}=10.58\,\mathrm{GeV}$ and then boosted and rotated to the Belle II
laboratory frame.

###  3.3 Signal selection

Figure 1: Schematic view of the Belle II detector ($xy$-plane) and example
displaced signature $e^{+}e^{-}\to\gamma A^{\prime},A^{\prime}\to
l^{+}l^{-}/h^{+}h^{-}$ ($l^{\pm}=e^{\pm},\mu^{\pm}$ and
$h^{\pm}=\pi^{\pm},K^{\pm}$).

Signal events consist of a high energy photon recoiling against a pair of high
energy charged particles of opposite charge (see Fig. 1). The two charged
particles originate from a common vertex and combine to an invariant mass of
the dark photon $A^{\prime}$. The two charged particles and the photon
together combine to the collision energy of the incoming $e^{+}e^{-}$ beams.
Experimentally, a vertex fit of the two electrons followed by a kinematic fit
of all three particles will improve the invariant mass resolution
significantly and allows exploiting the full potential of $e^{+}e^{-}$
colliders. For $A^{\prime}\to e^{+}e^{-}$ we expect that dominant background
events for displaced $A^{\prime}$ decays are $e^{+}e^{-}\to e^{+}e^{-}\gamma$
with a misreconstructed prompt decay vertex, and $e^{+}e^{-}\to\gamma\gamma$
followed by a pair conversion $\gamma\to e^{+}e^{-}$; for
$A^{\prime}\to\mu^{+}\mu^{-}/h^{+}h^{-}$ the dominant background is
$e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ and $e^{+}e^{-}\to\pi^{+}\pi^{-}\gamma$
with a misreconstructed prompt decay vertex, as well as the production and
decay of $K_{S}^{0}$ mesons.

As in Duerr:2019dmv  we select events based on the radial vertex position $R$
of the $A^{\prime}$ decay products (region selection), their trigger
signatures (trigger selection), and the final state kinematics (kinematic
selection). The region selections are used to select the following vertex
regions:

  * •

Decays with $R=\sqrt{x^{2}+y^{2}}<0.2\,\mathrm{cm}$ are very close to the
nominal interaction point and will be subject to large SM backgrounds from
$e^{+}e^{-}\to e^{+}e^{-}\gamma$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ from
vertex resolution tails and the geometrical spread of the interaction region.
We assume that this region is not suitable for a displaced vertex search.
However, this region is generally included in the complementary prompt
searches at $e^{+}e^{-}$ colliders BaBar:2014zli ; Kou:2018nap .

  * •

The region $0.2\,\mathrm{cm}<R<0.9\,\mathrm{cm}$ is within the vacuum of the
SuperKEKB beam pipe, but sufficiently separated from the interaction point to
expect negligible backgrounds from prompt SM backgrounds Bossi:2013lxa . While
we do not expect any true conversion backgrounds $\gamma\to e^{+}e^{-}$ from
$e^{+}e^{-}\to\gamma\gamma$ events, experience from past experiments has shown
that a small fraction of conversion events that happen at larger radii will be
misreconstructed with a vertex close to the nominal interaction point. Such
misreconstructed tracks are often tracks that do not have hits in the inner
vertex detector layers, but are extrapolated to a common vertex close to the
interaction point by the tracking algorithms. We expect that requiring hits in
all detector layers of the Belle II detector downstream of the reconstructed
vertex can be used to reduce these backgrounds for $A^{\prime}\to e^{+}e^{-}$
to small and maybe even negligible levels at the expense of some signal
efficiency. Since the number of conversion events is very large, even small
misreconstruction rates can lead to sizeable backgrounds. However, these
subtle reconstruction effects must be evaluated using full detector
simulations for both signal efficiency and background rejection.

  * •

The region $0.9\,\mathrm{cm}<R<17\,\mathrm{cm}$ includes the beam pipe, the
vertex detectors, support structures, and the inner wall of the CDC. This will
induce potentially large backgrounds from $\gamma\to e^{+}e^{-}$ from
$e^{+}e^{-}\to\gamma\gamma$ events. We expect that these backgrounds are
prohibitively large for $A^{\prime}\to e^{+}e^{-}$, but negligible for
$A^{\prime}\to\mu^{+}\mu^{-}/h^{+}h^{-}$. We therefore only consider
$\mu,\pi$, and $K$ in this region.

  * •

$17\,\mathrm{cm}<R<60\,\mathrm{cm}$ covers the region inside the CDC with
sufficiently high tracking efficiency and only small amounts of passive
material. We assume that the backgrounds for $A^{\prime}\to e^{+}e^{-}$ are
small, and that the backgrounds for $A^{\prime}\to\mu^{+}\mu^{-}/h^{+}h^{-}$
are negligible.

  * •

For $60\,\mathrm{cm}<R<150\,\mathrm{cm}$ there will be enough activity in the
outer parts of the detector to veto such final states in searches for
invisible final states, but not enough information to reconstruct displaced
vertices. We do not include this region in our study.

In addition, an event must pass at least one of the trigger selections that
are the same as in Duerr:2020muu . In this work we further assume that photons
behave like electrons in the calorimeter-based triggers. We note that for all
$A^{\prime}$ masses accessible in a displaced vertex search at Belle II, the
calorimeter cluster triggers are very efficient in triggering on the multi-GeV
photon recoiling against the $A^{\prime}$.

Table 1: Kinematic selections used in our analysis.  cut on | value
---|---
decay vertex | (i) $-55\,\mathrm{cm}\leq z\leq 140\,\mathrm{cm}$
(ii) $17^{\circ}\leq\theta_{\text{lab}}\leq 150^{\circ}$
electrons | (i) both $p(e^{+})$ and $p(e^{-})>0.1\,\mathrm{GeV}$
(ii) opening angle of pair $>0.025$ rad
(iii) invariant mass of pair $m_{ee}>0.03\,\mathrm{GeV}$
$\mu,\pi,K$ | (i) both $p_{\text{T}}(\mu^{+})$ and $p_{\text{T}}(\mu^{-})>0.05\,\mathrm{GeV}$
(ii) $m_{ll}<0.480\,\mathrm{GeV}$ or $m_{ll}>0.520\,\mathrm{GeV}$
photons | (i) $E_{\text{lab}}\geq 0.5\,\mathrm{GeV}$
(ii) $17^{\circ}\leq\theta_{\text{lab}}\leq 150^{\circ}$

Finally the events need to fulfil the kinematic selection from Table 1. The
requirements on the decay vertex ensure a decay within the tracking detectors
of Belle II. For electron final states, we assume that a minimal invariant
mass selection of $m_{e^{+}e^{-}}\gtrsim 0.03\,\text{GeV}$ and a minimal
opening angle of $0.025\,\text{rad}$ will be required to reject the majority
of conversion backgrounds events; for the non-electron final states, we assume
that particle identification alone is sufficient to reject conversions to a
negligible level, but that a mass region around the $K_{S}^{0}$ mass is needed
to be cut to reject backgrounds from displaced $K_{S}^{0}$ decays into two
pions.

While these selections are motivated by the performance shown in Kou:2018nap ,
we note that a full study of all possible backgrounds is beyond the scope of
this work. In particular, the background for electron final states requires
careful studies of reconstruction performance of the Belle II experiment. In a
real analysis, energy resolution effects will slightly reduce the signal
efficiency. We ignore this effect. We also assume that potentially increasing
beam background has a negligible effect on both high momentum track resolution
and high energy photon resolution Kou:2018nap .

##  4 Results

Figure 2: Expected sensitivity assuming 2.3 events of the search for displaced
non-electron final states at Belle II in the $\epsilon-m_{A^{\prime}}$
parameter plane for an integrated luminosity of $50\,\mathrm{ab^{-1}}$.
Existing limits are taken from Bauer:2018onh .

In this section we split our results into (i) non-electron final states and
(ii) final states with electrons. For the sensitivity predictions of non-
electron final states we assume zero background after selections. For the case
of electrons we discussed sources of background which will not be fully
eliminated by the requirements above and show a number of different
sensitivities for different assumptions.

Figure 3: Expected sensitivity of the search for displaced electron final
states at Belle II in the $\epsilon-m_{A^{\prime}}$ parameter plane for an
integrated luminosity of $50\,\mathrm{ab^{-1}}$. In the top panel we show the
individual sensitivities corresponding to the regions in which
$0.2\,\mathrm{cm}<R<0.9\,\mathrm{cm}$ (dash-dotted),
$17\,\mathrm{cm}<R<60\,\mathrm{cm}$ (dashed) and the sensitivity for both
regions combined (solid) for 10 events and an angular cut $\theta_{ee}=0.025$.
In the bottom panel we show the sensitivity for 10 events and different
angular cuts $\theta_{ee}$ (left) and a varying number of events for
$\theta_{ee}=0.025$ (right). Existing limits are taken from Bauer:2018onh .

###  4.1 Sensitivity for non-electron final states

Our main result for the non-electron case is presented in Fig. 2, where we
show the expected sensitivity of Belle II to a signature with a single high
energy photon in association with displaced non-electron final states in the
$\epsilon-m_{A^{\prime}}$ parameter plane for an integrated luminosity of
$50\,\mathrm{ab^{-1}}$ (black solid line) and zero backgrounds. We observe
that this search is expected to cover a dark photon mass range
$2m_{\mu}\lesssim m_{A^{\prime}}\lesssim 1$ GeV.222While we include charged
kaon final states in our analysis, it is evident from this mass range that
they do not contribute significantly to the sensitivity, which is dominated by
muon and pion final states. We find that both regions
$0.2\,\mathrm{cm}<R<0.9\,\mathrm{cm}$ and $0.9\,\mathrm{cm}<R<17\,\mathrm{cm}$
contribute similarly to the sensitivity, while larger displacements do not add
additional sensitivity. In total we expect at most around 50 events for
$50\,\mathrm{ab^{-1}}$. As a result, Belle II will become sensitive to this
signature only with an integrated luminosity exceeding $2\,\mathrm{ab^{-1}}$.

Figure 4: Overall expected sensitivity of the displaced search at Belle II in
the $\epsilon-m_{A^{\prime}}$ parameter plane compared with other projections
assuming an integrated luminosity of $0.5\,\text{ab}^{-1}$ (dashed),
$2\,\text{ab}^{-1}$ (dashed-dotted), and $50\,\text{ab}^{-1}$ (solid).
Existing limits as well as future projections are taken from Bauer:2018onh .
The Belle II projection for the prompt $A^{\prime}$ decays is taken from
Kou:2018nap .

###  4.2 Sensitivity for electron final states

The electron final states are experimentally significantly more challenging
than the muon final states. Pair conversion events are expected at invariant
masses well below 1 MeV. We assume that an invariant mass selection of
$m_{ee}\gtrsim 30$ MeV is needed to reject the largest fraction of correctly
reconstructed pair conversions due to finite tracking resolution effects.
Additional selections on the opening angle $\theta_{ee}$ of the daughter
electrons are likely needed to increase the tracking efficiency and momentum
resolution. In the top panel of Fig. 3 we show the individual sensitivities
corresponding to the regions in which $0.2\,\mathrm{cm}<R<0.9\,\mathrm{cm}$,
$17\,\mathrm{cm}<R<60\,\mathrm{cm}$, and the sensitivity for both regions
combined for an angular cut $\theta_{ee}=0.025$. We observe that a large
fraction of the overall sensitivity is due to displacements below
$0.9\,\mathrm{cm}$, but also large displacements contribute. As outlined in
Sec. 3.3, we expect that additional backgrounds from misreconstructed pair
conversions $e^{+}e^{-}\to\gamma\gamma$ cannot be reduced to zero by the
selections above. Stronger angular cuts will further reduce backgrounds and we
show the sensitivity for different angular cuts $\theta_{ee}$ in Fig. 3
(bottom-left). For more optimistic and more pessimistic assumptions on the
background levels, we also show the contours for 2.3, 10, 23, and 100 signal
events after all selections in Fig. 3 (bottom-right). This corresponds to 90%
C.L. sensitivity assuming Poisson statistics for 0, and approximately 50, 300,
and 6000 background events in the signal region. As discussed in Sec. 3.3,
additional selections may be developed to reduce background from
misreconstructed pair conversions which in turn will reduce the signal
efficiency. Specifically the sensitivity for 10 events could also be read as
an additional reduction of signal efficiency by a factor of 0.23 to reach zero
background events.

###  4.3 Combined sensitivity

In Fig. 4 we show the combined sensitivity taking into all displaced final
states, assuming $\theta_{ee}=0.025$ and 10 events for the electron final
state as an optimistic estimate. In addition we show all relevant projections
of other future experimental facilities for comparison. We see that Belle II
not only covers a large region of parameter space which is currently still
unexplored, but is also uniquely sensitive in particular for dark photon
masses close to 1 GeV. As we do not expect any background events in this
region, the underlying experimental uncertainties are rather small and our
projection therefore correspondingly robust.

##  5 Discussion

A kinetically mixed dark photon is one of the simplest and most well motivated
options to extend the SM of particle physics and has attracted a huge amount
of interest over the past couple of years. In this letter we point out that
Belle II will be able to probe a significant part of the uncovered parameter
space by looking for events with a single high energy photon and a pair of
displaced charged particles. The non-electron final states are experimentally
rather straight forward; the electron final states require more detailed
experimental studies to reject potentially large backgrounds from
misreconstructed pair conversions. We find that while generally a large
integrated luminosity will be required to achieve a sufficient sensitivity,
Belle II may be able to start probing so far uncovered dark photon parameter
space in the near future.

###### Acknowledgements.

We would like to thank Patrick Foldenauer for comments on the draft and for
providing the limits from Bauer:2018onh  as well as updated data. We thank
Christopher Hearty for discussions during early stages of this work, and
Isabel Haide for comments on the manuscript. This work is funded by the
Deutsche Forschungsgemeinschaft (DFG) through Germany’s Excellence Strategy –
EXC 2121 “Quantum Universe” – 390833306 and the Helmholtz (HGF) Young
Investigators Grant No. VH-NG-1303. C.G.C. has been supported by the Alexander
von Humboldt Foundation during the first stage of this project.

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