# HARQ Delay Minimization of 5G Wireless Network with Imperfect Feedback

Weihang Ding1 and Mohammad Shikh-Bahaei1  1Centre for Telecommunications
Research, Department of Engineering, King’s College London, London WC2R 2LS,
UK.

###### Abstract

5G new radio (NR) technology is introduced to satisfy more demanding services.
Ultra Reliable Low Latency Communication (URLLC) requires very low delay
compared with the previous techniques. This is hard to achieve when hybrid
automatic repeat request (HARQ) is applied and especially when the feedback
channel is erroneous. In this work, we consider various delay components in
incremental redundancy (IR) HARQ systems and minimize the average delay by
applying asymmetric feedback detection (AFD) and find the optimal transmission
length for each transmission attempt. A M/G/1 queuing model is used in this
work to analyze the queuing delay in 5G NR when there are multiple uses in the
system. Numerical results show that significant performance gains and lower
outage probability can be achieved by applying AFD.

###### Index Terms:

Hybrid automatic repeat request, rate adaptation, asymmetric feedback
detection

##  I Introduction

The fifth generation (5G) wireless mobile networks are developed to achieve
substantial performance improvements [1]. Low latency and high reliability are
key requirements in most scenarios, particularly in Ultra-Reliable Low Latency
Communication (URLLC). Ideally, 1ms end-to-end latency and 99.999% reliability
are expected, but there are still some hurdles to work out. For example,
hybrid automatic repeat request (HARQ) acts as a bottleneck to the
optimization of user-plane latency.

Combining Forward error correction (FEC) and automatic repeat request (ARQ),
HARQ is a high-efficiency technique for data transmission, which performs much
better than ordinary ARQ in poor channel conditions. Based on the soft-
combining method at the receiver, HARQ can be divided into chase-combining
HARQ (CC-HARQ) and incremental redundancy HARQ (IR-HARQ). In CC-HARQ, the same
codeword is transmitted in different transmissions, while in IR-HARQ,
retransmissions include additional parity-check bits. We only consider IR-HARQ
in this work as coding gain is more significant in general wireless
communication systems. In the conventional IR-HARQ protocols, the decoding
state is released by the receiver using feedback. If the feedback is binary,
the transmitter is only able to know whether the decoding was successful,
hence the length and power of each transmission attempt is fixed [2, 3]. With
longer feedback, the transmitter can acquire more information about the
decoding states, and based on which, the transmitter might be able to
adaptively select some of the parameters of the subsequent transmission.
Adaptive system is a promising technique to enhance the performance of various
systems by adjusting the system configuration based on the system parameters
[4, 5, 6, 7, 8, 9]. Adaptive ARQ in cognative radio networks can increase the
utilization of the resources, and somehow reduce overall delay [9, 10, 11,
12]. Similarly, adaptive HARQ is widely studied based on different
transmission parameters, including rate adaptation [13], [14], power
adaptation [15], [16], and adaptive modulation and coding (AMC) schemes[17].

The feedback has to be prolonged to convey the full decoding state. In [2],
the relationship between feedback resolution and the overall throughput is
studied. The results show that if the feedback includes more than 8 bits, the
performance is very close to an ideal feedback HARQ system. However,
transmitting 8-bit feedback is still unrealistic in reality due to the high
cost of the feedback channel. When the quality of the feedback channel is low,
applying extra feedback bits will increase the feedback error rate and make
the error harder to analyse.

In most of the previous works, the feedback channel is assumed to be
deterministic whereas the error probability of the HARQ feedback can not be
made arbitrarily low in reality. A feedback error rate of 1% is reasonable in
LTE [18] and 0.1%-1% in 5G NR[19]. With limited resource [20, 21], unreliable
feedback will impair the performance of the HARQ system. There are only a few
contributions on HARQ with unreliable feedback. In [22], [23], the feedback
channel is modeled as a Binary Symmetric Channel (BSC). In [24], a notion of
guaranteeable delay is introduced while the feedback channel is an erasure
channel. Different from conventional symmetric feedback detection, asymmetric
feedback detection (AFD) is introduced in [25] to provide a better protection
to negative acknowledgement (NACK) without assigning extra resources. In our
previous work [26], we apply AFD to rate-adaptive HARQ and find the optimal
thresholds and transmission rates to maximize throughput.

In this work, we study the long-term average delay of a HARQ process with
imperfect feedback. We consider multiple delay components including queuing
delay, transmission delay, and processing delay. We apply both AFD and rate-
adaptive HARQ in this work to minimize the long-term average delay of the
system. Another advantage of AFD is that it reduces the outage probability, so
we also study the performance of the systems with outage probability
constraints. Simulation results show that our proposed scheme can
significantly reduce the long-term average delay compared with traditional
methods.

##  II System model

The round-trip delay (RTD) of the system is the time it takes for the packet
to be sent and acknowledged. Among these delay components, the propagation
delay and the transmission delay of the feedback are negligible compared with
the other components. Therefore, the overall RTD can be expressed as the
summation of the queuing delay, the transmission delay of the packet, and the
processing delay at the receiver:

$T_{tot}=T_{queue}+T_{tran}+T_{proc}.$ (1)

Once a HARQ round begins, the source encodes the original packet of $N_{b}$
bits using a designated code rate. The feedback is generated by the
destination according to the decodability of the packet. If the transmitter
observes a NACK, it will transmit a certain amount of redundancy in the
subsequent transmission attempt. The current HARQ round can only terminate as
long as an ACK is observed, or the number of transmissions reached the maximum
limit $M$.

###  II-A Channel model

It is assumed that the packets are transmitted through block-fading channels
with additive Gaussian noise. The $k$-th received symbol can be written as:

$\bm{y}_{k}=\sqrt{\text{SNR}_{d}}h_{k}\bm{x}_{k}+\bm{z}_{k},$ (2)

where $\bm{x_{k}}$, $\bm{y_{k}}$, and $\bm{z_{k}}$ are the $k$-th transmitted
symbol, received symbol and additive noise respectively, $h_{k}$ is the
instantaneous channel fading coefficient, SNRd is the average signal to noise
ratio of the downlink channel.

We assume that the time between two transmission attempts is significantly
larger than the coherence time, so $h_{k}$ is independent identically
distributed and remains constant during a single transmission attempt. The
transmitter has no knowledge about $h_{k}$ and is not able to predict it using
the previous information before the transmission. $|h_{k}|$ is Rayleigh
distributed with unity power gain $\mathbb{E}[|h_{k}|^{2}]=1$ and zero mean.
$\bm{z}_{k}$ is a complex vector of zero mean, and unitary-variance Gaussian
variables representing the noise. It is assumed that SNRd remains constant and
is known by the transmitter. The decoding is based on the received sequence
$\bm{y}$, and uses maximum likelihood decoding.

###  II-B Queuing model

It is assumed that the packets arriving at the gNB follow a Poisson
distribution. The system can be modeled as a M/G/1 queue, since packets are
transmitted one-by-one, and the transmission time follows a general
distribution based on the length of different transmission attempts.

Unlike LTE, 5G NR supports multiple subcarrier spacings from 15kHz up to
240kHz. Different subcarrier spacings correspond to different slot durations,
as shown in Table I.

TABLE I: 5G NR subcarrier numerology $\mu$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
Sub-carrier spacing ($\mathrm{kHz}$) | 15 | 30 | 60 | 120 | 240
Slot duration ($\mathrm{\SIUnitSymbolMicro s}$) | 1000 | 500 | 250 | 125 | 62.5
Slots per subframe | 1 | 2 | 4 | 8 | 16
Symbols per slot | 14
OFDM symbol duration ($\mathrm{\SIUnitSymbolMicro s}$) | 66.67 | 33.33 | 16.67 | 8.33 | 4.17
Supported for data | Yes | Yes | Yes | Yes | No

The transmission time (service time) of the $i$-th transmission attempt in a
HARQ round $T_{i}$ can be calculated as $T_{i}=\frac{n_{i}}{14}T_{slot}$,
where $n_{i}$ is an integer denoting the number of slots required to transmit
this packet.

##  III Asymmetric feedback detection

If the feedback channel is perfect, an outage occurs only when the receiver is
still unable to recover the message after $M$ transmission attempts. If an
outage occurs, this transmission will be handed over to the higher layer
protocols and a new HARQ round will be initialized. Outages can cause
significant system performance degradation. For normal HARQ processes, an
outage probability of 0.1%-1% is tolerable [19]. If the feedback channel is
imperfect, NACK errors will cause outages, and ACK errors will only lead to
unnecessary retransmissions.

ACKNACK1

(a) SFD

$\alpha$ACKNACK1

(b) AFD

Figure 1: Constellation comparison between SFD and AFD. Figure 2: Comparison
between SFD and AFD

The feedback is binary quantized based on a feedback detection threshold. In
symmetric feedback detection (SFD), the feedback detection threshold is set as
Fig. 1(a) so that the areas of ACK and NACK on the constellation are equal.
However, in the case of AFD, the area of NACK on the constellation should be
larger than it of ACK to reduce NACK error rate, which is illustrated in Fig.
1(b). $\alpha\in\mathbb{R}$ is an index defined by the ratio of the minimum
Euclidean distance from the asymmetric decoding threshold to the origin on the
constellation diagram over the amplitude. The amplitude of the signal is
normalized in Fig. 1. This AFD scheme is distance-based. Other schemes based
on other parameters (such as phase shift) are also studied, but they result in
poorer performances. We assume that the feedback is transmitted with one
binary symbol through an additive Gaussian noise channel (AWGN). For the
$i$-th transmission, the ACK error rate $P_{A,i}$ and the NACK error rate
$P_{N,i}$ can be computed by:

$P_{N,i}=\frac{1}{2}erfc\left((1+\alpha_{i})\sqrt{\text{SNR}_{f}}\right),$ (3)
$P_{A,i}=\frac{1}{2}erfc\left((1-\alpha_{i})\sqrt{\text{SNR}_{f}}\right),$ (4)

where SNRf is the signal-to-noise ratio of the feedback channel and
$\alpha_{i}$ is the asymmetric detection index of the $i$-th feedback.

##  IV Problem formulation

At the receiver, a message can be considered decodable if the accumulated
mutual information (MI) is greater than $N_{b}$. The probability that the
decoding fails after the $i$-th transmission attempt $P_{i,f}$ can be written
as:

$\displaystyle P_{i,f}$
$\displaystyle=\mathbb{P}\left\\{\sum_{m=1}^{i}I_{m}<N_{b}\right\\},$ (5)

where $I_{m}$ is the normalized MI in the $m$-th transmission defined as:

$I_{m}=n_{m}T_{slot}W\log(1+|h_{m}|^{2}\text{SNR}_{d}),$ (6)

where $W$ is the available bandwidth.

To calculate $P_{i,f}$, we need to find the probability density function (pdf)
of the accumulated MI. Let $Z_{i}=\sum_{m=1}^{i}I_{l}$ denote the summation of
MI in the first $i$ transmissions. $P_{i,f}$ can be calculated as
$P_{i,f}=\int_{0}^{1}f(z_{i})dz_{i}$, where $f(z_{i})$ is the pdf of $Z_{i}$
computed through $I_{1},\dots,I_{i}$: $f(z_{i})=f(I_{1})*\dots*f(I_{i})$. To
get rid of convolution, the accumulated MI can be approximated by a Gaussian
variable which is accurate over low and moderate SNR regimes[27]. The
approximated pdf of accumulated MI $Z_{i}$ can be written as:

$f(Z_{i})=\frac{1}{\sqrt{2\pi\sum_{l=1}^{i}n_{l}T_{slot}W\sigma_{I}^{2}}}e^{-\frac{(Z_{k}-\sum_{l=1}^{i}n_{l}T_{slot}W)^{2}}{2\sum_{l=1}^{i}n_{l}T_{slot}W\sigma_{I}^{2}}},$
(7)

where $\bar{I}$ is the mean value of MI and $\sigma_{I}^{2}$ is the variance
of MI given by [28]:

$\bar{I}=\log_{2}(e)e^{\frac{1}{\text{SNR}_{d}}}\int_{1}^{\infty}t^{-1}e^{-\frac{t}{\text{SNR}_{d}}}dt$
(8)

$\displaystyle\sigma_{I}^{2}=\frac{2}{\text{SNR}_{d}}\log_{2}^{2}(e)e^{\frac{1}{\text{SNR}_{d}}}G^{4,0}_{3,4}\left(1/\text{SNR}_{d}|_{0,-1,-1,-1}^{0,0,0}\right)-\bar{I}^{2},$
(9)

where $G^{m,n}_{p,q}\left({}^{a_{1},\dots,a_{p}}_{b_{1},\dots,b_{q}}|z\right)$
is the Meijer G-function.

Define by $P_{i}$ the probability that the $i$-th transmission occurs in a
HARQ round, $P_{i}$ can be calculated via (10), where we define $P_{N,0}=0$.
Accordingly, the overall outage probability $P_{out}$ can be calculated as:

$P_{i}=\left\\{\begin{aligned} &1,\>\>\>\>\>\>\>\>\text{if i=1}\\\
&\left(P_{i-1,f}\prod_{j=1}^{i-1}(1-P_{N,j})+\sum_{j=1}^{i-1}\left((P_{i-j-1,f}-P_{i-j,f})\prod_{m=0}^{i-j-1}(1-P_{N,m})\prod_{m=i-j}^{i-1}P_{A,m}\right)\right),\>\>\>\>\>\text{if
i=2\ldots M}.\\\ \end{aligned}\right.$ (10)

$P_{out}=1-\left(P_{1,s}+\sum_{i=2}^{M}\left((P_{i-1,f}-P_{i,f})\prod_{j=1}^{i-1}(1-P_{N,j})\right)\right).$
(11)

The average transmission time (service time) of the effective transmissions
can be calculated as:

$\mathbb{E}[T_{tran}]=\frac{\sum_{i=1}^{M}T_{i}P_{i-1,f}}{\sum_{i=1}^{M}P_{i-1,f}}.$
(12)

The total arrival rate (including the packets and the retransmissions) is
$\lambda_{tot}=\lambda_{0}\frac{\sum_{i=1}^{M}P_{i}}{1-P_{out}}$. Therefore,
the average waiting and service time in the queue can be calculated as:

$\displaystyle\mathbb{E}[T_{queue}]=$
$\displaystyle\frac{\lambda_{tot}\mathbb{E}[T_{tran}^{2}]}{2(1-\lambda_{tot}\mathbb{E}[T_{tran}])}$
(13) $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{M}T_{i}^{2}P_{i}}{2\left(\frac{1-P_{out}}{\lambda_{0}}-\sum_{i=1}^{M}T_{i}P_{i}\right)}.$

The average service rate has to be greater than the packet arrival rate to
make the queuing system stable. Therefore, the follow requirement should
always be satisfied:

$\frac{\sum_{i=1}^{M}T_{i}P_{i-1,f}}{\sum_{i=1}^{M}P_{i-1,f}}\times\frac{\lambda_{0}\sum_{i=1}^{M}P_{i}}{1-P_{out}}\leq
1.$ (14)

According to [29], the UE has to provide corresponding HARQ feedback within
$k$ slots after the transmission, where $k$ is specified by the PDSCH-to-
HARQ_feedback timing indicator field in the downlink control information (DCI)
format. If we consider the propagation time to be negligible, the total RTT
for each transmission is upper bounded by:

$T_{tot}\leq T_{queue}+T_{tran}+kT_{slot}.$ (15)

The mean value of the upper bound can be written as:

$\displaystyle\bar{T}_{ub}=$
$\displaystyle\frac{\sum_{i=1}^{M}T_{i}^{2}P_{i}}{2\left(\frac{1-P_{out}}{\lambda_{0}}-\sum_{i=1}^{M}T_{i}P_{i}\right)}$
(16)
$\displaystyle+\frac{\sum_{i=1}^{M}T_{i}P_{i-1,f}}{\sum_{i=1}^{M}P_{i-1,f}}+kT_{slot}.$

The average delay of the HARQ process can be written in terms of the number of
RTTs:

$\mathbb{E}[D]=\frac{M+1-\sum_{i=1}^{M-1}P_{i,s}}{1-P_{out}}\times\bar{T}_{ub}.$
(17)

In our previous work [26], we have shown that the optimal performance of the
HARQ process is achieved when the outage probability is considerably higher
than expected. Therefore, the minimum delay with tight outage probability
limits $\epsilon$ must be achieved when the actual outage probability
$P_{out}=\epsilon$. Then, the optimization problem can be formulated as:

$\displaystyle\min_{n_{1},\dots,n_{M},\alpha_{1}\dots\alpha_{M-1}}\>\>\frac{M+1-\sum_{i=1}^{M-1}P_{i,s}}{1-\epsilon}\times$
(18)
$\displaystyle\left(\frac{\sum_{i=1}^{M}T_{i}^{2}P_{i}}{2\left(\frac{1-\epsilon}{\lambda_{0}}-\sum_{i=1}^{M}T_{i}P_{i}\right)}+\frac{\sum_{i=1}^{M}T_{i}P_{i-1,f}}{\sum_{i=1}^{M}P_{i-1,f}}+kT_{slot}\right)$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\rm{subject\>to:}\>\>P_{out}\leq\epsilon$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\frac{\sum_{i=1}^{M}T_{i}P_{i-1,f}}{\sum_{i=1}^{M}P_{i-1,f}}\times\frac{\lambda_{0}\sum_{i=1}^{M}P_{i}}{1-\epsilon}\leq
1$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>n_{i}\in\mathbb{N}^{+}\cup
n_{i}\leq n_{max},\>\>i=1,2,\dots,M$
$\displaystyle\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\alpha_{k}\in\mathbb{R},\>\>k=1,2,\dots,M-1.$

To solve this problem, we need to consider these two sets of variables
$\bm{n}=\\{n_{1},\dots,n_{M}\\}$ and
$\bm{\alpha}=\\{\alpha_{1},\dots,\alpha_{M-1}\\}$ separately. In reality, $M$
and $n_{max}$ are not very high values, making the size of the searching space
of $\bm{n}$ reasonable. Once $\bm{n}$ is fixed, we can use projected gradient
descent (PGD) to find the optimal $\bm{\alpha}$.

##  V Numerical results

In this section, we compare the long-term average day of our proposed AFD
scheme and conventional schemes. Some of the simulation parameters are listed
in Table II.

TABLE II: Simulation parameters Parameter | Value
---|---
Packet arrival rate $\lambda_{0}$ | 200 packets/$\mathrm{s}$
PDSCH-to-HARQ_feedback timing k | 1
Information length $N_{b}$ | 2816 bits
Slot duration $T_{slot}$ | 125 $\mathrm{\SIUnitSymbolMicro s}$
Number of transmissions in each HARQ round $M$ | 4
Figure 3: The long-term average delay of our proposed scheme compared with
conventional SFD when there is no outage probability limits.

First, we assume that there is no outage probability limits. We compare our
proposed AFD scheme with conventional SFD scheme in terms of long-term average
delay. Both schemes apply rate adaptation, and even then we can see that our
AFD scheme significantly outperforms SFD, especially when the quality of the
feedback channel is low. When the downlink channel is of low quality, the
average delay of SFD scheme increases sharply, while the impact of low-quality
downlink channel on AFD is not significant compared to the system with perfect
feedback channel. Compared with a system with perfect HARQ feedback, by
applying AFD, we can achieve almost the same performance at SNR${}_{f}=0$dB,
and only slightly higher delay at SNR${}_{f}=-5$dB.

Figure 4: The minimum achievable long-term average delay with outage
probability limits when the SNR${}_{f}=0$dB.

We also find the minimum achievable long-term average delay under strict
outage probability limits when SNR${}_{f}=0$dB (see Fig. 4). When 5% outage
probability is required, the performance suffers slightly, but still better
than the SFD scheme even without the limits. However, when the outage
probability is required to be under 1%, despite it is still achievable with
our AFD scheme, the delay is much higher than expected.

##  VI Conclusion

In this work, we optimize the long-term average delay of the HARQ process with
imperfect feedback. We analyze different delay components and jointly optimize
them by applying AFD and rate-adaptation. The results show that with our
proposed scheme, the overall delay of the HARQ process can be significantly
reduced, and the impact of low-quality feedback channel can be mitigated. In
addition, we can achieve much lower outage probability with AFD at a cost of
overall delay.

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# Measuring the Weak Mixing Angle in the DUNE Near Detector Complex

André de Gouvêa  Department of Physics & Astronomy, Northwestern University,
Evanston, IL 60208, USA  Pedro A. N. Machado  Theoretical Physics Department,
Fermilab, P.O. Box 500, Batavia, IL 60510, USA  Yuber F. Perez-Gonzalez
Department of Physics & Astronomy, Northwestern University, Evanston, IL
60208, USA  Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia,
IL 60510, USA  Colegio de Física Fundamental e Interdisciplinaria de las
Américas (COFI), 254 Norzagaray street, San Juan, Puerto Rico 00901  Zahra
Tabrizi  Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas
(UNICAMP), Rua Sérgio Buarque de Holanda, 777, Campinas, SP, 13083-859, Brazil

###### Abstract

The planned DUNE experiment will have excellent sensitivity to the vector and
axial couplings of the electron to the $Z$-boson via precision measurements of
neutrino–electron scattering. We investigate the sensitivity of DUNE-PRISM, a
movable near detector in the direction perpendicular to the beam line, and
find that it will qualitatively impact our ability to constrain the weak
couplings of the electron. We translate these neutrino–electron scattering
measurements into a determination of the weak mixing angle at low scales and
estimate that, with seven years of data taking, the DUNE near-detector can be
used to measure $\sin^{2}\theta_{W}$ with about 2% precision. We also discuss
the impact of combining neutrino–electron scattering data with neutrino
trident production at DUNE-PRISM.

###### pacs:

Valid PACS appear here

††preprint: FERMILAB-PUB-19-623-T, NUHEP-TH/19-17

The standard model of particle physics (SM) is a quantum field theory with a
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ gauge symmetry, corresponding to
the color, weak-isospin and hypercharge interactions, respectively, along with
a set of fermion and boson fields describing the particles observed in nature.
Free SM parameters – the gauge and Yukawa couplings, together with the scalar
potential parameters – need to be determined by comparing the results of
theoretical computations to a finite set of the experimental measurements.

The weak mixing angle $\theta_{W}$ (or, more precisely, its sine-squared,
$\sin^{2}\theta_{W}$) parameterizes several measurable quantities: the mass
ratio of the weak gauge bosons, some weak-interaction cross sections, and
parity-violating observables. It is a crucial ingredient of the _electroweak
precision observables_ , a set of experimental observables designed to test
the SM internal consistency.

The exact definition of the weak mixing angle depends on the renormalization
scheme, that is, the convention of which quantities are taken as input and
which are derived from these inputs, along with the recipe for handling
quantum corrections. As quantum corrections are relevant, $\sin^{2}\theta_{W}$
depends on the scale at which it is being measured. For example, in the
modified minimal subtraction scheme Erler and Ramsey-Musolf (2005); Erler and
Ferro-Hernández (2018), $\overline{\scalebox{0.8}{\text{MS}}}$, the weak
mixing angle is

$\sin^{2}\theta_{W}(\mu)\equiv\frac{g^{\prime 2}(\mu)}{g^{2}(\mu)+g^{\prime
2}(\mu)},$ (1)

where $g$ and $g^{\prime}$ are the $SU(2)_{L}$ and $U(1)_{Y}$ gauge coupling
constants, respectively, and $\mu$ is the scale of the physical process under
consideration. The SM predicts, under a specific renormalization scheme, a
unique scale dependence for $\sin^{2}\theta_{W}$. This dependence has been
confirmed by precise measurements at very different energy scales, including
atomic parity violation, electron-proton scattering, Möller scattering,
neutrino–nucleus and neutrino–electron scattering, electron deep-inelastic
scattering, and the $Z$\- and $W$-boson masses (see Ref. Tanabashi  _et al._
(2018) for a comprehensive review).

The NuTeV result Zeller  _et al._ (2002), the most precise measurement of
$\sin^{2}\theta_{W}$ using neutrino scattering, stands out from the other
measurements. Considering the ratios of neutral-current to charged-current and
neutrino–nucleus to antineutrino–nucleus cross sections, they find
$\sin^{2}\theta_{W}=0.2407\pm 0.0016$ (in the
$\overline{\scalebox{0.8}{\text{MS}}}$ scheme) at an average energy scale
$\langle\mu\rangle\simeq 4.5$ GeV. This measurement deviates from the SM
expectation anchored by the more precise measurements at LEP Schael  _et al._
(2006) at the $3\sigma$ level. Effects inherent to the intricacies of
neutrino-nucleus scattering, potentially unaccounted for or only partially
accounted for by the collaboration, have been identified as candidate sources
for the discrepancy Pumplin  _et al._ (2002); Kretzer  _et al._ (2004); Sather
(1992); Rodionov  _et al._ (1994); Martin  _et al._ (2004); Londergan and
Thomas (2003); Bentz  _et al._ (2010); Gluck  _et al._ (2005); Kumano (2002);
Kulagin (2003); Brodsky  _et al._ (2004); Hirai  _et al._ (2005); Miller and
Thomas (2005); Cloet  _et al._ (2009); Diener  _et al._ (2004); Arbuzov  _et
al._ (2005); Park  _et al._ (2009); Diener  _et al._ (2005); Dobrescu and
Ellis (2004). A definitive answer remains elusive. Regardless, it stands to
reason that other precise measurements of $\sin^{2}\theta_{W}$ using neutrino
scattering will help shed light on the situation.

Next-generation neutrino experiments like LBNF-DUNE Acciarri  _et al._ (2015)
and T2HK Abe  _et al._ (2011) will include very intense neutrino beams with
energies that range from several hundred MeV to several GeV. The
neutrino–nucleus scattering cross sections, at these energies, have large
uncertainties due to nuclear and non-perturbative effects Alvarez-Ruso  _et
al._ (2018), making it very challenging to use them to infer
$\sin^{2}\theta_{W}$. Neutrino–electron scattering, on the other hand,
provides a more promising environment Conrad  _et al._ (2005); de Gouvêa and
Jenkins (2006); Agarwalla and Huber (2011); Conrad  _et al._ (2014); Adelmann
_et al._ (2014). Even in this case, however, one still needs to address
significant challenges. First, the cross section for neutrino–electron
scattering is three orders of magnitude smaller than that for neutrino-nucleus
scattering, translating into poor statistics in most neutrino experiments.
Second, while the neutrino–electron cross section depends mostly on
$\sin^{2}\theta_{W}$, the neutrino beam originates from the in-flight decay of
charged mesons produced by high-energy protons hitting a fixed target. First-
principles computations of the meson production rate and kinematics are not
possible and one must rely on phenomenological models and experimental data;
uncertainties on the overall neutrino flux and energy distribution are at the
$5\%$ to $15\%$ level Abe  _et al._ (2013); Aliaga  _et al._ (2016); Marshall
_et al._ (2019).

Near detector complexes are designed to circumvent some of the large
uncertainties in the flux and cross sections and allow precision measurements
of neutrino oscillations Marshall  _et al._ (2019). DUNE-PRISM Pickering
(2019), currently part of the LBNF-DUNE proposal, is a near detector that is
capable of moving in the direction perpendicular to the neutrino-beam axis.
Although the neutrino flux has prohibitively large uncertainties, the ratios
of on-axis to off-axis fluxes are dictated only by meson-decay kinematics and
thus are much better understood. Therefore, measurements of the neutrino-
electron-scattering spectrum at different off-axis positions should allow an
unprecedented measurement of the weak mixing angle with neutrinos.

In general terms, the neutrino–electron scattering cross-section depends on
the vector and axial couplings, $g_{V}$ and $g_{A}$, between the $Z$-boson and
the electron (see CHARM-II Vilain  _et al._ (1994), LSND Auerbach  _et al._
(2001) and TEXONO Deniz  _et al._ (2010)). We will estimate the DUNE-PRISM
sensitivity to such parameters via neutrino-electron scattering data. A hidden
but very safe assumption is that the cross-section depends only on the
neutrino left-handed coupling to the $Z$-boson. The reason for this is that
all neutrinos and antineutrinos used in neutrino scattering are produced in
charged-current processes ($\pi^{+}\to\mu^{+}\nu_{\mu}$, $n\to
p\,e\,\bar{\nu}_{e}$, $D_{s}\to\tau^{+}\nu_{\tau}$, etc) and are, to a very
good precision, 100% polarized. Lepton-collider data, combined with those from
neutrino–electron scattering, for example, can be used to determine the right-
handed-coupling of neutrinos to the $Z$-boson Carena  _et al._ (2003).

The differential cross section for a neutrino with flavor $\alpha=e,\mu,\tau$
to scatter off an electron at rest is

$\displaystyle\frac{d\sigma}{dE_{R}}$
$\displaystyle=\frac{2G_{F}^{2}m_{e}}{\pi}\left\\{g_{1}^{2}+g_{2}^{2}\left(1-\frac{E_{R}}{E_{\nu}}\right)^{2}-g_{1}g_{2}\frac{m_{e}E_{R}}{E_{\nu}^{2}}\right\\}$
$\displaystyle\simeq 1.72\times
10^{-41}\left\\{g_{1}^{2}+g_{2}^{2}\left(1-\frac{E_{R}}{E_{\nu}}\right)^{2}\right\\}\frac{{\rm
cm}^{2}}{\rm GeV},$ (2)

where $G_{F}$ is the Fermi constant, $E_{\nu}$ is the incoming neutrino
energy, and $m_{e}$ and $E_{R}$ are the electron mass and recoil kinetic
energy, respectively.

The couplings $g_{1}$ and $g_{2}$ depend on the neutrino flavor and can be
written in terms of $g_{V}$ and $g_{A}$; thus, they can be expressed in terms
of $\sin^{2}\theta_{W}$, see Table 1. More generally, if $g_{V}$ and $g_{A}$
are considered to be free parameters, they can be independently extracted from
the recoil-electron energy spectrum so data from DUNE-PRISM are expected to
constrain nontrivial regions in the $g_{V}\times g_{A}$ plane.

Table 1: Couplings $g_{1}$ and $g_{2}$ (see Eq. (2)) as a function of the electron–$Z$-boson couplings $g_{V}$ and $g_{A}$, for each neutrino flavor, along with the corresponding SM value. $s^{2}_{W}\equiv\sin^{2}\theta_{W}$.  $\nu_{\alpha}$ | $g_{1}$ |  $g_{1}$(SM) | $g_{2}$ |  $g_{2}$(SM)
---|---|---|---|---
$\nu_{e}$ | $1\\!+\\!(g_{V}\\!+\\!g_{A})/2$ | $1/2\\!+\\!s_{W}^{2}$ | $(g_{V}\\!-\\!g_{A})/2$ | $s_{W}^{2}$
$\nu_{\mu,\tau}$ | $(g_{V}\\!+\\!g_{A})/2$ | $-1/2\\!+\\!s^{2}_{W}$ | $(g_{V}\\!-\\!g_{A})/2$ | $s^{2}_{W}$
$\bar{\nu}_{e}$ | $(g_{V}\\!-\\!g_{A})/2$ | $s^{2}_{W}$ | $1\\!+\\!(g_{V}+g_{A})/2$ | $1/2\\!+\\!s_{W}^{2}$
$\bar{\nu}_{\mu,\tau}$ | $(g_{V}\\!-\\!g_{A})/2$ | $s^{2}_{W}$ | $(g_{V}\\!+\\!g_{A})/2$ | $-1/2\\!+\\!s^{2}_{W}$

Strictly speaking, the neutrino–electron cross section is also subject to
quantum corrections that will introduce additional dependence on $Q^{2}\equiv
2E_{R}m_{e}$ Tomalak and Hill (2019); Hill and Tomalak (2019). Kinematics
dictates that the maximum recoil energy is approximately $E_{R}^{\rm
max}\simeq E_{\nu}-m_{e}/2$. Due to kinematics and the energy profile of
DUNE’s neutrino flux, most electron recoil events will lie within $0.2\lesssim
E_{R}\lesssim 10$ GeV. Therefore, the $Q^{2}$ values accessible to DUNE are,
roughly, in the range $(10-100~{}{\rm MeV})^{2}$, where loop corrections to
$\sin^{2}\theta_{W}$ have little scale dependence Tanabashi  _et al._
(2018)111We have checked numerically that flavor-dependent electroweak
corrections do not change our results at the 1$\sigma$ level.. Thus, by
analyzing the $Q^{2}$ distribution in detail, the couplings in Eq. (2) can be
interpreted as the renormalized couplings in the
$\overline{\scalebox{0.8}{\text{MS}}}$ scheme at an average scale $\langle
Q^{2}\rangle=(55~{}{\rm MeV})^{2}$.

Assuming the SM, the cross section for $\nu_{\mu}-e$-scattering is

$\frac{d\sigma}{dE_{R}}\propto\left(\frac{1}{4}-\sin^{2}\theta_{W}\right)+\sin^{4}\theta_{W}\left(2-\frac{2E_{R}}{E_{\nu}}+\frac{E_{R}^{2}}{E_{\nu}^{2}}\right).$
(3)

Since $\sin^{2}\theta_{W}$ is close to 1/4, the first term is suppressed
relative to the second one. This implies that the value of
$\sin^{2}\theta_{W}$, to leading order, modifies the overall normalization of
the $\nu_{\mu}-e$-scattering cross section and the effect of changing
$\sin^{2}\theta_{W}$ is nearly degenerate with that of changing the overall
normalization of the $\nu_{\mu}$ flux. The situation is different for
$\nu_{e}-e$ and $\bar{\nu}_{e}-e$ scattering; $\sin^{2}\theta_{W}$ has a
significant impact on the shape of the recoil-electron energy distributions.
It turns out, unfortunately, that, at DUNE, the neutrino flux is dominated by
$\nu_{\mu}$ and the $\nu_{e}$ contribution is relatively small, around a few
percent.

In this context, DUNE-PRISM is expected to provide nontrivial information. In
accelerator neutrino experiments, the $\nu_{\mu}$ comes predominantly from the
two-body decay $\pi^{+}\to\mu^{+}\nu_{\mu}$ (and $K^{+}\to\mu^{+}\nu_{\mu}$,
to a lesser extent) while the $\nu_{e}$ comes from the three-body decays of
kaons and muons. For the same parent energy, the flux of $\nu_{e}$ has a
larger angular spread than that of $\nu_{\mu}$ so the off-axis $\nu_{e}$ to
$\nu_{\mu}$ flux ratio is larger than the on-axis one.

To estimate how well DUNE-PRISM can contribute to the precision electroweak
physics program, we compute the sensitivity to $\sin^{2}\theta_{W}$ for both
on-axis and off-axis runnings. For concreteness, we assume seven years of data
taking equally divided between neutrino and antineutrino modes. We assume a 75
ton fiducial mass liquid argon time projection chamber (LArTPC) and a 1.2 MW
proton beam, as described in the DUNE Conceptual Design Report Acciarri  _et
al._ (2015). For the off-axis configuration, we assume the near detector will
take data at seven different positions, half of the time on-axis and half of
the time equally divided in the off-axis positions. The detector is assumed to
be 574 m away from the source in the beam axis direction while its transverse
distances to the beam axis are $6N$ meters, $N=0,\ldots 6$. The detector
experiences at each position a flux that is approximately $10N$ mrad off-axis,
respectively. Fig. 1 depicts the ratio of the number of events expected from
$\nu_{e}-e$ and $\bar{\nu}_{e}-e$-scattering to that of
$\nu_{\mu}-e$-scattering, in neutrino-mode running, as a function of the off-
axis distance. As expected, the relevance of the $\nu_{e}$ and
$\bar{\nu}_{e}$-initiated events grows significantly with the off-axis angle.
Note that, while the flux ratio is of order a few percent, the
$\nu_{e}-e$-scattering cross section is larger than the $\nu_{\mu}-e$ so, even
on-axis, the $\nu_{e}$ contribution is of order 10%.

Figure 1: Ratio of the number of events expected from
$\nu_{e}-e+\bar{\nu}_{e}-e$-scattering to that of $\nu_{\mu}-e$-scattering, in
neutrino-mode running, as a function of the off-axis distance.

To account for the energy-dependent neutrino-flux uncertainties and the
correlations between the fluxes at different off-axis angles, we make use of a
covariance matrix spanning all DUNE-PRISM positions and neutrino flavors,
derived from detailed simulations of hadron production in the beam target
followed by magnetic-horn focusing of charged particles Pickering . The
binning is performed in $E_{e}\theta_{e}^{2}$, where $E_{e}=E_{R}+m_{e}$ is
the total electron energy and $\theta_{e}$ is the electron scattering angle
relative to the beam direction (see Supplemental Material for details.). We
consider a threshold kinetic energy $E_{R}>50~{}$MeV and perform the analysis
in the range $(0.05<E_{R}<20)$ GeV.

The main backgrounds for neutrino–electron scattering are charged-current
quasi-elastic (CCQE) $\nu_{e}$-scattering events, $\nu_{e}A\to
e^{-}A^{\prime}$ and mis-identified $\pi^{0}$ events with no detectable
hadronic activity, $\nu A\to\nu\pi^{0}A$. Although the $\nu_{e}$ flux is only
a few percent of the total neutrino flux, the CCQE cross section is over 1000
times larger than that for neutrino–electron scattering. We simulate these
backgrounds using the NuWro event generator Golan  _et al._ (2012), and allow
a 10% normalization uncertainty of both of them. We cut any event which have
at least one protons with kinetic energy above 50 MeV. For the $\pi^{0}$
background, we also require one photon to be soft, below 30 MeV, to be
accepted.In principle, if the photons are sufficiently collinear, the two
showers could be mis-identified as an electron event. As the minimum photon-
photon opening angle is $\theta>15.5^{\circ}({\rm GeV}/E_{\pi^{0}})$, it is
unlikely that this poses a background and therefore we have neglected it.

Kinematics limit $E_{e}\theta_{e}^{2}<2m_{e}$ for neutrino–electron scattering
and thus we bin on $E_{e}\theta_{e}^{2}$ to improve background rejection.
LArTPCs have an exquisite angular resolution, of order $1^{\circ}$, for
electromagnetic showers Acciarri  _et al._ (2015). In the Supplemental
Material, we show how angular resolution affects the $E_{e}\theta_{e}^{2}$
spectrum and the sensitivity to $\sin^{2}\theta_{W}$.

Fig. 2 depicts the DUNE sensitivity to the vector and axial couplings, $g_{V}$
and $g_{A}$, in the on-axis LArTPC (dashed green) or the DUNE-PRISM
configuration (dark-blue). For comparison, we include existing measurements
from CHARM-II Vilain  _et al._ (1994) (gray), LSND Auerbach  _et al._ (2001)
(dotted light-brown) and TEXONO Deniz  _et al._ (2010) (dot-dashed light-
violet). Both the DUNE on-axis and CHARM-II measurements suffer from a four-
fold degeneracy; this is a consequence of the fact that the neutrino flux in
both these experiments is dominated by $\nu_{\mu}$. There is an exact
degeneracy in the differential cross section for $\nu_{\mu}-e$ scattering
under the transformations

$(g_{V},g_{A})\to(g_{A},g_{V})\,\,\,\text{ and
}\,\,\,(g_{V},g_{A})\to(-g_{V},-g_{A}),$ (4)

see Eq. (2) and Table 1, and hence an experiment with a pure $\nu_{\mu}$ beam
is intrinsically limited. The TEXONO experiment measured electron recoils from
electron anti-neutrinos produced in a nuclear reactor. The scattering cross
section, in this case, is proportional to $3g_{1}^{2}+g_{2}^{2}$, which
defines an oblique ellipse in the $(g_{V},g_{A})$ plane centered at
$(-0.5,-0.5)$. The TEXONO result in Fig. 2 reflects this fact, up to effects
related to information on the recoil energy spectrum. The LSND measurement can
also be understood by noticing that the flux of neutrinos consists of
$\nu_{\mu}$, $\bar{\nu}_{\mu}$, and $\nu_{e}$ with well-characterized energy
spectra from (mostly) pion decay at rest, followed by muon decay at rest.
Current data are not able to rule out very small $g_{A}$ and $g_{V}\sim-0.5$
(region on the left-hand part of Fig. 2)

Figure 2: Allowed regions in the plane $g_{V}\times g_{A}$ from CHARM-II
Vilain  _et al._ (1994) (gray, at 90 %C.L.), LSND Auerbach  _et al._ (2001)
(dotted light-brown, at 1$\sigma$ C.L.), and TEXONO Deniz  _et al._ (2010)
(dot-dashed light-violet, at 1$\sigma$ C.L.), and the estimated 90% C.L.
sensitivity from on-axis (7 years) DUNE electron scattering (dashed green) and
tridents (♆) (dashed purple), as well as DUNE-PRISM electron scattering (dark-
blue) and tridents (light-blue), assuming the SM value (red star).

In DUNE-PRISM, the presence of both $\nu_{\mu}$ and $\nu_{e}$, along with
their antiparticles, is a powerful tool for lifting degeneracies without
resorting to data from other experiments. To illustrate this point, Fig. 3
depicts the $E_{e}\theta_{e}^{2}$ spectra of neutrino-electron scattering
events for the first 5 off-axis positions without any angular or energy
resolution. For each position, histograms corresponding to three pairs of
vector and axial couplings $(g_{V},g_{A})$ are depicted: $(-0.04,-0.5)$, the
SM expectation (solid); $(-0.48,-0.04)$, the leftmost degenerate region
(dotted); and $(0.47,0.02)$, the rightmost degenerate region (dashed). It is
clear that the rightmost degeneracy is lifted due to the higher $\nu_{e}$
composition of the flux, as depicted in Fig. 1. Error bars illustrating the
statistical and systematic errors, are included for the SM case. DUNE-PRISM
neutrino–electron scattering data, alone, cannot fully distinguish the SM from
the leftmost degenerate region, as depicted in Fig. 2.

Figure 3: Neutrino-electron event rates as a function of $E_{e}\theta_{e}^{2}$
for the first 5 off-axis positions. For each position, the three histograms
correspond to three pairs of vector and axial couplings $(g_{V},g_{A})$:
$(-0.04,-0.5)$ (solid); $(-0.48,-0.04)$ (dotted); and $(0.47,0.02)$ (dashed).
Error bars illustrating the statistical and systematic errors, are included
for the SM case (solid histogram).

Neutrino-trident scattering, when a neutrino scatters off a nucleus producing
a charged lepton pair with same or different flavors, $\nu
A\to\nu\ell^{+}\ell^{-}\\!A$, is also sensitive to $g_{V}$ and $g_{A}$.222Here
we assume that the electron and muon couplings to the $Z$-boson are identical.
This scattering can be coherent off the electromagnetic field of the nucleus
or diffractive off the nucleons themselves. Although the trident cross section
is quite involved (see e.g. Refs Magill and Plestid (2017); Ballett  _et al._
(2019); Altmannshofer  _et al._ (2019)), in the limit where the final state
leptons are massless, it is proportional to the electroweak parameters
$(C_{V}^{2}+C_{A}^{2})$. For a $\nu_{\mu}$ beam, these couplings are Magill
and Plestid (2017); Ballett  _et al._ (2019)

$\displaystyle C_{V}$ $\displaystyle=g_{V}$ $\displaystyle C_{A}$
$\displaystyle=g_{A}\quad$ $\displaystyle(e^{+}e^{-}\,{\rm trident}),$ (5)
$\displaystyle C_{V}$ $\displaystyle=g_{V}+1$ $\displaystyle C_{A}$
$\displaystyle=g_{A}+1\quad$ $\displaystyle(\mu^{+}\mu^{-}\,{\rm trident}).$
(6)

The processes that lead to $\mu^{\mp}e^{\pm}$ tridents are pure-charged-
current and do not contribute to this discussion. Hence, measurements of
$e^{+}e^{-}$ $\nu_{\mu}$-tridents – the statistically-dominant mode –
constrain $g_{V}^{2}+g_{A}^{2}$ while those of $\mu^{+}\mu^{-}$
$\nu_{\mu}$-tridents constrain $(g_{V}+1)^{2}+(g_{A}+1)^{2}$ in the limit of
vanishing muon mass. A similar behavior is expected of $\nu_{e}$-tridents,
with $e\leftrightarrow\mu$, and those associated to antineutrinos. It is easy
to see that, in the limit where the muon mass vanishes, all cross sections are
invariant under $g_{V}\leftrightarrow g_{A}$. A finite muon mass, however,
breaks the $g_{V}\leftrightarrow g_{A}$ symmetry.

Due to the very high intensity of the DUNE neutrino beam, this rare process is
accessible. Fig. 2 also depicts the measurement of $(g_{V},g_{A})$ from both
$\mu^{+}\mu^{-}$ and $e^{+}e^{-}$ neutrino-trident events in DUNE on-axis
(dashed purple) and DUNE-PRISM (light-blue), considering the efficiencies from
Ballett  _et al._ (2019), which range from $48-66\%$ ($17-39\%$) for coherent
(diffractive) trident processes. These efficiencies stem from cuts on hadronic
activity and kinematical variables in order to make backgrounds negligible.
Improvements on the reconstruction of di-electron or di-muon events would
benefit the $(g_{V},g_{A})$ couplings determination. The allowed region is not
symmetric under $g_{V}\leftrightarrow g_{A}$ since, as highlighted earlier,
for DUNE energies, the mass of the muon is not negligible. Indeed, we checked
that the subleading $\mu^{+}\mu^{-}$ neutrino-trident event sample plays the
decisive role here. Hence, the combination of neutrino–electron scattering and
neutrino trident data in the DUNE near detector complex, assuming these are
consistent with the SM, lifts all degeneracies in the $g_{V}\times g_{A}$
plane even if one chooses to exclude information from outside data.

Assuming the SM, our results can be translated into a measurement of
$\sin^{2}\theta_{W}$ at $\langle Q^{2}\rangle=(55~{}{\rm MeV})^{2}$. Fig. 4
depicts the value of $\sin^{2}\theta_{W}$ in the
$\overline{\scalebox{0.8}{\text{MS}}}$ scheme as a function of $Q$, obtained
from a fit to existing data, together with our estimate for the expected DUNE
and DUNE-PRISM sensitivities. The former is slightly better, but we emphasize
that the on-axis measurement of $\sin^{2}\theta_{W}$ depends more strongly on
the neutrino-flux modeling, while the DUNE-PRISM sensitivity depends more on
the relative on- to off-axis flux uncertainties. The main systematic
uncertainty for this analysis comes from hadron production in the beam target,
and extra running time would further improve the determination of
$\sin^{2}\theta_{W}$ (see Supplemental Material). Note that current
experiments like NA61/SHINE Abgrall  _et al._ (2011) or the future experiment
EMPHATIC Akaishi  _et al._ (2019) may achieve a better knowledge of the hadron
production mechanism leading to reduced systematic uncertainties and thus
improving the determination of the weak mixing angle. Regardless, both
measurements are estimated to be competitive with existing results.

Figure 4: $\sin^{2}\theta_{W}$ in the $\overline{\rm MS}$ scheme (light blue
line) as a function of $Q$, obtained from a fit to existing data (gray data
points), together with the DUNE on-axis (dark blue data point) and DUNE-PRISM
(green data point) sensitivities to this angle. The horizontal error bars
indicate the range of $Q$ values accessible to DUNE neutrino–electron
scattering. Note that the Tevatron, LHC and SLC data points where slightly
shifted from $Q=M_{Z}$ to improve readability.

In summary, we estimated that the future DUNE experiment will have excellent
sensitivity to the vector and axial couplings of the electron to the
$Z$-boson, and thus to the weak mixing angle $\sin^{2}\theta_{W}$, via
precision measurements of neutrino–electron scattering. The sub-dominant
$\nu_{e}$ beam component in DUNE-PRISM, as well as neutrino trident events,
play an important role in resolving degeneracies currently present in the
world data.

###### Acknowledgements.

We are extremely grateful to Luke Pickering for providing us with the flux
covariance matrix, and we thank Laura Fields and Oleksandr Tomalak for useful
discussions. The work of AdG is supported in part by the DOE Office of Science
award #DE-SC0010143. This manuscript has been authored by Fermi Research
Alliance, LLC under Contract No. DE-AC02- 07CH11359 with the U.S. Department
of Energy, Office of Science, Office of High Energy Physics. ZT is supported
by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under
contract 2018/21745-8.

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##  Appendix A Supplemental Material

In this Supplemental Material, we provide technical details that may be
relevant to experts.

In Fig. 5, we present the neutrino fluxes considered in the analysis at the
near detector facility for the on-axis and 5 off-axis positions and for both
running – neutrino and antineutrino – modes. The number of expected neutrino-
electron scattering events in the DUNE liquid argon near detector, for each
neutrino flavor in each position, is shown in Tab. 2. The numbers in the top
rows (bottom rows) correspond to neutrino (antineutrino) mode runs. We have
assumed 3.5 years of data taking in each mode, half of the time spent in the
on-axis position, and the other half equally distributed in each off-axis
position.

Channel | 0 m | 6 m | 12 m | 18 m | 24 m | 30 m | 36 m
---|---|---|---|---|---|---|---
$\nu_{\mu}e\to\nu_{\mu}e$ | 13,624 | 1,576 | 565 | 238 | 122 | 72 | 47
| 2,025 | 269 | 129 | 65 | 37 | 25 | 18
$\bar{\nu}_{\mu}e\to\bar{\nu}_{\mu}e$ | 1,238 | 165 | 80 | 40 | 23 | 15 | 11
| 10,135 | 1,144 | 397 | 162 | 83 | 49 | 32
$\nu_{e}e\to\nu_{e}e$ | 1,447 | 184 | 96 | 48 | 26 | 17 | 11
| 629 | 83 | 46 | 25 | 15 | 10 | 7
$\bar{\nu}_{e}e\to\bar{\nu}_{e}e$ | 192 | 25 | 15 | 8 | 5 | 3 | 2
| 415 | 52 | 27 | 14 | 8 | 5 | 3
Total | 16,501 | 1,950 | 756 | 334 | 176 | 107 | 71
| 13,204 | 1,548 | 599 | 266 | 143 | 89 | 60

Table 2:  Total expected number of $\nu-e$ events at different off-axis
positions for the neutrino mode (top row) and anti-neutrino mode (bottom row)
assuming $3.5$ years of data taking in each mode. Half of the time is spent in
the on-axis position while the other half is divided equally in each off-axis
position.

We also present in Tab. 3, the expected number of trident events for the same
running plan aforementioned.

Channel | 0 m | 6 m | 12 m | 18 m | 24 m | 30 m | 36 m
---|---|---|---|---|---|---|---
Total $e^{\pm}\mu^{\mp}$ | 760 | 91 | 31 | 12 | 5 | 3 | 2
| 651 | 76 | 25 | 10 | 5 | 3 | 2
Total $e^{+}e^{-}$ | 180 | 21 | 8 | 3 | 2 | 1 | 0.6
| 166 | 19 | 7 | 3 | 1 | 1 | 0.6
Total $\mu^{+}\mu^{-}$ | 93 | 12 | 4 | 1 | 0.6 | 0.3 | 0.2
| 77 | 10 | 3 | 1 | 0.5 | 0.3 | 0.2

Table 3: Total expected number of trident events at different off-axis
positions for the neutrino mode (top row) and anti-neutrino mode (bottom row)
assuming $3.5$ years of data taking in each mode. Half of the time is spent in
the on-axis position while the other half is divided equally in each off-axis
position.  Figure 5: Neutrino fluxes considered in the on-axis and five off-
axis positions. We show the fluxes for $\nu_{\mu}$ (purple),
$\overline{\nu_{\mu}}$ (green), $\nu_{e}$ (red) and $\overline{\nu_{e}}$
(light blue) in neutrino (full) and antineutrino (dashed) running modes.

The test statistics considered in the analysis is defined as,

$\displaystyle\chi^{2}=(\mathbf{D}-\mathbf{T}-\alpha_{CC}\mathbf{B}^{\rm
CCQE}-\alpha_{NC}\mathbf{B}^{\rm\pi^{0}})^{T}C^{-1}(\mathbf{D}-\mathbf{T}-\alpha_{CC}\mathbf{B}^{\rm
CCQE}-\alpha_{NC}\mathbf{B}^{\rm\pi^{0}})+\left(\frac{\alpha_{CC}}{\sigma_{\alpha}}\right)^{2}+\left(\frac{\alpha_{NC}}{\sigma_{\alpha}}\right)^{2},$
(7)

where $\mathbf{D}$ and $\mathbf{T}$ are vectors of Asimov data and expectation
values, respectively, spanning all positions and bins. The binning is
performed in $E_{e}\theta_{e}^{2}$, $E_{e}$ the total electron energy and
$\theta_{e}$ the scattered electron angle with respect to the beam direction.
$C$ is the covariance matrix derived from detailed simulations of hadron
production. The vectors $\mathbf{B}^{\rm CCQE}$ and $\mathbf{B}^{\rm\pi^{0}}$
correspond to the simulated background events from CCQE interactions and mis-
identified $\pi^{0}$ events, respectively. $\alpha_{CC}$ and $\alpha_{NC}$ are
systematic uncertainties related to the normalization of such charged-current
and neutral-current backgrounds. We include a penalty term for these
systematic uncertainties with $\sigma_{\alpha}=10\%$

Regarding the angular resolution and its impact on the determination of the
weak mixing angle, we note that the key to reject backgrounds in this study is
the kinematic limit $E_{e}\theta_{e}^{2}<2m_{e}$, where $E_{e}$, $\theta_{e}$
and $m_{e}$ are the outgoing electron energy, angle with respect to the
neutrino beam, and mass. The detector capability to measure energy and angle
of recoiled electrons has a direct effect on the $E_{e}\theta_{e}^{2}$
spectrum. To exemplify that, we show in Fig. 6 the signal (solid), CCQE
background (dashed, magenta) and $\pi^{0}$ mis-identified background (dashed,
cyan) spectra for four assumptions on the angle resolution $\sigma_{\theta}$,
as indicated in the figure for half year of data taking at the on-axis
position in the neutrino mode. We have checked that energy resolution plays a
small role in the spectral distortion. The kinematic limit can be seen as the
sharp feature in the perfect resolution histogram around 1 MeV. Clearly, the
signal-to-background ratio in the low $E_{e}\theta_{e}^{2}$ bins are worse for
worse angular resolution.

Figure 6: Neutrino-electron event rates as function of $E_{e}\theta_{e}^{2}$
for different values of the angular resolution. The pink dashed (light blue
dot-dashed) lines correspond to the CCQE (misID $\pi^{0}$) estimated
backgrounds.

To see how the this affects the sensitivity to $\sin^{2}\theta_{W}$, we
present in the left panel of Fig. 7 the $\Delta\chi^{2}$ as a function of
$\sin^{2}\theta_{W}$ for several angular resolutions $\sigma_{\theta}$ as
indicated in the figure for the on-axis only, 7 years running. The $1\sigma$
uncertainty for $\sin^{2}\theta_{W}$ for each assumed angular resolution are
$\\{1.27\%,2.07\%,2.75\%,2.88\%\\}$ for
$\sigma_{\theta}=0^{\circ},\,1^{\circ},\,3^{\circ},\,5^{\circ}$, respectively.
This indicates that it is important to achieve a good angular resolution in
order for DUNE to provide a competitive measurement of the weak mixing angle.

Figure 7: Comparison of the $\Delta\chi^{2}$ for different values of the
resolution of the electron angle $\sigma_{\theta}$ (left), running plans
(center), systematic uncertainties (right).

We also show, in the middle panel of Fig. 7 how more statistics or different
running plans could affect DUNE’s sensitivity to $\sin^{2}\theta_{W}$. The
purple, green and magenta lines represent three different running plans: only
on-axis; time equally divided in each position (running plan 0 - RP0); and
half of the time on-axis and half of the time divided equally in each off-axis
position (unning plan 1 - RP1), respectively. The solid lines correspond to a
total run of 7 years while the dashed lines correspond to a total run of 14
years. It is clear from the comparison of solid to dashed lines, that the
measurement of $\sin^{2}\theta_{W}$ is still limited by statistics. Due to
that, spending more time in positions which lead to higher statistics (e.g.,
on-axis), is beneficial to this measurement.

Finally, to estimate what is the dominant systematic uncertainty that hinders
the determination of $\sin^{2}\theta_{W}$, we perform the analysis singling
out specific systematic uncertainties. The uncertainties in the neutrino flux
come from hadron production, focusing of the beam, the alignment of the beam
itself, the number of protons on target (POT), and the horn alignment. The
results of the analysis for a single uncertainty at a time are shown in the
right panel of Fig. 7. It is clear that the major uncertainty comes from the
hadron production model, while all other sources of uncertainty affect the
measurement marginally.


# Analysis of Robocode Robot Adaptive Confrontation Based on Zero-Sum Game*
††thanks: National Natural Science Foundation of China.No. 62076028

st Xiangri LU  Automation college
Beijing University of Technology
Beijing, China

###### Abstract

The confrontation of modern intelligences is to some extent a non-complete
information confrontation, where neither side has access to sufficient
information to detect the deployment status of the adversary, and then it is
necessary for the intelligences to complete information retrieval adaptively
and develop confrontation strategies in the confrontation environment. In this
paper, seven tank robots, including TestRobot, are organized for $1V1$
independent and mixed confrontations. The main objective of this paper is to
verify the effectiveness of TestRobot’s Zero-sum Game Alpha-Beta pruning
algorithm combined with the estimation of the opponent’s next moment motion
position under the game round strategy and the effect of releasing the
intelligent body’s own bullets in advance to hit the opponent. Finally, based
on the results of the confrontation experiments, the natural property
differences of the tank intelligences are expressed by plotting histograms of
1V1 independent confrontations and radar plots of mixed confrontations.

###### Index Terms:

Zero-sum Game, Robocode, Alpha-Beta Pruning Algorithm, Game Circle

##  I Introduction:Research background

Zero-sum game is a concept of game theory, which is a non-cooperative game[1].
It means that the gain of one party must mean the loss of the other party, and
the sum of gain and loss of each party is always ”zero”, and there is no
possibility of cooperation between the two parties [2, 3, 4, 5]. The result of
a zero-sum game is that what one side gains is exactly what the other side
loses, and the benefit to society as a whole does not increase by a single
point.

The zero-sum game is now an analogue in the sense of strict competition in
human society. Nowadays, the two sides of social competition are abstractly
represented as the process of zero-sum game. Nowadays, the social competition
is complex and changeable, and the two sides confrontation relies more and
more on artificial intelligence algorithms to analyze the competitive
situation and thus form objective judgments. Machine gaming is an important
research direction in the field of artificial intelligence[6].In 1997, Deep
Blue defeated Kasparov, the king of chess at that time, by one point of total
score, after which computers gradually ruled most of chess games except Go[7,
8, 9], Deep Blue only has 12 levels of search depth. The basic principle of
Deep Blue is similar to the method introduced in this paper, but since the
computing speed of Deep Blue is much higher than our current personal
computer[10]. 2016 Google’s Alpha Go victory over Korean Go master Lee Sedol
made the human-computer game a household name, and then the upgraded Alpha Go
Zero defeated the world’s top Go player Ke Jie by 4:0 and other events The
next upgraded version of Alpha Go Zero with 4:0 victory over the world’s top
Go player Ke Jie and other events reveal the importance of human-computer
gaming. Machine games can be divided into complete information and non-
complete information games according to the transparency of information in the
game process[11]. The information of both sides of the complete information
game is completely transparent, and the two sides of the game are fully aware
of each other’s game information. The background of this paper is the
hypothesis that both sides of the zero-sum game have game inducing and
fraudulent behaviors due to the opaqueness of the information of the non-
complete information game, i.e., the game confrontation between the generator
and the discriminator in the adversarial neural network[12, 13].

Then, in this paper, we simulate the tank robot adversarial environment by
Robocode adversarial platform and make the tank robot with Zero-sum Game
against the sample robot in Robocode platform, and observe the adversarial
data to analyze the adversarial effect produced by Zero-sum Game thinking.

##  II Technical support

###  II-A Software platform support

Robocode is a tank robot combat simulation engine released in July 2001 on
IBM’s Web alpha Works in the U.S. Robocode is a robot combat simulation engine
created by Mat Nelson, an IBM engineer, in the Java language[14]. Robocode is
programmed to allow tanks to move, attack, defend, dodge, and fire, while
example robots of varying levels of skill are selected from the Robocode
counter platform to fight against each other.

###  II-B Algorithm support

Minimalized maximal search is the core idea of zero-sum games, and minimalized
maximal search includes a special Alpha-Beta pruning algorithm[15, 16,
17].Alpha Beta pruning algorithm is a safe pruning strategy, that is, it does
not have any negative impact on the Robocode platform and the natural
properties of the tank robot.Alpha Beta pruning algorithm is based on the fact
that tank robots do not take decisions that are detrimental to themselves. If
a node in an adversarial environment is clearly a node that is unfavorable to
itself, then it can simply prune that node. the Robocode platform tank robot
will choose the maximum node at the MAX level, while the newly coded tank
intelligence will choose the minimum node at the MIN level. The choice that is
unfavorable to both sides then directly cuts this node, i.e. firstly, at the
MAX layer, assuming that the current layer has searched for the maximum value,
if it finds that the next layer of the next node (which is the MIN layer) will
produce a value smaller than the maximum value; secondly, at the MIN layer,
assuming that the current layer has searched for the minimum value, if it
finds that the next layer of the next node (which is the MAX layer) will
produce a value smaller than the minimum value value that is even larger.

According to the analysis of simulation experiments on Robocode platform, when
both sides play Zero-sum Game, the opponent always wants to be close to one
side to strike so that it can strike the opponent most effectively. Then the
process of both tanks playing against each other is represented in the alpha-
beta pruning algorithm below[18]. First of all, both sides confront the
environment as shown in the figure. The objective of both sides of the game is
to destroy the opponent, and both sides design the idea to move a very small
part of the distance to the quadrant where the enemy is located to approach
the opponent, but not yet allow the opponent to detect the native tank. Then
it is necessary to construct the game circumference and find the optimal path
value according to the alpha-beta pruning algorithm of the zero-sum game[19].
The attacking posture of the attacker is countered.

Figure 1: Game confrontation environment.

If the red tank robot wants to attack the blue robot, it must first advance to
the white area within a very small area in the figure, then the path to reach
the white area is assumed to have N paths in the figure and their curvature
varies, and the resulting radius are 20,30,40,70,80,100,120,140,160.

Then for the situation in the figure at this time, the $\alpha-\beta$ pruning
algorithm is used to analyze the following process: where the red block
represents the red tank maximum Max selection game, and the blue block
represents the blue tank minimum Min selection game, then

Figure 2: $\alpha-\beta$ pruning algorithm based game tree structure.

If it is known that both tanks are confronted in a region of the map and the
backpropagation values of all subnodes of the possible path radius of the
confrontation between the two sides can be pushed, the red block represents
the red tank maximum Max selection game and the blue block represents the blue
tank minimum Min selection game.

If some of the possible path radius sub-nodes of a confrontation region are
known, although the backpropagation value of the node cannot be calculated,
the range of values of the backpropagation value of the node can be
calculated. At the same time, using the range of the backpropagation value of
the node, it is not necessary to search for the remaining subnodes if it has
been determined that there is no better path when searching for its subnodes.
That is, the redundant child nodes are partially cut off.

Specify the positive direction from the bottom to the top, respectively, for
1-5 times the game, as shown in the figure, six nodes in the second layer,
four nodes in the third layer, and two nodes in the second layer. Let V be the
backward value of the node and $\alpha<V<\beta$, that is, $\alpha$ is the
maximum lower bound and $\beta$ is the minimum upper bound. When
$\alpha>=\beta$, the remaining branches of the node need not continue
searching.

Figure 3: $\alpha-\beta$ pruning game based on the first starting point of the
second level.

First, observe the first game extrapolation between the two sides before the
confrontation. Initialization, so that $\alpha=-\infty,\beta=+\infty$, that
is, $-\infty<V<+\infty$, to the second layer of the first node, due to the
left child node of the backpropagation value of 30, and the node is MIN node,
try to find the backpropagation value of the small path, so the $\beta$ value
is modified to 30, this is because 30 is less than the current $\beta$ value.
Then the backpropagation value of the right child of the node is 40, and the
value of 30 of the node is not modified at this time, because 40 is larger
than the current $\beta$ value. After all the children of the node are
searched, the backpropagation value of the node is calculated as 30.

Figure 4: $\alpha-\beta$ pruning game based on the second node of the second
layer.

The first node of the second layer is the child of the first node of the third
layer, and after calculating the backpropagation value of the first node of
the second layer, we can update the backpropagation value range of the first
node of the third layer. Since the first node of the third layer is the Max
node, we try to find the path with large backpropagation value, so we modify
the $\alpha$ value to 30, because 30 is larger than the current $\alpha$
value. After that, the right child node of the first node in the third layer
is searched and the $\alpha$ and $\beta$ values of the first node in the third
layer are passed to the right child node of the first node in the third layer.

Figure 5: $\alpha-\beta$ pruning game based on the first node of the three
layers.

For the second node of the second layer, since there is only one child node,
and this node is a Min node, to find the minimum value, the range value of the
second node of the second layer is changed to $\alpha$=30 and $\beta$=20, this
node violates the pruning algorithm logic, so the right nodes of this node
will be cut off. However, the second node in the second layer in the figure
does not have a right node, so this step of the operation execution procedure
can be omitted. So it can be introduced that the first node path selection in
the third layer is 30, and because the second layer is looking for Min path
point, then the $\beta$ value will be changed to 30 and passed to the right
child node.

Figure 6: $\alpha-\beta$ pruning game based on the third starting point of the
second level.

The third node of the second layer belongs to the search for Min value, then
the third node of the second layer has a range value of $\alpha$=$-\infty$ and
$\beta$=80, but it can be seen from the figure that the right child node value
of the fifth node of the first layer is in the range of
$-\infty<\mathrm{V}<80$, then the third node of the second layer takes the
value of 70. the second node of the second layer belongs to the search for Max
value, then the second node of the third layer has a range value of
$\alpha$=70 and $\beta$=30, then the first node in the fourth layer is finally
determined to be 30.

Figure 7: $\alpha-\beta$ pruning game based on the fourth starting point of
the second level.

The top node belongs to the search for Max value, then the range value of the
top node is $\alpha$=30 and $\beta$=$+\infty$. According to the right node
passing law of pruning algorithm, the fifth node of the second layer has a
range of $30<\mathrm{V}<+\infty$, while the fifth node of the second layer is
replaced with $\beta$ value of 120 according to the passing of the sixth node
of the first layer, and the second node of the fourth layer gets the range
value of $\alpha$=30 and $\beta$=120 similarly. according to the two nodes of
the fourth layer, we can judge that the final value of the top node is 30. at
this time, the second node of the fourth layer can be completely pruned. For
system integrity, the path traversal tree is analyzed as follows.

Figure 8: Complete game tree structure based on $\alpha-\beta$ pruning.

The fourth node of the third layer belongs to the search for Max value, then
inherit the second node of the fourth layer range value $\alpha$=30,
$\beta$=120, similarly the fifth node of the second layer range value inherit
$\alpha$=30, $\beta$=120, because this point belongs to the search for Min
value, then $\beta$ value is 100, range value $\alpha$=30, $\beta$=100; the
fourth node of the third layer belongs to the search for Max value, then
$\alpha$ value is 100 The sixth node of the second layer inherits the fourth
node of the third layer range value $\alpha$=100, $\beta$=120, the sixth node
of the second layer belongs to the search for Min value, then $\beta$=140,
according to the pruning rule to select the sixth node of the second layer
path value of 140, the fourth node of the third layer path value of 100, the
second node of the fourth layer path value is 120.

Through the $\alpha$-$\beta$ pruning algorithm of the Zero-sum Game, the
radius of the arc trajectory that the opponent tank chooses to move to a
certain area can be initially determined, i.e., the opponent tank robot will
most likely choose an arc trajectory with a radius of 30. If corresponding
measures are taken, the trajectory of the adversary tank has to be calculated
and the strike route of the unmanned tank is designed in advance. For the game
problem of nonlinear complex system, a Zero-sum Game against circular
algorithm is specially designed.

Figure 9: $\alpha-\beta$ pruning based game against rounding.

In summary, the process of $\alpha$-$\beta$ pruning algorithm is only a game
selection of the above game circumference 1, game circumference 2 and game
circumference 3 still need to repeat the above process of $\alpha$-$\beta$
pruning algorithm, it should be noted that in each game circumference of
$\alpha$-$\beta$ pruning algorithm of the first layer of path value selection
The upper layer node process is a dynamic change process, then there will be a
variety of a game circumference multiple path values of the situation.

##  III Zero-sum Game against rounding algorithm

As opposed to the attacking side, the defending side needs to construct the
corresponding mathematical model for defense, then the construction rules need
to be followed as follows.

  * •

Determine the body of the tank robot, the gun and radar and scanning arc need
to be marked with color.

  * •

Determine the separation of the tank robot’s vehicle gun and radar, so that
they do not affect each other.

  * •

Determine the algorithm of tank robot movement.

  * •

Determine the algorithm of the tank robot to select the firepower.

  * •

Determine the algorithm for the tank robot to lock on to other robots.

  * •

Determine the algorithm for the tank robot to calculate and adjust the muzzle
of the gun to the enemy.

$(1)$ and $(2)$ belong to the basic rules, which can be defined by calling
statements inside the Robocode platform, and $(3)$, $(4)$, $(5)$ and $(6)$
belong to the quick calculation design. As shown in the figure

Figure 10: Adversarial coordinate transformation model in games.

The red tank robot is in the third quadrant and the defending robot has to
define the angle of the sensor, the positive direction of the tank and the
angle between the positive direction of the tank and the positive direction of
the red tank in order to build an accurate model.

According to the analysis of the above figure,When the radar scans the enemy,
it may get the enemy’s pinch angle, and in addition the robot can get its own
positive direction angle and the radar’s positive direction angle at any time.
After the radar has swept the enemy, it will sweep the enemy again to trigger
the radar scan event and get the information of the enemy again, and at the
same time calculate the angle of the radar back to sweep. Because the enemy’s
position in the left side of the heading, so the bearing angle is negative,
according to the above chart can be analyzed algorithm for $radarHeading$ \-
$heading$ \+ $bearing$ can get the robot radar should be back to sweep the
angle.

Figure 11: Adversarial circular analytic geometry model of the game process.

If the red tank robot wants to strike the blue tank robot, the length of arc
BC is known according to the $\alpha$-$\beta$ pruning algorithm, then the blue
side needs to get the radius of circle A if it wants to determine the position
of the red tank’s movement at the next moment. Then the geometric model is
constructed as described below.

In the circle $A$ at the center of the circle $A$ as the origin to establish a
plane right angle coordinate system, red tank at point $B$, will be moving
along the arc $BC$ to point $C$, let $BE$ is the tangent of circle $A$ and
extended to $D$, $EC$ is the tangent of circle $A$ and extended to $F$.Connect
$AB$ and $AC$ and prove that $\angle\mathrm{BAC}=\angle\mathrm{CED}$.

Proof

$\because$ both $BE$ and $EC$ are tangents to circle $A$

$\therefore\angle\mathrm{ABE}=\angle\mathrm{ACE}=\frac{\pi}{2}$

$\therefore\angle\mathrm{BAC}+\angle\mathrm{BEC}=\pi$

Also

$\because\angle\mathrm{CED}+\angle\mathrm{BEC}=\pi$

$\therefore\angle\mathrm{BAC}=\angle\mathrm{CED}$

The number of radians of $\angle\mathrm{BAC}$, which is the angle of the tank
turning out in the positive direction, is measured by the tank robot
sensor.According to the formula $R$ = $arc$ $BC$ / $\angle\mathrm{BAC}$, and
according to the tank robot sensor measurement of $AC$ and $X$-axis angle that
can be known $C$ point coordinates.

$X=R*\operatorname{COS}\angle CAX$ (1)
$\mathrm{Y}=\mathrm{R}*\mathrm{SIN}\angle\mathrm{CAX}$ (2)

Then the blue tank robot can determine the direction of the bullet in advance
for the next moment.

##  IV Experimental design and analysis of results

The experiments in this paper are based on hardware $Intel(R)$ $Core(TM)$
$i5-9400$<EMAIL_ADDRESS>$RAM$ $8GB$, Eclipse IDE for Java Developers and
$robocode-1.9.4.3$ tank adversarial platform. Firstly the alpha-beta pruning
path algorithm for zero-sum game of red and blue tanks, the natural properties
of tanks and the evasive strike algorithm were compiled and constructed using
$EclipseIDE$ for Java Developers. Secondly $robocode-1.9.4.3$ specifies the
specifications of the adversarial environment as well as the ammunition rate
of fire, etc. The final results are observed using the alpha-beta pruning path
algorithm of the zero-sum game against the example robot in
$robocode-1.9.4.3$.

###  IV-A Independent confrontation

In the $1V1$ tank battle mode, the maximum number of rounds for each type of
tank battle is set to 30, and the tank positions are randomly assigned at the
beginning of each round, and the initial tank life value is 100. In the
following, TestRobot tank robots are constructed to confront six groups of
typical example robots and observe the results, as shown in Table 1, where the
graphs indicate that TestRobot confronts other six types of tank intelligences
in five parts: Total Score, Survival, Bullet Damage, Bullet Bonus, and WINS,
respectively. The comparative observation of TestRobot tank robot’s
confrontation status.

Then set the natural properties in $robocode-1.9.4.3$, such as Number of
Rounds:30, Gun Cooling Rate:0.1, Inactivity Time:450, Sentry Border Size:100.

TABLE I: Comparison of TestRobot and other Six Types of Tank Intelligences State Confrontation Robot Name | Total Score | Survival | Bullet Damage | Bullet Bonus | WINS
---|---|---|---|---|---
TestRobot |  V S | Crazy | 4573 | 231 | 1400 | 50 | 2388 | 136 | 459 | 2 | 29 | 2
Fire | 5531 | 380 | 1500 | 0 | 3006 | 376 | 612 | 0 | 30 | 0
My-Robot | 5144 | 326 | 1500 | 0 | 2730 | 317 | 531 | 0 | 30 | 0
V-Robot | 4192 | 1105 | 1150 | 350 | 2299 | 518 | 412 | 25 | 23 | 7
SpinBot | 3314 | 1943 | 800 | 700 | 1943 | 933 | 269 | 89 | 16 | 14
Walls | 5145 | 300 | 1500 | 0 | 2782 | 290 | 557 | 0 | 30 | 0

Figure 12: Independent adversarial comparison data between smart bodies.

Among them, it can be observed from the graph that SpinBot can be comparable
to TestRobot in terms of strength against Robot tank intelligences, but still
cannot beat TestRobot, and the strength of the other four types of tank robots
are all far from TestRobot. Through the table, we can calculate the relative
strength of TestRobot and Crazy, Fire and other six types of robots, and the
relative values of Total Score are $TestRobot$: $Crazy$ = 19.80; $TestRobot$:
$Fire$ = 14.56; $TestRobot$: $My-Robot$ = 15.78. $TestRobo$t: $V-Robot$ =
3.79; $TestRobot$: $SpinBot$ = 1.71; $TestRobot$: $Walls$ = 17.15. From the
Total Score relative values, it is clear that the TestRobot tank robots have a
relative advantage in the adaptive learning environment.

###  IV-B Mixed Confrontation

In order to avoid the chance of experimental results, seven tank counter
robots are put into the counter environment at the same time to observe their
counter results and analyze them. As shown in Fig.

Figure 13: Intelligent body hybrid adversarial gaming environment.

After 30 rounds of confrontation with the seven tank robots in the
confrontation environment, the table shows that the $TestRobot$ tank robot can
still achieve a good Total Score, the $TestRobot$ and $SpinBot$ tank robots
are comparable in terms of survivability, the $Fire$ and $My-Robot$ have the
weakest survivability compared to the $TestRobot$, resulting in a lower
confrontation score compared to the other tank intelligences. According to the
analysis of the Bullet Bonus attribute, $TestRobot$’s hit rate is also very
high among the seven tank robots, which leads to a high Bullet Damage score.
At the same time, $TestRobot$ was in the top three games 21 times, accounting
for 70 percent of the overall.

Figure 14: Intelligent body hybrid adversarial data radar map. TABLE II: Seven types of robots mixed confrontation comparison table Rank | Robot Name | Total Score | Survival | Bullet Damage | Bullet Bonus | 1sts | 2nds | 3rds
---|---|---|---|---|---|---|---|---
1st | TestRobot | 15667 (28%) | 6600 | 6885 | 820 | 14 | 5 | 2
2nd | SpinBot | 10933 (20%) | 6100 | 3781 | 234 | 10 | 2 | 5
3rd | Walls | 8963 (16%) | 5550 | 2861 | 146 | 5 | 8 | 6
4th | V-Robot | 6351 (11%) | 3600 | 1863 | 41 | 1 | 6 | 2
5th | Crazy | 4855 (9%) | 3750 | 942 | 5 | 1 | 1 | 7
6th | Fire | 4460 (8%) | 2650 | 1719 | 64 | 0 | 2 | 4
7th | My-Robot | 4099 (7%) | 3200 | 853 | 9 | 0 | 5 | 4

Combining the above experimental results of independent and mixed
confrontation, it can be concluded that $TestRobot$ can overcome the effects
of external environmental changes on the intelligent body itself in an
adaptive confrontation environment.

##  V Conclusion

In the unmanned adversarial environment, especially when the adversarial
environment is complex and the access to information from both sides is
narrow, and the intelligences themselves are required to try to explore the
surrounding adversarial environment, the self-adaptation ability of the
intelligences based on the zero-sum game adversarial algorithm is verified to
be higher than other intelligences through the $Robocode$ tank robot
adversarial platform, specifically to design a $TestRobot$ tank intelligences
The robot, firstly, uses the $\alpha$-$\beta$ pruning algorithm to select a
small range of moving paths, after that, judges the moving direction of the
tank robot based on the Zero-sum Game circumference, and finally solves the
coordinate points where the opponent may move through the game circumference,
making its own intelligent body hit the opponent with advance bullets in
advance.

## Acknowledgment

Project supported by the National Natural Science Foundation of China (No.
62076028)

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Appendix

Due to the diversity of path choices in a small range of the attacker’s tank
intelligence, nine path choices are allowed when the simulation can be
achieved and is not general, then there are multiple expressions of the game
tree for both sides of the confrontation, here to simplify the model, while
starting with the bottom branch on the left side, the left starting game
branch on the second level lists three $\alpha$-$\beta$ pruning algorithm
cases, which are sequential ( from small to large), inverse order (from large
to small), and single-branch maximum starting game.

Figure 15: Starting game branching paths from small to large $\alpha$-$\beta$
pruning games.

First of all the second layer of the left side of the starting game branch is
the order (from small to large), according to the principle of the two sides
of the confrontation to carry out three game rounds constructed the game tree,
it can be observed that after screening by $\alpha$-$\beta$ pruning algorithm,
the attacker’s game path selection range has been clearly marked in the
figure, the path value of the second game branch point of the four layers not
in the range of the right branch cut off, this game tree final path selection
is 100.

Figure 16: Starting game branching paths from large to small $\alpha$-$\beta$
pruning games.

The left starting game branch of the second layer of this game tree is in
reverse order (from largest to smallest), and the game tree is constructed
according to the principle that the two sides confront each other for three
game rounds, and it can be observed that after screening by $\alpha$-$\beta$
pruning algorithm, the game path selection range of the attacker is clearly
marked in the figure, and because of the contradictory path range of the
second game branch point of the second layer of this game tree and all branch
points of the fourth layer. The final path selection of this game tree is 100
by cutting off their right branches.

Figure 17: $\alpha$-$\beta$ pruning game with single branch maxima of the
starting game branching path.

The left starting game branch of the second layer of this game tree is a
single branch maximum, and the game tree is constructed according to the
principle that both sides confront each other for three game rounds, and it
can be observed that after screening by $\alpha$-$\beta$ pruning algorithm,
the game path selection range of the attacker is clearly marked in the figure,
and because of the contradictory path range of the fourth game branch point of
the second layer of this game tree and the second branch point of the fourth
layer, their The final path choice of this game tree is 160.