# RL meets Multi-Link Operation in IEEE 802.11be: Multi-Headed Recurrent Soft-
Actor Critic-based Traffic Allocation

Pedro Enrique Iturria-Rivera1, , Marcel Chenier2, Bernard Herscovici2,
Burak Kantarci1, and Melike Erol-Kantarci1, 2NetExperience., Ottawa, Canada
Emails:{pitur008, burak.kantarci<EMAIL_ADDRESS>{marcel,
<EMAIL_ADDRESS>1School of Electrical Engineering and Computer
Science, University of Ottawa, Ottawa, Canada

###### Abstract

IEEE 802.11be —Extremely High Throughput—, commercially known as Wireless-
Fidelity (Wi-Fi) 7 is the newest IEEE 802.11 amendment that comes to address
the increasingly throughput hungry services such as Ultra High Definition
(4K/8K) Video and Virtual/Augmented Reality (VR/AR). To do so, IEEE 802.11be
presents a set of novel features that will boost the Wi-Fi technology to its
edge. Among them, Multi-Link Operation (MLO) devices are anticipated to become
a reality, leaving Single-Link Operation (SLO) Wi-Fi in the past. To achieve
superior throughput and very low latency, a careful design approach must be
taken, on how the incoming traffic is distributed in MLO capable devices. In
this paper, we present a Reinforcement Learning (RL) algorithm named Multi-
Headed Recurrent Soft-Actor Critic (MH-RSAC) to distribute incoming traffic in
802.11be MLO capable networks. Moreover, we compare our results with two non-
RL baselines previously proposed in the literature named: Single Link Less
Congested Interface (SLCI) and Multi-Link Congestion-aware Load balancing at
flow arrivals (MCAA). Simulation results reveal that the MH-RSAC algorithm is
able to obtain gains in terms of Throughput Drop Ratio (TDR) up to $35.2\%$
and $6\%$ when compared with the SLCI and MCAA algorithms, respectively.
Finally, we observed that our scheme is able to respond more efficiently to
high throughput and dynamic traffic such as VR and Web Browsing (WB) when
compared with the baselines. Results showed an improvement of the MH-RSAC
scheme in terms of Flow Satisfaction (FS) of up to $25.6\%$ and $6\%$ over the
the SCLI and MCAA algorithms.

Index Terms — Multi-Link Operation, Reinforcement Learning, WiFi, Soft-Actor
Critic, traffic allocation

##  I Introduction

W ireless Fidelity (Wi-Fi) technology has experienced a rapid evolution in the
last four years with the introduction of Wi-Fi $6$ and Wi-Fi $6$E in $2019$
and $2021$, respectively. This technological development momentum seems to
keep going and IEEE 802.11be —Extremely High Throughput (EHT)— known
commercially as Wi-Fi $7$ is expected to become reality by $2024$. Even more,
IEEE 802.11 has started in $2022$ the first conversations about devising the
next generation of Wi-Fi (named Wi-Fi 8) believed to be released by $2028$
[1]. The still under development IEEE 802.11be amendment is considered with
the goal of dealing with highly congested network environments with stringent
requirements in terms of high throughput and low latency. To do so, the
amendment proposes to increase the channel bandwidth up to $320$ MHz, a higher
modulation rate up to 4096 QAM and different than all other Wi-Fi generations,
Wi-Fi 7 introduces Multi-Link Operation (MLO) and Multiple Resource Units
(MRU) capabilities.

In this work, we focus our interest towards MLO in 802.11be. Specifically, MLO
allows Multi-Link Devices (MLD)s to concurrently use their available
interfaces for multi-link communications. In this context, we intend to
optimize the traffic allocation policy over the available interfaces with the
aid of Reinforcement Learning (RL). RL has proven its effectiveness to deal
with challenging problems in wireless networks[2]. Thus, in this paper we
propose to utilize a Soft-Actor Critic (SAC) algorithm named Multi-Headed
Recurrent SAC (MH-RSAC). We choose the SAC algorithm over other Actor-Critic
methods such as Advantage Actor Critic (A2C) or Proximal Policy Optimization
(PPO) due its consistent performance in many RL challenging tasks [3].
Differently from others, the SAC algorithm maximizes its rewards altogether
with the entropy which benefits the agent’s exploration. We propose for the
first time, to the best of our knowledge, a novel SAC-based traffic-to-link
allocation policy in 802.11be MLO capable networks. In Fig. 1, we show the
scenario and system overview of our proposed scheme. Here, each agent residing
in the Access Point (AP) equipment, decides the incoming traffic distribution
percentage (a1, a2, a3) according to the MLO capabilities. For instance, if
the number of interfaces of the end station corresponds to $n_{f}>1$, the
agent will propose a set of actions for the case of two and three interfaces,
respectively. Finally, the function $U(n_{f})$ selects the final action based
on the actual interface number. Note that, when $n_{f}=1$ the agent is not
involved since all the traffic is passed to the only available interface. This
design allows to have one agent per AP instead of two. In addition, we
consider the non-Markovian behavior of the scenario and include a recurrent
neural network in the model design. Moreover, we improve even further the
performance of our agent by utilizing a modified reward function named rewards
with hindsight that consider the goals of the baselines. Finally, we utilize
in our simulations, three traffic flow types: Web Browsing (WB), High
Definition (HD)/ Ultra High Definition (4K) and Virtual Reality (VR). For the
latter, we derive the VR CDFs based on [4] and present the given equations to
be utilized in any wireless simulator.

We compare our results with two non-machine learning type baselines presented
in [5] named Single Link Less Congested Interface (SLCI), Multi Link
Congestion-aware Load balancing at flow arrivals (MCAA), respectively. Our
results show an improvement in terms of Throughput Drop Ratio (TDR) with a
gain of up to $35.2\%$ and $6\%$ when compared with SLCI, MCAA non-RL
baselines, respectively. TDR is defined as the percentage of traffic in terms
of throughput dropped after a traffic allocation decision. Finally, we
observed that our scheme is able to respond more efficiently to high
throughput and dynamic traffic such as VR and Web Browsing (WB) with an
improvement in terms of Flow Satisfaction (FS) up to $25.6\%$ and $6\%$ for
the SCLI and MCAA, respectively.

The rest of this paper is organized as follows. Section II presents the
existing works related to MLO in 802.11be. System model is demonstrated in
Section III. Section IV provides a description of the MH-RSAC scheme and the
considerations taken on its design, including the Markov Decision Process
(MDP) and the description of the baselines. Section V depicts the traffic
considerations in this work. Section VII presents our proposed scheme’s
performance evaluation and comparison with the baselines. Finally, section
VIII concludes the paper.

Figure 1: Overview of the MH-RSAC based traffic allocation policy in IEEE
802.11be MLO: Upon an incoming flow the agent will decide the percentage (a1,
a2, a3) of traffic flow allocated to each available interface (l1, l2, l3)
based on the observed state.

##  II Related work

To the best of our knowledge, this is the first work that propose a
Reinforcement Learning-based traffic allocation policy in 802.11be Multi-Link
Operation. However, 802.11be MLO has been recently of great interest among
researchers. Some of these works are referred bellow.

In [6], the authors present a study of MLO in Wi-Fi 802.11be in the context of
a sub-standard of 802.1, Time Sensitive Networking (TSN). Results show that
MLO is able to improve latency over a single link, however the usage of
uncorrelated channels are recommended to increase the average SNR perceived by
the stations. In addition, the authors in [7] perform an experimental study
using a real dataset measurement; a dataset containing 5 GHz spectrum
occupancy measurements. In the before mentioned study, it is proved that when
MLO is available in symmetrical occupied channels, latency can be reduced by
one order of magnitude over single link operation. Interestingly, it is
observed that when the channels do not fulfill the symmetry condition, the
latency performance worsens. The authors solve the previous issue by adding a
backoff counter in each interface to avoid packet collision. Related to our
current work, the authors in [5] propose three non-dynamic flow traffic
allocation policies in Wi-Fi 802.11be: SLCI, MCAA and Multi Link Same Load to
All interfaces (MLSA), respectively. They prove that MLO performance is
tightly related to the efficiency of the flow distribution over the available
interfaces in a Multi-Link Device. The authors present in [8], a dynamic flow
traffic allocation policy that is able to adapt its traffic policy not only
upon the arrival of the traffic flow but every $1$ second. Finally, surveys
can be found throughout the literature that delve into the challenges and
perspective of 802.11be and MLO such as in [9, 10].

##  III System Model

In this work, we utilize an IEEE 802.11be network with a predefined set of
$\mathcal{M}$ APs and $\mathcal{N}$ stations attached per AP. In addition, we
consider all APs to be MLO-capable with a number of available interfaces
$n_{f}=3$ whereas in the case of the stations the MLO capability can vary.
This design decision is taken based on the fact that single device and multi-
device terminals will coexist in real scenarios due to terminal manufacturer
diversity. Furthermore, stations are positioned in two ways with respect their
attached AP: $80\%$ of the users are positioned randomly in a radius
$r\sim[1-8]$ m and the rest with a radius $r\sim[1-3]$ m. All APs and stations
support up to a maximum of 16-SU multiple-input multiple-output (MIMO) spatial
streams. The path loss model corresponds to the enterprise model described in
[11]. In addition, we consider an adaptive control rate based on Signal to
Noise Ratio (SNR) with a maximum 4096 QAM modulation rate.

##  IV Multi-Headed Recurrent Soft-Actor Critic

In the current section we discuss the details and considerations taken in the
design of the Multi-Headed Recurrent Soft-Actor Critic (MH-RSAC) agent.

###  IV-A Soft-Actor Critic

The SAC agent introduced initially in [3] is a maximum entropy, model-free and
off-policy actor-critic method that outperformed most of the state-of-the-art
RL algorithms such as Twin Delayed Deep Deterministic Policy Gradient (TD3)
and PPO. The success of SAC relays on the inclusion of a policy entropy term
into the reward function to encourage exploration. In addition, it reutilizes
the successful experience of the minimum operator in the selection of the
double Q-Functions that comprise the critic in the Deep Deterministic Policy
Gradient (DDPG) algorithm [12].

In this work, we utilize a discrete SAC agent presented in [13] that provides
an adaptation of the original SAC agent to the discrete action space.
Moreover, we take into consideration the results obtained recently in [14]. In
such work, the authors study the discrete SAC and find that such algorithm is
affected by Q-Value underestimation due to the usage of the minimum operator.
In consequence, they propose to substitute the minimum operator by an average
operator that allowed to reduce the bias of the lower bound of the double
critics. In addition to the previous considerations we modify the structure of
the critic and the actor of the discrete SAC to allow multi-output. To do so,
we diverge the model in two heads as indicated by Fig. 2: one head (Head 1)
producing the actions when two interfaces are available at the station end and
another (Head 2) when three are available. The reason behind this design
decision corresponds to the practical assumption, briefly mentioned in Section
III, that terminals with diverse MLO capabilities will coexist. Thus, instead
of utilizing two agents handling each individual case, we propose to utilize
one agent capable of deciding in both circumstances. In such way, we can
reduce the computational load required to train and run two agents
simultaneously.

1

2

3Initialize actor’s policy $\phi$, critic’s Q-function parameters
$\theta_{1}$, $\theta_{2}$ and experience replay buffer $\mathcal{D}$. Set
critic’s Q-target function equal to main parameters
$\hat{\theta}_{1}\leftarrow\theta_{1}$ and
$\hat{\theta}_{2}\leftarrow\theta_{2}$. $\bm{op}\in\\{\texttt{min, avg}\\}$
corresponds to the operator, $L_{per}$ the update periodicity, $n_{up}$ number
of updates per learning step and $\rho\in[0,1]$ is the polyak factor.

4 for  _environment step $t\leftarrow 1$ to $T$_ do

5

6 Observe state $s=[C_{1}^{o},C_{2}^{o},C_{3}^{o},O_{f},T_{id}]$ and select
action for each head $a^{h_{1,2}}\sim\pi_{\phi}^{h_{1,2}}(\cdot|s)$

7 Execute $a$ in the environment

8 Observe next state $s^{\prime}$ and reward $r$

9 Store $(s,a,r,s^{\prime})$ in experience replay buffer $\mathcal{D}$

10 if  _$t$ mod $L_{per}=0$_ then

11 for  _$n\leftarrow 1$ to $n_{up}$ _ do

12 Randomly sample a batch of transitions with size $B$ from $\mathcal{D}$

13 Compute targets for the Q-functions (Critic) for each head $h_{1,2}$:

14 where $\tilde{a}^{\prime
h_{1,2}}\sim\pi_{\phi}^{h_{1,2}}(\cdot|s^{\prime})$

15

16 Update Q-functions (Critic) considering each head $h_{1,2}$:

17 $L_{1}^{h_{1,2}}(\theta_{1},\mathcal{D})=E_{(s,a,r,s^{\prime})\in
B}[(Q^{h_{1,2}}_{\theta_{1}}(s,a^{h_{1,2}})-y_{1}(r,s^{\prime}))^{2}]$

18

19 $L_{2}^{h_{1,2}}(\theta_{2},\mathcal{D})=E_{(s,a,r,s^{\prime})\in
B}[(Q^{h_{1,2}}_{\theta_{2}}(s,a^{h_{1,2}})-y_{2}(r,s^{\prime}))^{2}]$

20

21
$L(\theta^{h_{1,2}},\mathcal{D})=L_{1}^{h_{1,2}}(\theta_{1},\mathcal{D})+L^{h_{1,2}}_{2}(\theta_{2},\mathcal{D})$

22 Update policy (Actor) considering each head $h_{1,2}$:

23

24 $L^{h_{1,2}}_{1}(\phi^{h_{1}},\mathcal{D})=E_{(s)\in
B}[\bm{op}_{k=1,2}Q_{\hat{\theta}_{1},k}^{h_{1,2}}(s,\tilde{a}^{\prime
h_{1,2}})-\alpha\log\pi_{\phi}^{h_{1,2}}(\tilde{a}^{\prime
h_{1,2}}|s^{\prime})]$

25

26 $L^{h_{1,2}}_{2}(\phi^{h_{2}},\mathcal{D})=E_{(s)\in
B}[\bm{op}_{k=1,2}Q_{\hat{\theta}_{1},k}^{h_{1,2}}(s,\tilde{a}^{\prime
h_{1,2}})-\alpha\log\pi_{\phi}^{h_{1,2}}(\tilde{a}^{\prime
h_{1,2}}|s^{\prime})]$

27

28
$L(\phi^{h_{1,2}},\mathcal{D})=L^{h_{1,2}}_{1}(\phi^{h_{1}},\mathcal{D})+L^{h_{1,2}}_{2}(\phi^{h_{2}},\mathcal{D})$

29

30 Finally, update Q-functions target networks with Polyak averaging:

31 $\hat{\theta}_{i}\leftarrow\rho\hat{\theta}_{i}+(1-\rho)\theta_{i}$ for
$i=1,2$

32 end for

33

34 end if

35

36 end for

Algorithm 1 Multi-Headed Recurrent Soft-Actor Critic

###  IV-B Dealing with Non-Markovian environments in RL

Reinforcement Learning faces some challenges when dealing with non-Markovian
environments. The reason is that RL main goal is to maximize the reward
$R_{t+1}$ given a certain state $S_{t}$ and action $A_{t}$ as:
$R=E[R_{t+1}|S_{t},A_{t}]$. When the previous relationship is not fulfilled
due partial observability of the Markovian state, the observation state should
include more that one input and utilize a portion of the interaction
history[15, 16].

Figure 2: Network structure used in the Multi-Headed Recurrent Soft-Actor
Critic agent. Figure 3: Non-Markovian nature of the scenario: Rewards are not
only dependant on the current state but affected by other arrival flows.

In Fig. 3, we show an example of the non-Markovian behavior of our
environment. For instance, the reward obtained upon the arrival of Flow $1$,
$r1$ is affected by the arrival of Flow $2$. That means that the reward
obtained given the action taken upon Flow $1$ arrival is affected as well by
the action taken upon Flow $2$ arrival as well. Such behavior violates the
assumption of being on the presence of an MDP. To solve the beformentioned
issue, we include as the first layer of the model’s structure of the MH-RSAC a
Long Short-Term Memory (LSTM) neural network as shown in Fig. 2. In addition,
we add two more linear layers after the LSTM layer.

###  IV-C State space selection

As discussed in previous subsections, the environment is modeled as a
Partially Observable Markov Decision Process (POMDP), thus a sequence of the
observation space is considered instead one instance. The action space is
defined as:

$\begin{split}\bm{s}_{t}=[\\{C_{1,t-W}^{o},C_{2,t-W}^{o},C_{3,t-W}^{o},O_{f,t-W},T_{id,t-W}\\},\\\
\\{C_{1,t-W-1}^{o},C_{2,t-W-1}^{o},C_{3,t-W-1}^{o},O_{f,t-W-1},\\\
T_{id,t-W-1}\\},...,\\{C_{1,t}^{o},C_{2,t}^{o},C_{3,t}^{o},O_{f,t},T_{id,t}\\}],\end{split}$
(1)

where $W$ corresponds to the size of the interaction window,
$C_{1}^{o},C_{2}^{o},C_{3}$ the occupancy of $2.4$ GHz, $5$ GHz and $6$ GHz
interface, respectively, $O_{f}$ the ratio of active flows and $T_{id}$
corresponds to the type of the upcoming flow. Furthermore, we define $O_{f}$
and $T_{id}$, respectively as:

$O_{f}=A_{f}/N_{A},$ (2)

where $A_{f}$ corresponds to the number stations receiving traffic flows and
$N_{A}$ the number of stations attached to the corresponding AP.

$T_{id}=\begin{cases}0.33&\text{if }\textit{UHD/4K traffic},\\\ 0.66&\text{if
}\textit{VR traffic},\\\ 1&\text{else }\textit{WB traffic}\end{cases}$ (3)

###  IV-D Action space selection

In this subsection, we proceed to describe the action space of the MH-RSAC
scheme. As discussed in subsection IV-A, the MH-RSAC structure is comprised by
two heads: one providing the action output when two interfaces are available
and another for the case when three are. Thus, the action space can be defined
as:

$\displaystyle\bm{a}_{t}=\begin{cases}\bm{a}_{h1,t}=[\\{a_{1,t-W},a_{2,t-W}\\},\\\
\\{a_{1,t-W-1},a_{2,t-W-1}\\}\\\ ,...,\\{a_{1,t},a_{2,t}\\}],&\text{if
}n_{f}=2,\\\ \bm{a}_{h2,t}=[\\{a_{1,t-W},a_{2,t-W},a_{3,t-W}\\},\\\
\\{a_{1,t-W-1},a_{2,t-W-1},a_{3,t-W-1}\\}\\\
,...,\\{a_{1,t},a_{2,t},a_{3,t}\\}],&\text{if }n_{f}=3\end{cases}$ (4)

where $\bm{a}_{k\in\\{1,2,3\\}}$ refers to the fraction of the total flow
traffic to be allocated to each available interface.

Consequently, the action space size of each head is calculated as the number
of permutations of the possible fractions that sum to one. The previous number
is obtained using the well-known combinatronics “stars and bars”
technique[17]. The size of each head can be calculated as:

$|A|_{h2}=\frac{(n+1)!}{n!},$ (5)

$|A|_{h3}=\frac{(n+2)!}{2!n!},$ (6)

where $n=10$. The previous $n$ value is chosen based on the discretization of
the maximum flow traffic distribution value using a $0.1$ interval. After the
substitution of $n$, we obtain $|A|_{h2}=11$ and $|A|_{h3}=66$.

###  IV-E Reward function

The reward function in the t-$th$ episode is defined as:

$R_{t}=1-D_{t}^{avg},$ (7)

where $D_{i}^{avg}$ corresponds to the average throughput drop ratio observed
by the $m^{th}$ AP. In addition, we scale the reward to $[-1,1]$.

###  IV-F Reward function with hindsight

In [18], the authors present a buffer technique called Hindsight Experience
Replay (HER) that allows the successful application of model-free RL
algorithms in environments characterized by sparse rewards. The idea involves
the usage of goals states to improve the sample efficiency and convergence.
Differently from the previous work, we propose to utilize goals not in the
state space but in the reward function itself. Thus, we define the term goal
reward as the threshold in which given a reward from the environment a
penalization is applied to the composite reward function as described as
follows:

$R_{t}^{h}=\begin{cases}R_{t}&\text{if }D_{t}^{avg}<D_{TOL},\\\
-1&\textit{otherwise}\end{cases}$ (8)

where $D_{TOL}$ corresponds to a hindsight reward based on baselines results
or on expert knowledge. The intuition behind the before mentioned formulation
relays on the more number of times the agent obtains a reward worse than the
goal reward given an action $\bm{a}$, the stronger the indication that the
action taken was not positive given an observed state. Eventually, the agent
will be able to learn from those undesired actions and start learning from the
desired ones.

###  IV-G Complexity Analysis

The complexity of our proposed MH-RSAC scheme corresponds to the complexity of
the neural network comprising the SAC base algorithm. Thus, the training
complexity is $O(\mathcal{H}\cdot\mathcal{W}\cdot\frac{|\mathcal{D}|}{B}\cdot
T)$, where $\mathcal{H}$ represents the learning update periodicity,
$\mathcal{W}$ the number of updates per session, $\mathcal{D}$ the buffer
size, $B$ the batch size and $O(T)$ denotes the time complexity of a single
iteration.

###  IV-H Baselines: SLCI and MCAA

In this work, we compare our proposed scheme with two non-RL approaches
described in [5].

Single Link Less Congested Interface (SLCI): The traffic is allocated to the
less congested interface.

Multi-Link Congestion-aware Load balancing at flow arrivals (MCAA): The
traffic is distributed among the available interfaces given the observed
occupancy per interface at the AP.

Note that, in a recent letter [9765523] the authors have presented a dynamic
traffic allocation proposal. However, different than the baselines and our
proposed approach, such policy could modify the flow distribution over the
available interfaces periodically without the constrain of only doing so upon
an incoming flow.

##  V Traffic considerations

As previously mentioned, IEEE 802.11be aims to respond to the increasing
demand of high-throughput services. Among them, we encounter Virtual Reality
(VR). In [4], the authors present an empirical model based on two campaigns of
VR gaming data measurements. The first campaign is run using a VR game (“Beat
Saber”) in a local server and the second in a cloud server, respectively. In
both cases, the frame size and frame inter-arrival time are measured using
Wireshark. Results showed that the frame size behaved as a loglogistic
distribution and the frame inter-arrival time as a Burr distribution. In this
work, we utilize the cloud server model and derive the Cumulative Distribution
Functions (CDF) to be utilized as one of our simulation traffic flows. The
derivation of the inverse of the CDF for the frame inter-arrival time and
frame size is described below:

Figure 4: Identified frame inter-arrival time CDF Figure 5: Identified frame
size CDF

$F_{fs}(x)=\int_{0}^{\infty}\frac{e^{\frac{\ln(x)-\mu}{\sigma}}}{\sigma
x(1+e^{\frac{\ln{x}-\mu}{\sigma}})^{2}}\,dx=-\frac{1}{e^{\frac{\ln(x)-\mu}{\sigma}}+1}+C,$
(9)

Figure 6: Reward and TDR convergence for the “avg”, “min” operator of the MH-
RSAC’s variants in the U2 scenario: (a) Reward with hindsight $(R_{h})$, (b)
Reward without hindsight $(R_{nh})$ and (c) TDR convergence.

where $F_{fs}$ corresponds to the frame size CDF, $\mu$ and $\sigma$ are the
statistical model parameters of the loglogistic distribution and $C$ a
constant related to the mathematical integration. Finally, the inverse of the
CDF in bytes is:

$F_{fs}^{-1}(y)=e^{\mu}(-\frac{y}{y-1})^{\sigma}[bytes],$ (10)

The frame inter-arrival time can be described as follows:

$F_{ft}(x)=\int_{0}^{\infty}\frac{\frac{kc}{a}(\frac{x}{a})^{c-1}}{\left(1+(\frac{x}{a})^{c}\right)^{k+1}}\,dx=-\frac{1}{\left((\frac{x}{a})^{c}+1\right)^{k}}+C,$
(11)

where $F_{ft}$ corresponds to the frame inter-arrival time CDF, $k$, $a$ and
$c$ are the statistical model parameters of the Burr distribution. The inverse
of the CDF in ms is:

$F_{ft}^{-1}(y)=a\sqrt[c]{\sqrt[k]{\frac{1}{1-y}}-1}[ms],$ (12)

Finally the VR flow size in Mbps can be calculated as:

$f_{VR}(y)=\frac{F_{fs}^{-1}(y)}{F_{ft}^{-1}(y)}[Mbps],$ (13)

where $y$ is randomly sampled as $y\sim{\mathcal{U}(0,1]}$. In Fig. 4 and 5,
we present the VR CDFs for the frame inter-arrival and frame size considering
AP bandwidth throttling of $54$ Mbps, $40.5$ Mbps and $27$ Mbps and no AP
throttling (Normal), respectively. Note that only downlink traffic is
considered and all traffic flows are modeled as a single Constant Bit Ratio
(CBR) flow. For this work purpose the VR traffic generated corresponds to the
non-throttling case (Normal). Besides the VR traffic, we assume the
coexistence of two more traffic flows. One depicting an HD/4K flow with a size
of $f_{4K}\sim{\mathcal{U}(7,25)}$ Mbps and a web browsing traffic with a size
of $f_{WB}\sim{\mathcal{U}(1,3)}$ Mbps. At the beginning of each simulation,
the data flow types are distributed among the stations attached to their AP as
follows: {WB: $0.8$, V4K: $0.1$, VR: $0.1$}.

##  VI Performance evaluation

Simulations are performed using the flow-level simulator Neko 802.11be[5] and
Pytorch-based RL agents. The communication between the simulator and the RL
agents is done using the ZMQ broker library.

###  VI-A Simulation Settings

Simulation settings and RL parameters utilized in this work are depicted in
Table I and II, respectively. Furthermore, we consider two load distribution
scenarios: U1 with number of users attached to its corresponding AP,
$N_{A}\sim\mathcal{U}(15,20)$ and U2 with $N_{A}\sim\mathcal{U}(20,25)$. In
the next subsection we will discuss the performance results of the proposed
scheme.

TABLE I: Network Settings

Parameter | Value
---|---
IEEE protocol | 802.11be
Channel Bandwidth |  $20$ MHz/$40$ MHz/$80$ MHz/$160$ MHz
Carrier Frequency ($f_{c}$) |  $2.437$ GHz/$5.230$ GHz/$6.295$ GHz
Max Modulation/Max Modulation Coding Scheme | 4096 QAM/MCS 13
Number of APs | 5
Number of Stations per AP |  U1 : $N_{A}\sim\mathcal{U}(15,20)$, U2 : $N_{A}\sim\mathcal{U}(20,25)$
Max Spatial Streams | 16
Propagation Loss Model |  $P_{l}(d)=40.05+20\text{log}(f_{c}/2.4)+20\text{log}(\text{min}(d,10))+$
| $(d>10)*35\text{log}(d/10)+7W$
Data Flow Types | Web browsing (WB), Video 4K (V4K), Virtual Reality (VR)
Data Flow User Distribution |  WB: $0.8$, V4K: $0.1$, VR: $0.1$
AP/STA Noise Figure |  $7$ dB
AP/STA Transmission Power |  $20$/$15$ dBm
CCA threshold | -82 dBm
Packet Error Rate |  $10$%

Figure 7: (a) Throughput Drop Ratio (TDR) per load distribution U1 and U2 and
Flow Satisfaction (FS) per traffic for Video HD/4K (V4K), Virtual Reality (VR)
and Web Browsing (WB) per load distribution (b) U1 and (c) U2

###  VI-B Simulation Results

We present the performance results of our proposed scheme in terms of
convergence, FS per interface and TDR. Figure 6 shows the convergence behavior
of the U2 scenario for the SAC variants proposed in this work. The figures
that depict the scenario U1 are not shown in this work, as they show similar
information. Figure 6 (a) shows how the “avg” operator outperforms the “min”
operator when the reward with hindsight is considered. Conversely, the
previous behavior does not repeat when the rewards without hindsight are
utilized as shown in Fig. 6 (b). Finally, we show the convergence in terms of
TDR for the four variants in Fig. 6 (c) where the best convergence is achieved
when the “avg”operator with rewards with hindsight are used. The previous
results confirm that the goal rewards strategy proposed in this work affects
positively to the learning process in non-Markovian scenarios.

TABLE II: Reinforcement Learning Settings  Parameter | Value
---|---
Actor learning rate |  $1\mathrm{e}{-4}$
Actor final layer activation function | Softmax
Critic learning rate |  $1\mathrm{e}{-3}$
Critic final layer activation function | None
Linear hidden layers |  $[64,64]$
Window ($W$) | 10
Tau ($\tau$) |  $5\mathrm{e}{-3}$
Discount rate | $0.99$
Gradient clipping norm | $1$
Weight initializer | Xavier
Batch size |  $512$
Learning updates periodicity ($\mathcal{H}$) |  $50$ steps
Learning updates per learning session ($\mathcal{W}$) |  $10$
Automatic entropy tuning | Enabled

Furthermore, in Fig. 7 we present the results in terms of TDR and FS per
traffic type for U1 and U2 scenarios, respectively. In Fig. 7 (a), we show the
TDR performance of the MH-RSAC scheme with “avg” operator and rewards with
hindsight and the two baselines: SLCI and MCAA. Consequently, we observe that
the MH-RSAC scheme offers a gain of $34.2\%$ and $35.2\%$ when compared with
the SLCI algorithm and $2.5\%$ and $6\%$ when compared with the MCAA algorithm
for the U1 and U2 scenarios, respectively. Additionally, we present a
comparison in terms of FS per traffic type in Fig. 7(b) and (c). For instance,
in Fig. 7 (b) we can see that the MH-RSAC scheme’s gains for the V4K traffic
around $1\%$ but when it comes to the VR traffic which is a more throughput
hungry service SLCI is not capable to perform positively with a reduction of
$25.6\%$ in terms of FS. Moreover, for the dynamic WB traffic the MH-RSAC
scheme shows an improvement of $2.5\%$ and $4\%$ over the SLCI and MCAA
algorithms. In a similar fashion in Fig. 7 (c), we obtain a gain of $21\%$ and
$3\%$ for the VR traffic and $3.2\%$ and $6\%$ for the WB traffic of the MH-
RSAC algorithm over the two baselines SLCI and MCAA, respectively. To sum, we
can see that MH-RSAC is capable of distributing high throughout traffic over
multiple links and also respond efficiently to more dynamic traffic types like
Web Browsing.

##  VII Conclusions

In this paper, we presented a Soft-Actor Critic (SAC) Reinforcement Learning
(RL) agent to improve performance metrics such as Throughput Drop Ratio (TDR)
and Flow Satisfaction (FS) in IEEE 802.11be MLO capable networks. More
specifically, we proposed an agent named Multi-Headed Recurrent Soft-Actor
Critic (MH-RSAC) that is capable to deal with different types of Multi-Link
Devices (MLD)s. We included the modifications required in its design such as
the “avg” operator to reduce underestimation in the SAC algorithm. In
addition, we acknowledged the non-Markovian nature of the scenario under study
and included two main techniques that involve the usage of Long Short-Term
Memory (LSTM) neural networks and rewards with hindsight. Moreover, we
presented the derivation of a Virtual Reality (VR) traffic to be further
utilized in any wireless simulator. We compared the proposed RL variants in
terms of convergence and observed the best performance for the case of the MH-
RSAC with “avg” operator and utilization of rewards with hindsight (MH-
RSAC$(Q_{avg},R_{h})$). Furthermore, we presented the simulation results of
our proposed scheme and two previously defined baselines named Single Link
Less Congested Interface (SLCI) and Multi Link Congestion-aware Load balancing
at flow arrivals in terms of TDR and FS per traffic type. Results showed that
MH-RSAC$(Q_{avg},R_{h})$ outperforms in terms of TDR the SLCI baseline with an
average gain of $34.2\%$ and $35.2\%$ and the MCAA baseline with $2.5\%$ and
$6\%$ in the U1 and U2 proposed scenarios, respectively. Finally, we observed
that our scheme is able to respond more efficiently to high throughput and
dynamic traffic such as VR and Web Browsing (WB) when compared with the
baselines. Results showed an improvement of the MH-RSAC scheme in terms of FS
of up to $25.6\%$ and $6\%$ for the SCLI and MCAA, respectively. As future
work we intend to study with more details how to tackle more effectively non-
Markovian scenarios in wireless networks and more specifically in IEEE
802.11be MLO capable networks.

##  VIII Acknowledgment

This research is supported by Mitacs Accelerate Program and NetExperience Inc.

## References

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Present address: ]Universisdad Santo Tomás, 15001 Tunja, Colombia. Present
address: ]Department of Physics and Astronomy, Michigan State University, East
Lansing, MI 48824-1321, USA. Present address: ]Institutionen för Fysik,
Chalmers Tekniska Högskola, 41296 Göteborg, Sweden. Present address:
]Department of Physics, Central Michigan University, Mount Pleasant, MI 48858,
USA. Present address: ]EC-JRC, Institute for Reference Materials and
Measurements, Retieseweg 111, B-2440 Geel, Belgium. Present address: ]National
Superconducting Cyclotron Laboratory, Michigan State University, East Lansing,
MI 48824-1321, USA. Present address: ]Niigata University, 8050 Ikarashi 2-no-
cho, Nishi-ku, 950-2181 Niigata, Japan. Present address: ]Niigata University,
8050 Ikarashi 2-no-cho, Nishi-ku, 950-2181 Niigata, Japan.

# Systematic study of $\Delta$(1232) resonance excitations using single
isobaric charge-exchange reactions induced by medium-mass projectiles of Sn

J. L. Rodríguez-Sánchez<EMAIL_ADDRESS>IGFAE, Universidad
de Santiago de Compostela, E-15782 Santiago de Compostela, Spain  GSI-
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  J.
Benlliure  IGFAE, Universidad de Santiago de Compostela, E-15782 Santiago de
Compostela, Spain  I. Vidaña  INFN, Sezione di Catania, Dipartimento di Fisica
“Ettore Majorana”, Università di Catania, I-95123 Catania, Italy  H. Lenske
Institut für Theoretische Physik der Justus-Liebig Universität Giessen,
D-35392 Giessen, Germany  J. Vargas  [  Universidad de Santiago de Compostela,
E-15782 Santiago de Compostela, Spain  C. Scheidenberger  GSI-Helmholtzzentrum
für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  H. Alvarez-Pol
IGFAE, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela,
Spain  J. Atkinson  GSI-Helmholtzzentrum für Schwerionenforschung GmbH,
D-64291 Darmstadt, Germany  T. Aumann  GSI-Helmholtzzentrum für
Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  Institut für
Kernphysik, Technische Universityät Darmstadt, D-64289 Darmstadt, Germany  Y.
Ayyad  IGFAE, Universidad de Santiago de Compostela, E-15782 Santiago de
Compostela, Spain  S. Beceiro-Novo  [  Universidad de Santiago de Compostela,
E-15782 Santiago de Compostela, Spain  K. Boretzky  GSI-Helmholtzzentrum für
Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  M. Caamaño  IGFAE,
Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain
E. Casarejos  CINTECX, Universidad de Vigo, E-36200 Vigo, Spain  D. Cortina-
Gil  IGFAE, Universidad de Santiago de Compostela, E-15782 Santiago de
Compostela, Spain  P. Díaz Fernández  [  Universidad de Santiago de
Compostela, E-15782 Santiago de Compostela, Spain  A. Estrade  [  GSI-
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
Saint Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada  H. Geissel
GSI-Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
E. Haettner  GSI-Helmholtzzentrum für Schwerionenforschung GmbH, D-64291
Darmstadt, Germany  A. Kelić-Heil  GSI-Helmholtzzentrum für
Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  Yu. A. Litvinov  GSI-
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  C.
Paradela  [  D. Pérez-Loureiro  [  Universidad de Santiago de Compostela,
E-15782 Santiago de Compostela, Spain  S. Pietri  GSI-Helmholtzzentrum für
Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  A. Prochazka  [  GSI-
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  M.
Takechi  [  GSI-Helmholtzzentrum für Schwerionenforschung GmbH, D-64291
Darmstadt, Germany  Y. K. Tanaka  GSI-Helmholtzzentrum für
Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  High Energy Nuclear
Physics Laboratory, RIKEN, Saitama 351-0198, Japan  H. Weick  GSI-
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  J.
S. Winfield  GSI-Helmholtzzentrum für Schwerionenforschung GmbH, D-64291
Darmstadt, Germany

###### Abstract

The fragment separator FRS has been for the first time used to measure the
$(n,p)$ and $(p,n)$-type isobaric charge-exchange cross sections of stable
112,124Sn isotopes accelerated at 1$A$ GeV with an uncertainty of $3\%$ and to
separate quasi-elastic and inelastic components in the missing-energy spectra
of the ejectiles111This work comprises part of the Ph.D. thesis of J. Vargas..
The inelastic contribution can be associated to the excitation of isobar
$\Delta$(1232) resonances and to the pion emission in s-wave, both in the
target and projectile nuclei, while the quasi-elastic contribution is
associated to the nuclear spin-isospin response of nucleon-hole excitations.
The data lead to interesting results where we observe a clear quenching of the
quasi-elastic component and their comparisons to theoretical calculations
demonstrate that the baryonic resonances can be excited in the target and
projectile nuclei. To go further in this investigation, we propose to study
the excitation of baryonic resonances taking advantage of the combination of
high-resolving power magnetic spectrometers with the WASA calorimeter. These
new measurements will allow us to determine the momenta of the ejectiles and
pions emitted in coincidence after the single isobaric charge-exchange
collisions, providing us unique opportunities to study the evolution of the
baryonic resonance dynamics with the neutron-proton asymmetry through the use
of exotic radioactive ion beams.

††preprint: APS/123-QED

##  I Introduction

The understanding of the nucleon and its excited internal structure, forming
the so-called baryonic resonances, still remains a fundamental challenge in
hadronic physics. The experimental and theoretical study of the baryon excited
states helps to improve our knowledge of Quantum Chromodynamics (QCD)
Greenberg1964  in the non-perturbative regime, especially for understanding
the composition of a baryon as a superposition of hadronic and quarkonic
components. Experimental information about the mass, width, and decay modes of
baryonic resonances in free space and in nuclear environments is crucial for
validating theoretical models describing the internal structure of the nucleon
and its excited states. Moreover, the investigations of baryon resonances in
nuclear matter are a natural extension of nuclear physics at the intersection
of nucleonic and subnuclear degrees of freedom, elucidating open problems as
the role of resonances in three-body interactions Krebs2018 ; Feng2016  and
neutron star matter Li2018 ; Li2019 .

In the literature, the lowest-lying baryon resonance is the $\Delta$(1232)
state. This $\Delta$-resonance is the first spin-isospin excited mode of the
nucleon corresponding to a $\Delta S=1$; $\Delta I=1$ transition, seen
prominently as an elastic p-wave resonance in pion-nucleon scattering. That
state is closely connected with the excitation of the well known GT resonance
as already seen in nuclear collisions Sakai1999 ; Liang2018 . Charge-exchange
excitations of the nuclear GT-states and the nucleonic $\Delta$-resonance are
both related to the action of the same spin-isospin transition operators
Oset1979 ; Arima2001 . Actually, a long standing problem of nuclear structure
physics is to understand the coupling of these two modes. Here, we would like
to remark that in the last decade a large effort has been carried out to
understand the quenching problem of nuclear GT-states at low momentum
transfer, leading to the development of state-of-the art model calculations
Gysbers2019  that provide a reasonable description of the experimental data.
However, at high momentum transfer the quenching of the GT strength Sakai2004
remains still unsolved, which is mainly attributed to a large redistribution
of transition strength due to the coupling of $\Delta$-particle–nucleon-hole
and purely nucleonic particle–hole excitations Lenske2018 ; Lenske2019 .
Similar mechanisms, although not that clearly seen, are present in the high-
energy flank of the Fermi-type spectral distribution, i.e. the non-spin flip
charge-exchange excitations mediated by the isospin transition operator alone,
caused by the coupling to the Roper resonance Morsch1992 . Concerning its
properties, modifications of the mass and width of the $\Delta$-resonance have
been observed in central heavy-ion collisions leading to dense and heated
hadron matter. In Ref. Hjort1997 , for example, the mass shift and the width
were determined as functions of the centrality, both showing a substantial
reduction with decreasing impact parameter. The modifications of the $\Delta$
properties has been interpreted in terms of the values of hadronic density,
temperature, and various non-nucleon degrees of freedom in nuclear matter
Hennino1982 . However, many experimental findings on baryon resonance
production are obtained after non-trivial analysis that usually result in
ambiguities with their theoretical interpretations. Partly, the persisting
uncertainties are related to difficulties in extracting the spectral
structures from data. As discussed in Ref. Trzaska1991 , in heavy-ion
collisions a major obstacle for the reconstruction of the resonance spectral
distributions is the large background of uncorrelated p$\pi$ pairs coming from
other sources. This problem has been found to be less severe in peripheral
reactions.

A renewed interest on the in-medium properties of baryon resonances is
motivated by the challenge to understand the composition of neutron stars. The
use of realistic equations of state with updated values for the density
derivative of the symmetry energy yields a threshold density for the
appearance of $\Delta$-isobars similar to the one for hyperons Drago14 ; Cai15
; Drago16 . Recently, the study of magnetic field effects on neutron star
matter has pointed out that the vector-isovector self-interaction for the
mesons in combination with a strong magnetic field enhances the
$\Delta$-baryon population in dense matter, while decreasing the relative
density of hyperons Dexheimer2021 . These findings also allow for a better
agreement with the observations of neutron star masses, indicating that likely
magnetic neutron stars also contain $\Delta$-baryons in their interior.

Pioneering experiments showing for the first time the in-medium excitation of
the $\Delta$-resonance in peripheral single isobaric charge-exchange reactions
were performed in the 1980’s at the Laboratoire National SATURNE in Saclay
(France). In these experiments, the ejectiles produced in collisions of stable
relativistic projectiles of 3He, 12C, 20Ne, and 40Ar on different target
materials were identified and momentum analyzed with the magnetic spectrometer
SPES-IV Grorud1981 ; Ellegaard1985 ; Contardo1986 ; Roy1988 . The high-
resolving power of the spectrometer made it possible to prove the excitation
of the $\Delta$-resonance from the velocity recoil experienced by the
ejectiles due to the pion emission. Slightly later, similar experiments were
also perfomed in the Synchro-Phasetron at the JINR Dubna facility as discussed
in Ref. Lenske2018 .

Following SATURNE’s research, the first investigations at the GSI accelerator
facility in Darmstadt (Germany) were performed in the 1990’s by Sümmerer and
collaborators Summerer1995  with medium-mass projectiles of 129Xe at a kinetic
energy of 790$A$ MeV. This experiment was carried out at the fragment
separator FRS Geissel1992 , in which the individual charge-exchange products
were identified by charge and mass number. Their inclusive cross sections were
used to investigate the effects due to the $\Delta$-resonance production
through the comparison to model calculations, showing that in the $(n,p)$-type
isobaric charge-exchange reaction the presence of this resonance can increase
the cross section by a factor of two. Latter, more experiments were perfomed
by the CHARMS collaboration employing medium-mass and heavy projectiles of
136Xe Benlliure2008  and 209Bi Kelic2004 , measuring simultaneously the cross
sections and missing-energy distributions of the single isobaric charge-
exchange products with resolutions of around 50 MeV Vargas2013 . More
recently, the missing-energy resolution has been improved down to 10 MeV
Rodriguez2020  by using time-projection chambers (TPCs) Janik11  with a
position resolution of 300 $\mu m$ full width at half maximum (FWHM) and owing
to the reduction of layers of matter in the beam line.

In the present work, we use single isobaric charge-exchange reactions produced
with medium-mass projectiles of 112,124Sn impinging on different target
nuclei, which allow us to cover a large range in neutron excess. The
experiment and setup employed for the measurements are described in Sec. II,
while the experimental results are shown in Sec. III. For the data
interpretation we use a theoretical model decribed in Sec. IV and the
discussion is presented in Sec. V. Finally, the main conclusions and the
future perspectives are given in Sec. VI.

##  II Experimental approach

The experiment has been carried out at the GSI facility using the SIS-18
synchrotron combined with the FRS spectrometer Geissel1992 . The FRS is a
zero-degree high-resolving power spectrometer with a typical resolving power
of $B\rho/\Delta B\rho=1500$ and with an angular acceptance of $\pm$15 mrad
around the central trajectory. Beams of 112Sn and 124Sn, accelerated up to
kinetic energies of 1$A$ GeV, were delivered by the SIS-18 with an intensity
around 108 ions per spill and guided up to the FRS entrance to produce the
single isobaric charge-exchange reactions in different target nuclei. The FRS
spectrometer was used in the achromatic mode, where the reaction target is
placed at the FRS entrance and the full spectrometer is utilized to separate
and to identify the projectile residues, as shown in Fig. 1.

The measurements were performed using polyethylene (CH2), Carbon, Copper, and
Lead targets with thicknesses of 95, 103, 373 and 255 mg$/$cm2, respectively.
The experimental data with the carbon target was also used to subtract the
contribution of the carbon-induced reactions from the CH2 target and obtain
the proton contribution. An additional measurement was performed without
target in order to obtain the background of reactions coming from other layers
of matter located at the FRS target area, which was of around $1\%$.

Figure 1: (Color online) Schematic view of the FRS experimental setup used in
the present work with the reaction target located at the entrance of the FRS
spectrometer to induce the single isobaric charge-exchange reactions.

The magnetic ridigity $B\rho$ of each reaction product passing through the FRS
can be obtained with the TPC detectors Janik11  located at the FRS focal
planes on an event-by-event basis. For the intermediate focal plane (S2) the
magnetic ridigity ($B\rho^{S2}$) was determinated as follows:

$B\rho^{S2}=B\rho_{0}^{ref}\left(1-\frac{x_{2}}{D_{S2}}\right),$ (1)

where $x_{2}$ is the horizontal position at the intermediate focal plane,
$D_{S2}$ is the value of the dispersion from the production target up to the
focal plane S2, and $B\rho_{0}^{ref}$ is the magnetic ridigity of a central
trajectory along the first section of the FRS.

In the case of the final focal plane (S4), the magnetic ridigity
($B\rho^{S4}$) was obtained according to:

$B\rho^{S4}=B\rho^{ref}_{0}\left(1-\frac{x_{4}-Mx_{2}}{D_{S4}}\right),$ (2)

where $x_{4}$ is the position at the final focal plane S4, $D_{S4}$ is the
dispersion between both focal planes, S2 and S4, and $M$ is the magnification.

In Eqs. (1) and (2), the magnetic rigidity of the central trajectory as well
as the dispersions and the magnification were determined by measuring the
trajectory of 112Sn projectiles at 1$A$ GeV for different values of the
magnetic fields in the dipoles of the spectrometer. These measurements allowed
us to calibrate the FRS optics, obtaining for the middle and final focal plane
dispersions $D_{S2}=-7.48$ cm/$\%$ and $D_{S4}=7.56$ cm/$\%$, respectively,
with an uncertainty of $0.02$ cm/$\%$ in both cases. These quantities also
provided us the magnification of the spectrometer according to
$M=D_{S4}/D_{S2}$, resulting in $M$=($-1.011\pm 0.004$).

Nuclei transmitted through the FRS were identified in mass over charge ($A/q$)
through the determination of the magnetic ridigity at the final focal plane
(see Eq. (2)) and the fragment velocity ($v$) according to Refs. Teresa14 ;
JL2017 :

$\frac{A}{q}=\frac{eB\rho^{S4}}{u\gamma\beta c},$

where $q$ is the atomic number if we assume that the fragments were completely
stripped ($q=Z$), $u$ is the atomic mass unit, $e$ is the elementary charge,
$c$ is the speed of the light, $\beta=v/c$ and $\gamma=1/\sqrt{1-\beta^{2}}$.
The velocity of fragments was obtained by combining the S2-S4 path length with
the time-of-flight (ToF) measurements performed with the plastic scintillators
located at the focal planes of the FRS, which provided the ToF with a
resolution of around 150 ps (FWHM). Additionally, the measurement of the
atomic number $Z$ was provided by two multi-sampling ionization chambers
(MUSIC) located in the focal planes of the FRS.

Figure 2: (Color online) Identification matrix at the final focal plane of the
FRS obtained from reactions induced by 112Sn impinging on the carbon target
where one can identify the single isobaric charge-exchange residues 112Sb and
112In. In addition, the single electron charge state of 112Sn is marked by a
dotted ellipse.

As a result, Fig. 2 displays the identification matrix of the fragmentation
residues produced by the projectiles of 112Sn impinging on the carbon target.
The figure was obtained by overlapping three magnetic settings of the FRS
centered on the isotopes 112Sb, 112Sn, and 112In. The calibration in atomic
and mass number was performed with respect to the signals registered for the
beam of 112Sn. In the figure, one can see the good resolution in atomic and
mass number achieved in this measurement: $\Delta Z/Z$=6 $\times$ 10-3 and
$\Delta A/A$=1 $\times$10-3 (FWHM), respectively.

##  III Results

Total isobaric charge-exchange cross sections were accurately determined by
normalizing the production yield of the charge exchange residual nuclei to the
number of projectiles and target nuclei. The number of incoming projectiles
was obtained by using a secondary electron monitor (SEETRAM) Junghans1996
placed at the entrance of the FRS, as shown in Fig. 1, which was calibrated
with a reference plastic scintillator Voss1995 . The uncertainty in the
determination of the incoming projectile intensity was of 3$\%$. The number of
isobaric charge exchange residues was obtained from the corresponding
identification matrix at the final focal plane by gating on the fragment of
interest, as shown in Fig. 2 for the single isobaric charge-exchange residues
of 112Sb and 112In. The number of counts for these residues was then corrected
by the detection efficiency according to the procedure detailed in Refs.
Teresa14 ; JL2017 . The first correction factor is related to the use of high-
intensity beams, which affect the determination of the cross sections because
of the acquisition dead time. This correction was evaluated from the ratio
between the number of accepted and total triggers registered by the data
acquisition system. This value was kept below 20$\%$ during the experiment by
reducing the optical transmission of other fragments. This was carried out by
closing the slits located between the first and second FRS dipole (see Fig.
1), leaving a hole of 40 mm. The second correction is due to secondary
reactions that occur in the different layers of matter along the beam line,
which were taken into account by calculating with the KAROL code karol75  the
total probability of interaction in each layer of matter. The obtained
correction factors were less than 2$\%$. The third correction is to take into
account the population of atomic charge states due to electromagnetic
interactions of the nuclear residues with the different layers of matter along
the beam line, which can alter the measurement of the $A/q$ ratio and
therefore affects the number of counts used in the determination of the cross
section Alcantara15 ; Paradela2017 . In order to determine these losses we
used the code GLOBAL global , being this correction less than $1\%$. Finally,
we remark that the FRS ion-optical transmission for the isobaric charge-
exchange residues investigated in this work was of 100$\%$. The final values
obtained for the total isobaric charge-exchange cross sections are listed in
Table 1.

The missing-energy spectra of the recoiling $(p,n)$ and $(n,p)$-type charge-
exchange residues were obtained by measuring the ejectile kinetic energies
with respect to the projectile one determined in the middle of the target,
being the kinetic energy ($E_{k}$) calculated as:

$E_{k}=\sqrt{P+M}-M,$

where $M$ is the nuclear mass and $P$ is the momentum, obtained as follows:

$P=B\rho^{S2}Z,$

where $Z$ is the atomic number and $B\rho^{S2}$ is determinated according to
Eq. (1), taking also into account the beam-extraction time correction
Tanaka2018  to reach the best FRS resolution.

Table 1: Total, quasi-elastic (qe), and inelastic (in) cross sections of the single isobaric charge-exchange reactions measured in the present work. The uncertainties include statistical and systematic contributions. Reaction |  $\sigma_{Total}$ [$\mu$b] |  $\sigma_{qe}$ [$\mu$b] |  $\sigma_{in}$ [$\mu$b]
---|---|---|---
p(112Sn,112In)X | 291 $\pm$ 9 | - | 291 $\pm$ 9
p(112Sn,112Sb)X | 604 $\pm$ 19 | 235 $\pm$ 10 | 369 $\pm$ 15
12C(112Sn,112In)X | 705$\pm$ 22 | 223 $\pm$ 9 | 432 $\pm$ 20
12C(112Sn,112Sb)X | 718 $\pm$ 22 | 276 $\pm$ 16 | 442 $\pm$ 26
Cu(112Sn,112In)X | 809 $\pm$ 25 | 307 $\pm$ 18 | 502 $\pm$ 30
Cu(112Sn,112Sb)X | 758 $\pm$ 23 | 235 $\pm$ 16 | 523 $\pm$ 33
Pb(112Sn,112In)X | 1000 $\pm$ 31 | 465 $\pm$ 28 | 535 $\pm$ 31
Pb(112Sn,112Sb)X | 793 $\pm$ 25 | 198 $\pm$ 14 | 595 $\pm$ 39
p(124Sn,124Sb)X | 585 $\pm$ 19 | 203 $\pm$ 9 | 382 $\pm$ 17
C(124Sn,124Sb)X | 705 $\pm$ 22 | 220 $\pm$ 10 | 485 $\pm$ 22

The results are displayed in Fig. 3, where we show the missing-energy spectra
obtained from the $(n,p)$ (left panels) and $(p,n)$ (right panels) isobaric
charge-exchange reactions induced by ions of 112Sn in the Pb, Cu, 12C and
proton targets at projectile kinetic energies of 1$A$ GeV, which were
normalized to the cross section of each reaction (see Table 1). The
contribution of the proton target was obtained from the CH2 spectrum by
subtracting the carbon contribution, according to the equation:
$\sigma_{p}=0.5\times[\sigma_{CH_{2}}-\sigma_{C}]$, applied on a bin-by-bin
basis. For all the spectra, the background was subtracted on a bin-by-bin
basis as well. The same methodology was applied to the $(n,p)$-type isobaric
charge-exchange measurements performed with the primary beam of 124Sn
impinging on the 12C and proton targets, displayed in the upper and lower
panels of Fig. 4, respectively. Thanks to the high-resolving power of the FRS
together with a sizeable reduction of matter along the beam line, we succeed
to measure these spectra with a resolution of 10 MeV, which is a factor of two
better than the one reached in the SATURNE experiments Roy1988 . Because these
spectra are affected by the energy and angular straggling of ions passing
through the targets, as observed in previous experiments Kelic2004 , an
unfolding procedure based on the Richardson-Lucy’s technique Lucy1974
together with a regularization method to optimize the stability of the
solution against statistical fluctuations Vargas2013  was used to improve the
sharpness of the spectra. This unfolding approach was applied to both Sn
projectiles using for the FRS response function the missing-energy spectrum of
the primary beam passing through the corresponding target after correcting the
local energy-straggling due to the exchange of a single proton with the target
nucleus.

Figure 3: (Color online) Missing-energy spectra obtained from the Pb, Cu, 12C
and proton targets for the single isobaric charge-exchange reactions
(112Sn,112Sb) and (112Sn,112In). The quasi-elastic and inelastic contributions
are displayed with gray and brown histograms, respectively.  Figure 4: (Color
online) Missing-energy spectra obtained from the 12C (a) and proton (b)
targets for the single isobaric charge-exchange reaction (124Sn,124Sb). The
quasi-elastic and inelastic contributions are displayed with gray and brown
histograms, respectively.

These spectra show that the spin-isospin response of our projectile nuclei is
concentrated in two energy domains, in which the momentum transfer is larger
than 100 MeV/c. The first group of excitations starting at missing energies
close to zero is of quasi-elastic origin. This region is dominated by quasi-
free charge-exchange processes between projectile and target nucleons,
primarily reflecting global intrinsic properties of the colliding nuclei like
Fermi motion and, to a lesser extent, isovector mean-field dynamics.
Collective isovector multipole modes could contribute as well, but their
identification requires a multipole decomposition of the spectra as perfomed
for light ion reactions PhysRevC.86.014304 . In this region, nucleon-nucleon
correlations could also contribute to the spectra by increasing the kinetic
energy of the outgoing isobaric charge-exchange residues Litvinova2016 ,
resulting in positive missing-energy tails observed to the right of the quasi-
elastic peaks. Note that the quasi-elastic contribution for the (112Sn,112In)
reaction with the proton target disappears, as expected, because this
transition needs a target containing neutrons. At lower missing energies, the
observed bump corresponds to a nucleon being excited into a $\Delta$-resonance
in the target or projectile nuclear systems. This twofold spectrum is a common
feature of all isobaric charge-exchange reactions induced at high kinetic
energies (more than 0.4$A$ GeV) and illustrates clearly that the $\Delta$ is
the first spin-isospin excited mode of the nucleon, corresponding to a $\Delta
S=1$ and $\Delta I=1$ spin-isospin change as forementioned above Lenske2018 .
In the spectra shown in Figs. 3 and 4, one can also see a clear energy shift
of around 63 MeV for the inelastic peak between the proton and the other
targets. This will be discussed later in Sec. V.

Figure 5: (Color online) Total, quasi-elastic (qe), and inelastic cross
sections as a funtion of the neutron excess of target nuclei for the
(112Sn,112In) (a) and (112Sn,112Sb) (b) reactions. We also compare these data
to the cross sections obtained for the (124Sn,124Sb) reaction. Lines represent
the best data fit.

The dependence on the target neutron excess of the total isobaric charge-
exchange cross section is illustrated in Figs. 5 and 5 for the (112Sn,112In)
and (112Sn,112Sb) charge-exchange reactions, respectively. As expected, the
cross section of the isobaric charge-exchange reactions increases with the
size of the target nuclei, like it happens with the total interaction cross
section Ozawa2001 . In these two figures, we also display the cross sections
for the quasi-elastic and inelastic components listed in Table 1, where we
have fitted the quasi-elastic peak in the spectra to a generalized asymmetric
Breit-Wigner function in order to extract its cross section, following the
methodology proposed in Ref. Seitz1970 . On the one hand, we can see that the
quasi-elastic and inelastic cross sections for the (112Sn,112In) reaction
increase with the target size, although this increase is steeper for the
quasi-elastic peak. The best fit of the data also allows us to extrapolate the
measurements to larger target neutron excess, where one can see that for very
neutron-rich target nuclei, $N/Z$ larger than 1.7, the predominant
contribution to the total cross section would be the quasi-elastic channel. On
the other hand, for the (112Sn,112Sb) reaction we can observe that the quasi-
elastic cross section decreases with the target neutron excess, nosediving to
zero for the case of very neutron-rich target nuclei. This reduction could be
attributed to the quenching of GT transitions as observed in other
measurements performed with target nuclei with $A>12$ Gaarde1983 , which
pointed out a quenching of around $50\%$. However, many collective isovector
multipole modes are also contributing to both target and projectile
excitations PhysRevC.86.014304  and, therefore, it is very unlikely that this
suppression can be only caused by the quenching of GT transitions.
Additionally, the comparison to (124Sn,124Sb) in Fig. 5 also shows the
relevance of the projectile neutron excess because the cross sections decrease
systematically around $4\%$ (see Table 1), which could be attributed to the
lower neutron separation energy in 124Sn JL2017 .

##  IV Model calculations

The model employed to analyze the missing-energy spectra measured in the
experiment was already introduced in our previous study of Ref. Rodriguez2020
. We emphasize that charge-exchange reactions with nuclei require to account
in parallel for $np^{-1}$ transitions in one ion (projectile or target) in
combination with the complementary $pn^{-1}$ ones in the other nuclear system.
A brief review of the model is given in the following.

The nuclear charge-exchange response functions of projectile and target for
the quasi–elastic $np^{-1}$ and the $pn^{-1}$ components, respectively, are
obtained together with the $N^{*}N^{-1}$ spectral distributions in Random
Phase Approximation (RPA) by solution of a system of coupled Dyson–equations.
Since this approach is discussed in detail in Ref. Lenske2018 , here we only
discuss briefly the essential details. The Dyson–equations are solved in
infinite asymmetric matter, accounting for blocking effects by using proton
and neutron Fermi–spheres of the proper particle numbers. The non–elastic
resonance components cover particle states given by the $\Delta_{33}(1232)$
resonance and the Roper $P_{11}(1440)$ resonance. The nucleonic RPA
polarization propagators are obtained by mean–field and Landau–Migdal
interactions derived consistently from the Giessen Energy Density Functional
(GiEDF) Lenske:2019ubp ; Adamian:2021gnm , where interactions are extracted
from a Brueckner G–matrix and supplemented by three–body interactions of the
Urbana–type. Interactions between the $N^{*}N^{-1}$ configurations are
described by $\pi$ and $\rho$–meson exchange. Short range nuclear effects are
also taken into account by Landau parameters $g^{\prime}_{N,N^{*}}$. All
interactions include anti-symmetrization. The calculations also include
dissipation effects through polarization self–energies for nucleons and
resonances. As discussed in Ref. Lenske2018 , the $N^{*}$ in–medium
self–energies are corrected for Pauli–blocking effects in matter by using the
Oset–Salcedo method Oset:1987re . The set of coupled RPA Dyson–equations is
solved first in infinite nuclear matter. The resulting response functions,
depending on the local proton and neutron densities $\rho_{p,n}$, are
integrated over the nuclear volumes by replacing
$\rho_{p,n}\mapsto\rho^{(HFB)}_{p,n}(r)$. In summary, response functions are
obtained in local density approximation accounting for in–medium interactions,
anti-symmetrization, and Pauli-blocking effects at all levels.

Table 2: Elementary processes included in the description of the isobaric
charge-exchange reactions p(112,124Sn,112,124Sb)X and
AZ(112,124Sn,112,124Sb)X.  p(112,124Sn,112,124Sb)X
---
quasi-elastic | $p(n,p)n$
inelastic (target) | inelastic (projectile)
$p(n,p)\Delta^{0}\rightarrow p(n,p)n\pi^{0}$ | $p(n,\Delta^{0})p\rightarrow p(n,p\pi^{-})p$
$p(n,p)\Delta^{0}\rightarrow p(n,p)p\pi^{-}$ | $p(n,\Delta^{+})n\rightarrow p(n,p\pi^{0})n$
$p(n,p)n\pi^{0}$ (s-wave) |  $p(n,p\pi^{-})p$ (s-wave)
$p(n,p)p\pi^{-}$ (s-wave) |  $p(n,p\pi^{0})n$ (s-wave)
AZ(112,124Sn,112,124Sb)X
quasi-elastic | $p(n,p)n$
inelastic (target) | inelastic (projectile)
$p(n,p)\Delta^{0}\rightarrow p(n,p)n\pi^{0}$ | $p(n,\Delta^{0})p\rightarrow p(n,p\pi^{-})p$
$p(n,p)\Delta^{0}\rightarrow p(n,p)p\pi^{-}$ | $p(n,\Delta^{+})n\rightarrow p(n,p\pi^{0})n$
$p(n,p)n\pi^{0}$ (s-wave) |  $p(n,p\pi^{-})p$ (s-wave)
$p(n,p)p\pi^{-}$ (s-wave) |  $p(n,p\pi^{0})n$ (s-wave)
$n(n,p)\Delta^{-}\rightarrow n(n,p)n\pi^{-}$ | $n(n,\Delta^{0})n\rightarrow n(n,p\pi^{-})n$
$n(n,p)n\pi^{-}$ (s-wave) |  $n(n,p\pi^{-})n$ (s-wave)
Table 3: Elementary processes included in the description of the isobaric
charge-exchange reactions p(112,124Sn,112,124In)X and
AZ(112,124Sn,112,124In)X.  p(112,124Sn,112,124In)X
---
inelastic (target) | inelastic (projectile)
$p(p,n)\Delta^{++}\rightarrow p(p,n)p\pi^{+}$ | $p(p,\Delta^{+})p\rightarrow p(p,n\pi^{+})p$
$p(p,n)p\pi^{+}$ (s-wave) |  $p(p,n\pi^{+})p$ (s-wave)
AZ(112,124Sn,112,124In)X
quasi-elastic | $n(p,n)p$
inelastic (target) | inelastic (projectile)
$p(p,n)\Delta^{++}\rightarrow p(p,n)p\pi^{+}$ | $p(p,\Delta^{+})p\rightarrow p(p,n\pi^{+})p$
$p(p,n)p\pi^{+}$ (s-wave) |  $p(p,n\pi^{+})p$ (s-wave)
$n(p,n)\Delta^{+}\rightarrow n(p,n)n\pi^{+}$ | $n(p,\Delta^{+})n\rightarrow n(p,n\pi^{+})n$
$n(p,n)\Delta^{+}\rightarrow n(p,n)p\pi^{0}$ | $n(p,\Delta^{0})n\rightarrow n(p,n\pi^{0})p$
$n(p,n)n\pi^{+}$ (s-wave) |  $n(p,n\pi^{+})n$ (s-wave)
$n(p,n)p\pi^{0}$ (s-wave) |  $n(p,n\pi^{0})p$ (s-wave)

The reaction model takes into account the direct and exchange contributions to
the missing-energy spectrum from quasi-elastic and inelastic $(n,p)$ and
$(p,n)$ elementary processes, which are described in terms of the exchange of
virtual $\pi$\- and $\rho$-mesons between the interacting nucleons. The list
of elementary processes included in the description of the $(n,p)$\- and
$(p,n)$-type charge exchange reactions is given in Tables 2 and 3,
respectively. The Feynman diagrams illustrating the direct contributions to
these elementary processes are shown in Fig. 6. The exchange contributions can
be simply obtained by interchanging the lines of the incoming nucleon
$\tau_{P}$ of the projectile and the nucleon $\tau_{T}$ of the target. Note
that the incoming projectile nucleon $\tau_{P}$ can only be a proton in the
case of the p(112,124Sn,112,124In)X reactions but it can be either a neutron
or a proton in all other reactions considered. Diagram (a) gives the direct
contribution to the quasi-elastic $(n,p)$ and $(p,n)$ elementary processes
whereas diagram (b) takes into account the effect of short-range correlations
by means of the Landau–Migdal parameter $g^{\prime}$ taken here to be $0.7$.
The excitation of the $\Delta$ resonance and its subsequent decay into a
nucleon and a pion is considered both in the target (diagram c) and in the
projectile (diagram d). Note that the label $\Delta$ in diagrams (c) and (d)
indicates schematically the isospin states $\Delta^{++},\Delta^{+},\Delta^{0}$
or $\Delta^{-}$ depending on the particular elementary process (see Tables 2
and 3). We note that contributions from the excitation of other resonances,
such as for instance the Roper $P11(1440)$, have been ignored in the present
model. Diagrams (e) and (f) show the contributions where the emitted pion is
produced in s-wave. The basic ingredients of the model are the $NN\pi$,
$NN\rho$, $N\Delta\pi$ and $N\Delta\rho$ vertices Ericson87 ; Fernandez92  and
the $\pi N\rightarrow\pi N$ s-wave amplitude Fernandez92 ; Oset89 . The
inclusive cross section for the reaction $A(a,b)B$ describing the differential
missing-energy spectrum of the outgoing projectile-like ion is given by the
sum of a quasi-elastic (qe) and an inelastic (in) contribution

$\frac{d\sigma}{dE_{b}}=\frac{d\sigma}{dE_{b}}\Big{|}_{qe}+\frac{d\sigma}{dE_{b}}\Big{|}_{in}\
.$ (3)

The quasi-elastic contribution is obtained as

$\frac{d\sigma}{dE_{b}}\Big{|}_{qe}=\langle
N_{\tau_{P}\tau_{T}}\rangle|\mathcal{M}_{qe}|^{2}$ (4)

being $\langle N_{\tau_{P}\tau_{T}}\rangle$ (see Eq. (11) below) the average
number of neutron-proton ($\tau_{P}=n,\tau_{T}=p$) and proton-neutron
($\tau_{P}=p,\tau_{T}=n$) elementary processes contributing, respectively, to
the $(n,p)$\- and the $(n,p)$-type reactions, and

$\displaystyle\left|\mathcal{M}_{qe}\right|^{2}=\frac{2|{\vec{p}}_{b}|}{(2\pi)^{2}}\frac{m^{4}}{\lambda^{1/2}(s,m^{2},m^{2})}$
$\displaystyle\times\int\frac{d\vec{q}}{(2\pi)^{3}}\frac{d\Omega_{b}}{E_{B}}R^{(A)}_{N}(\sqrt{s}-E_{b},\vec{q})\langle\left|M_{qe}(\vec{q})\right|^{2}\rangle\
,$ (5)

where $s$ is the total energy in the center-of-mass frame, $m$ is the nucleon
mass, $\langle\cdot\rangle$ indicates the average and sum over the initial and
final spins, $R^{(A)}_{N}(\omega,\vec{q})$ is the spectral distribution of
nucleon hole-nucleon $N^{\prime}N^{-1}$ target transitions (see Refs.
Lenske2018 ; Lenske18b ; Lenske18c ; Lenske2019  for detailed discussions and
explicit expressions), $M_{qe}$ is the scattering amplitude of the elementary
process $p+n\rightarrow n+p$ and
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$ is the so-called Källen
function.

The inelastic contribution is given by

$\displaystyle\frac{d\sigma}{dE_{b}}\Big{|}_{in}=\left\\{\begin{array}[]{lll}\langle
N_{nn}\rangle|\mathcal{M}_{in}^{(nn\rightarrow pn\pi^{-})}|^{2}+\langle
N_{np}\rangle|\mathcal{M}_{in}^{(np\rightarrow pp\pi^{-})}|^{2}+\langle
N_{np}\rangle|\mathcal{M}_{in}^{(np\rightarrow pn\pi^{0})}|^{2}\,,&&\mbox{for
$(n,p)$-type reactions}\\\ \\\ \langle
N_{pp}\rangle|\mathcal{M}_{in}^{(pp\rightarrow np\pi^{+})}|^{2}+\langle
N_{pn}\rangle|\mathcal{M}_{in}^{(pn\rightarrow nn\pi^{+})}|^{2}+\langle
N_{pn}\rangle|\mathcal{M}_{in}^{(pn\rightarrow np\pi^{0})}|^{2}\,,&&\mbox{for
$(p,n)$-type reactions}\end{array}\right.$ (9)

with $\langle N_{nn}\rangle$ and $\langle N_{pp}\rangle$, similarly to
$\langle N_{np}\rangle$ and $\langle N_{pn}\rangle$, being the average number
of neutron-neutron and proton-proton elementary processes contributing to the
$(n,p)$ and $(p,n)$ reactions, defined also through Eq. (11) and

$\displaystyle\left|\mathcal{M}_{in}^{(\tau_{P}\tau_{T}\rightarrow\tau^{\prime}_{P}\tau^{\prime}_{T}\pi)}\right|^{2}=\frac{1}{S}\frac{|{\vec{p}}_{b}|}{(2\pi)^{5}}\frac{m^{4}}{\lambda^{1/2}(s,m^{2},m^{2})}$
$\displaystyle\times\int\frac{d\vec{q}}{(2\pi)^{3}}\frac{d\vec{p}_{\pi}d\Omega_{b}}{E_{B}E_{\pi}}$
$\displaystyle\times
R^{(A)}_{\Delta}(\sqrt{s}-E_{b}-E_{\pi},\vec{q}-\vec{p}_{\pi})$
$\displaystyle\times\langle\left|M_{in}^{(\tau_{P}\tau_{T}\rightarrow\tau^{\prime}_{P}\tau^{\prime}_{T}\pi)}(\vec{q}-\vec{p}_{\pi})\right|^{2}\rangle\
.$ (10)

Here $R^{(A)}_{\Delta}(\omega,\vec{q})$ is the spectral distribution of
$\Delta$ hole-nucleon $\Delta N^{-1}$ target transitions,
$M_{in}^{(\tau_{P}\tau_{T}\rightarrow\tau^{\prime}_{P}\tau^{\prime}_{P}\pi)}$
is the scattering amplitude of the elementary process
$\tau_{P}+\tau_{T}\rightarrow\tau^{\prime}_{P}+\tau^{\prime}_{P}+\pi$ and $S$
is a symmetry factor given by

$S=\prod_{l}k_{l}!$

for $k_{l}$ identical particles of species $l$ in the final state. In our case
$S$ can be 1 or 2 depending on the particular reaction channel (see Tables 2
and 3).

The average number of elementary processes contributing to the $(n,p)$\- and
$(p,n)$-type reactions is calculated as

$\langle N_{\tau_{P}\tau_{T}}\rangle=\langle N_{\tau_{P}}\rangle\times\langle
N_{\tau_{T}}\rangle\ ,\,\,\,\,\tau_{P}=n,p\ ,\,\,\tau_{T}=n,p$ (11)

where $\langle N_{\tau_{P}}\rangle$ and $\langle N_{\tau_{T}}\rangle$ are,
respectively, the average number of neutrons or protons in the projectile and
in the target participating in the reaction. These numbers are calculated as

$\displaystyle\langle N_{n_{P}}\rangle$ $\displaystyle=$
$\displaystyle(A_{P}-Z_{P})\frac{\sigma_{T}}{\sigma_{PT}}$
$\displaystyle\langle N_{p_{P}}\rangle$ $\displaystyle=$ $\displaystyle
Z_{P}\frac{\sigma_{T}}{\sigma_{PT}}$ $\displaystyle\langle N_{n_{T}}\rangle$
$\displaystyle=$ $\displaystyle(A_{T}-Z_{T})\frac{\sigma_{P}}{\sigma_{PT}}$
$\displaystyle\langle N_{p_{T}}\rangle$ $\displaystyle=$ $\displaystyle
Z_{T}\frac{\sigma_{P}}{\sigma_{PT}}\ ,$

where $A_{P}$ $(Z_{P})$ and $A_{T}$ $(Z_{T})$ are the mass (atomic) numbers of
the projectile and target nuclei, respectively, and $\sigma_{P(T)}$ and
$\sigma_{PT}$ the total nucleon-nucleus and nucleus-nucleus cross section
determined by using the Glauber model

$\sigma_{P(T)}=\int
d\vec{b}\Big{(}1-(1-T_{P(T)}(b)\sigma_{NN})^{A_{P(T)}}\Big{)}$
$\sigma_{PT}=\int
d\vec{b}\Big{(}1-(1-T_{PT}(b)\sigma_{NN})^{A_{P}A_{T}}\Big{)}$

with $\sigma_{NN}$ the nucleon-nucleon cross section, for which we take here a
value of 40 mb, and

$T_{P(T)}(b)=\int dz\rho_{P(T)}(b,z)$ $T_{PT}(b)=\int
d\vec{s}T_{P}(|\vec{s}-\vec{b}|)T_{T}(s)\ ,$

with the projectile and target densities $\rho_{P}$ and $\rho_{T}$ obtained
from a relativistic mean field model calculation using the FSU model Todd05 .

Figure 6: Direct contributions to the quasi-elastic (diagrams a and b), and
inelastic (diagrams from c to f) $(n,p)$ and $(p,n)$ elementary processes. The
emitted pion in the final state is assumed to come either from the decay
$\Delta\rightarrow N\pi$ (diagrams c and d) or to be produced in s-wave
without the excitation of any resonance (diagrams e and f). Exchange
contributions are obtained by interchanging the lines of the incoming neutron
or proton of the projectile and the nucleon $\tau$ of the target.

##  V Discussion

The comparison of the experimental data for the single isobaric charge-
exchange reactions (112Sn,112Sb) and (112Sn,112In) with the model calculations
is shown in Fig. 7. The separate contributions from the quasi-elastic and
inelastic (target and projectile excitation) processes are shown, as well as
the interference between the target and projectile excitation processes. We
should mention that in the calculation for the $(p,n)$-type reactions these
contributions have been rescaled separately to compare with the data. For the
$(n,p)$-type ones, however, a common rescaling factor for these contributions
has been considered. The reason for this is to check whether the model can
explain the quenching of the quasi-elastic peak observed in the data of the
(112Sn,112Sb) reactions without adjusting it ad hoc. We note that although the
shape of the experimental spectrum is in general reasonably well reproduced by
the model, the quenching of the quasi-elastic peak cannot be described. Since
many multipoles are contributing to both target and projectile excitations, it
is very unlikely that the suppression is caused by the quenching of GT
transitions as observed in Ref. Gaarde1983 . From the theoretical point of
view this effect is not fully understood, in particular because the response
functions account correctly for the differences in the available configuration
spaces of the $pn^{-1}$ and $np^{-1}$ transitions for the projectile and
target nuclei, respectively. In the following, we will see that the model
allows us to reveal a large amount of information on the different reaction
mechanism that lead to the inclusive spectra measured.

Figure 7: (Color online) Missing-energy spectra obtained from the Pb, Cu, 12C
and proton targets for the single isobaric charge-exchange reactions
(112Sn,112Sb) and (112Sn,112In) compared to the theoretical calculations
described in Ref. Rodriguez2020 . See the legend displayed in the upper left
panel for explanation.

We consider first the spectra obtained with the proton target. The first
observation is, as expected, the absence of the quasi-elastic peak at missing-
energies close to zero in the case of the p(112Sn,112In)X reaction whereas it
is clearly present in the p(112Sn,112Sb)X one. At lower missing energies the
spectrum of both reactions can be explained as a result of the superposition
of the four (see Table 3) and eight (see Table 2) contributing elementary
processes, respectively. As it can be seen in Fig. 7, the spectrum of the
p(112Sn,112In)X reaction is clearly dominated by the excitation of the
$\Delta^{++}$ in the target (see Table 3), although the contribution of the
other three elementary processes, particularly the $\Delta^{+}$ excitation in
the projectile, is necessary to reproduce its shape. This is not the case for
the p(112Sn,112Sb)X reaction where the contribution due to the excitation of
the $\Delta^{0}$ in the target (see Table 2) and the excitation of the
$\Delta^{+}$ and $\Delta^{0}$ in the projectile are similar as reflected by
the double-peak structure seen in the data. Note, in addition, that the small
shoulder observed in the p(112Sn,112In)X reaction at the right of the peak can
be understood mainly due to the contribution of the $\Delta^{+}$ excitation in
the projectile. In both reactions, the contribution from the processes where
the pion is emitted in a s-wave is small in comparison to those of the
excitation of the $\Delta$ isobar. Moreover, one can see that the shape and
the position of the resonance is different if it is excited in the target or
in the projectile nucleus. This is because its invariant mass is different in
both cases. When the $\Delta$-resonance is excited in the target, its
invariant mass does not depend on the momentum of the emitted pion and,
therefore, the scattering amplitude can be taken out of the integral in Eq.
(10). In this case, the shape of the resonance appears almost symmetric and it
has a peak at approximately -345 MeV. On the contrary, when the $\Delta$ is
excited in the projectile, its invariant mass depends explicitly on the
momentum of the emitted pion and, therefore, it is clear here that the
scattering amplitude must also be integrated. Consequently, the resonance
acquires an apparent asymmetric shape and its position is shifted to the right
by $\sim$63 MeV. This is precisely the shift in the position of the $\Delta$
peak that seems to be observed between the measurements with the proton and
heavier targets.

Figure 8: (Color online) Inelastic cross sections as a function of the neutron
excess of target nuclei for the (112Sn,112In) (a) and (112Sn,112Sb) (b)
reactions. The lines represent the contributions from np, pp, pn, and nn
interactions predicted by our theoretical model (see Tables 2 and 3). (c)
Quasi-elastic cross sections for the (112Sn,112In) and (112Sn,112Sb) reactions
compared to our calculations.

Let us finally analyze the results of the (112Sn,112Sb) and (112Sn,112In)
reactions on heavier targets. We observe first, as mentioned before, that
whereas the quasi-elastic peak is reasonably well reproduced by the model in
the case of the $(^{112}$Sn,112In) reactions, its quenching in the
(112Sn,112Sb) ones cannot be described, likely due to the uncertainties in the
momentum distributions of the nucleons involved in the collision. These
uncertainties would be attributed to the re-scattering of the interacting
nucleons with the rest of nucleons (final state interactions) Diaz2020 , or
short range nucleon-nucleon correlations CLAS . The strength in the lower
missing-energy region of the spectrum is once more the result of the
superposition of elementary processes where the excitation ($\Delta$ resonance
or pion production in s-wave) occurs either in the target or in the
projectile. Note that in this case there are additional elementary processes
(see Tables 2 and 3) contributing that were absent in the case of the
reactions with the proton target, explaining the larger magnitude of the
spectrum, apart from the trivial scaling by the increased number of
participating target nucleons. In addition, there is a change of the relative
magnitude between the excitation in target and projectile, with the latter one
dominating now in both (112Sn,112Sb) and (112Sn,112In) reactions. The reason
is that in a complex target nucleus only part of the energy transferred goes
into the excitation of the $\Delta$ resonance or the direct s-wave production
of the pion. Besides the excitations of $N^{-1}\Delta$ configurations, another
part of the energy loss is employed in reactions such as, for instance, the
knockout of nucleons from the target or its fragmentation. Consequently, this
contribution is reduced with respect to that of the projectile excitation and
is now observed simply as a shoulder at $\sim$ 63 MeV to the left of the
dominant peak corresponding to the excitation of the $\Delta$ in the
projectile. This seems to indicate that the apparent shift in the position of
the $\Delta$-resonance peak, observed in targets heavier than the proton, can
be simply interpreted as a change in the relative magnitude between the
contribution of the excitation of the resonance in the target and in the
projectile nuclei. Of course, only future exclusive measurements, allowing the
separate identification of the resonance excitation in target and projectile
nuclei, will be able to confirm this conclusion. We would like to emphasize
that a similar interpretation was already pointed out by Oset, Shiino, and
Toki Oset89  in their analysis of the (3He,t) reaction on proton, deuteron,
and carbon targets.

Finally, we also compare the measured inelastic cross sections of the
(112Sn,112In) and (112Sn,112Sb) reactions to our theoretical calculations in
Figs. 8 and 8, respectively, showing the np, nn, and pp contributions from the
reactions listed in Tables 2 and 3. This allows us to investigate the neutron
abundance effects in the $\Delta$-resonance excitation. For the reaction
(112Sn,112In), we can see that the pn contribution to the $\Delta$ excitation
increases linearly with the target neutron excess while the pp contribution
decreases. This is expected since pn collisions are most likely when the
target nuclei contain more neutrons at their nuclear surface. In the case of
(112Sn,112Sb) reactions we observe that the np contribution is more or less
constant with the target neutron excess while the nn collisions increase
linearly. The same evolutions with the neutron excess are also observed for
the np collisions involved in the quasi-elastic excitations, as shown in Fig.
8. However, in the case of (112Sn,112Sb) only the inelastic cross sections are
well described. The decreasing value of the measured quasi-elastic cross
sections is not reproduced by the model calculations as mentioned afore.

##  VI Conclusions and perspectives

Single isobaric charge-exchange reactions induced by stable projectiles of
112,124Sn at energies of $1A$ GeV on different targets have been investigated
at the GSI facility by using the magnetic spectrometer FRS. The experimental
setup allowed us to measure the cross sections of these charge-exchange
reactions with an accuracy of around $3\%$ and to determine simultaneously the
missing-energy spectra of the corresponding ejectiles. Thanks to the FRS high-
resolving power we can clearly identify in the missing-energy spectra the
quasi-elastic and inelastic components, corresponding to the nuclear spin-
isospin response of nucleon-hole and $\Delta$-resonance excitations,
respectively.

For both Sn projectiles, we observe an apparent energy shift for the
$\Delta$-resonance peak of about (63$\pm$5) MeV when comparing the missing-
energy spectra obtained from the measurements with the proton target to the
other ones. This observation is consistent with the results obtained in the
SATURNE experiments Contardo1986 . However, the detailed analysis with our
theoretical model indicates that this energy shift can be simply interpreted
as a change in the relative magnitude between the $\Delta$-resonance
excitation in target and projectile nuclei. We have also seen that for the
$(p,n)$-type isobaric charge-exchange channel the quasi-elastic and inelastic
cross sections increase with the target neutron excess, which can be explained
by the increase of the interaction cross section with the size of target
nuclei. However, for the $(n,p)$ channel we find a decrease of the quasi-
elastic cross section with the target neutron excess that clearly indicates a
quenching of this transition. At first glance, the quenching factor of about
$50\%$ seems to indicate an effect similar to the widely observed suppression
of GT-strength Gaarde1983 . However, this explanation seems rather unlikely in
this case because a multitude of multipolarities contributes to the reaction.

Further investigations require the combination of the magnetic spectrometer
with pion detection systems, like the WASA (Wide Angle Shower Apparatus)
calorimeter designed originally to measure light ion collisions and $\eta$
rare decays at the CELSIUS storage ring in Uppsala (Sweden) Bargholtz2008
and, more recently, at COSY in Jülich (Germany) Cosy2013 . In 2019, this
calorimeter was moved from Jülich to GSI and is now being installed at the
intermediate focal plane of the FRS to study the properties of light
hypernuclei and mesonic nuclei YT2020 . In the future it will also be
installed at the intermediate focal plane of the Super-FRS Winfield2021  as
shown in Fig. 9. This calorimeter would be used to measure the emitted pions
from the baryonic resonance decays and non-resonant excitation channels in
coincidence with the isobaric charge-exchange residues. For producing the
reactions, the targets can be located inside the calorimeter. Surrounding the
target we will have the WASA detectors and other passive elements whose
details can be found in Ref. Bargholtz2008 . The WASA detectors will provide
us the momenta of the pions that combined with the momenta of the isobaric
charge-exchange residues will give us the possibility of reconstructing the
resonance invariant mass with resolutions of $\Delta M/M\sim 10^{-3}$.

Figure 9: (Color online) Schematic illustration of the WASA calorimeter at the
intermediate focal plane of Super-FRS. This setup consists of a mini-drift
chamber made of straw tubes, a counter barrel made of plastic scintillators, a
superconducting solenoid magnet, and an external iron yoke. Upstream of the
WASA device we will install tracking detectors to identify the incoming
projectiles.

Finally, these exclusive measurements tagging the pions will allow us to
distinguish between target and projectile excitations unambiguously, giving
access to the baryon invariant mass reconstruction and confirming whether the
apparent shift in the position of the $\Delta$-resonance peak, observed in
targets heavier than the proton, can indeed be interpreted as a change in the
relative magnitude between the processes where the $\Delta$-resonance is
excited in the target or in the projectile. Moreover, we will also study the
in-medium modifications of the mass and width of the resonances, as well as to
perfom missing mass spectroscopy of neutral decay channels involving neutron
or $\pi^{0}$ emissions. This methodology was already proposed by Hennino and
collaborators at the SATURNE facility where they observed a clear target mass
dependence of the invariant mass Hennino1992 , resulting in a $\Delta$-mass
reduction of $\approx$25 MeV with respect to the elementary pn process.
Additionally, this research will also offer the possibility of studying the
in-medium decay properties of other resonances, like the Roper P11(1440)
Roper1964 ; Morsch1992 ; Alvarez1998 . In conclusion, these kinds of
experiments with radioactive ion beams represent unique opportunities to
understand in detail the formation of baryonic resonances in the nuclear
medium and will give us for the first time access to systematically study
their dynamics and properties with the nuclear isospin, covering large ranges
in neutron-proton asymmetry.

###### Acknowledgements.

The authors are grateful to the GSI accelerator staff for providing intense
and stable beams of 112,124Sn. This work was partially supported by the
Spanish Ministry for Science and Innovation under grants RTI2018-101578-B-C21
and PGC2018-099746-B-C22, and by the Regional Government of Galicia under the
program “Grupos de Referencia Competitiva” ED431C-2021-38. This work has
received financial support from Xunta de Galicia (Centro singular de
investigación de Galicia accreditation 2019-2022). I.V. thanks the support
from the COST Action CA16214. H.L. acknowledges support by DFG, Grant
Le439$/$16-2. J.L.R.S. thanks the support from Xunta de Galicia under the
program of postdoctoral fellowships ED481B-2017-002 and ED481D-2021-018.

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