# ActiveDP: Bridging Active Learning and Data Programming

Naiqing Guan  University of TorontoTorontoCanada<EMAIL_ADDRESS>and  Nick Koudas  University of TorontoTorontoCanada<EMAIL_ADDRESS>
###### Abstract.

Modern machine learning models require large labelled datasets to achieve good
performance, but manually labelling large datasets is expensive and time-
consuming. The data programming paradigm enables users to label large datasets
efficiently but produces noisy labels, which deteriorates the downstream
model’s performance. The active learning paradigm, on the other hand, can
acquire accurate labels but only for a small fraction of instances. In this
paper, we propose ActiveDP, an interactive framework bridging active learning
and data programming together to generate labels with both high accuracy and
coverage, combining the strengths of both paradigms. Experiments show that
ActiveDP outperforms previous weak supervision and active learning approaches
and consistently performs well under different labelling budgets.

Dataset Curation, Data programming, Active learning

††ccs: Information systems Data cleaning††ccs: Computing methodologies Active
learning settings

##  1\. Introduction

Modern machine learning models require large training datasets to achieve good
accuracy, yet manual labelling and curation of large datasets are both
expensive and time-consuming. Thus, acquiring labelled datasets has become one
of the main bottlenecks in applying machine learning in practical scenarios.
Data programming (DP) (Ratner et al., 2016, 2017), a recent paradigm for weak
supervision, provides an approach to automatically label large datasets
without manually annotating specific instances. In the data programming
paradigm, users represent weak supervision sources in the form of label
functions (LFs), which are rules that provide noisy labels to a subset of
data. For example, suppose the task is to classify the sentiment of customer
reviews. In that case, the users can write code snippets that label the review
as positive or negative based on specific keywords. Since the label functions
have varying accuracy and may exhibit ad-hoc correlations, a generative model
(also called the label model) is designed to aggregate noisy, weak labels into
probabilistic labels. Optionally, a downstream model (also referred to as the
end model) can be trained with the probabilistic labels and utilized for
downstream tasks.

While the data programming paradigm can label large datasets rapidly, the
generated labels are usually noisy, and as a result this may deteriorate the
performance of the downstream model. Active learning (AL) (Settles, 2009,
2012), on the other hand, identifies and selects a subset of data for manual
annotation and utilizes the subset of data for training the downstream model.
As the annotation is instance-specific, users can provide cleaner labels to
the selected instances. While active learning excels in label quality, it
falls short in label quantity, as users can only annotate a small fraction of
data with limited effort.

Table 1. Approaches briding active learning with data programming.

Method | Query Type | Train AL Model? | Step to Improve Data Quality
---|---|---|---
Nemo (Hsieh et al., 2022) | LF Creation | No | LF Generation
IWS (Boecking et al., 2020) | LF Validation | No | LF Generation
Revising LF (Nashaat et al., 2018) | Instance Labelling | No | LF Revision
Active WeaSuL (Biegel et al., 2021) | Instance Labelling | No | Label Model Training
ActiveDP (ours) | LF Creation | Yes | Post Training

It is evident that the merits of data programming and active learning
compensate for each other. In this paper, we address the question: how can we
combine the advantages of both methods to label large datasets rapidly while
maintaining superior label quality?  Researchers have investigated several
approaches to bridge data programming and active learning, such as leveraging
active learning to develop label functions (Hsieh et al., 2022; Boecking et
al., 2020), revise label functions (Nashaat et al., 2018) or tune label model
parameters (Biegel et al., 2021). However, these methods do not fully leverage
the merits of both paradigms. Nemo (Hsieh et al., 2022) and IWS (Boecking et
al., 2020) leverage active learning to select instances for LF development.
However, they only use LFs to annotate the training dataset instead of
combining the LFs with fine-grained instance labels, which impedes them from
achieving high-quality labels. Nashaat et al. (Nashaat et al., 2018) correct
the LF outputs on labelled data. However, as pointed out by Biegel et al.
(Biegel et al., 2021), the revision does not always benefit predictions on
unlabelled data, as revising the LFs on the labelled subset may mislead the
label model to believe they are also accurate on the unlabeled subset. Biegel
et al. (Biegel et al., 2021) use the labelled subset to tune label model
parameters. However, it only leverages label functions to generate training
labels and thus has a similar drawback to Nemo and IWS. Besides, the trade-off
factor between the original model loss and the loss term induced on the
labelled subset is hard to select.

Figure 1. Workflow of ActiveDP. Left: iterative LF creation at training phase.
Right: label aggregation at inference phase.

To address the abovementioned limitations, we propose ActiveDP, a novel
interactive framework bridging active learning and data programming. Table 1
highlights the difference between ActiveDP and previous works (Nashaat et al.,
2018; Biegel et al., 2021; Boecking et al., 2020; Hsieh et al., 2022). The
main differences are: (1) ActiveDP combines weak supervision with active
learning model predictions to balance label quality and quantity; (2) ActiveDP
improves the label quality in the post-training phase. This enables ActiveDP
to treat the label model as a black box and neglect the intricate effect of
revising LFs on label model training.

Figure 1 illustrates the workflow of the ActiveDP framework. In the training
phase, ActiveDP leverages a novel ADP sampler to select instances for users to
develop LFs. Apart from collecting the LFs returned by users, ActiveDP also
creates a small pseudo-labelled dataset based on user-returned LFs to train an
active learning model. Besides, ActiveDP learns the dependency structure
between LFs and the labels and chooses a diverse and informative set of LFs to
improve the end-to-end classification accuracy. In the inference phase,
ActiveDP aggregates the prediction results of the label model in the DP
paradigm and the active learning model using a novel label aggregation method
named ConFusion to generate labels with high accuracy and coverage.

To summarize, we make the following main contributions:

  * •

We propose a novel interactive framework named ActiveDP that explores the
design space between active learning and data programming and combines their
strengths.

  * •

We design various novel strategies to improve the efficiency of ActiveDP and
the quality of generated labels, including the ConFusion method for label
aggregation, the ADP sampler for query instance selection, and the LabelPick
method for LF filtering.

  * •

We conduct extensive experiments to compare ActiveDP with baseline methods,
demonstrating its efficiency in providing labels with high coverage and
accuracy and enhancing the performance of the downstream model.

##  2\. Related Works

###  2.1. Data Programming

Data programming (DP) is a weak supervision paradigm for automatically
labelling large datasets. Given an unlabeled dataset
$D=\\{x_{i}\\}_{i=1}^{n}$, the users design a series of label functions
$\Lambda=\\{\lambda_{j}\\}_{j=1}^{m}$, each provide weak labels to a subset of
data. We use $\lambda_{j}(x_{i})$ to denote the output of the $\lambda_{j}$ on
$x_{i}$. For a class classification problem with $Y=\\{0,1,...,C-1\\}$ as the
label space, the weak label $\lambda_{j}(x_{i})\in Y\cup\\{-1\\}$ where $-1$
means $\lambda_{j}$ does not make prediction on $x_{i}$. We say $\lambda_{j}$
is activated on $x_{i}$ if $\lambda_{j}(x_{i})\neq-1$ and $\lambda_{j}$
abstains on $x_{i}$ otherwise. The generated weak labels form a label matrix
$W$ where $W_{ij}=\lambda_{j}(x_{i})$, which is used to train a label model
$f_{l}$. We use $f_{l}(x,\Lambda)\in R^{C}$ to denote the probabilistic labels
predicted by the label model for $x$. Finally, the probabilistic labels (or
hard labels by selecting the most likely class) will be used to train the
downstream machine learning model.

The works in the DP area can be briefly categorized into developing more
advanced label models (Ratner et al., 2016, 2017, 2019; Awasthi et al., 2019;
Biegel et al., 2021) and designing label functions efficiently (Varma and Ré,
2018; Boecking et al., 2020; Hsieh et al., 2022; Denham et al., 2022; Hancock
et al., 2018; Smith et al., 2022; Huang et al., [n. d.]). Snorkel (Ratner et
al., 2016, 2017) uses a factor graph to model the accuracy and pairwise
dependencies between label functions. MeTaL (Ratner et al., 2019) extends the
framework to multi-task settings and proposes to estimate LF accuracy by
solving a matrix completion problem. Another line of work focuses on
automating or aiding LF design. IWS (Boecking et al., 2020) incorporates human
experts in an iterative process by letting experts decide whether a candidate
LF is accurate. Nemo (Hsieh et al., 2022) actively selects the instances given
to human experts, and experts return LFs based on them.

###  2.2. Active Learning

Active learning (Settles, 2009, 2012) is an established technique in machine
learning literature, which aims to maximize model accuracy while annotating
the fewest samples possible. There are abundant selectors designed for active
learning, such as uncertainty sampling (Lewis, 1995), query-by-committee
(Seung et al., 1992), core-set approaches (Sener and Savarese, 2018), density-
based approaches (Settles and Craven, 2008) and hybrid approaches (Li and Guo,
2013; Gao et al., 2020). We refer interested readers to recent surveys (Ren et
al., 2021; Liu et al., 2022) for a comprehensive description of active
learning methods.

Recently, there have been emerging works that leverage active learning to
improve the performance of PWS pipelines. Nashaat et al. (Nashaat et al.,
2018, 2020) leverage uncertainty sampling to correct LF outputs. Active WeaSuL
(Biegel et al., 2021) actively selects a subset of instances for labelling to
guide label model training. IWS (Boecking et al., 2020) and Nemo (Hsieh et
al., 2022) also leverages active learning for sample selection.

##  3\. ActiveDP Framework

In this section, we introduce and design the ActiveDP framework. We first
provide an overview of the ActiveDP framework and illustrate its workflow
using a running example; then, we discuss each component of ActiveDP in
detail.

###  3.1. Framework Overview

As illustrated in Figure 1, the ActiveDP framework contains a training phase
and an inference phase. In the training phase, it iteractively selects query
instances for users to inspect and design LFs, and trains the label model and
active learning model based on user responses. In the inference phase, it
aggregates the prediction results of the label model and the active learning
model to improve label quality.

In the training phase, during the $t$-th iteration, the system leverages the
ADP Sampler (Section 3.3) to actively pick an example $x_{l_{t}}\in D$ as the
query instance for the user to inspect. The user designs a LF $\lambda_{t}$
based on $x_{l_{t}}$. In the running example in Figure 1 for spam detection,
the sampler picks query instance ”check out my channel” and asks the user to
suggest a LF based on the query, and the user responds with a LF ”check”$\to$
SPAM(1), indicating that a comment is likely to be spam when it contains the
keyword ”check”. The LFs will be collected into the LF set
$\Lambda_{t}=\\{\lambda_{1},...,\lambda_{t}\\}$. We further leverage the
LabelPick method (Section 3.4) to select a subset of helpful LFs
$\Lambda_{t}^{*}\subset\Lambda_{t}$ , and use $\Lambda_{t}^{*}$ to train the
label model $f_{l}^{t}$.

Apart from collecting the LFs, we also curate a pseudo-labelled subset of the
training data to train an active learning model. While the user does not
explicitly provide labels to data, since we have the LFs designed by the user
based on the query instances, these LFs should be at least accurate on the
corresponding query instances. In the running example, as the user designs a
LF ”check”$\to$ SPAM(1) when they observe the query instance ”check out my
channel”, we can infer the label of the query instance should be SPAM. Based
on that observation, we create the pseudo-labelled subset as
$L_{t}=\\{(x_{l_{t}},\tilde{y}_{l_{t}})\\}_{t=1}^{T}$, where
$\tilde{y}_{l_{t}}=\lambda_{t}(x_{l_{t}})$ being the pseudo-label inferred
from LFs, and use $L_{t}$ to train the active learning model $f_{a}^{t}$.

In the inference phase, we leverage the ConFusion method (Section 3.2) to
aggregate the prediction results of the label model and active learning model
based on a confidence threshold $\tau_{t}$. We would follow the prediction of
the active learning model if its confidence exceeds the confidence threshold.
Otherwise, we would follow the prediction of the label model as long as the
instance has non-abstain LFs. In the running example in Figure 1, the
unlabeled instance is ”just check the reviews”. Suppose the active learning
model predicts it to be HAM with confidence score 0.8, and the confidence
threshold is 0.7. Since the confidence score exceeds the threshold, we will
follow the active learning model’s prediction for that unlabeled instance.

###  3.2. Label Aggregation

In this section, we introduce ConFusion, a confidence-based label aggregation
approach. While the label model and the active learning model are both trained
on the training dataset, their predictions differ because they are trained
using different features and their model structures are different. This makes
label aggregation helpful in combining the strengths of both models and
improves label quality. We use $f_{l}^{t}(x,\Lambda^{*}_{t})$ and
$f_{a}^{t}(x)$ to denote the soft labels predicted by the label model and the
active learning model after the t-th iteration respectively. The confidence
score of the active learning model is defined as the predicted probability for
the top-1 candidate class, i.e. $max\\{f_{a}^{t}(x)\\}$. For example, the
confidence score for the unlabeled instance in Figure 1 is 0.8. The aggregated
labels are defined as follows:

(1)
$\hat{y}^{t}=\begin{cases}f_{a}^{t}(x)&max\\{f_{a}^{t}(x)\\}\geq\tau_{t}\\\
f_{l}^{t}(x,\Lambda_{t}^{*})&max\\{f_{a}^{t}(x)\\}<\tau_{t}\wedge\Lambda_{t}^{*}(x)\neq\mathbf{-1}\\\
\emptyset&otherwise.\end{cases}$

In other words, when the confidence of the active learning model is above the
threshold $\tau_{t}$, we adopt its prediction results. Otherwise, if at least
one label function is activated on the unlabeled instance (i.e.
$\exists\lambda\in\Lambda_{t}^{*}\ s.t.\lambda(x)\neq-1$), we adopt the
prediction results of the label model. If all label functions abstain on the
unlabeled instance and the confidence of the active learning model is below
$\tau_{t}$, we reject making predictions and discard the instance when
training the downstream model. Such an approach balances label accuracy and
label coverage because we leverage data programming to label a significant
fraction of data while using the active learning model to improve label
accuracy.

The confidence threshold $\tau_{t}$ determines the relative importance of the
active learning model and the label model. We dynamically tune the confidence
threshold $\tau_{t}$ using a holdout validation set $D_{V}$ drawn I.I.D. from
the underlying distribution of training data (with hidden labels). Let
$C^{t}=\\{0.0,c_{1}^{t},c_{2}^{t},...,c^{t}_{k},1.0\\}$ be the unique
confidence scores of the active learning model $f_{a}^{t}$ on instances in
$D_{V}$ together with two boundary values 0.0 and 1.0. We evaluate every
candidate threshold in $C^{t}$, use Equation 1 to aggregate labels and compute
the accuracy of the aggregated labels on the validation set. Note that when we
evaluate the label accuracy, we only consider the part of validation data not
rejected by the ConFusion method due to low confidence. Finally, we choose
$\tau_{t}\in C^{t}$ to maximize the aggregated label’s accuracy on the
validation set.

Both high label accuracy and coverage are desirable for training the
downstream models, thus it is also possible to select $\tau_{t}\in C^{t}$ that
maximize the aggregated label’s coverage on the validation set following a
similar process. However, selecting $\tau_{t}=0.0$ would always lead to
maximal coverage as the active learning model would be active on every
instance, making ActiveDP fall back to active learning. In our experiments, we
observe that improving label accuracy is more important in enhancing the end
model performance than improving label coverage, which also motivates us to
maximize label accuracy during threshold tuning.

###  3.3. Sample Selection Strategy

As our framework combines the prediction results of the active learning model
and the label model, the sample selector we design needs to balance between
two goals: improving the performance of the active learning model and guiding
experts to design helpful LFs. To trade-off between these goals, we use
entropy (Shannon, 1948) to evaluate the uncertainty of both models, and
propose the ADP sampler that selects the most uncertain point based on the
prediction of both models:

(2)
$x_{t+1}^{*}=argmax_{x}[Ent(f_{a}^{t}(x))^{\alpha}*Ent(f_{l}^{t}(x,\Lambda_{t}^{*}))^{1-\alpha}]$

Where $Ent()$ is the entropy function defined as follows.

(3) $Ent(p)=-\sum_{i}{p_{i}*log(p_{i})}$

The trade-off factor $\alpha\in[0,1]$ balances between the active learning
model and the label model. We set $\alpha=0.5$ for textual datasets and
$\alpha=0.99$ for tabular datasets in our experiments. We give higher weights
to the active learning model in tabular datasets because these datasets are
relatively easy to classify, thus the active learning model can reach good
performance with a small labelling budget.

###  3.4. Label Function Selection

Figure 2. Workflow of the LF selection module in ActiveDP.

In this section, we introduce the LabelPick method for LF selection, which
selects a subset of helpful LFs $\Lambda_{t}^{*}\subset\Lambda_{t}$ to train
the label model. Figure 2 illustrates the LabelPick method. The main
contribution of the LabelPick method is reducting the LF selection problem
into feature selection in supervised settings. In LabelPick, we first evaluate
the accuracy of candidate LFs using a holdout validation set $D_{V}$ drawn
I.I.D. from the underlying distribution of training data (with hidden labels)
and prune out LFs performing worse than random on the validation set; then we
create a small labelled dataset
$L_{\Lambda}^{t}=\\{(\Lambda_{t}(x_{l_{i}}),\tilde{y}_{l_{i}}\\}_{i=1}^{t}$.
The table in Figure 2 illustrates the labelled dataset $L_{\Lambda}^{t}$ we
created, which consists the weak labels for four query instances and their
pseudo labels inferred from user feedback. We leverage the graphical lasso
method (Friedman et al., 2008) to infer the dependency structure between the
LFs and the label based on $L_{\Lambda}^{t}$, and select the subset of LFs
adjacent to the class label in the dependency structure, corresponding to
choosing the Markov Blanket (Pearl, 1988) of the class label. Formally, if a
subset of LFs $\Lambda_{t}^{*}\subset\Lambda_{t}$ forms the Markov Blanket of
label $Y$, then we have
$Y\perp\\!\\!\\!\perp\Lambda_{t}/\Lambda_{t}^{*}|\Lambda_{t}^{*}$. In other
words, the Markov Blanket contains all the helpful information for inferrring
the label, making other LFs unecessary once the LFs corresponding to the
Markov Blanket are selected. Therefore, we can prune out the other LFs without
sacrificing the label model’s performance.Figure 2 shows that in the running
example, the LFs $\lambda_{1}$ and $\lambda_{3}$ forms the Markov Blanket of
$Y$, thus we select these two LFs and discard $\lambda_{2}$ as it is redundant
given $\lambda_{1}$ and $\lambda_{3}$. $\lambda_{4}$ was discarded in the
previous step due to low accuracy.

##  4\. Experiments

In this section, we evaluate the ActiveDP framework using the classification
performance of the downstream model, compare it with other interactive
frameworks for data labelling, and conduct extensive studies to investigate
the benefits of each component.

###  4.1. Experiment Setup

####  4.1.1. Datasets.

We conduct experiments with six textual datasets that have been used to
evaluate previous data programming works (Hsieh et al., 2022; Boecking et al.,
2020): Youtube Spam (Alberto et al., 2015), IMDB Review (Maas et al., 2011),
Amazon Review (He and McAuley, 2016), Yelp Review (Zhang et al., 2015), and
two subsets of the BiasBios dataset (De-Arteaga et al., 2019) that distinguish
between professor and teacher (marked as Bios-PT), and between journalist and
photographer (marked as Bios-JP) respectively. These datasets cover three
tasks: spam classification, sentiment analysis and biography classification.
Besides, we also evaluated our framework on two tabular datasets: Occupancy
(Candanedo and Feldheim, 2016), aiming predicting office room occupancy, which
has been used to evaluate the Revising LF baseline (Nashaat et al., 2018);
Census (Kohavi et al., 1996), which aims at predicting whether the income of a
person exceeds 50K or not. For each dataset, we randomly partition the data
into 80%/10%/10% for the train-validation-test split. The detailed information
of the datasets is listed in Table 2.

Table 2. Datasets used in Evaluation.

Name | Task | #Train | #Valid | #Test
---|---|---|---|---
Youtube | Spam classification | 1,566 | 195 | 195
IMDB | Sentiment analysis | 20,000 | 2,500 | 2,500
Yelp | Sentiment analysis | 20,000 | 2,500 | 2,500
Amazon | Sentiment analysis | 20,000 | 2,500 | 2,500
Bios-PT | Biography classification | 19,672 | 2,458 | 2,458
Bios-JP | Biography classification | 25,808 | 3,225 | 3,225
Occupancy | Occupancy prediction | 14,317 | 1,789 | 1,789
Census | Income classification | 25,541 | 3,192 | 3,192

####  4.1.2. Baselines.

We compare the ActiveDP framework with the following baseline methods.

  * •

Nemo (Hsieh et al., 2022): Nemo111https://github.com/ChengYuHsieh/Nemo is a
representative framework of the IDP paradigm. Nemo is also the current state-
of-the-art work in automatically labelling large datasets. As Nemo only
supports textual datasets and its SEU strategy is designed for textual data,
we compare ActiveDP with Nemo on the six textual datasets only.

  * •

IWS (Boecking et al., 2020): IWS222https://github.com/benbo/interactive-weak-
supervision actively selects a candidate label function for human verification
in each iteration. We evaluate IWS under the unbounded setting (IWS-LSE-a in
(Boecking et al., 2020)), where the final LF set includes all LFs that the
system predicts as accurate.

  * •

Revising LF (RLF) (Nashaat et al., 2018): Nashaat et al. propose to
iteratively select the instances where the current label model is most
uncertain for human labelling and use the labels to correct LF outputs on
selected instances.

  * •

Uncertainty Sampling (US) (Lewis, 1995): Uncertainty Sampling is a classical
method for active learning, where the system selects the data instance with
the highest predictive entropy for manual labelling.

####  4.1.3. Evaluation Protocol.

We evaluate all frameworks by simulating 300 iterations of manual annotation,
assess the performance of the downstream model every ten iterations, and plot
the performance curve for the downstream model. We use MeTaL (Ratner et al.,
2019) as the label model to aggregate the label functions. We train a logistic
regression model as the active learning model in ActiveDP for label quality
improvement. When training the downstream model, we extract the TF-IDF
representation of the input text and trains a logistic regression model as the
downstream ML model for classification. We adopt classification accuracy on
the test set for the downstream model performance evaluation metric and report
the average test accuracy during the run, corresponding to the area under the
performance curve. We repeat each experiment 5 times using different seeds and
report average results.

As different frameworks require different human supervision types, we set each
iteration corresponding to labelling a single instance in uncertainty sampling
and Revising LF, verifying a single LF in IWS, and designing a single LF in
Nemo and ActiveDP. While the user response time may differ in each framework,
such an evaluation protocol lets us compare the frameworks intuitively.
333Nemo (Hsieh et al., 2022) provides a user study on the response time for
different supervision types. Asking users to give an LF takes slightly more
time for users to label a single instance or verify a given LF.  Besides,
Revising LF (Nashaat et al., 2018) requires a pre-specified set of LFs, which
is not required by other frameworks. For a fair comparison, we use the same
method applied in ActiveDP to create a set of LFs during the interaction and
use $\Lambda_{t}$ as the input LF set for Revising LF when evaluating it at
the t-th iteration.

####  4.1.4. Simulated User.

Following previous works (Hsieh et al., 2022; Boecking et al., 2020), we use
ground truth examples to simulate user responses when evaluating all
frameworks. We apply the following simulation process for ActiveDP. For
textual datasets, we first consider the candidate LF space with all LFs
$\lambda_{w,y}$ that return a specific class label $y$ when observing keyword
$w$ and abstain otherwise. Given a query instance $x$, we first build a
candidate LF set $\Lambda^{c}=\\{\lambda_{w,y}:w\in x\wedge
acc(\lambda_{w,y})>t\\}$, which contains all LFs with their keywords inside
the instance $x$ and with accuracy above threshold $t$. For tabular datasets,
we consider the candidate LF space with all decision stumps
$\lambda_{j,v,op,y}$ that returns a specific class label $y$ for instance $x$
if $x_{j}\geq y(op=^{\prime}\geq^{\prime})$ or $x_{j}\leq
y(op=^{\prime}\leq^{\prime})$, and returns $-1$ otherwise. Given a query
instance $x$, we build the candidate LF set as
$\Lambda^{c}=\\{\lambda_{j,v,op,y}:1\leq j\leq m_{feat}\wedge
op\in\\{^{\prime}\leq^{\prime},^{\prime}\geq^{\prime}\\}\wedge v=x_{j}\wedge
acc(\lambda_{j,v,op,y})>t\\}$, where $m_{feat}$ is the number of features in
the dataset. In other words, we consider the decision stumps based on a single
feature with $x$ lying on the boundary. We set the accuracy threshold
$\tau_{acc}=0.6$ in our experiments. Then we filter out LFs returned in
previous iterations and select an LF from the filtered set with probability
proportional to the LF coverages. For IWS, the simulated user marks a
candidate LF as accurate when its accuracy exceeds $\tau_{acc}$. For
uncertainty sampling and Revising LF, the simulated user returns the correct
label of the selected instance.

###  4.2. System Performance

Figure 3. End-to-end Performance comparison between ActiveDP and Baseline
Methods.

Figure 3 shows the performance curve of ActiveDP and other baseline methods.
The X-axis indicates the number of queries given to the simulated user, and
the Y-axis shows the downstream model’s test set classification accuracy.
Across all datasets, ActiveDP improves the downstream model’s test set
accuracy by an average of 4.4% compared to Nemo, 13.5% compared to IWS, 2.6%
compared to Revising LF, and 6.5% compared to uncertainty sampling.

Looking into the labelling process in detail, when the label budget is small,
ActiveDP, Nemo and Revising LF outperform uncertainty sampling and IWS in
general. These three methods leverage data programming to efficiently label
large datasets, giving them advantages at the early stage of the labelling
process. On the other hand, uncertain sampling can only use a small labelled
subset of data to train the downstream model, making it perform worse than
data programming methods. IWS, although following the data programming
paradigm, does not perform well in the early steps. This is likely because the
system fails to provide good candidate LFs for human experts to verify when
the labelled data is scarce. It underperforms Nemo and ActiveDP, which ask
human experts to provide LFs instead.

When the label budget increases, the performance of uncertain sampling and
ActiveDP improves steadily, as they both train an active learning model which
benefits from more labelled data. However, for Nemo and IWS, we observe a
performance drop in some datasets. These methods only use label functions for
prediction. As they omit fine-grained instance labels for supervision, their
performance will be surpassed by active learning paradigms when the label
budget is sufficient. Further, as these methods do not apply label function
selection or only use a simple accuracy-based filter, their performance will
be impeded when some harmful LFs are used to train the label model.

For tabular datasets, the performance of uncertain sampling and ActiveDP
improves with more labelling budget, the performance of Revising LF grows
slowly, and the performance of IWS does not improve. This is because the LFs
for tabular datasets are based on decision stumps, which can only provide
coarse-grained supervision to the data. Thus developing more LFs does not
significantly improve the data quality. On the other hand, these tabular
datasets are relatively easy to classify using a small labelled dataset, thus
the performance of uncertain sampling exceeds IWS and Revising LF after around
100 queries. By combining active learning and data programming, ActiveDP
maintains good performance with only a few queries, and similar to uncertain
sampling, it has steady performance improvement with increased labelling
budgets.

Note that while the performance of Revising LF is close to ActiveDP in some
textual datasets, the performance gaps in tabular datasets is significant.
This is because the performance of Revising LF benefits from both aggregating
more LFs and revising the LFs with user feedback. In textual datasets, both
factors contribute to its performance improvement. However, in tabular
datasets, the benefit of aggregating more LFs diminishes due to the reason
explained above, leaving the effect of revising LFs to be dominant. As the LFs
are only revised on queried instances, the performance improvement for
Revising LF is much slower than ActiveDP or uncertain sampling, which trains
an active learning model to predict a larger fraction of data.

In summary, ActiveDP has good performance under both small and large labelling
budget scenarios, and its performance improves steadily with more labels,
making it a more general and effective solution in combining data programming
and active learning compared to previous works.

###  4.3. Comparative Studies

####  4.3.1. Ablation study.

Table 3. Performance of Ablated Versions of ActiveDP.

Method | Dataset |  |
---|---|---|---
Youtube | IMDB | Yelp | Amazon | Bios-PT | Bios-JP | Occupancy | Census
Baseline | 0.8130 | 0.7826 | 0.5776 | 0.7096 | 0.8527 | 0.8892 | 0.8881 | 0.7651
LabelPick | 0.8598 | 0.7850 | 0.6564 | 0.7176 | 0.8572 | 0.9027 | 0.8881 | 0.7652
ConFusion | 0.8708 | 0.7843 | 0.7538 | 0.7182 | 0.8671 | 0.8902 | 0.9906 | 0.8060
ActiveDP | 0.8894 | 0.8016 | 0.7834 | 0.7198 | 0.8785 | 0.9078 | 0.9905 | 0.8096

We conduct ablation studies to investigate the benefit of the ConFusion and
LabelPick methods. Table 3 evaluates the performance of the downstream model
using different ablated versions of ActiveDP. The baseline method uses all
user-returned LFs to train the label model; the LabelPick and ConFusion
approach only apply the LF selection and confidence-based label aggregation
techniques, respectively, and the ActiveDP method combines the two techniques.
Compared to the baseline method, the LabelPick strategy improves the test set
accuracy on average by 1.9%, and the ConFusion strategy improves the test
accuracy on average by 5.0%. ActiveDP, combining both strategies, improves the
test accuracy on average by 6.3%, showing that both strategies benefit the
performance of ActiveDP.

####  4.3.2. Sampler Performance.

Table 4. Performance of ActiveDP with different sample selectors.

Sampler | Dataset |  |
---|---|---|---
Youtube | IMDB | Yelp | Amazon | Bios-PT | Bios-JP | Occupancy | Census
Passive | 0.8699 | 0.7992 | 0.7607 | 0.7263 | 0.8692 | 0.9030 | 0.9823 | 0.7919
US | 0.8746 | 0.7997 | 0.7660 | 0.7212 | 0.8703 | 0.9058 | 0.9889 | 0.7959
LAL | 0.8858 | 0.7889 | 0.7596 | 0.7242 | 0.8657 | 0.9002 | 0.9852 | 0.8033
SEU | 0.8701 | 0.7566 | 0.7632 | 0.7259 | 0.8682 | 0.9023 | 0.9837 | 0.7927
ADP | 0.8894 | 0.8016 | 0.7834 | 0.7198 | 0.8785 | 0.9078 | 0.9905 | 0.8096

To test the sensitivity of ActiveDP to the sample selector choice, we
evaluated ActiveDP using different sample selection strategies: passive
sampling, uncertainty sampling (US) (Lewis, 1995), learning active learning
(LAL) (Konyushkova et al., 2017), select by expected utility (SEU) (Hsieh et
al., 2022), and the ADP sampler proposed by us. For the samplers designed for
active learning (US, LAL), we use the implementation in the AliPy (Tang et
al., 2019) package with default parameters.

Table 4 presents the downstream model performance using different samplers.
Our ADP sampler outperforms other evaluated samplers on 7 out of 8 datasets.
The ADP sampler is specifically designed to consider both active learning and
data programming, making it a good fit for our ActiveDP framework.

Table 5. Performance of ActiveDP with different simulated Label Noise Rates.

Label Noise | Dataset |  |
---|---|---|---
Youtube | IMDB | Yelp | Amazon | Bios-PT | Bios-JP | Occupancy | Census
0% | 0.8894 | 0.8016 | 0.7834 | 0.7198 | 0.8785 | 0.9078 | 0.9905 | 0.7948
5% | 0.8671 | 0.7902 | 0.7449 | 0.7172 | 0.8692 | 0.9049 | 0.9905 | 0.7959
10% | 0.8628 | 0.7846 | 0.7303 | 0.7091 | 0.8692 | 0.9036 | 0.9888 | 0.7922
15% | 0.8474 | 0.7793 | 0.6999 | 0.7040 | 0.8583 | 0.9002 | 0.9884 | 0.7756

####  4.3.3. Label Noise

Label noise occurs when the LF has accuracy above the threshold but misfires
on the corresponding query instances. We simulate label noise by randomly
selecting a certain fraction of query instances and letting the simulated user
generate LFs for the flipped label. The LF generation process is identical to
the process described in Section 4.1.4, except that as the label is reversed,
the generated LFs will not be accurate on the query instance (their accuracy
on the training set is still above the threshold of 0.6). As ActiveDP applies
the LFs on related query instances to generate labelled subsets, such label
noise will deteriorate the performance of the active learning model in
ActiveDP.

Table 5 illustrates the performance under various label noise levels. While
injecting label noise deteriorates the performance of ActiveDP, the
performance degradation is not significant with a low label noise level. The
average test set accuracy degradation is 1.1% with 5% label noise, 1.6% with
10% label noise, and 2.7% with 15% label noise.

##  5\. Conclusions

In this paper, we propose ActiveDP, a novel interactive framework that
combines active learning and data programming to automatically generate labels
for training ML models. ActiveDP selects a subset of helpful LFs for training
the label model and leverages active learning to improve data quality.
Extensive experiments show that ActiveDP outperforms previous weak supervision
approaches and performs well under different labelling budgets.

## References

  * (1)
  * Alberto et al. (2015) Túlio C Alberto, Johannes V Lochter, and Tiago A Almeida. 2015.  Tubespam: Comment spam filtering on youtube. In _2015 IEEE 14th international conference on machine learning and applications (ICMLA)_. IEEE, 138–143.
  * Awasthi et al. (2019) Abhijeet Awasthi, Sabyasachi Ghosh, Rasna Goyal, and Sunita Sarawagi. 2019.  Learning from Rules Generalizing Labeled Exemplars. In _International Conference on Learning Representations_.
  * Biegel et al. (2021) Samantha Biegel, Rafah El-Khatib, Luiz Otavio Vilas Boas Oliveira, Max Baak, and Nanne Aben. 2021.  Active WeaSuL: Improving Weak Supervision with Active Learning.  _arXiv preprint arXiv:2104.14847_ (2021).
  * Boecking et al. (2020) Benedikt Boecking, Willie Neiswanger, Eric Xing, and Artur Dubrawski. 2020.  Interactive weak supervision: Learning useful heuristics for data labeling.  _arXiv preprint arXiv:2012.06046_ (2020).
  * Candanedo and Feldheim (2016) Luis M Candanedo and Véronique Feldheim. 2016.  Accurate occupancy detection of an office room from light, temperature, humidity and CO2 measurements using statistical learning models.  _Energy and Buildings_ 112 (2016), 28–39.
  * De-Arteaga et al. (2019) Maria De-Arteaga, Alexey Romanov, Hanna Wallach, Jennifer Chayes, Christian Borgs, Alexandra Chouldechova, Sahin Geyik, Krishnaram Kenthapadi, and Adam Tauman Kalai. 2019.  Bias in bios: A case study of semantic representation bias in a high-stakes setting. In _proceedings of the Conference on Fairness, Accountability, and Transparency_. 120–128.
  * Denham et al. (2022) Benjamin Denham, Edmund MK Lai, Roopak Sinha, and M Asif Naeem. 2022.  Witan: unsupervised labelling function generation for assisted data programming.  _Proceedings of the VLDB Endowment_ 15, 11 (2022), 2334–2347.
  * Friedman et al. (2008) Jerome Friedman, Trevor Hastie, and Robert Tibshirani. 2008.  Sparse inverse covariance estimation with the graphical lasso.  _Biostatistics_ 9, 3 (2008), 432–441.
  * Gao et al. (2020) Mingfei Gao, Zizhao Zhang, Guo Yu, Sercan Ö Arık, Larry S Davis, and Tomas Pfister. 2020.  Consistency-based semi-supervised active learning: Towards minimizing labeling cost. In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part X 16_. Springer, 510–526.
  * Hancock et al. (2018) Braden Hancock, Paroma Varma, Stephanie Wang, Martin Bringmann, Percy Liang, and Christopher Ré. 2018.  Training Classifiers with Natural Language Explanations.  arXiv:1805.03818 [cs.CL]
  * He and McAuley (2016) Ruining He and Julian McAuley. 2016.  Ups and downs: Modeling the visual evolution of fashion trends with one-class collaborative filtering. In _proceedings of the 25th international conference on world wide web_. 507–517.
  * Hsieh et al. (2022) Cheng-Yu Hsieh, Jieyu Zhang, and Alexander Ratner. 2022.  Nemo: Guiding and Contextualizing Weak Supervision for Interactive Data Programming.  _Proc. VLDB Endow._ 15, 13 (sep 2022), 4093–4105.  https://doi.org/10.14778/3565838.3565859
  * Huang et al. ([n. d.]) Tzu-Heng Huang, Catherine Cao, Spencer Schoenberg, Harit Vishwakarma, Nicholas Roberts, and Frederic Sala. [n. d.].  SCRIPTORIUMWS: ACode GENERATION ASSISTANT FOR WEAK SUPERVISION.  ([n. d.]).
  * Kohavi et al. (1996) Ron Kohavi et al. 1996\.  Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid.. In _Kdd_ , Vol. 96. 202–207.
  * Konyushkova et al. (2017) Ksenia Konyushkova, Raphael Sznitman, and Pascal Fua. 2017.  Learning active learning from data.  _Advances in neural information processing systems_ 30 (2017).
  * Lewis (1995) David D Lewis. 1995.  A sequential algorithm for training text classifiers: Corrigendum and additional data. In _Acm Sigir Forum_ , Vol. 29. ACM New York, NY, USA, 13–19.
  * Li and Guo (2013) Xin Li and Yuhong Guo. 2013.  Adaptive active learning for image classification. In _Proceedings of the IEEE conference on computer vision and pattern recognition_. 859–866.
  * Liu et al. (2022) Peng Liu, Lizhe Wang, Rajiv Ranjan, Guojin He, and Lei Zhao. 2022.  A survey on active deep learning: from model driven to data driven.  _ACM Computing Surveys (CSUR)_ 54, 10s (2022), 1–34.
  * Maas et al. (2011) Andrew Maas, Raymond E Daly, Peter T Pham, Dan Huang, Andrew Y Ng, and Christopher Potts. 2011.  Learning word vectors for sentiment analysis. In _Proceedings of the 49th annual meeting of the association for computational linguistics: Human language technologies_. 142–150.
  * Nashaat et al. (2020) Mona Nashaat, Aindrila Ghosh, James Miller, and Shaikh Quader. 2020.  Asterisk: Generating large training datasets with automatic active supervision.  _ACM Transactions on Data Science_ 1, 2 (2020), 1–25.
  * Nashaat et al. (2018) Mona Nashaat, Aindrila Ghosh, James Miller, Shaikh Quader, Chad Marston, and Jean-Francois Puget. 2018.  Hybridization of active learning and data programming for labeling large industrial datasets. In _2018 IEEE International Conference on Big Data (Big Data)_. IEEE, 46–55.
  * Pearl (1988) Judea Pearl. 1988.  _Probabilistic reasoning in intelligent systems: networks of plausible inference_.  Morgan kaufmann.
  * Ratner et al. (2017) Alexander Ratner, Stephen H Bach, Henry Ehrenberg, Jason Fries, Sen Wu, and Christopher Ré. 2017.  Snorkel: Rapid training data creation with weak supervision. In _Proceedings of the VLDB Endowment. International Conference on Very Large Data Bases_ , Vol. 11. NIH Public Access, 269.
  * Ratner et al. (2019) Alexander Ratner, Braden Hancock, Jared Dunnmon, Frederic Sala, Shreyash Pandey, and Christopher Ré. 2019.  Training complex models with multi-task weak supervision. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 33. 4763–4771.
  * Ratner et al. (2016) Alexander J Ratner, Christopher M De Sa, Sen Wu, Daniel Selsam, and Christopher Ré. 2016.  Data programming: Creating large training sets, quickly.  _Advances in neural information processing systems_ 29 (2016), 3567–3575.
  * Ren et al. (2021) Pengzhen Ren, Yun Xiao, Xiaojun Chang, Po-Yao Huang, Zhihui Li, Brij B Gupta, Xiaojiang Chen, and Xin Wang. 2021.  A survey of deep active learning.  _ACM computing surveys (CSUR)_ 54, 9 (2021), 1–40.
  * Sener and Savarese (2018) Ozan Sener and Silvio Savarese. 2018.  Active Learning for Convolutional Neural Networks: A Core-Set Approach. In _International Conference on Learning Representations_.
  * Settles (2009) Burr Settles. 2009.  Active learning literature survey.  (2009).
  * Settles (2012) Burr Settles. 2012.  Active learning.  _Synthesis lectures on artificial intelligence and machine learning_ 6, 1 (2012), 1–114.
  * Settles and Craven (2008) Burr Settles and Mark Craven. 2008.  An analysis of active learning strategies for sequence labeling tasks. In _proceedings of the 2008 conference on empirical methods in natural language processing_. 1070–1079.
  * Seung et al. (1992) H Sebastian Seung, Manfred Opper, and Haim Sompolinsky. 1992.  Query by committee. In _Proceedings of the fifth annual workshop on Computational learning theory_. 287–294.
  * Shannon (1948) Claude Elwood Shannon. 1948.  A mathematical theory of communication.  _The Bell system technical journal_ 27, 3 (1948), 379–423.
  * Smith et al. (2022) Ryan Smith, Jason A. Fries, Braden Hancock, and Stephen H. Bach. 2022.  Language Models in the Loop: Incorporating Prompting into Weak Supervision.  arXiv:2205.02318 [cs.LG]
  * Tang et al. (2019) Ying-Peng Tang, Guo-Xiang Li, and Sheng-Jun Huang. 2019.  _ALiPy: Active Learning in Python_.  Technical Report. Nanjing University of Aeronautics and Astronautics.  https://github.com/NUAA-AL/ALiPy available as arXiv preprint https://arxiv.org/abs/1901.03802.
  * Varma and Ré (2018) Paroma Varma and Christopher Ré. 2018.  Snuba: Automating weak supervision to label training data. In _Proceedings of the VLDB Endowment. International Conference on Very Large Data Bases_ , Vol. 12. NIH Public Access, 223.
  * Zhang et al. (2015) Xiang Zhang, Junbo Zhao, and Yann LeCun. 2015.  Character-level convolutional networks for text classification.  _Advances in neural information processing systems_ 28 (2015).


# Green’s function knockout formalism

C. Hebborn<EMAIL_ADDRESS>Facility for Rare Isotope Beams, Michigan
State University, East Lansing, MI 48824, USA  Lawrence Livermore National
Laboratory, P.O. Box 808, L-414, Livermore, California 94551, USA
<EMAIL_ADDRESS>G. Potel<EMAIL_ADDRESS>Lawrence Livermore
National Laboratory, P.O. Box 808, L-414, Livermore, California 94551, USA

###### Abstract

Knockout nuclear reactions, in which a nucleon is removed from a nucleus as a
result of the collision with another nucleus, have been widely used as an
experimental tool, both to populate isotopes further removed from stability,
and to obtain information about the single-particle nature of the nuclear
spectrum. In order to fully exploit the experimental information, theory is
needed for the description of both the structure of the nuclei involved, and
the dynamics associated with the nucleon removal mechanisms. The standard
approach, using theoretical shell-model spectroscopic factors for the
structure description coupled with an eikonal model of reaction, has been
successful when used in the context of the removal of valence nucleons in
nuclei close to stability. However, it has been argued that the reaction
theory might need to be revisited in the case of exotic nuclei, more
specifically for highly asymmetric nuclei in which the deficient species
(neutrons or protons) is being removed. We present here a new formalism for
the nucleon-removal and -addition reaction through knockout and transfer
reactions, that treats consistently structure and reaction properties using
dispersive optical potentials. In particular, our formalism includes the
dynamical effects associated with the removal of a nucleon from the
projectile, which might explain the long standing puzzle of the quenching of
spectroscopic factors in nuclei with extreme neutrons-to-protons ratios.

##  I Introduction

Radioactive Ion Beams (RIBs) facilities are transforming the field of low-
energy nuclear physics by setting short-lived, exotic isotopes within
experimental reach. The availability of new experimental data has been matched
by theoretical efforts towards the description of nuclear systems away from
the stability valley, and, more generally, towards an understanding of nuclear
structure in an exotic context [1]. The corresponding paradigm shift in the
theory of nuclear structure has to be complemented by a revision of nuclear
reaction theory, needed for the description of the experiments in which
radioactive ions are involved. Within this context, a considerable effort has
been devoted recently to the description of reactions with weakly-bound nuclei
(see, e.g. [2, 3, 4, 5], see also [6] and references therein).

However, the study of neutron-rich (resp. proton-rich) raises a complementary
question about the behaviour of the deeply-bound protons (resp. neutrons)
belonging to the same nucleus. This question has been highlighted in
publications by Gade and collaborators [7, 8, 9], in which they present a
review of results of one-neutron (resp. one-proton) knockout experiments
expressed as a function of the difference $\Delta S=S_{n}-S_{p}$ between the
neutron $S_{n}$ and proton separation $S_{p}$ energies (resp. $\Delta
S=S_{p}-S_{n}$). In this work, they arrive at the puzzling conclusion that
theory is unable to account for as much as 80% of the quenching of single-
particle strength when the knocked out particle belongs to the deficient
species in systems with a large value of $\Delta S$. Several authors have
suggested that nuclear structure calculations might fail to fully account for
short-range correlations between neutrons and protons in highly asymmetric
systems, leading to an overestimation of the single-particle content of the
states populated in knockout reactions [10, 11].

On the other hand, the fact that the strong dependence on $\Delta S$ of the
spectroscopic factor quenching is not observed in transfer and quasifree
scattering experiments has led some authors to suggest that the issue might be
in the theory associated with the description of the reaction process in the
case of knockout experiments [12, 13, 14, 15, 16] (see Ref. [17] for a recent
review). Knockout experiments are often described within the eikonal model
[18, 19], assumed to be valid for high beam energies. The sudden and the core
spectator approximations, in which the nuclear degrees of freedom are frozen
during the collision process, are used to describe the one-nucleon removal
from the projectile. The core spectator approximation is based on the
asumption that the characteristic decay times of the core states populated in
the nucleon removal process are large compared to the collision time. This
seems reasonable when the nucleon is removed from a state not too far away
from the Fermi energy on a stable nucleus, since the narrow associated energy
width is small, the corresponding damping (decay) time being therefore rather
large.

However, it has been well known since knockout experiments were performed in
the early 60’s (see, e.g., [20]) that hole states associated with the removal
of deeply-bound nucleons have much larger widths, sometimes of the order of
tens of MeV. In other words, hole states resulting from the removal of a
deeply-bound nucleon can decay into complicated many-body nuclear states
rather quickly, in times of the order of $10^{-23}$-$10^{-22}$ s, which is
comparable to the short collision times associated with fast knockout
experiments. This damping process results in dissipative effects associated
with the dynamics of hole states in nuclei, not taken into account within the
core spectator approximation. These effects have already been estimated in
Ref. [21] within the intra-nuclear cascade approach, showing that they can
excite the core above the particle emission threshold, leading to particle
evaporation and a loss of flux in the outgoing channel that could account for
the observed spectroscopic factor quenching. This discussion applies to the
cases in which the residual core is the only species detected in the
experiment, such as in most standard knockout reactions.

The above parlance highlights the general need to integrate in a consistent
theoretical framework the structure and dynamics of many nucleons in a
nucleus, and the description of reactions used to study them in an
experimental context. An attempt to account for the dynamics between a nucleus
and a transferred nucleon has already been implemented in the the Green’s
Function Transfer (GFT) formalism, albeit in a one-nucleon addition context
[22, 23] (see also equivalent theories in Refs. [24, 25, 26]). We present here
an extension of this idea, the Green’s Function Knockout (GFK), which
describes one-nucleon removal processes.

In Sec. II, we derive the general formalism for one-nucleon removal reactions,
such as knockout and $(p,d)$, and we show in Sec. III its connection to one-
neutron addition processes and the GFT formalism. In Sec. IV we discuss
different approximations that can be made in the context of the GFK, before
concluding in Sec. V with a summary and outlook of future developments.

##  II Green’s function knockout formalism

Let us introduce the GFK by considering a reaction involving a projectile
nucleus $P$ of mass $m_{P}$ impinging on a target $T$ of mass $m_{T}$. This
system is described by the solution $\Psi$ of the many-body Hamiltonian
$\hat{H}$

$\displaystyle\hat{H}\ket{\Psi}=E\ket{\Psi},$ (1)

where $E$ is the energy of the system in the center of mass frame. We are
interested in the reaction channel in which a nucleon $N$ is knocked out from
the projectile and only the residual core $c$ is detected with a kinetic
energy $E_{f_{cT}}$. Since the final state of the $N$-$T$ system is often not
measured, we focus here on deriving the inclusive cross section in both the
core $c$ and $N$-$T$ systems, i.e., summed over all energy-conserving final
states $f_{{NT}}$ of the $N$-$T$ system and over all the energetically
available states $f_{c}$ of the core $c$ [27, 19, 28]. When expressed as a
function of the deflection angle of the core $\Omega$ and its final kinetic
energy $E_{f_{cT}}$, the cross section in the prior representation reads

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=\frac{2\pi\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})$
$\displaystyle\times\sum_{f_{NT},f_{c}}\left|\left\langle\left.\Psi(f_{NT},f_{c})\,\right|\hat{V}_{prior}|\phi_{T}^{(0)}\phi_{P}^{(0)}\mathcal{F}\right\rangle\right|^{2}$
$\displaystyle\times\delta(E-E_{f_{cT}}-E_{f_{c}}-E_{f_{NT}}),$ (2)

where $\Psi(f_{NT},f_{c})$ is the solution of (1) subject to the boundary
condition consisting in having an outgoing wave containing the core in a state
$f_{c}$ moving with kinetic energy $E_{f_{cT}}$ with respect to the target,
and the $N$-$T$ system is in a state $f_{NT}$. The many-body wave functions
$\phi_{T}^{(0)}$ and $\phi_{P}^{(0)}$ correspond to the ground state of the
target and the projectile, respectively, while $\mathcal{F}$ describes the
relative motion between the projectile and the target in the incoming channel.
The term
$\rho(E_{f_{cT}})=\mu_{cT}k^{f}_{cT}/\left[(2\pi)^{3}\hbar^{2}\right]$ is the
asymptotic phase space factor (density of states) of the core fragment,
$k^{f}_{cT}$ and $k_{PT}$ are respectively the final $c$-$T$ and initial
$P$-$T$ wave numbers, $\mu_{cT}$ and $\mu_{PT}$ being the $c$-$T$ and $P$-$T$
reduced masses.

In its exact form, the prior potential $\hat{V}_{prior}$ depends on the
$c$-$T$ and $N$-$T$ many-body potentials, respectively $\hat{V}_{cT}$ and
$\hat{V}_{NT}$, and on the potential $\hat{V}_{i}$ used to compute the
incoming scattering function $\mathcal{F}$

$\displaystyle\hat{V}_{prior}=\hat{V}_{cT}+\hat{V}_{NT}-\hat{V}_{i}.$ (3)

The exact prior potential therefore depends on the $N$-$T$ and $c$-$T$
relative coordinates, respectively $\mbox{\boldmath$R$}_{NT}$ and
$\mbox{\boldmath$R$}_{cT}$ (see Fig. 1), and the intrinsic coordinates of the
target and the core, respectively $\mbox{\boldmath$\xi$}_{T}$ and
$\mbox{\boldmath$\xi$}_{c}$. In this formalism, we neglect the dependence of
the prior potential on the core intrinsic coordinates
$\mbox{\boldmath$\xi$}_{c}$, although the role of reaction channels in
exciting the core is accounted for with the inclusion of an imaginary part. In
Sec. IV, we will discuss different possible choices of $\mathcal{F}$ and
$\hat{V}_{i}$.

We now approximate the exact wave function as

$\Psi(f_{NT},f_{c})\approx\chi_{cT}^{(f_{cT})}(\mbox{\boldmath$R$}_{cT})\psi_{NT}^{(f_{NT})}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$\xi$}_{T})\psi_{c}^{(f_{c})}(\mbox{\boldmath$\xi$}_{c})$
(4)

where $\chi_{cT}^{(f_{cT})}$ is the wave function describing the final $c$-$T$
relative motion, and the final wave functions of the core and the $N$-$T$
systems, respectively $\psi^{(f_{c})}_{c}$ and $\psi_{NT}^{(f_{NT})}$, are
many-body objects. These functions satisfy the Schrödinger equations

$\displaystyle\left(E_{f_{c}}-\hat{h}_{c}\right)\,\psi_{c}^{(f_{c})}(\mbox{\boldmath$\xi$}_{c})=0,$
(5)
$\displaystyle\left(E_{f_{NT}}-\hat{T}_{NT}-\hat{V}_{NT}-\hat{h}_{T}\right)\,\psi_{NT}^{(f_{NT})}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$\xi$}_{T})=0,$
(6)

where we define the many-body core ($\hat{h}_{c}$) and target ($\hat{h}_{T}$)
Hamiltonian operators, and the kinetic energy operator $\hat{T}_{NT}$. Then,

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=\frac{2\pi\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})$
$\displaystyle\times\sum_{f_{NT},f_{c}}\left|\left\langle\chi_{cT}^{(f_{cT})}\,\psi_{NT}^{(f_{NT})}\psi_{NT}^{(f_{c})}\,\right|\hat{V}_{prior}\left|\phi_{T}^{(0)}\phi_{P}^{(0)}\mathcal{F}\right\rangle\right|^{2}$
$\displaystyle\times\delta(E-E_{f_{cT}}-E_{f_{c}}-E_{f_{NT}}).$ (7)

$T$$N$$c$$P$$R$$R_{cT}$$R_{NT}$$r$$\mbox{\boldmath$k$}_{PT}$$\widehat{Z}$
Figure 1: Set of coordinates used in this article: the $c$-$N$, $c$-$T$ and
$N$-$T$ relative coordinates $r$, $\mbox{\boldmath$R$}_{cT}$ and
$\mbox{\boldmath$R$}_{NT}$, respectively.

We now write the sum over delta functions in Eq. (II), which enforces energy
conservation, in terms of the imaginary part of the Green’s function [29]

$\displaystyle\sum_{f_{NT},f_{c}}$
$\displaystyle\left|\psi_{NT}^{(f_{NT})}\psi_{c}^{(f_{c})}\right\rangle\left\langle\psi_{c}^{(f_{c})}\psi_{NT}^{(f_{NT})}\right|$
$\displaystyle\times\delta(E-E_{f_{cT}}-E_{f_{c}}-E_{f_{NT}})=-\frac{1}{\pi}\text{Im
}\hat{G}(E-E_{f_{cT}}),$ (8)

which expresses the relationship between the spectral function and the many-
body Green’s function

$\displaystyle\hat{G}(\mathcal{E})$ $\displaystyle=\lim_{\eta\to
0}\frac{1}{\mathcal{E}-\hat{h}_{c}-\hat{T}_{NT}-\hat{V}_{NT}-\hat{h}_{T}+i\eta}.$
(9)

Making use of the complete set of product states $\psi_{NT}^{(f_{NT})}$ and
$\psi_{c}^{(f_{c})}$, the Green’s function can be written in the Lehmann
representation,

$\displaystyle\hat{G}(\mathcal{E})=\lim_{\eta\to
0}\sum_{f_{NT},f_{c}}\frac{\left|\psi_{NT}^{(f_{NT})}\psi_{c}^{(f_{c})}\right\rangle\left\langle\psi_{c}^{(f_{c})}\psi_{NT}^{(f_{NT})}\right|}{\mathcal{E}-E_{f_{c}}-E_{f_{NT}}+i\eta},$
(10)

or in two other equivalent representations,

$\displaystyle\hat{G}_{h}(\mathcal{E})=\lim_{\eta\to
0}\sum_{f_{NT}}\frac{\left|\psi_{NT}^{(f_{NT})}\right\rangle\left\langle\psi_{NT}^{(f_{NT})}\right|}{\mathcal{E}-E_{f_{NT}}-\hat{h}_{c}+i\eta},$
(11) $\displaystyle\hat{G}_{NT}(\mathcal{E})=$ $\displaystyle\lim_{\eta\to
0}\sum_{f_{c}}\frac{\left|\psi_{c}^{(f_{c})}\right\rangle\left\langle\psi_{c}^{(f_{c})}\right|}{\mathcal{E}-E_{f_{c}}-\hat{T}_{NT}-\hat{V}_{NT}-\hat{h}_{T}+i\eta}.$
(12)

Using Eq. (10) we can rewrite (II) as

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=-\frac{2\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})$
$\displaystyle\times\left\langle\phi_{T}^{(0)}\phi_{P}^{(0)}\mathcal{F}\,\right|\hat{V}_{prior}\left|\,\chi^{(f_{cT})}_{cT}\right\rangle$
$\displaystyle\times\text{Im
}\hat{G}(E-E_{f_{cT}})\left\langle\chi^{(f_{cT})}_{cT}\,\right|\hat{V}_{prior}\left|\phi_{T}^{(0)}\phi_{P}^{(0)}\mathcal{F}\right\rangle,$
(13)

or, equivalently,

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=-\frac{2\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})$
$\displaystyle\times\left\langle\phi_{T}^{(0)}\phi_{P}^{(0)}\mathcal{F}\,\right|\hat{V}_{prior}\left|\,\chi^{(f_{cT})}_{cT}\right\rangle$
$\displaystyle\times\text{Im
}\hat{G}_{h}(E-E_{f_{cT}})\left\langle\chi^{(f_{cT})}_{cT}\,\right|\hat{V}_{prior}\left|\phi_{T}^{(0)}\phi_{P}^{(0)}\mathcal{F}\right\rangle.$
(14)

This expression is still difficult to handle as it contains the many-body
operator $\hat{G}_{h}$.

In order to reduce the dimensionality of the problem, we average the Green’s
function over the ground state of the projectile

$\displaystyle\left\langle\phi_{P}^{(0)}\right|\hat{G}_{h}(E-E_{f_{cT}})\left|\phi_{P}^{(0)}\right\rangle$
$\displaystyle=\lim_{\eta\to
0}\sum_{f_{NT}}\left|\psi_{NT}^{(f_{NT})}\right\rangle\left\langle\psi_{NT}^{(f_{NT})}\right|$
$\displaystyle\times\,\left\langle\phi_{P}^{(0)}\right|\frac{1}{E-E_{f_{cT}}-E_{f_{NT}}-\hat{h}_{c}+i\eta}\left|\phi_{P}^{(0)}\right\rangle.$
$\displaystyle=\lim_{\eta\to
0}\sum_{f_{NT}}\left|\psi_{NT}^{(f_{NT})}\right\rangle\left\langle\psi_{NT}^{(f_{NT})}\right|\hat{G}^{opt}_{h}(E-E_{f_{cT}}-E_{f_{NT}})$
(15)

where we have defined the optical reduction of the Green’s function

$\displaystyle\hat{G}^{opt}_{h}(E_{h})$ $\displaystyle=\lim_{\eta\to
0}\left\langle\phi_{P}^{(0)}\right|\left(E_{h}-\hat{h}_{c}+i\eta\right)^{-1}\left|\phi_{P}^{(0)}\right\rangle$
$\displaystyle=\lim_{\eta\to
0}\left(E_{h}-\hat{T}_{h}-\hat{U}_{h}+i\eta\right)^{-1},$ (16)

which is a one-body operator. In this equation, we have defined the one-
particle $\hat{T}_{h}$ kinetic and $\hat{U}_{h}$ hole potential operators,
which in general is non-local, complex and energy-dependent [30]. The
eigenstates of the hole Hamiltonian correspond to discrete overlap functions

$\psi_{h}^{(f_{c})}(\mbox{\boldmath$r$})=\innerproduct{\phi_{P}^{(0)}}{\psi_{c}^{(f_{c})}},$
(17)

solutions to the Schrödinger equation [29, 30]

$\displaystyle\left(E_{h}-\hat{T}_{h}-\hat{U}_{h}\right)\,\psi_{h}^{(f_{c})}(\mbox{\boldmath$r$})=0.$
(18)

The energy needed to promote a nucleon of the projectile $P$ to a zero-energy
state, leaving the core in its ground state, is the nucleon separation energy
$S_{N}^{(P)}$, which we define positive for particle-bound systems, according
to the standard convention. Therefore, the core ground state is obtained by
creating a hole of energy $E_{h}=-S_{N}^{(P)}$ in the projectile. An excited
state of the core with energy $E_{f_{c}}$ is obtained by delivering the
corresponding additional energy, thus creating a deeper hole,

$E_{h}=-E_{f_{c}}-S_{N}^{(P)}.$ (19)

Note that Eq. (18) does not constrain the proper normalization of
$\psi_{h}^{(f_{c})}$, namely its spectroscopic factor $S$. Since a dispersive
optical potential can be identified with the self-energy of the nucleon in the
nuclear medium, the spectroscopic factor is connected with the energy
dependence of the hole potential,

$S(E_{f_{c}})=\left(1-\left.\frac{\partial U_{h}(E)}{\partial
E}\right|_{E_{h}}\right)^{-1}.$ (20)

The above equation is verified only for dispersive potentials, and expresses
the relationship between the optical potential and the energy distribution of
single particle strength [30, 29]. It highlights the importance of the use of
dispersive potentials in the present formalism, where it is essential to have
an accurate description of the spectral function, including the verification
of sum rules based on the conservation of the number of particles.

By also defining the transition amplitudes

$\displaystyle
T^{(f_{NT})}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT})=\int\mathcal{F}(\mbox{\boldmath$R$}_{cT},\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$\xi$}_{T})\phi_{T}^{(0)}(\mbox{\boldmath$\xi$}_{T})\,\psi^{{(f_{NT})}*}_{NT}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$\xi$}_{T})\hat{V}_{prior}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT},\mbox{\boldmath$\xi$}_{T})d\mbox{\boldmath$\xi$}_{T},$
(21)

and the source term

$\displaystyle\rho_{h}^{(f_{NT})}(\mbox{\boldmath$r$})=\int\chi^{(f_{cT})*}_{cT}(\mbox{\boldmath$R$}_{cT})\,T^{(f_{NT})}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT})\,d\mbox{\boldmath$R$}_{cT},$
(22)

Eq. (14) becomes

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=-\frac{2\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})$
$\displaystyle\times\,\sum_{f_{NT}}\left\langle\rho_{h}^{(f_{NT})}\right|\text{Im
}\hat{G}^{opt}_{h}(E_{h})\left|\rho_{h}^{(f_{NT})}\right\rangle,$ (23)

where the argument of the Green’s function reflects energy conservation, i.e.

$\displaystyle E_{f_{c}}$ $\displaystyle=E-E_{f_{cT}}-E_{f_{NT}},$ (24)

complemented with (see Eq. (19))

$\displaystyle E_{h}=E_{f_{cT}}+E_{f_{NT}}-E-S_{N}^{(P)}.$ (25)

Eq. (II) can be interpreted in terms of the source term (22) expressing the
probability of the production of a hole in the projectile $P$ at a position
$r$, while the Green’s function describes the dynamical evolution of the core-
hole system. Dissipative effects associated with core excitation are connected
with the imaginary part of the optical potential, and are fully accounted for.
When these excitations take place at energies above the nucleon emission
threshold they can result in particle evaporation, and thus reduce the
knockout cross section of deeply-bound nucleons [21].

In order to highlight the effect of the absorption of the core by the hole, we
will express Eq. (II) in terms of $\hat{U}_{h}$ by first defining the free
hole propagator $\hat{G}_{h}^{opt,0}$ as

$\hat{G}_{h}^{opt,0}(E_{h})=\lim_{\eta\to
0}\left(E_{h}-\hat{T}_{h}+i\eta\right)^{-1},$ (26)

so that we can write

$\displaystyle(\hat{G}_{h}^{opt,0})^{-1}-(\hat{G}^{opt}_{h})^{-1}=\hat{U}_{h}.$
(27)

By manipulating this expression, we obtain

$\displaystyle\hat{G}_{h}^{opt,0}\left[(\hat{G}_{h}^{opt,0})^{-1}-(\hat{G}^{opt}_{h})^{-1}\right]\hat{G}^{opt}_{h}$
$\displaystyle=\hat{G}^{opt}_{h}-\hat{G}_{h}^{opt,0}$
$\displaystyle=\hat{G}_{h}^{opt,0}\hat{U}_{h}\hat{G}^{opt}_{h},$ (28)

which leads to the Dyson equation

$\displaystyle\hat{G}^{opt}_{h}=\hat{G}_{h}^{opt,0}+\hat{G}_{h}^{opt,0}\hat{U}_{h}\hat{G}^{opt}_{h}.$
(29)

We can now rewrite this propagator as

$\displaystyle\hat{G}^{opt}_{h}$
$\displaystyle=\left(1+\hat{G}^{opt\dagger}_{h}\hat{U}_{h}^{\dagger}\right)\hat{G}_{h}^{opt,0}\left(1+\hat{U}_{h}\hat{G}^{opt}_{h}\right)$
$\displaystyle-\hat{G}^{opt\dagger}_{h}\hat{U}_{h}^{\dagger}\hat{G}^{opt}_{h}.$
(30)

The first term of Eq. (II) corresponds to scattering states of the core-hole
system. Since the hole Hamiltonian describes the removal of a bound nucleon
from the projectile ground state, it does not have any scattering solutions
and this first term vanishes. We therefore obtain

${\rm Im}\,\hat{G}^{opt}_{h}=\hat{G}^{opt\dagger}_{h}{\rm
Im}\,\hat{U}_{h}\hat{G}^{opt}_{h}.$ (31)

By defining the hole wave function

$\displaystyle\phi^{(f_{NT})}_{h}(\mbox{\boldmath$r$})$
$\displaystyle=\hat{G}^{opt}_{h}(E_{h})\,\rho^{(f_{NT})}_{h}(\mbox{\boldmath$r$})$
(32)

we can then write the cross section (II) as

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=-\frac{2\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})$
$\displaystyle\times\,\sum_{f_{NT}}\left\langle\phi^{(f_{NT})}_{h}\right|{\rm
Im}\,\hat{U}_{h}(E_{h})\left|\phi_{h}^{(f_{NT})}\right\rangle.$ (33)

If we assume that for excitation energies above the first particle emission
threshold $S_{x}^{(c)}$ the core will evaporate particles, the experimental
cross section for observing the core $c$ is restricted to
$0<E_{f_{c}}<S_{x}^{(c)}$ and therefore the hole energy is restricted to
$-S_{x}^{(c)}-S_{N}^{(P)}<E_{h}<-S_{N}^{(P)}$. The cross section therefore
reads

$\displaystyle\frac{d\sigma}{dE_{f_{cT}}d\Omega}=-\frac{2\mu_{PT}}{\hbar^{2}k_{PT}}\,\rho(E_{f_{cT}})\sum_{E_{h}=-S_{x}^{(c)}-S_{N}^{(P)}}^{-S_{N}^{(P)}}\left\langle\phi_{h}^{(f_{NT})}\right|{\rm
Im}\,\hat{U}_{h}(E_{h})\left|\phi_{h}^{(f_{NT})}\right\rangle.$ (34)

Let us stress that the above expression is inclusive in both the $N$-$T$ and
the final states of the core. The sum over all energy-accessible final states
of the core is implied in the imaginary part of the hole potential ${\rm
Im}\,\hat{U}_{h}$, while it is explicit in the $N$-$T$ channel, as we sum over
the hole states $\phi_{h}^{(f_{NT})}$, which contain the $N$-$T$ transition
amplitudes in the source term (22). The only approximations made in the
derivation of Eq. (34) are the assumption that the prior potential does not
depend explicitly on the core internal coordinates, and the factorization of
the many-body wave function (4). In particular, no sudden or core spectator
approximations has been made concerning the dynamics of the $N$-$T$ and
$N$-$c$ systems. This is in contrast with the standard eikonal framework [19,
27, 7], where dynamical, non-sudden effects associated with the nucleon
extraction from the projectile (or hole creation) are neglected. It is
reasonable to expect that these effects will be particularly important for the
knockout of deeply-bound nucleons, creating deeply-bound holes, which could
give rise to core excitation above particle emission thresholds and therefore
to particle evaporation. The quenching of spectroscopic strength is further
enhanced in systems with high neutron-proton asymmetry by the fact that the
emission threshold of the deficient species tends to be low, and the sum in
Eq. (34) is severely restricted.

##  III Application to $A(p,d)B$ and $B(d,p)A$ transfer reactions

Let us now consider a pickup reaction, in which a neutron is transferred from
the projectile $A(\equiv B+n)$ to the proton target $p$, resulting in the
formation of a deuteron $d$ and the core nucleus $B$. In these reactions, the
deuteron is detected with kinetic energy $E_{f_{d}}$, and the residual nucleus
$B$ is not observed. While this process is still inclusive in the core ($B$)
channel, it is exclusive in the $n$-$p$ channel, in which the deuteron has
been detected in the only available bound state, namely its ground state.
According to the notation of the previous section,

$\displaystyle c\equiv B\quad T\equiv p\quad N\equiv n\quad c+N\equiv A\quad
T+N\equiv d.$ (35)

In this case, the deuteron ground state, with binding energy
$\epsilon_{d}=-2.2246$ MeV, is the only one to be kept in the sum appearing in
Eq. (34). The final $N$-$T$ wave function corresponds to the ground state of
the deuteron $\psi_{NT}^{(f_{NT})}\equiv\phi_{d}$, and the differential cross
section becomes

$\frac{d\sigma}{dE_{f_{d}}d\Omega}=-\frac{2\mu_{pA}}{\hbar^{2}k_{pA}}\,\rho(E_{f_{d}})\left\langle\phi_{h}\right|{\rm
Im}\,\hat{U}_{h}(E_{h})\left|\phi_{h}\right\rangle,$ (36)

where $k_{pA}$ is the initial $p$-$A$ wave number, $\mu_{pA}$ is the $p$-$A$
reduced mass,
$E_{h}={E_{f_{d}}}+\epsilon_{d}-E-S_{n}^{(A)}=-E_{f_{B}}-S_{n}^{(A)}$ is the
$B$-hole energy, ${E_{f_{B}}}$ is the energy of the final state of the
residual nucleus $B$ and $S_{n}^{(A)}$ is the neutron separation energy of the
$A$ nucleus. The GFK formalism describes non-sudden, dissipative processes in
the core $B$ system, and takes into account the quantum many-body dynamics
(encoded here in the imaginary part of the hole optical potential, ${\rm
Im}\,\hat{U}_{h}$) to describe its final state. Thanks to the use of
dispersive optical potentials, the GFK formalism allows a consistent
description of pickup reactions leaving the residual nucleus in a bound and
resonant state. Typically, ${\rm Im}\,\hat{U}_{h}$ will be spin- and parity-
dependent. Within this context, this framework might be useful in situations
in which one is interested in the determination of the energy, spin, and
parity of the final state of the residual nucleus, as in, e.g., $(p,d)$
surrogate reactions [31, 32].

In reactions where a neutron is transferred from the target to the projectile,
such as $B(d,p)A$, the state of the nucleus $A$ is not measured. However, as
for pickup reactions, the energy of the state of the nucleus $A$ can be
deduced from the energy of the final proton $E_{f_{p}}$. In this case, we have

$\frac{d\sigma}{dE_{f_{p}}d\Omega}=-\frac{2\mu_{d}}{\hbar^{2}k_{d}}\,\rho(E_{f_{p}})\left\langle\phi_{n}\right|{\rm
Im}\,\hat{U}_{n}(E_{n})\left|\phi_{n}\right\rangle,$ (37)

where $k_{d}$ is the initial wave number of the $d$-$B$ system, $\mu_{d}$ is
the $d$-$B$ reduced mass, $E_{n}$ is the excitation energy relative to the
ground state of the final nucleus $A$ and $\hat{U}_{n}$ is the $n$-$B$ optical
potential. Note that if the incoming scattering wave function $\mathcal{F}$ is
obtained within the Distorted Wave Born Approximation (DWBA) described in Sec.
IV.2, this expression is similar to Eq. (25) of Ref. [22], and the definition
of the $n$-$B$ wave function $\phi_{n}$ is analogous to the one in Sec. II.
The framework presented here represents therefore a generalization of the GFT
[22] to describe inclusive measurements of both one-nucleon addition and
removal reactions. Compared to previous models, the GFK is therefore
applicable to both transfer and knockout reactions. This general applicability
of the GFK is a great advantage, as it provides a theoretical framework to
compare their analysis and gives insights on what are the reaction mechanisms
at play.

Consequently, the GFK might allow to shed some light on the discrepancy
between the analyses of transfer and knockout experiments for projectile with
large neutron-to-proton asymmetry [7, 8, 9, 12, 13, 17]. Since in transfer
reactions, the knocked-out nucleon is measured (in the case of ($p,d$), it
forms a deuteron with the proton target), any dissipative effects associated
with the extraction of the neutron from the projectile $A$, will not impact
the cross sections (except if the excited residual nucleus is able to emit
deuterons). In knockout reactions, since it is the residual nucleus that is
detected, the cross section is therefore directly impacted by these particle
emissions of the residual nucleus. Since these effects are expected to be more
important for the removal of deeply-bound nucleons, this suggests that that
these dissipative effects might contribute to explain the discrepancy between
the analyses of transfer and knockout data as discussed in Ref. [21].

##  IV Computation of the incoming scattering wave

To implement the GFK formalism, it is useful to approximate the incoming
scattering wave $\mathcal{F}$. We present below two possible approaches, which
will also determine the choice of the potential $\hat{V}_{prior}$ used in the
transition amplitudes (21).

###  IV.1 Eikonal approximation

Reactions measured at energies above 60 MeV/nucleon, such as breakup
reactions, are accurately described by the eikonal model [19, 33, 34, 35, 36].
This approximation [18] assumes that the projectile-target relative motion
does not differ much from the initial plane wave $\chi_{i}$, strongly
simplifying the three-body problem (more details can be found in Ref. [3]).
The eikonal incoming distorted wave is given by

$\displaystyle\mathcal{F}\approx
S_{NT}(\mbox{\boldmath$R$}_{NT})\,S_{cT}(\mbox{\boldmath$R$}_{cT})\,\chi_{i}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT}).$
(38)

where $S_{NT}$ and $S_{cT}$ are respectively the $N$-$T$ and $c$-$T$ eikonal
$S$-matrices. The eikonal model exhibit cylindrical symmetry, it is therefore
often expressed in terms of the transverse $(b_{(N,c)T})$ and longitudinal
$(Z_{(N,c)T})$ coordinates

$\displaystyle R_{(N,c)T}^{2}=b_{(N,c)T}^{2}+Z_{(N,c)T}^{2}.$ (39)

In these coordinates, the eikonal $S$-matrices can then be written as

$\displaystyle S_{NT}(Z_{NT},b_{NT})=\exp[i\delta^{\rm
eik}_{NT}(Z_{NT},b_{NT})\Big{]},$ (40) $\displaystyle
S_{cT}(Z_{cT},b_{cT})=\exp[i\delta^{\rm eik}_{cT}(Z_{cT},b_{cT})\Big{]},$ (41)

in terms of the eikonal phases $\delta^{\rm eik}_{NT}$ and $\delta^{\rm
eik}_{cT}$

$\displaystyle\delta^{\rm
eik}_{NT}(Z_{NT},b_{NT})=-\frac{\mu_{PT}}{\hbar^{2}k_{PT}}\int_{-\infty}^{Z_{NT}}U_{NT}(Z^{\prime}_{NT},b_{NT})\,dZ^{\prime}_{NT},$
(42) $\displaystyle\delta^{\rm
eik}_{cT}(Z_{cT},b_{cT})=-\frac{\mu_{PT}}{\hbar^{2}k_{PT}}\int_{-\infty}^{Z_{cT}}U_{cT}(Z^{\prime}_{cT},b_{cT})\,dZ^{\prime}_{cT},$
(43)

with $\hat{U}_{NT}$ and $\hat{U}_{cT}$ the $N$-$T$ and $c$-$T$ optical
potentials.

By assuming that the nucleon and the core have the same initial velocity as
the projectile, i.e.,

$\displaystyle k_{NT}=\frac{\mu_{NT}}{\mu_{PT}}k_{PT};\quad
k_{cT}=\frac{\mu_{cT}}{\mu_{PT}}k_{PT},$ (44)

the $\chi_{i}$ eikonal scattering wave is simply given by incoming plane waves

$\displaystyle\chi_{i}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT})=\exp[i\left(k_{NT}Z_{NT}+k_{cT}Z_{cT}\right)\Big{]}.$
(45)

In the eikonal model, the prior potential $V_{prior}$ is chosen as the sum of
the $c$-$T$ and $n$-$T$ interactions [27, 37]

$\hat{V}_{prior}=\hat{V}_{cT}+\hat{V}_{NT}.$ (46)

As discussed in Ref. [37], this allows to take into account breakup effects in
the entrance channel.

In order to further simplify the calculation, one can also approximate the
$N$-$T$ and $c$-$T$ final wave functions within the eikonal model

$\displaystyle\chi^{(f_{NT})*}_{NT}(\mbox{\boldmath$R$}_{NT})$
$\displaystyle=S_{NT}^{f*}(\mbox{\boldmath$R$}_{NT})e^{-i\mbox{\boldmath$k$}^{f}_{{NT}}\mbox{\boldmath$R$}_{NT}},$
(47) $\displaystyle\chi^{(f_{cT})*}_{cT}(\mbox{\boldmath$R$}_{cT})$
$\displaystyle=S_{cT}^{f*}(\mbox{\boldmath$R$}_{cT})e^{-i\mbox{\boldmath$k$}^{f}_{{cT}}\mbox{\boldmath$R$}_{cT}}.$
(48)

where $S_{{(N,c)}T}^{f*}$ is the eikonal $S$-matrix and
$\mbox{\boldmath$k$}^{f}_{(N,c)T}$ are the $(N,c)$-$T$ final wave vectors.

Then, by approaching the integral over the target degrees of freedom
$\mbox{\boldmath$\xi$}_{T}$ in Eq. (21) in terms of an eikonal $N$-$T$ final
wave function, $U_{NT}$ and $U_{cT}$, the transition amplitude reads

$\displaystyle
T^{(f_{NT})}(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT})$
$\displaystyle\approx
S_{NT}(\mbox{\boldmath$R$}_{NT})S_{{N}T}^{f*}(\mbox{\boldmath$R$}_{NT})\,S_{cT}(\mbox{\boldmath$R$}_{cT})\,$
$\displaystyle\times\,[U_{NT}(\mbox{\boldmath$R$}_{NT})+U_{cT}(\mbox{\boldmath$R$}_{cT})]e^{-i\mbox{\boldmath$q$}_{NT}(\mbox{\boldmath$R$}_{NT}-Z_{NT}\hat{Z})}e^{ik_{cT}Z_{cT}},$
(49)

and the source term to be used within this eikonal framework is

$\displaystyle\rho_{h}^{(f_{NT})}(\mbox{\boldmath$r$})$ $\displaystyle\approx$
$\displaystyle\int
S_{NT}(\mbox{\boldmath$R$}_{NT})S_{{N}T}^{f*}(\mbox{\boldmath$R$}_{NT})\,S_{cT}(\mbox{\boldmath$R$}_{cT})S_{cT}^{f*}(\mbox{\boldmath$R$}_{cT})\,$
$\displaystyle\times[U_{NT}(\mbox{\boldmath$R$}_{NT})+U_{cT}(\mbox{\boldmath$R$}_{cT})]$
$\displaystyle\times
e^{-i\mbox{\boldmath$q$}_{NT}(\mbox{\boldmath$R$}_{NT}-Z_{NT}\hat{Z})}e^{-i\mbox{\boldmath$q$}_{cT}(\mbox{\boldmath$R$}_{cT}-Z_{cT}\hat{Z})}d\mbox{\boldmath$R$}_{cT}$
(50)

where we define the transferred momenta
$\mbox{\boldmath$q$}_{cT}=\mbox{\boldmath$k$}^{f}_{{cT}}-k_{cT}\hat{Z}$ and
$\mbox{\boldmath$q$}_{NT}=\mbox{\boldmath$k$}^{f}_{{NT}}-k_{NT}\hat{Z}$. Note
that this expression can be further simplifying assuming
$k^{f}_{(c,N)T}\approx k_{(c,N)T}$, neglecting the dynamics of the reaction,
as done in the usual eikonal model.

Since the wave function (38) takes into account breakup effects [37], the
eikonal approximation is able to describe physical process in which the hole
or nucleon absorption takes place before as well as after (or simultaneously
to) the breakup process. However, it should only be applied when the
kinematical conditions are suitable for an eikonal approximation, i.e., when
the bombarding energy is large enough. It is important to note that, contrary
to the usual eikonal description of knockout reactions [19, 27], the cross
section (34) accounts for non-sudden effects in the breakup of the projectile,
by treating explicitly the dynamics of the core-hole system in terms of the
hole optical potential $\hat{U}_{h}$.

Finally, let us stress that the extension of the eikonal approximation to
treat explictly non-local $N$-$T$ and $c$-$T$ optical potentials is not
straighforward [38]. The issue lies in the fact that non-local interactions
depend on a integral over the whole space of the wave function and the non-
local potential, while the eikonal wave function is not accurate at short
distances. One way to avoid this issue is to derive the local-equivalent
potentials, (i.e, local potentials producing the same elastic phase shifts as
the original non-local ones), and to use them to compute the eikonal phase
shifts (42)–(43).

###  IV.2 Distorted Wave Born Approximation

In what we will call the DWBA approximation to the GFK formalism, the
distorted wave $\mathcal{F}$ is approximated by

$\displaystyle\mathcal{F}\approx\chi_{PT}(\mbox{\boldmath$R$}),$ (51)

where $\chi_{PT}$ is the solution of the Schrödinger equation with the
projectile-target optical potential $\hat{U}_{PT}$

$\displaystyle\left(\hat{T}_{PT}+\hat{U}_{PT}-E\right)\chi_{PT}(\mbox{\boldmath$R$})=0.$
(52)

The potential $\hat{V}_{prior}$ will now include the remnant term associated
with the standard DWBA,

$\displaystyle\hat{V}_{prior}=\hat{V}_{cT}+\hat{V}_{NT}-\hat{U}_{PT}.$ (53)

In a similar spirit as the one we adopted in the eikonal approach, one can
also approximate the $N$-$T$ and $c$-$T$ final wave functions
$\chi^{(f_{NT})}_{NT}$ and $\chi^{(f_{cT})}_{cT}$ by the solutions of the
Schrödinger equations with the optical potentials $U_{NT}$ and $U_{cT}$,
respectively. The integral over the target degrees of freedom
$\mbox{\boldmath$\xi$}_{T}$ in Eq. (21) can then be evaluated in terms of a
$N$-$T$ final wave function, $U_{NT}$ and $U_{cT}$

$\displaystyle T^{(f_{NT})}$
$\displaystyle(\mbox{\boldmath$R$}_{NT},\mbox{\boldmath$R$}_{cT})\approx\chi_{PT}(\mbox{\boldmath$R$})\chi^{(f_{NT})*}_{NT}(\mbox{\boldmath$R$}_{NT})$
$\displaystyle\times\left[U_{NT}(\mbox{\boldmath$R$}_{NT})+U_{cT}(\mbox{\boldmath$R$}_{cT})-U_{PT}(\mbox{\boldmath$R$})\right],$
(54)

and the source term becomes

$\displaystyle\rho_{h}^{(f_{NT})}$
$\displaystyle(\mbox{\boldmath$r$})\approx\int\chi^{(f_{cT})*}_{cT}(\mbox{\boldmath$R$}_{cT})\chi_{PT}(\mbox{\boldmath$R$})\chi^{(f_{NT})*}_{NT}(\mbox{\boldmath$R$}_{NT})$
$\displaystyle\times\left[U_{NT}(\mbox{\boldmath$R$}_{NT})+U_{cT}(\mbox{\boldmath$R$}_{cT})-U_{PT}(\mbox{\boldmath$R$})\right]d\mbox{\boldmath$R$}_{cT}.$
(55)

Let us emphasize that $\chi^{(f_{NT})}_{NT}$ can be calculated with the help
of the optical reduction of the Green’s function $\hat{G}_{NT}$ (12),

$\displaystyle\hat{G}^{opt}_{NT}(E_{f_{NT}})$ $\displaystyle=\lim_{\eta\to
0}\left\langle\phi_{T}^{(0)}\right|\left(E_{f_{NT}}-\hat{T}_{NT}-\hat{V}_{NT}-\hat{h}_{T}+i\eta\right)^{-1}\left|\phi_{T}^{(0)}\right\rangle$
$\displaystyle=\lim_{\eta\to
0}\left(E_{f_{NT}}-\hat{T}_{NT}-\hat{U}_{NT}-\epsilon_{T}^{(0)}+i\eta\right)^{-1},$
(56)

where $\epsilon_{T}^{(0)}$ is the target ground-state energy. This single-
particle Green’s function can be calculated numerically with Lagrange mesh
techniques [39]. Although straighforward for local potentials, computing
Green’s functions for non-local potentials is not trivial and will be reported
in another contribution [40]. The overlap can then be obtained making use of
the relation

$\displaystyle\chi_{NT}^{(f_{NT})*}(\mbox{\boldmath$r$};E_{f_{NT}})\chi_{NT}^{(f_{NT})}(\mbox{\boldmath$r$};E_{f_{NT}})=$
$\displaystyle\hskip 85.35826pt-\frac{1}{\pi}\,\text{Im
}G^{opt}_{NT}(\mbox{\boldmath$r$},\mbox{\boldmath$r$};E_{f_{NT}}).$ (57)

As mentioned in Sec. II, this procedure enforces the proper normalization of
the overlap as the spectroscopic factor is directly encoded in the energy
dependence of the Green’s function [see Eq. (20)]. Since the cross section
(34) is a functional of $\chi_{NT}^{(f_{NT})*}\chi_{NT}^{(f_{NT})}$, it is
unchanged under an arbitrary phase change $\chi_{NT}^{(f_{NT})}\to
e^{i\varphi}\chi_{NT}^{(f_{NT})}$, and the expression (IV.2) is enough to
provide $\chi_{NT}^{(f_{NT})}$. Once the overlap has been determined, the
source term (IV.2) and the hole wave function (32) can be computed by
numerical integration.

As for the eikonal model, the DWBA formulation of the GFK accounts for
processes in which breakup has been induced. The main difference here is that
the approximation is valid for low bombarding energies, for which the eikonal
approximation may not be accurate.

##  V Conclusions

One-nucleon knockout and transfer reactions are key probes of the single-
particle structure of nuclei away from stability. The standard theoretical
approach associated with these observables rely on spectroscopic factors
derived within some nuclear structure formalism, and reaction cross sections,
and the discrepancy between theory and experiment is often associated with
missing correlations in the structure description [17]. A striking feature of
the comparison between the theoretical and experimental knockout observables
is a marked neutron-to-proton asymmetry dependence which is not observed in
the analysis of transfer and quasifree reactions [17, 7, 8, 9, 16, 13, 12]. In
order to understand what causes this discrepancy, it is pressing to describe
both of these reaction processes within the same framework, providing a
unified description of structure and reaction.

In this work, we introduce a new formalism, the GFK, which describes one-
neutron knockout and transfer reactions making use of dispersive optical
potentials, hence treating on the same footing bound and scattering states.
For one-nucleon addition transfer reactions, which are typically measured at
low to medium energies (from few MeVs to 50 MeV/nucleon), the use of a DWBA
incoming scattering function leads to the GFT formalism [22]. Moreover, the
GFK can also predict one-nucleon removal reactions, such as $(p,d)$ and
knockout reactions, thus allowing the description of transfer and knockout
reactions within the same framework.

Because the GFK relies on Green’s functions, the link between the few-body
problem and the underlying nuclear structure of the ground states of the
target and the projectile is made explicit. Moreover, no core spectator or
sudden approximation is made, which allows to include dynamical effects
associated with the nucleon extraction from the projectile, such as excitation
of the core above the particle emission threshold during the collision. Our
analysis suggests that the discrepancy observed in the analysis of knockout
and transfer data might arise from these dynamical effects, that are neglected
in the usual eikonal model.

The main approximations made in the GFK are the assumptions that the prior
potential does not depend on the intrinsic coordinates of the core and the
factorization of the many-body wave function (4), in which the effect of the
projectile-target interaction is described by a incoming scattering function
$\mathcal{F}$. The choice of this function reflects the approximation of the
few-body problem we are willing to make, and can be adapted to a specific
energy regime. In particular, we discuss an eikonal and a DWBA approximation.
We plan to test the validity of these approximations and verify their
applicability for different systems, i.e., with various beam energies and for
nuclei ranging from the valley of stability to the proton and neutron
driplines.

In a future publication we plan to compare knockout and transfer observables
obtained within a standard reaction model, with the GFK calculation along an
isotopic chain. For this, we plan to use the dispersive optical model
developed in Refs. [30, 41, 42, 43], which provides a description of structure
and reaction properties for nuclei exhibiting different neutron-to-proton
asymmetry. This study will provide quantitative estimates of the dynamical
effects associated with the extraction of the nucleon, and might help to
explain the systematic discrepancy observed by Gade _et al._ [7, 8, 9].

###### Acknowledgements.

The authors thank P. Capel, W. H. Dickhoff, J. E. Escher, T. Frederico and C.
D. Pruitt for insightful discussions and J. E. Escher and C. D. Pruitt for
their careful rereading of this manuscript. C. H. acknowledges the support of
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics,
under the FRIB Theory Alliance award no. DE-SC0013617 and under Work Proposal
no. SCW0498. This work was performed under the auspices of the U.S. Department
of Energy by Lawrence Livermore National Laboratory under Contract No. DE-
AC52-07NA27344.

## References

  * Otsuka et al. [2020] T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, and Y. Utsuno, Rev. Mod. Phys. 92, 015002 (2020).
  * Hagino et al. [2022] K. Hagino, K. Ogata, and A. Moro, Prog. Part. Nucl. Phys. 125, 103951 (2022), ISSN 0146-6410.
  * Baye and Capel [2012] D. Baye and P. Capel, _Breakup reaction models for two- and three-cluster projectiles_ (Springer, Heidelberg, 2012), vol. 848 of _Lecture Notes in Physics_ , pp. 121–163.
  * Bonaccorso [2018] A. Bonaccorso, Prog. Part. Nucl. Phys. 101, 1 (2018), ISSN 0146-6410.
  * Bertulani and Bonaccorso [2022] C. A. Bertulani and A. Bonaccorso, arXiv:2201.00433 (2022).
  * Nunes et al. [2020] F. Nunes, G. Potel, T. Poxon-Pearson, and J. Cizewski, Annual Review of Nuclear and Particle Science 70, 147 (2020).
  * Gade et al. [2008] A. Gade, P. Adrich, D. Bazin, M. D. Bowen, B. A. Brown, C. M. Campbell, J. M. Cook, T. Glasmacher, P. G. Hansen, K. Hosier, et al., Phys. Rev. C 77, 044306 (2008).
  * Tostevin and Gade [2014] J. A. Tostevin and A. Gade, Phys. Rev. C 90, 057602 (2014).
  * Tostevin and Gade [2021] J. A. Tostevin and A. Gade, Phys. Rev. C 103, 054610 (2021).
  * Jensen et al. [2011] O. Jensen, G. Hagen, M. Hjorth-Jensen, B. A. Brown, and A. Gade, Phys. Rev. Lett. 107, 032501 (2011).
  * Wylie et al. [2021] J. Wylie, J. Okołowicz, W. Nazarewicz, M. Płoszajczak, S. M. Wang, X. Mao, and N. Michel, Phys. Rev. C 104, L061301 (2021).
  * Tsang et al. [2009] M. B. Tsang, J. Lee, S. C. Su, J. Y. Dai, M. Horoi, H. Liu, W. G. Lynch, and S. Warren, Phys. Rev. Lett. 102, 062501 (2009).
  * Lee et al. [2010] J. Lee, M. B. Tsang, D. Bazin, D. Coupland, V. Henzl, D. Henzlova, M. Kilburn, W. G. Lynch, A. M. Rogers, A. Sanetullaev, et al., Phys. Rev. Lett. 104, 112701 (2010).
  * Flavigny et al. [2012] F. Flavigny, A. Obertelli, A. Bonaccorso, G. F. Grinyer, C. Louchart, L. Nalpas, and A. Signoracci, Phys. Rev. Lett. 108, 252501 (2012).
  * Flavigny et al. [2013] F. Flavigny, A. Gillibert, L. Nalpas, A. Obertelli, N. Keeley, C. Barbieri, D. Beaumel, S. Boissinot, G. Burgunder, A. Cipollone, et al., Phys. Rev. Lett. 110, 122503 (2013).
  * Gómez-Ramos and Moro [2018] M. Gómez-Ramos and A. M. Moro, Phys. Lett. B 785, 511 (2018).
  * Aumann et al. [2021] T. Aumann, C. Barbieri, D. Bazin, C. Bertulani, A. Bonaccorso, W. Dickhoff, A. Gade, M. Gómez-Ramos, B. Kay, A. Moro, et al., Prog. Part. Nucl. Phys. 118, 103847 (2021).
  * Glauber [1959] R. J. Glauber, in _Lecture in Theoretical Physics_ , edited by W. E. Brittin and L. G. Dunham (Interscience, New York, 1959), vol. 1, p. 315.
  * Hansen and Tostevin [2003] P. G. Hansen and J. A. Tostevin, Ann. Rev. Nucl. Part. Sc. 53, 219 (2003).
  * Jacob and Maris [1966] G. Jacob and T. A. J. Maris, Rev. Mod. Phys. 38, 121 (1966).
  * Louchart et al. [2011] C. Louchart, A. Obertelli, A. Boudard, and F. Flavigny, Phys. Rev. C 83, 011601 (2011).
  * Potel et al. [2015] G. Potel, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 92, 034611 (2015).
  * Potel et al. [2017] G. Potel, G. Perdikakis, B. V. Carlson, and _et al._ , Eur. Phys. J. A 53, 178 (2017).
  * Lei and Moro [2015a] J. Lei and A. M. Moro, Phys. Rev. C 92, 044616 (2015a).
  * Lei and Moro [2015b] J. Lei and A. M. Moro, Phys. Rev. C 92, 061602 (2015b).
  * Carlson et al. [2016] B. V. Carlson, R. Capote, and M. Sin, Few-Body Syst. 57, 307 (2016).
  * Hussein and McVoy [1985] M. S. Hussein and K. W. McVoy, Nucl. Phys. A 445, 124 (1985).
  * Ichimura et al. [1985] M. Ichimura, N. Austern, and C. M. Vincent, Phys. Rev. C 32, 431 (1985).
  * Dickhoff and Neck [2005] W. H. Dickhoff and D. V. Neck, _Many–Body Theory Exposed: Propagator description of quantum mechanics in many–body systems_ (World Scientific, Singapore, 2005).
  * Dickhoff and Charity [2019] W. H. Dickhoff and R. J. Charity, Prog. Part. Nucl. Phys. 105, 252 (2019), ISSN 0146-6410.
  * Escher et al. [2012] J. E. Escher, J. T. Burke, F. S. Dietrich, N. D. Scielzo, I. J. Thompson, and W. Younes, Rev. Mod. Phys. 84, 353 (2012).
  * Escher et al. [2018] J. E. Escher, J. T. Burke, R. O. Hughes, N. D. Scielzo, R. J. Casperson, S. Ota, H. I. Park, A. Saastamoinen, and T. J. Ross, Phys. Rev. Lett. 121, 052501 (2018).
  * Baye et al. [2005] D. Baye, P. Capel, and G. Goldstein, Phys. Rev. Lett. 95, 082502 (2005).
  * Baye et al. [2009] D. Baye, P. Capel, P. Descouvemont, and Y. Suzuki, Phys. Rev. C 79, 024607 (2009).
  * Pinilla et al. [2012] E. C. Pinilla, P. Descouvemont, and D. Baye, Phys. Rev. C 85, 054610 (2012).
  * Hebborn and Capel [2021] C. Hebborn and P. Capel, Phys. Rev. C 104, 024616 (2021).
  * Gómez-Ramos, M. et al. [2021] Gómez-Ramos, M., Gómez-Camacho, J., Lei, Jin, and Moro, A. M., Eur. Phys. J. A 57, 57 (2021).
  * Hebborn and Nunes [2021] C. Hebborn and F. M. Nunes, Phys. Rev. C 104, 034624 (2021).
  * Descouvemont and Baye [2010] P. Descouvemont and D. Baye, Rep. Prog. Phys. 73, 036301 (2010).
  * [40] C. Hebborn and G. Potel, _in preparation_.
  * Pruitt et al. [2020a] C. D. Pruitt, R. J. Charity, L. G. Sobotka, J. M. Elson, D. E. M. Hoff, K. W. Brown, M. C. Atkinson, W. H. Dickhoff, H. Y. Lee, M. Devlin, et al., Phys. Rev. C 102, 034601 (2020a).
  * Pruitt et al. [2020b] C. D. Pruitt, R. J. Charity, L. G. Sobotka, M. C. Atkinson, and W. H. Dickhoff, Phys. Rev. Lett. 125, 102501 (2020b).
  * Atkinson and Dickhoff [2019] M. Atkinson and W. Dickhoff, Phys. Lett. B 798, 135027 (2019), ISSN 0370-2693.