# CAMEO: Curiosity Augmented Metropolis for Exploratory Optimal Policies

Simo Alami Chehboune  Ecole Polytechnique/IRT SystemX
Palaiseau, France
<EMAIL_ADDRESS>Fernando Llorente  Universidad
Carlos III de Madrid
Leganés, Spain
<EMAIL_ADDRESS>Rim Kaddah  IRT SystemX
Palaiseau, France
<EMAIL_ADDRESS>Luca Martino  Universidad Rey Juan Carlos
Fuenlabrada, Spain
<EMAIL_ADDRESS>Jesse Read  LIX, Ecole Polytechnique, IP-Paris
Palaiseau, France
<EMAIL_ADDRESS>
###### Abstract

Reinforcement Learning has drawn huge interest as a tool for solving optimal
control problems. Solving a given problem (task or environment) involves
converging towards an optimal policy. However, there might exist multiple
optimal policies that can dramatically differ in their behaviour; for example,
some may be faster than the others but at the expense of greater risk. We
consider and study a distribution of optimal policies. We design a curiosity-
augmented Metropolis algorithm (CAMEO), such that we can sample optimal
policies, and such that these policies effectively adopt diverse behaviours,
since this implies greater coverage of the different possible optimal
policies. In experimental simulations we show that CAMEO indeed obtains
policies that all solve classic control problems, and even in the challenging
case of environments that provide sparse rewards. We further show that the
different policies we sample present different risk profiles, corresponding to
interesting practical applications in interpretability, and represents a first
step towards learning the distribution of optimal policies itself.

###### Index Terms:

Reinforcement Learning, Curiosity model, Metropolis, MCMC

##  I Introduction

Reinforcement Learning (RL) is a sequential decision framework where a model
(agent) has to solve a task in multiple steps and find the optimal strategy
(policy, $\pi$) to do so. The agent interacts directly with its environment
and learns through this experience. After each time step $t$ that the agent
performs an action $a_{t}$, the environment returns a new state $s_{t+1}$ and
a reward $r_{t}$. The agent then seeks to maximise the expected return i.e.
the discounted sum of rewards received over time: $G=\mathbb{E}[r_{0}+\gamma
r_{1}+\gamma^{2}r_{2}+\dots\gamma^{t}r_{t}+\dots|s_{0}]$; where gamma is a
discount factor of future rewards. We define the value of state $s$,
$V^{\pi}(s)$, as the expected return obtained when following policy $\pi$ from
state $s$; and the Q-value $Q^{\pi}(s,a)$ as the state value obtained after
performing action $a$ in state $s$.
$V^{\pi}(s)=\mathbb{E}_{\pi}\left[\sum_{k=0}^{\infty}\gamma^{k}r(s_{t+k+1})|s\right]=\sum_{a}\pi(a|s)Q^{\pi}(s,a)$

An optimal policy $\pi^{*}$ is a policy that reaches the maximal value
possible for all states. Such a policy always exists but might not be unique
[14]. For instance, given policies $\pi_{\theta_{i}}$ parameterized by
$\theta_{i}$, in parameter space $\Omega$, there are many $\theta_{i}$ that
represent an optimal policy. Most of the literature focuses on finding models
and algorithms that learn policies iteratively and converge towards a unique
optimal policy [16]. But considering that many optimal policies exist, we
propose a process to generate different optimal policies.

Indeed, even-though all desired policies may be optimal, they may differ in
their behaviour towards solving the task. We can argue that an optimal policy
is the fastest and most consistent one that solves the given task. However,
these two concepts can be antagonist depending on the policy’s risk profile.
Two policies can be optimal but one might take more risk than the other and
solve the task faster. Sutton and Barto [16] give an example of policies
obtained using Q-learning or SARSA and show that in the cliff problem (see
table I), both methods solve the task but Q-learning takes more risk by
walking near the edge while SARSA takes a longer, safer, path.

As many optimal policies exist, then a distribution of optimal policies also
exists. Learning such a distribution would allow to sample different optimal
policies and choose the one which profile suits the best the task needs. For
instance, the mode of the learnt distribution corresponds to the minimum
variance optimal policy. However, to learn a distribution, many samples are
needed. This paper addresses the question of whether or not it is possible to
sample many different optimal policies. While all existing approaches focus on
converging towards one optimal policy, we try to find a solution to generate
many different ones that could correspond to different profiles.

We adapted the Metropolis algorithm [6] for RL in order to obtain a Markov
chain of optimal policies whose stationary distribution corresponds to the
distribution of optimal policies. We thus propose to sample a suitable
distribution featuring intrinsic and extrinsic rewards (see Eq. (8)). We show
that the obtained policies effectively adopt different behaviours while
solving the desired tasks.

##  II Related work and brief introduction to Reinforcement Learning

We cited Q-learning [17] and SARSA [16] above as classic methods used in RL.
These two methods build Q-tables that store for every state-action pair the
mean reward obtained afterwards until converging towards good estimations.
After convergence, the optimal policy is the one that always follows the
action that returns the highest value. While these methods try to evaluate Q
values directly, other parameterize the policy itself like policy gradient
methods [15]. The advantage of the latter being that these are capable to
learn stochastic policies but also perform in continuous action spaces.

Deep Learning had a massive impact on RL and had been naturally incorporated
into the cited approaches. This led to the emergence of Deep Q-learning [11],
which uses deep neural networks to evaluate Q-values in large state spaces, or
to Deep Policy Gradients that use a neural network parameters as policy
parameters. Some other methods bridged the gap between these approaches like
Actor Critic methods that use 2 neural networks, one that acts as policy and
an other that evaluates state values [2].

Generative Adversarial Networks (GAN) [5] are closely related to Actor Critic
where the actor network can be assimilated to a generator and the critic to a
discriminator [13]. Indeed, GAN have been used in imitation learning for
instance where expert demonstrations are used to learn a policy [7]. In this
case the generator has to generate trajectories while the discriminator has to
decide whether the trajectory111In RL literature, a trajectory is a
successions of state action pairs or states alone. A trajectory can be longer
than one episode. In this work, we constraint a trajectory to not exceed the
length of an episode. Therefore, in this context, the words episode and
trajectory are equivalent comes from the available demonstration or not.
However instead of learning a new policy or outputting many different ones,
the aim is to mimic the expert behaviour. Yet again, there is a close
connection between this approach and GAN [4].

However these methods converge towards a unique optimal policy, rather than
generate different optimal ones. All these advances show that using generative
approaches to learn an optimal policy is possible but for our knowledge, there
is yet no generative model that is used to generate different optimal policies
that output different behaviours and different risk profiles. This work is a
first step towards generating such samples in the hope of being then able to
learn a distribution of optimal policies in future work.

##  III Metropolis and Direct Policy Search

Our main objective is to sample from the distribution of optimal policies. In
this section we present a simple adaptation of Metropolis algorithm to RL
problems.

###  III-A Metropolis Algorithm

Let $\theta$ denote the parameters of a policy $\pi_{\theta}$. Rather than
searching directly for optimal $\theta$, several works consider sampling
policy parameters $\theta_{i}$ using Markov Chain Monte Carlo algorithms [1,
8]. For instance, in [8], Hoffman et al. showed that for direct policy search,
sampling directly from a trans-dimensional distribution that is proportional
to the reward performs better than classic simulations methods. Therefore, we
model a distribution $f(\theta)$, i.e the target distribution, that is
proportional to the expectation of a monotonically increasing function $U$ of
the empirical return, i.e., it should be maximal when the return is maximal; a
utility function $U(\tau)$.

$f(\theta)\propto p(\theta)\eta(\theta).$ (1)

where $p(\theta)$ is a prior over $\theta$ (in the simple case, uniform) and
$\eta$ the performance of $\theta$ wrt the environment:

$\displaystyle\eta(\theta)=\mathbb{E}_{p(\tau|\theta)}[U(\tau)]=\int
U(\tau)p(\tau|\theta)d\tau,$ (2)

where $\tau$ denotes a trajectory (i.e. the succession of states visited),
$p(\tau|\theta)$ is the distribution of $\tau$ (i.e., the one induced by
following policy $\pi_{\theta}$), and $U(\tau)$ the utility of $\tau$.
Provided that $\eta(\theta)$ is non-negative, we can sample it via a
Metropolis-Hastings (MH) algorithm.

Denoting the empirical return by
$\widetilde{G}(\tau)=\sum_{t=0}^{L-1}\gamma^{t}r_{t}$, a usual choice is
$U(\tau)=\exp(\widetilde{G}(\tau)/T)$. Therefore:

$f(\theta;T)\propto
p(\theta)\eta_{T}(\theta)=p(\theta)\mathbb{E}_{p(\tau|\theta)}[e^{\widetilde{G}(\tau)/T}]$
(3)

where $T$ denotes an inverse temperature parameter.

In practice, in order to apply the MH algorithm on $f(\theta)$, the evaluation
of $\eta(\theta)$ must be substituted with an unbiased estimate over $N$
episodes,

$\displaystyle\eta(\theta)$
$\displaystyle=\mathbb{E}_{p(\tau|\theta)}[U(\tau)]$
$\displaystyle\approx\bar{U}_{N}(\theta)=\frac{1}{N}\sum_{i=1}^{N}U(\tau_{i}),\quad\tau_{i}\sim
p(\tau|\theta),$ (4)

where $\bar{U}_{N}(\theta)$ denotes the empirical utility for $\theta$ over
$N$ episodes $\tau_{i}$. Algorithm 1 consider the noisy evaluation of the
utility function in Eq. (III-A), and is called Monte Carlo-within-Metropolis
[9, 10].

Algorithm 1 Monte Carlo-within-Metropolis for RL

$K$: the number of iterations, $N$: number of episodes

Initialise Agent $\pi$

$\theta_{0}\sim\mathcal{N}(\mathbf{0},\sigma_{p}^{2}{\bf I}_{D})$

for $k$ from $1$ to $K$ do

${\theta^{\prime}}\sim\mathcal{N}(\theta_{k-1},\sigma_{p}^{2}{\bf I}_{D})$

Run $N$ episodes with $\pi_{\theta_{k-1}}$ and compute
$\bar{U}_{N}(\theta_{k-1})$

Run $N$ episodes with $\pi_{\theta^{\prime}}$ and compute
$\bar{U}_{N}(\theta^{\prime})$

$\beta\leftarrow\dfrac{p(\theta^{\prime})\bar{U}_{N}(\theta^{\prime})}{p(\theta_{k-1})\bar{U}_{N}(\theta_{k-1})}$

$\alpha\leftarrow\min(1,\beta)$

$\epsilon\sim\mathcal{U}_{[0,1]}$

if $\epsilon<\alpha$ then

$\theta_{k}\leftarrow\theta^{\prime}$

else

$\theta_{k}\leftarrow\theta_{k-1}$

end if

end for

return $\\{\theta_{k}\\}_{k=1}^{K}$

##  IV Curiosity Augmented Metropolis

###  IV-A Prior Bias

In section V-A, we showed that $\\{\theta_{k}\\}_{k=1}^{K}$ are highly
correlated; due to a scaling factor among them. Indeed, as we considered that
$p(\theta)\sim\mathcal{U}[-a,a]$, there is no bound to $\theta_{i}$ values and
therefore we obtain $\theta_{k+1}=\lambda\theta_{k}$, with $\lambda$ a
constant. A solution is to constraint $\theta_{k}\in[-1,1]^{D}$ but it is not
possible to use $p(\theta)\sim\mathcal{U}[-1,1]$, as it would cancel out
anyway during the calculation of $\beta$ (see algorithm 1). However it is
possible to measure the variance of $\theta_{k}$ from $[-1,1]$:

$\sigma^{2}_{\theta_{k}}=\frac{1}{D}\sum_{j=0}^{D}\mathbbm{1}_{\theta_{kj}\notin[-1,1]}(\theta_{kj}^{2}-1)^{2},$
(5)

and therefore we define: $p(\theta)=e^{-\sigma^{2}_{\theta}}$

###  IV-B Trajectory bootstrap

In Algorithm 1, we had to run $N$ episodes twice to estimate the mean return
of $\theta_{k-1}$ and $\theta^{\prime}$. We could store the return of
$\theta_{k-1}$ and use it in subsequent runs. However, the method shown in
Algorithm 1 was preferred to penalise the possible variance of the returns
obtained using $\theta_{k-1}$.

An alternative to still penalise variance while avoiding to run the $N$
episodes twice is to make use of importance sampling and the advantage
function [16]. Intuitively, it corresponds to the extra reward that the agent
could obtain by taking action $a$ over a random action:
$A_{\theta}(s,a)=Q_{\theta}(s,a)-V_{\theta}(s)$. More specifically, by
defining $G$ as a function of $\theta$:

$G(\theta)=\int\widetilde{G}(\tau)p(\tau|\theta)d\tau$ (6)

the _advantage_ of $\theta^{\prime}$ over $\theta_{k-1}$:

$\begin{split}G(\theta^{\prime})&=G(\theta_{k-1})+\mathbb{E}_{\theta^{\prime}}\left[\sum_{t=0}\gamma^{t}A_{\theta^{\prime}}(s_{t},a_{t})\right]\\\
&=G(\theta_{k-1})+\sum_{s}\rho_{\theta^{\prime}}(s)\sum_{a}\pi_{\theta^{\prime}}(a|s)A_{\theta^{\prime}}(s_{t},a_{t})\\\
&\approx
G(\theta_{k-1})+\sum_{s}\rho_{\theta_{k-1}(s)}\sum_{a}\pi_{\theta^{\prime}}(a|s)A_{\theta^{\prime}}(s_{t},a_{t})\\\
&=G(\theta_{k-1})+\mathbb{E}_{s\sim\rho(\theta_{k-1})}\left[\sum_{a}\pi_{\theta^{\prime}}(a|s)A_{\theta^{\prime}}(s_{t},a_{t})\right]\\\
&=\medmath{G(\theta_{k-1})+\mathbb{E}_{s\sim\rho(\theta_{k-1})}\mathbb{E}_{a\sim\pi_{\theta_{k-1}}}\left[\frac{\pi_{\theta^{\prime}}(a|s)}{\pi_{\theta_{k-1}}(a|s)}A_{\theta^{\prime}}(s_{t},a_{t})\right]}\\\
&\approx G(\theta_{k-1})\\\
&\quad+\medmath{\mathbb{E}_{s,a\sim\rho(\theta_{k-1}),\pi_{\theta_{k-1}}}\left[\Pi(r+\gamma(V_{\pi_{\theta^{\prime}}}(s^{\prime})-V_{\pi_{\theta_{k-1}}}(s))\right]},\end{split}$
(7)

where $\Pi=\frac{\pi_{\theta^{\prime}}(a|s)}{\pi_{\theta_{k-1}}(a|s)}$. $\rho$
is the state visitation frequency and we used
$\rho_{\theta^{\prime}}(s)\approx\rho_{\theta_{k-1}}(s)$ because we already
have trajectories sampled from $\pi_{\theta_{k-1}}$, so it is easier to obtain
$\rho_{\theta_{k-1}}(s)$ than $\rho_{\theta^{\prime}}(s)$. Finally, TD Error
[16] was used as an estimator for the advantage in place of Q-values in order
reduce variability. This way, the return obtained using $\theta^{\prime}$ can
be estimated using the behaviour of the agent parameterised by
$\theta^{\prime}$ on the trajectories obtained under $\theta_{k-1}$, noted
$\pi_{\theta^{\prime}}(\tau_{\theta_{k-1}})$.

###  IV-C Curiosity Module

As shown in section V-A, the previous implementation fails on Griworld and
Cliff environments. One of the main reason explaining this failure is that the
rewards in these environments are sparse. This means that a reward of -1 at
each time step does not bring much information about the state of the agent
and how close it is from the goal. The only valuable information are obtained
either on the pit or when reaching the goal. In a gradient free approach like
Metropolis where there is no criteria to drive the search, it means that the
agent should first reach the goal by luck before improving. Moreover, the fact
that there is no valuable information gathered at each run makes the process
get stuck at some points in parameter space as all $\theta^{\prime}$ are drawn
from $\mathcal{N}(\theta_{k},\sigma^{2}_{p}\textbf{I}_{D})$.

It is therefore necessary to implement a mechanism that drives exploration
dynamically. Inspired by the approach in [12], we propose a curiosity module
that consists of a Neural Network that learns to predict the next state given
the last state of an agent. This way, if the network learns to predict the
trajectory of $\theta_{k}$ and $\theta^{\prime}$ tries something new, it will
fail and output a large prediction error, called the intrinsic reward. The
intrinsic reward is then added to the extrinsic reward, i.e the reward
returned by the environment:

$R(\tau)=\mu\widetilde{G}(\tau)+(1-\mu)\mathcal{L}^{(k)}(\tau),$ (8)

with $\mathcal{L}^{(k)}$ the prediction error (i.e. the loss) at step $k$ and
$\mu\in[0,1]$. The utility is thus computed on this modified reward

$\displaystyle\widetilde{U}_{N}(\theta)=\frac{1}{N}\sum_{i=1}^{N}U(\tau_{i})\quad\tau_{i}\sim
p(\tau|\theta),$ (9)

where, if we take $U$ to be an exponential as above, we have

$\displaystyle U(\tau_{i})$
$\displaystyle=\exp\\{\mu\widetilde{G}(\tau_{i})+(1-\mu)\mathcal{L}(\tau_{i})\\}.$
(10)

The output $\theta_{k}$ are all stochastic optimal policies that fool the
curiosity module and that explore different trajectories. Algorithm 2 details
the procedure.

Algorithm 2 CAMEO

$K$: the number of iterations, $N$: number of episodes, $\mu$ a constant

Initialise Agent $\pi$ and $\Phi$ the curiosity neural network

$\theta_{0}\sim\mathcal{N}(\mathbf{0},\sigma_{p}^{2}{\bf I}_{D})$

for $k$ from $1$ to $K$ do

${\theta^{\prime}}\sim\mathcal{N}(\theta_{k-1},\sigma_{p}^{2}{\bf I}_{D})$

Run $N$ episodes with $\pi_{\theta_{k-1}}$ and store trajectories
$\tau_{\theta_{k-1}}$

for each trajectory $\tau_{\theta_{k-1}}^{(i)}$ do

$\mathcal{L}(\tau_{\theta_{k-1}}^{(i)})\leftarrow
MSE(\Phi(\tau_{\theta_{k-1}}^{(i)}),\tau_{\theta_{k-1}}^{(i)})$

$\mathcal{L}(\tau_{\theta^{\prime}}^{(i)})\leftarrow
MSE(\Phi(\pi_{\theta^{\prime}}(\tau_{\theta_{k-1}}^{(i)})),\pi_{\theta^{\prime}}(\tau_{\theta_{k-1}}^{(i)}))$

Train $\Phi$ using $\mathcal{L}(\tau_{\theta^{\prime}}^{(i)})$

$\mathcal{L}(\tau_{\theta_{k-1}})\leftarrow\mathcal{L}(\tau_{\theta_{k-1}})+\mathcal{L}(\tau_{\theta_{k-1}}^{(i)})$

$\mathcal{L}(\tau_{\theta^{\prime}})\leftarrow\mathcal{L}(\tau_{\theta^{\prime}})+\mathcal{L}(\tau_{\theta^{\prime}}^{(i)})$

end for

Compute $\widetilde{G}(\tau_{\theta_{k-1}})$

$\widetilde{G}(\tau_{\theta^{\prime}})\leftarrow\widetilde{G}(\pi_{\theta^{\prime}}(\tau_{\theta_{k-1}}))$

$R(\tau_{\theta_{k-1}})\leftarrow\mu\cdot\widetilde{G}(\tau_{\theta_{k-1}})+(1-\mu)\cdot\mathcal{L}(\tau_{\theta_{k-1}})$

$R(\tau_{\theta^{\prime}})\leftarrow\mu\cdot\widetilde{G}(\tau_{\theta^{\prime}})+(1-\mu)\cdot\mathcal{L}(\tau_{\theta^{\prime}})$

$\widetilde{U}(\theta_{k-1})\leftarrow\dfrac{1}{N}\cdot\left(U(R(\tau_{\theta_{k-1}}))\right)$

$\widetilde{U}(\theta^{\prime})\leftarrow\dfrac{1}{N}\cdot\left(U(R(\tau_{\theta^{\prime}}))\right)$

$\beta\leftarrow\dfrac{p(\theta^{\prime})\widetilde{U}_{N}(\theta^{\prime})}{p(\theta_{k-1})\widetilde{U}_{N}(\theta_{k-1})}$

$\alpha\leftarrow\min(1,\beta)$

$\epsilon\sim\mathcal{U}_{[0,1]}$

if $\epsilon<\alpha$ then $\theta_{k}\leftarrow\theta^{\prime}$

else $\theta_{k}\leftarrow\theta_{k-1}$

end if

end for

return $\\{\theta_{k}\\}_{k=1}^{K}$

##  V Experiments and Results

Our proposed framework has been tested in Classic Control Gym environments,
Cartpole, Acrobot and cliff [3] as well as on a gridworld. Table I shows
environments details.

Environments Cliff Cartpole Acrobot Gridworld Snapshot State Space $s_{t}\in$
$\\{1,\ldots,48\\}$ $\mathbb{R}^{4}$
$[-1,1]^{4}\times[-4\pi,4\pi]\times[-9\pi,9\pi]$ $\\{1,\ldots,16\\}$ Action
Space $a_{t}\in$ $\\{0,1,2,3\\}$ $\\{0,1\\}$ $\\{0,1,2\\}$ $\\{0,1,2,3\\}$
Reward $r_{t}=$ -1 per move, 10 for the goal and -10 for the pit +1 per time
step -1 per time step -1 per move, 10 for the goal and -10 for the pit

TABLE I: Specifications of Environments

###  V-A Simple Metropolis implementation

In this section we present the results obtained using the simple
implementation detailed in algorithm 1. The agent is a neural network composed
of 1 hidden layer of 8 neurons and a ReLU activation function. The average
return was estimated on 20 episodes for every $\theta_{i}$. The metropolis
algorithm was done on 200 timesteps for Cartpole and 500 hundred for Acrobot.

Figure 1 presents the results obtained on Cartpole and Acrobot environments.
Cartpole is solved while we converge towards a mean average return of -80
which is on par with great implementations according to gym leaderboard. Best
implementation achieves a score of -40; we believe that our implementation can
reach this score with enough iterations, which is the main drawback of MCMC
methods.

Figure 1: Average return for every new $\theta$ (in blue) and incremental
count of the number of $\theta_{i}$ retained (in orange) on Cartpole (left)
and Acrobot (right) using simple implementation.

When visualizing the succeeding agents on Cartpole and Acrobot, it is not
obvious if they effectively adopt different behaviours. We therefore plot the
cosine similarity between all pairs of retained $\theta_{i}$ in figure 2. It
appears that succeeding $\theta_{i}$ are heavily correlated.

Figure 2: Cosine similarity between pairs of retained $\theta_{i}$ on Cartpole
(left) and Acrobot (right) using simple implementation.

However, the simple approach fails when confronted to Gridworld or Cliff. In
both cases, the agent remains stuck, always performing the same action. The
reasons for this failure are explained and tackled in section IV-C.

###  V-B Results for CAMEO

In this section we present the results obtained for CAMEO on Gridworld and
Cliff. We recall that for these environments, the rewards are sparse, and
therefore the prior adaptation, as well as the curiosity module are necessary.
Here we kept the same architecture for the agent while the curiosity module is
a neural network of 1 hidden layer of size 150 with a ReLU activation
function. As shown in figure 3, the model succeeds in solving those two
environments. We do not present the results on Cartpole and Acrobot as they
are similar to those obtained without curiosity augmentation.

Figure 3: CAMEO Results on Cliff (above) and Gridworld (below). The figure
presents the mean return, the Prediction error and the count of $\theta_{i}$
retained over time steps Figure 4: Cosine similarity between pairs of retained
$\theta_{i}$ on Gridworld (left) and Cliff (right) using CAMEO implementation.

It is interesting to note in figure 3 that the prediction error peaks when the
model tries something new, even if it is not a good move but also that it
recovers quickly afterwards. Moreover, we can note that the rate of
$\theta_{i}$ retained is nearly linear, which shows that the ratio of
rejection is low and therefore that the sampling process is efficient.
Finally, figure 4 shows that the $\theta_{i}$ retained are less correlated
than in the simple implementation, which suggests that the curiosity module
and the prior are effective. However non correlated weights do not always
imply a different behaviour. Figure 5 shows the aggregated state visitation
frequency of 100 different policies that solve the problems. The most
efficient (shortest) paths are the most taken but the state visitation
frequencies are non negligible for other paths. Therefore, the learned
policies effectively correspond to different behaviours.

Figure 5: State visitation frequency aggregated on 100 policies obtained using
CAMEO on Gridworld and Cliff. Less visited states are in light blue and most
visited ones in dark shade

##  VI Conclusion

In the context of reinforcement learning, we were able to sample optimal
policies on the fly and showed that the resulting behaviours are diverse. We
also showed that our approach succeeded to output optimal policies even when
the rewards structure of our problems were sparse, by using a curiosity module
that encouraged exploration dynamically.

Our approach still bears some limitations as the policy spaces of studied
environments are discrete. As is generally the case with Monte Carlo methods,
convergence is more challenging in high dimensions. To mitigate this, it is
possible to replace the standard proposal distribution with a policy guided
proposal that draws educated samples. The new distribution could therefore
explore larger spaces by focusing on areas of interest. This will be subject
to future work.

## References

  * [1] Vahid Tavakol Aghaei, Ahmet Onat, and Sinan Yıldırım.  A markov chain monte carlo algorithm for bayesian policy search.  Systems Science & Control Engineering, 6(1):438–455, 2018.
  * [2] Shalabh Bhatnagar, Richard Sutton, Mohammad Ghavamzadeh, and Mark Lee.  Natural Actor-Critic Algorithms.  Automatica, 45(11), July 2009.
  * [3] Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba.  Openai gym, 2016.
  * [4] Chelsea Finn, Paul Christiano, Pieter Abbeel, and Sergey Levine.  A connection between generative adversarial networks, inverse reinforcement learning, and energy-based models, 2016.
  * [5] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.  Generative adversarial networks, 2014.
  * [6] W.K. Hastings.  Monte carlo sampling methods using markov chains and their applications.  Biometrika, 57(1):97–109, 1970.
  * [7] Jonathan Ho and Stefano Ermon.  Generative adversarial imitation learning, 2016.
  * [8] Matthew Hoffman, Arnaud Doucet, Nando Freitas, and Ajay Jasra.  Bayesian policy learning with trans-dimensional mcmc.  In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems, volume 20. Curran Associates, Inc., 2008.
  * [9] F. Llorente, L. Martino, J. Read, and D. Delgado.  A survey of Monte Carlo methods for noisy and costly densities with application to reinforcement learning.  arXiv preprint arXiv:2108.00490, 2021.
  * [10] F. Llorente, L. Martino, J. Read, and D. Delgado.  Optimality in Noisy Importance Sampling.  Signal Processing, page 108455, 2022.
  * [11] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller.  Playing atari with deep reinforcement learning, 2013.
  * [12] Deepak Pathak, Pulkit Agrawal, Alexei A. Efros, and Trevor Darrell.  Curiosity-driven exploration by self-supervised prediction, 2017.
  * [13] David Pfau and Oriol Vinyals.  Connecting generative adversarial networks and actor-critic methods, 2017\.
  * [14] Martin L. Puterman.  Markov Decision Processes: Discrete Stochastic Dynamic Programming.  John Wiley and Sons, Inc., New York, NY, USA, 1st edition, 1994.
  * [15] Richard Sutton, David Mcallester, Satinder Singh, and Yishay Mansour.  Policy gradient methods for reinforcement learning with function approximation.  Adv. Neural Inf. Process. Syst, 12, 02 2000.
  * [16] Richard S. Sutton and Andrew G. Barto.  Reinforcement Learning: An Introduction.  The MIT Press, second edition, 2018.
  * [17] Christopher J. C. H. Watkins and Peter Dayan.  Q-learning.  Machine Learning, 8(3):279–292, May 1992.


# The $B$ Anomalies and non-SMEFT New Physics

David London  Physique des Particules, Université de Montréal,
1375 Avenue Thérèse-Lavoie-Roux, Montréal, QC, Canada H2V 0B3

###### Abstract

The modern viewpoint is that the Standard Model is the leading part of an
effective field theory that obeys the symmetry $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$. Since the discovery of the Higgs boson, it is
generally assumed that this symmetry is realized linearly (SMEFT), but a
nonlinear realization (e.g., HEFT) is still possible. The two differ in their
predictions for the size of certain low-energy dimension-6 four-fermion
operators: for these, HEFT allows $O(1)$ couplings, while in SMEFT they are
suppressed by a factor $v^{2}/\Lambda_{\rm NP}^{2}$, where $v$ is the Higgs
vev. In this talk, I argue that (i) such non-SMEFT operators contribute to
both $b\to s\ell^{+}\ell^{-}$ and $b\to c\,\tau^{-}{\bar{\nu}}_{\tau}$,
transitions involved in the present-day $B$ anomalies, (ii) the contributions
to $b\to s\ell^{+}\ell^{-}$ are constrained to be small, at the SMEFT level,
and (iii) the contribution to $b\to c\,\tau^{-}{\bar{\nu}}_{\tau}$ can be
sizeable. I show that the angular distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$ contains
enough information to extract the coefficients of all new-physics operators.
The measurement of this angular distribution can tell us if non-SMEFT new
physics is present.

††preprint: UdeM-GPP-TH-22-xxx

##  I SMEFT vs. HEFT

The Standard Model (SM) of particle physics explains almost all experimental
data to date. There is no doubt that it is correct. This said, the SM is not
complete: for example, it has no explanation for neutrino masses, dark matter
and the baryon asymmetry of the universe. There must exist physics beyond the
SM. And since no new particles have been seen at the LHC, this new physics
(NP) must be heavy.

The modern view is that the SM is just the leading part of an effective field
theory (EFT) (see, e.g., Refs. EFTBook ; Burgess:2007pt ), obtained when this
NP is integrated out at a scale of $O({\rm TeV})$). This EFT must respect the
SM symmetry group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$.

But this raises the question: is the SM symmetry realized linearly [SMEFT
(see, e.g., Refs. Buchmuller:1985jz ; Brivio:2017vri )] or nonlinearly [e.g.,
HEFT (see, e.g., Refs. Feruglio:1992wf ; Bagger:1995mk ; Burgess:1999ha ;
Grinstein:2007iv ; Buchalla:2012qq ; Alonso:2012px ; Alonso:2012pz ;
Buchalla:2013rka ; Cohen:2020xca )]? Since the discovery of the Higgs boson,
the SMEFT is the default assumption, but HEFT is still possible.

The question of whether the symmetry is realized linearly or nonlinearly can
only be answered experimentally. To see how this can be done, one must compare
power counting in SMEFT and in HEFT.

Consider a non-standard $Z{\bar{u}}_{R}u_{R}$ coupling,
$g_{z}\,Z_{\mu}(\bar{u}\gamma^{\mu}P_{R}u)$. Within HEFT, it has mass-
dimension 4, so $g_{z}\sim O(1)$. But within SMEFT, it arises at dimension 6
from $\Lambda_{h}^{-2}(H^{\dagger}D_{\mu}H)(\bar{u}\gamma^{\mu}P_{R}u)$. Thus,
$g_{z}\sim v^{2}/\Lambda_{h}^{2}$. There are other dimension-6 operators that
do not involve the Higgs field (e.g., 4-fermion operators). Their coefficients
$\sim\Lambda^{-2}$, where $\Lambda$ is the scale of NP, $\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}O({\rm TeV})$.
Within SMEFT, the assumption is that $\Lambda_{h}=\Lambda$. This implies that
SMEFT predicts that $g_{z}$ is considerably smaller than the value allowed by
HEFT.

In order to test the SMEFT assumption, we must (i) identify operators whose
power counting is different in SMEFT and HEFT, and (ii) find ways of measuring
these operators. If it is found that the coefficient of such an operator is
larger than what is predicted by SMEFT, this points to non-SMEFT NP.

##  II LEFT

As the title of the talk indicates, we want to test SMEFT with $B$ decays. At
this energy scale, $O(m_{b})$, the EFT is the LEFT (low-energy effective field
theory, also known as WET, weak effective field theory), obtained when the
heavier SM particles ($W^{\pm}$, $Z^{0}$, $H$, $t$) are also integrated out.
(This is like the Fermi theory.)

In Ref. Jenkins:2017jig , a complete and non-redundant basis of LEFT operators
up to dimension 6 is presented. We focus on those operators that conserve
lepton and baryon number. All dimension-6 LEFT operators must respect
$U(1)_{em}$. Most of them are also invariant under $SU(2)_{L}\times U(1)_{Y}$,
and can be generated from dimension-6 SMEFT operators. However, a handful of
dimension-6 LEFT operators are not invariant under $SU(2)_{L}\times U(1)_{Y}$.
That is, they are not generated by dimension-6 SMEFT operators. These “non-
SMEFT operators” are the ones that interest us.

In Ref. Burgess:2021ylu , my collaborators and I examine the LEFT operators
and identify eleven types of non-SMEFT four-fermion operators: there are six
four-quark operators, one four-lepton operator, and four (semileptonic)
operators with two quarks and two leptons.

But this is just the first step; we must also find ways of measuring them.

##  III The $B$ Anomalies

At the present time, there are discrepancies with the SM in measurements of a
number of observables involving the semileptonic transitions $b\to
s\ell^{+}\ell^{-}$ ($\ell=e,~{}\mu)$ and $b\to c\,\tau^{-}{\bar{\nu}}_{\tau}$
decays. These are the $B$ anomalies (for a review, see Ref. London:2021lfn ).
Our list of non-SMEFT four-fermion operators includes contributions to these
two types of decays. It is only natural to explore whether these can be
measured in tests of the $B$ anomalies.

There are two dimension-6 non-SMEFT operators that contribute to $b\to
s\ell^{+}\ell^{-}$. Along with the dimension-8 SMEFT operators to which they
are mapped at tree level, they are

$\displaystyle\mathcal{O}_{ed}^{S,RR}\equiv(\overline{e}_{Lp}e_{Rr})(\overline{d}_{Ls}d_{Rt})$
$\displaystyle~{}~{}~{}~{}\longrightarrow Q_{\ell
eqdH^{2}}^{(3)}\equiv(\overline{\ell}_{p}e_{r}H)(\overline{q}_{s}d_{t}H)~{},$
$\displaystyle\mathcal{O}_{ed}^{T,RR}\equiv(\overline{e}_{Lp}\sigma^{\mu\nu}e_{Rr})(\overline{d}_{Ls}\sigma_{\mu\nu}d_{Rt})$
$\displaystyle~{}~{}~{}~{}\longrightarrow Q_{\ell
eqdH^{2}}^{(4)}\equiv(\overline{\ell}_{p}\sigma_{\mu\nu}e_{r}H)(\overline{q}_{s}\sigma^{\mu\nu}d_{t}H)~{}.$
(1)

Here, the $\mathcal{O}$s are dimension-6 four-fermion LEFT operators
Jenkins:2017jig , while the $Q$s are dimension-8 SMEFT operators, taken from
Ref. Murphy:2020rsh . The subscripts $p,r,s,t$ are generation indices; they
are suppressed in the operator labels. In the LEFT operators, the chiralities
of $e$ and $d$ are given explicitly, while in the SMEFT operators, $\ell$ and
$q$ denote left-handed $SU(2)_{L}$ doublets, and $e$, $u$ and $d$ denote
right-handed $SU(2)_{L}$ singlets.

Both of these LEFT operators contribute to $b\to s\mu^{+}\mu^{-}$. However,
they contribute in a chirally unsupressed way, unlike the SM. Because of this,
the addition of either of these operators can dramatically change the
prediction for ${\cal B}(B_{s}^{0}\to\mu^{+}\mu^{-})$. But the present
measured value is ${\cal B}(B_{s}^{0}\to\mu^{+}\mu^{-})=(2.9\pm 0.4)\times
10^{-9}$ ParticleDataGroup:2020ssz , close to the SM prediction. We find that
the coefficients of these non-SMEFT operators cannot be larger than
$O(10^{-4})$, which is consistent with SMEFT expectations. Things are similar
for the analogous operators that contribute to $b\to se^{+}e^{-}$: the upper
limit of ${\cal B}(B_{s}^{0}\to e^{+}e^{-})<9.4\times 10^{-9}$
ParticleDataGroup:2020ssz  constrains the coefficients of these operators to
be $<O(10^{-3})$, again consistent with SMEFT. We therefore conclude that
there is no indication that non-SMEFT NP contributes to $b\to
s\ell^{+}\ell^{-}$.

The non-SMEFT (dimension-6) four-fermion operator that contributes to $b\to
c\,\ell^{-}{\bar{\nu}}_{\ell}$ is

$\mathcal{O}_{\nu
edu}^{V,LR}\equiv(\overline{\nu}_{Lp}\gamma^{\mu}e_{Lr})(\overline{d}_{Rs}\gamma_{\mu}u_{Rt})+\mathrm{h.c.}$
(2)

It maps to SMEFT operators as follows:

$\displaystyle\longrightarrow$ $\displaystyle Q_{Hud}\equiv
i(\tilde{H}^{\dagger}D_{\mu}H)(\overline{u}_{p}\gamma^{\mu}d_{r})+\mathrm{h.c.}$
(3) $\displaystyle~{}~{}~{}~{}~{}({\hbox{dimension 6}})~{},$
$\displaystyle\longrightarrow$ $\displaystyle
Q_{\ell^{2}udH^{2}}\equiv(\overline{\ell}_{p}d_{r}H)(\tilde{H}^{\dagger}\overline{u}_{s}\ell_{t})+\mathrm{h.c.}$
$\displaystyle~{}~{}~{}~{}~{}({\hbox{dimension 8}})~{},$

where $\tilde{H}=i\sigma_{2}H^{*}$. Now, the fact that there is a mapping to
the dimension-6 SMEFT operator $Q_{Hud}$ seems to suggest that
$\mathcal{O}_{\nu edu}^{V,LR}$ is itself a SMEFT operator after all. But one
must be careful here: $Q_{Hud}$ is a lepton-flavour-universal operator that
generates equal effective couplings for the operators $\mathcal{O}_{\nu\ell
bc}^{V,LR}$, $\ell=e$, $\mu$, $\tau$ Bernard:2006gy . But we are interested in
an effective operator that generates only $\mathcal{O}_{\nu\tau bc}^{V,LR}$
without the other two, i.e., it violates lepton-flavour universality. This can
be generated only by the dimension-8 SMEFT operator $Q_{\ell^{2}udH^{2}}$
given above.

Consider the decay $b\to c\,\tau^{-}{\bar{\nu}}_{\tau}$. Assuming only left-
handed neutrinos, the most general LEFT effective Hamiltonian describing this
decay involves five four-fermion operators:

$\displaystyle{\cal H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}\,V_{cb}\,O_{V}^{LL}-\frac{C_{V}^{LL}}{\Lambda^{2}}\,O_{V}^{LL}\,-\frac{C_{V}^{LR}}{\Lambda^{2}}\,O_{V}^{LR}\,,$
(4)
$\displaystyle-~{}\frac{C_{S}^{LL}}{\Lambda^{2}}\,O_{S}^{LL}\,-\frac{C_{S}^{LR}}{\Lambda^{2}}\,O_{S}^{LR}-\frac{C_{T}}{\Lambda^{2}}\,O_{T}~{},$

with

$\displaystyle O_{V}^{LL,LR}$ $\displaystyle\equiv$
$\displaystyle({\bar{\tau}}\gamma^{\mu}P_{L}\nu)\,({\bar{c}}\gamma_{\mu}P_{L,R}b)~{},$
$\displaystyle O_{S}^{LL,LR}$ $\displaystyle\equiv$
$\displaystyle({\bar{\tau}}P_{L}\nu)\,({\bar{c}}P_{L,R}b)~{},$ $\displaystyle
O_{T}$ $\displaystyle\equiv$
$\displaystyle({\bar{\tau}}\sigma^{\mu\nu}P_{L}\nu)\,({\bar{c}}\sigma_{\mu\nu}P_{L}b)~{}.$
(5)

But note: $O_{V}^{LR}$ is the non-SMEFT operator of Eq. (2); its coefficient
$C_{V}^{LR}$ is suppressed by $v^{2}/\Lambda^{2}$ in SMEFT. For this reason,
it is usually ignored when looking for NP in $b\to
c\,\tau^{-}{\bar{\nu}}_{\tau}$.

For example, fits to the $b\to c\,\tau^{-}{\bar{\nu}}_{\tau}$ data are
performed in Refs. Blanke:2018yud ; Blanke:2019qrx , but with $O_{V}^{LR}$
excluded (precisely because it is a non-SMEFT operator). In Ref.
Burgess:2021ylu , we redo the fit, including $O_{V}^{LR}$. The results are
shown in Table 1. Although the best fit is provided by the scenario in which
only $C_{V}^{LL}$ is added, the fit remains acceptable when both $C_{V}^{LL}$
and $C_{V}^{LR}$ are allowed to be nonzero. And in this scenario, the best fit
has $C_{V}^{LR}=O(1)$. Such a large value is allowed within HEFT, but not
SMEFT.

Table 1: Fit results for the scenarios in which $C_{V}^{LL}$, $C_{V}^{LR}$ or both $C_{V}^{LL}$ and $C_{V}^{LR}$ are allowed to be nonzero. New-physics coeff. | Best fit |  $p$ value (%)
---|---|---
$C_{V}^{LL}$ | $-3.1\pm 0.7$ | 51
$C_{V}^{LR}$ | $2.8\pm 1.2$ | 0.3
$(C_{V}^{LL},C_{V}^{LR})$ | $(-3.0\pm 0.8,0.6\pm 1.2)$ | 35

The bottom line is that a non-SMEFT value of $C_{V}^{LR}$ is allowed by the
present data. Furthermore – and this is the point of this talk – it can be
measured.

##  IV Measuring Non-SMEFT Operators

Consider ${\bar{B}}\to D^{*}\tau^{-}{\bar{\nu}}_{\tau}$. Because ${\bar{B}}$
is a pseudoscalar meson, while $D^{*}$ is a vector, there are only four NP
operators that contribute: $O_{V}^{LL}$, $O_{V}^{LR}$, $O_{LP}\equiv
O_{S}^{LR}-O_{S}^{LL}$, $O_{T}$. This implies that there are seven NP
parameters (four magnitudes, three relative phases).

In Ref. Bhattacharya:2020lfm , it is proposed to measure the angular
distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$. The
differential decay rate is a function of five parameters: $q^{2}$ (the
momentum2 of the $\tau^{-}{\bar{\nu}}_{\tau}$ pair), $E_{\pi}$ (the energy of
the pion in $\tau^{-}\to\pi^{-}\nu_{\tau}$), and the three angles
$\theta^{*}$, $\theta_{\pi}$, $\chi_{\pi}$ (see Fig. I). The data is separated
into $q^{2}$-$E_{\pi}$ bins, and an angular analysis is performed in each bin.

Figure 1: Definition of the angles in the
$\bar{B}\to D^{*}(\to D\pi)\,\tau^{-}(\to\pi^{-}\nu_{\tau})\bar{\nu}_{\ell}$
distribution.

The angular distribution can be written as

$\displaystyle\sum^{9}_{i=1}f^{R}_{i}(q^{2},E_{\pi})\Omega^{R}_{i}(\theta^{*},\theta_{\pi},\chi_{\pi})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\sum^{3}_{i=1}f^{I}_{i}(q^{2},E_{\pi})\Omega^{I}_{i}(\theta^{*},\theta_{\pi},\chi_{\pi})~{}.$
(6)

The nine $f^{R}_{i}\Omega^{R}_{i}$ terms are CP-conserving and are present in
the SM. The 3 $f^{I}_{i}\Omega^{I}_{i}$ terms are CP-violating.

The key point is the following. There are twelve observables (angular
functions) in each $q^{2}$-$E_{\pi}$ bin. (And there may well be several bins
– the exact number will be decided by experiment.) But there are only seven NP
parameters. This implies that all NP parameters can be extracted. If the value
of $|C_{V}^{LR}|$ is found to be larger than that predicted by SMEFT, then not
only will NP have been discovered, but it will have been determined that this
is new, non-SMEFT NP.

Finally, we note that Ref. Bobeth:2021lya  argues that there are hints of NP
in $b\to c\,\mu^{-}{\bar{\nu}}_{\mu}$. As with $b\to
c\,\tau^{-}{\bar{\nu}}_{\tau}$, there are several dimension-6 four-fermion NP
operators that contribute to $b\to c\,\mu^{-}{\bar{\nu}}_{\mu}$, including one
non-SMEFT operator. This decay can be analyzed similarly to $b\to
c\,\tau^{-}{\bar{\nu}}_{\tau}$: the angular distribution for $b\to
c\,\mu^{-}{\bar{\nu}}_{\mu}$ described in Ref. Bhattacharya:2019olg  provides
enough observables to fit for the coefficients of all dimension-6 NP
operators, including the non-SMEFT one.

##  V Recap

It is generally thought that the SM is the leading part of an EFT produced
when the heavy NP is integrated out. Since the discovery of the Higgs boson,
this EFT is usually taken to be the SMEFT. However, this is an assumption; the
identity of the EFT must be determined experimentally.

One difference between HEFT and SMEFT is power counting: the coefficients of
certain low-energy four-fermion operators are predicted to be considerably
smaller in SMEFT than in HEFT. This suggests a simple test: find such an
operator and measure its coefficient. If the value of the coefficient is found
to be larger than that predicted by SMEFT, this imples that the NP operator is
of non-SMEFT type.

At present, there are anomalies in $b\to s\ell^{+}\ell^{-}$ and $b\to
c\,\tau^{-}{\bar{\nu}}_{\tau}$ decays. Both of these receive contributions
from non-SMEFT NP operators. In the case of $b\to s\ell^{+}\ell^{-}$, the
coefficients of such operators are constrained to be small, of SMEFT size. But
the coefficient of the non-SMEFT $b\to c\,\tau^{-}{\bar{\nu}}_{\tau}$ operator
$O_{V}^{LR}$ is still allowed by present data to be sizeable.

The angular distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$ provides
enough observables to extract all NP parameters from a fit. If the coefficient
of $O_{V}^{LR}$ is found to be larger than the SMEFT prediction, this will
point to the presence of non-SMEFT NP.

###### Acknowledgements.

I thank Cliff Burgess, Serge Hamoudou and Jacky Kumar for collaboration on
Ref. Burgess:2021ylu . This work was partially financially supported by funds
from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

## References

  * (1) C. P. Burgess, “Introduction to Effective Field Theory,” Cambridge University Press (December 2020).
  * (2) C. P. Burgess, “Introduction to Effective Field Theory,” Ann. Rev. Nucl. Part. Sci. 57 (2007), 329-362 doi:10.1146/annurev.nucl.56.080805.140508 [arXiv:hep-th/0701053 [hep-th]].
  * (3) W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions and Flavor Conservation,” Nucl. Phys. B 268, 621-653 (1986) doi:10.1016/0550-3213(86)90262-2
  * (4) For a review, see I. Brivio and M. Trott, “The Standard Model as an Effective Field Theory,” Phys. Rept. 793, 1-98 (2019) [arXiv:1706.08945 [hep-ph]].
  * (5) F. Feruglio, “The Chiral approach to the electroweak interactions,” Int. J. Mod. Phys. A 8, 4937-4972 (1993) doi:10.1142/S0217751X93001946 [arXiv:hep-ph/9301281 [hep-ph]].
  * (6) J. Bagger, V. D. Barger, K. m. Cheung, J. F. Gunion, T. Han, G. A. Ladinsky, R. Rosenfeld and C. P. Yuan, “CERN LHC analysis of the strongly interacting W W system: Gold plated modes,” Phys. Rev. D 52, 3878-3889 (1995) doi:10.1103/PhysRevD.52.3878 [arXiv:hep-ph/9504426 [hep-ph]].
  * (7) C. P. Burgess, J. Matias and M. Pospelov, “A Higgs or not a Higgs? What to do if you discover a new scalar particle,” Int. J. Mod. Phys. A 17 (2002), 1841-1918 [arXiv:hep-ph/9912459 [hep-ph]].
  * (8) B. Grinstein and M. Trott, “A Higgs-Higgs bound state due to new physics at a TeV,” Phys. Rev. D 76, 073002 (2007) doi:10.1103/PhysRevD.76.073002 [arXiv:0704.1505 [hep-ph]].
  * (9) G. Buchalla and O. Cata, “Effective Theory of a Dynamically Broken Electroweak Standard Model at NLO,” JHEP 07, 101 (2012) [arXiv:1203.6510 [hep-ph]].
  * (10) R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin and J. Yepes, “The Effective Chiral Lagrangian for a Light Dynamical ”Higgs Particle”,” Phys. Lett. B 722, 330-335 (2013) [erratum: Phys. Lett. B 726, 926 (2013)] doi:10.1016/j.physletb.2013.04.037 [arXiv:1212.3305 [hep-ph]].
  * (11) R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin and J. Yepes, “Flavor with a light dynamical ”Higgs particle”,” Phys. Rev. D 87, no.5, 055019 (2013) doi:10.1103/PhysRevD.87.055019 [arXiv:1212.3307 [hep-ph]].
  * (12) G. Buchalla, O. Catà and C. Krause, “Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO,” Nucl. Phys. B 880, 552-573 (2014) [erratum: Nucl. Phys. B 913, 475-478 (2016)] [arXiv:1307.5017 [hep-ph]].
  * (13) T. Cohen, N. Craig, X. Lu and D. Sutherland, “Is SMEFT Enough?,” JHEP 03, 237 (2021) doi:10.1007/JHEP03(2021)237 [arXiv:2008.08597 [hep-ph]].
  * (14) E. E. Jenkins, A. V. Manohar and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,” JHEP 03, 016 (2018) [arXiv:1709.04486 [hep-ph]].
  * (15) C. P. Burgess, S. Hamoudou, J. Kumar and D. London, “Beyond the standard model effective field theory with $B\rightarrow c\tau^{-}\overline{\nu}$,” Phys. Rev. D 105, no.7, 073008 (2022) [arXiv:2111.07421 [hep-ph]].
  * (16) D. London and J. Matias, “$B$ Flavour Anomalies: 2021 Theoretical Status Report,” [arXiv:2110.13270 [hep-ph]].
  * (17) C. W. Murphy, “Dimension-8 operators in the Standard Model Eective Field Theory,” JHEP 10, 174 (2020) [arXiv:2005.00059 [hep-ph]].
  * (18) P. A. Zyla et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2020, no.8, 083C01 (2020)
  * (19) This observation was first made in V. Bernard, M. Oertel, E. Passemar and J. Stern, “K(mu3)**L decay: A Stringent test of right-handed quark currents,” Phys. Lett. B 638, 480-486 (2006) [arXiv:hep-ph/0603202 [hep-ph]].
  * (20) M. Blanke, A. Crivellin, S. de Boer, T. Kitahara, M. Moscati, U. Nierste and I. Nišandžić, “Impact of polarization observables and $B_{c}\to\tau\nu$ on new physics explanations of the $b\to c\tau\nu$ anomaly,” Phys. Rev. D 99, no.7, 075006 (2019) [arXiv:1811.09603 [hep-ph]].
  * (21) M. Blanke, A. Crivellin, T. Kitahara, M. Moscati, U. Nierste and I. Nišandžić, “Addendum to “Impact of polarization observables and $B_{c}\to\tau\nu$ on new physics explanations of the $b\to c\tau\nu$ anomaly”,” [arXiv:1905.08253 [hep-ph]].
  * (22) B. Bhattacharya, A. Datta, S. Kamali and D. London, “A measurable angular distribution for $\overline{B}\to{D}^{\ast}{\tau}^{-}{\overline{v}}_{\tau}$ decays,” JHEP 07, no.07, 194 (2020) [arXiv:2005.03032 [hep-ph]].
  * (23) C. Bobeth, M. Bordone, N. Gubernari, M. Jung and D. van Dyk, “Lepton-flavour non-universality of ${\bar{B}}\rightarrow D^{*}\ell{{\bar{\nu}}}$ angular distributions in and beyond the Standard Model,” Eur. Phys. J. C 81, no.11, 984 (2021) [arXiv:2104.02094 [hep-ph]].
  * (24) B. Bhattacharya, A. Datta, S. Kamali and D. London, “CP Violation in ${\bar{B}}^{0}\to D^{*+}\mu^{-}{\bar{\nu}}_{\mu}$,” JHEP 05, 191 (2019) [arXiv:1903.02567 [hep-ph]].


# Explaining Explainability: Towards Deeper Actionable Insights into Deep
Learning through Second-order Explainability

E. Zhixuan Zeng 10*, Hayden Gunraj10*, Sheldon Fernandez2, Alexander Wong1
1 University of Waterloo, Canada 2 DarwinAI Corp.
{ezzeng, hayden.gunraj<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Explainability plays a crucial role in providing a more comprehensive
understanding of deep learning models’ behaviour. This allows for thorough
validation of the model’s performance, ensuring that its decisions are based
on relevant visual indicators and not biased toward irrelevant patterns
existing in training data. However, existing methods provide only instance-
level explainability, which requires manual analysis of each sample. Such
manual review is time-consuming and prone to human biases. To address this
issue, the concept of second-order explainable AI (SOXAI) was recently
proposed to extend explainable AI (XAI) from the instance level to the dataset
level. SOXAI automates the analysis of the connections between quantitative
explanations and dataset biases by identifying prevalent concepts. In this
work, we explore the use of this higher-level interpretation of a deep neural
network’s behaviour to allows us to "explain the explainability" for
actionable insights. Specifically, we demonstrate for the first time, via
example classification and segmentation cases, that eliminating irrelevant
concepts from the training set based on actionable insights from SOXAI can
enhance a model’s performance.

**footnotetext: These authors contributed equally to this work.

##  1 Introduction

(a)

(b)

Figure 1: SOXAI visualizations of a classification model on chainsaws 1a and a
segmentation model on hand drills 1b. Different regions show groupings of
related quantitative explanations via first-order XAI, with significance
discussed in Section 3.

Although quantitative performance metrics such as accuracy are essential
indicators of a deep neural network’s performance, they do not offer insights
into the decision-making process. To fill this gap in the performance
analysis, explainable AI (XAI) can facilitate the auditing of model behaviour.
This auditing helps ensure that the decisions are based on relevant visual
indicators. Additionally, it can uncover potential biases in the training
data, which may then be used to guide improvements to the training framework.

First-order explainability techniques such as Grad-CAM [9],
integrated/expected gradients [10, 3], LIME [8], GSInquire [6], and SHAP [7]
yield per-instance visualizations of explanations. However, reviewing these
visualizations can be time-consuming, particularly for large-scale datasets
with multiple classes or high intra-class variability. In addition, human
biases can impact manual review.

In this work, we explore the concept of second-order explainable AI (SOXAI)
[4] for obtaining actionable insights and demonstrate, for the first time,
that such insights can be used to enhance model performance. SOXAI extends XAI
from the instance level to the dataset level to enable the auditing of the
model and dataset during development. Rather than relying on manual reviews of
visual explanations to explore patterns in a model’s decision-making behavior,
SOXAI seeks to automatically unveil these patterns through the analysis of the
relationships between quantitative explanations. This expedites the
identification of the shared visual concepts utilized by a model during
inference and can uncover apparent model and dataset biases. Furthermore, this
improves transparency by uncovering problematic patterns that exist among a
groupings of examples in the dataset, which can adversely impact the model’s
decision-making process. In essence, SOXAI enables us to "explain the
explainability" by providing higher-level interpretations of model behaviour
for actionable insights.

##  2 Methods

The concept of SOXAI takes first-order instance-level quantitative
explanations of samples in a dataset and groups similar embeddings of these
explanations to generate a user-friendly visualization that enables the
uncovering of patterns among different groupings of data to unveil trends.

Here, we employ GSInquire [6] to generate first-order quantitative
explanations of a neural network’s decision-making process across a dataset.
GSInquire examines the network’s activation signals in response to the input
image and employs them to identify critical features within the sample that
quantitatively led to the network’s decision.

###  2.1 Second-order explainability

Second-order explainability is treated as an embedding problem: given an image
$I$ and the corresponding quantitative explanation $\alpha$ for the trained
model $M$, we define the $n$th element of the embedding
$f:(I,\alpha)\rightarrow\mathbb{R}^{N}$ as:

$f(I,\alpha)_{n}=\frac{\sum_{i=1}^{H}\sum_{j=1}^{W}M(I)_{ijn}\alpha_{ij}}{\sum_{i=1}^{H}\sum_{j=1}^{W}\alpha_{ij}},$
(1)

producing an $N$-dimensional vector embedding from the regions of $I$ weighted
by $\alpha$. Notably, $M$ is truncated such that its output is a convolutional
feature map of size $H\times W\times N$, and $\alpha$ is resized to $H\times
W$ to match. Equation 1 ignores regions not identified as critical and only
considers regions with higher weighting score provided by $\alpha$ – in
essence, $f$ performs a weighted average of $M(I)$ with weights $\alpha$.

Here, we use t-distributed stochastic neighbour embedding (t-SNE) [11] to
group the resulting embeddings across a full dataset [4]. In addition,
embeddings were reduced to 50 dimensions via principal component analysis
before applying t-SNE to map them to a 2D space for visualization.

##  3 Experimental Results and Discussion

We present two example cases of SOXAI visualization: image classification and
foreground instance segmentation, discuss the actionable insights gained from
each, and demonstrate how such actionable insights can be used to enhance
model performance.

Chainsaw classification: To explore SOXAI for classification, we apply it to a
ResNet-50 trained on ImageNet 1k [2]. An example result for the chainsaw class
can be seen in Figure 1a, which also highlights four groupings of interest.
Groupings 1 and 2 show the frontal part of chainsaws (_i.e_., the cutting
chain and guide bar) and the handle, respectively, demonstrating that the
model has learned important features representing the target class. However,
smaller groupings highlighted in areas 3 and 4 also reveal biases that the
model has learned over time. In grouping 3, we see that the model has learned
a relationship between earmuffs commonly worn when using chainsaws and the
actual class prediction. Grouping 4 shows images of logs and even wooden
sculptures instead of chainsaws directly.

Through the use of SOXAI, we were able to quickly identify reoccurring biases
learned by the model towards objects that commonly appear in the same frame as
the target class. This was accomplished without the need to manually inspect
each example in the validation set, as would be necessary for first-order XAI
algorithms. Based on the identified biases, enhanced model performance may be
achieved by better-targeted elimination of biases in future training and data
collection or cleaning.

(a)

(b)

Figure 2: Example of a drill with incomplete segmentation 2a, and with the
label filled in 2b. Model prediction is outlined in green.

Drill segmentation: Here, we apply SOXAI to a MaskRCNN model [5] trained on
the MetaGraspNet dataset [1] to detect foreground objects. As an example, we
analyze the segmentation of drills, an object category not seen in the
training set, chosen for its geometric and textural complexity. Figure 1b
presents the SOXAI result, highlighting two groupings representing different
faces of the drill.

The face shown in grouping 1 exhibits a high level of focus on the large logo.
Since the model was not explicitly trained to recognize drills, some other
foreground object must have biased it towards recognizing letters. We observe
that the large logo is over-represented in the grouping, while the frontal
black head of the drill is underrepresented.

To investigate further, we evaluate the prevalence of incomplete segmentations
of the drill when each face is visible, such as the incomplete segmentation
shown in Figure 2a. We find that 37% of predictions for drills with the large
logo facing up are incomplete segmentations, with much of the frontal black
segment missing, while only 14% of segmentation predictions on the other face
are incomplete.

To confirm the model’s bias towards text, we mask out the logo (see Figure
2b), and evaluate the mAP score. We observe an increase from 0.592 to 0.618,
suggesting that allowing the model to ignore its learned bias and focus on a
fuller representation of the object improves its performance. These example
cases demonstrate the usefulness of SOXAI for unveiling actionable insights
into model biases that can be used to enhance a model’s performance.

## References

  * [1] Yuhao Chen, E. Zhixuan Zeng, Maximilian Gilles, and Alexander Wong.  Metagraspnet: a large-scale benchmark dataset for vision-driven robotic grasping via physics-based metaverse synthesis.  arXiv preprint arXiv:2112.14663, 2021.
  * [2] J. Deng, W. Dong, R. Socher, L. Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In 2009 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 248–255, 2009.
  * [3] Gabriel Erion, Joseph D. Janizek, Pascal Sturmfels, Scott M. Lundberg, and Su-In Lee.  Improving performance of deep learning models with axiomatic attribution priors and expected gradients.  Nature Machine Intelligence, 3:620–631, 2021.
  * [4] Hayden Gunraj, Paul Guerrier, Sheldon Fernandez, and Alexander Wong.  SolderNet: Towards trustworthy visual inspection of solder joints in electronics manufacturing using explainable artificial intelligence.  In 35th Annual Conference on Innovative Applications of Artificial Intelligence (IAAI-23). Association for the Advancement of Artificial Intelligence (AAAI), 2023.
  * [5] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick.  Mask r-cnn.  In Proceedings of the IEEE international conference on computer vision, pages 2961–2969, 2017.
  * [6] Zhong Qiu Lin, Mohammad Javad Shafiee, Stanislav Bochkarev, Michael St. Jules, Xiao Yu Wang, and Alexander Wong.  Do explanations reflect decisions? a machine-centric strategy to quantify the performance of explainability algorithms.  arXiv preprint. arXiv:1910.07387, 2019.
  * [7] Scott Lundberg and Su-In Lee.  A unified approach to interpreting model predictions.  In 30th International Conference on Neural Information Processing Systems (NIPS 2017), page 768–4777, 2017.
  * [8] Marco Ribeiro, Sameer Singh, and Carlos Guestrin.  “why should I trust you?”: Explaining the predictions of any classifier.  In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations, pages 97–101, San Diego, California, June 2016. Association for Computational Linguistics.
  * [9] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra.  Grad-cam: Visual explanations from deep networks via gradient-based localization.  In 2017 IEEE International Conference on Computer Vision (ICCV), pages 618–626, 2017.
  * [10] Mukund Sundararajan, Ankur Taly, and Qiqi Yan.  Axiomatic attribution for deep networks.  In Proceedings of the 34th International Conference on Machine Learning - Volume 70, ICML’17, page 3319–3328. JMLR.org, 2017.
  * [11] Laurens van der Maaten and Geoffrey Hinton.  Visualizing data using t-sne.  Journal of Machine Learning Research, 9(86):2579–2605, 2008.


#  Cluster-shell competition and effect of adding hyperons

Naoyuki Itagaki  Department of Physics, Osaka Metropolitan University, Osaka
558-8585, Japan  Nambu Institute for Theoretical and Experimental Physics,
Osaka Metropolitan University, Osaka 558-8585, Japan  Emiko Hiyama  Department
of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578,
Japan  RIKEN Nishina Center for Accelerator-Based Science, Wako 351-0198,
Japan

###### Abstract

Background

The fundamental question is how the hyperon plays a role in the nuclear
structure. It is of particular importance, especially in the light mass
region, to verify the structure change when $\Lambda$ particle(s) is added to
normal nuclei.

Purpose

The ground state of ${}^{8}\mathrm{Be}$ has been know to have a well-developed
$\alpha$–$\alpha$ cluster structure, whereas ${}^{12}\mathrm{C}$ has a mixed
structure of three $\alpha$ clusters and $jj$-coupling shell model, where
$\alpha$ clusters are partially broken. Adding $\Lambda$ particle(s) could
induce the structure change. We compare the Be and C cases.

Methods

Using the antisymmetrized quasi-cluster model (AQCM), the $\alpha$-cluster
states and $jj$-coupling shell-model states of ${}^{8}\mathrm{Be}$ and
${}^{12}\mathrm{C}$ are prepared on the same footing, and we add $\Lambda$
particles. The cluster-shell competition in the ground state can be well
described with this model. Using AQCM, we calculate 8Be, ${}^{9}_{\Lambda}$Be,
${}^{10}_{\Lambda\Lambda}$Be, 12C, ${}^{13}_{\Lambda}$C, and
${}^{14}_{\Lambda\Lambda}$C.

Results

By adding one or two $\Lambda$ particle(s), the ground state of
${}^{12}\mathrm{C}$ approaches the $jj$-coupling shell model side. On the
other hand, in the Be case, although the $\Lambda$ particle(s) shrinks the
$\alpha$–$\alpha$ distance, the breaking effect of the cluster structure is
rather limited.

Conclusions

The spin-orbit interaction is the driving force of breaking the $\alpha$
clusters, and whether the glue-like effect of $\Lambda$ particle(s) attracts
the cluster inside the range of this interaction is crucial. In
${}^{14}_{\Lambda\Lambda}$C, the breaking of $\alpha$ clusters in 12C is much
enhanced by the addition of the $\Lambda$ particles than the case of free 12C.
We also found that breaking $\alpha$ clusters in the ground state of
${}^{14}_{\Lambda\Lambda}$C affects the excited state with the pure cluster
structure.

##  I Introduction

One of the most intriguing phenomena of nuclear structure physics is the
competition of the shell and cluster structures [1]. This is attributed to the
effect of the spin-orbit interaction, which strengthens the symmetry of the
$jj$-coupling shell model. It is well known that this interaction is vital in
explaining the observed magic numbers of $28$, $50$, $82$, and $126$ [2]. The
spin-orbit interaction also has the effect of breaking clusters [1], where
some of the strongly correlated nucleons are spatially localized.

Nevertheless, the $\alpha$ cluster structure is known to be important in the
light mass region. The Be isotopes are known to have the $\alpha$–$\alpha$
cluster structure; ${}^{8}\mathrm{Be}$ decays into two $\alpha$ clusters, and
the molecular-orbital structure of valence neutrons appears in the neutron-
rich Be isotopes [3, 4, 5], which is confirmed by the recent $ab\ initio$
shell-model calculation [6]. This persistence of the $\alpha$–$\alpha$ cluster
structure is owing to the $\alpha$–$\alpha$ distance, which is about 3–4 fm
and large enough compared with the range of the spin-orbit interaction.

In light nuclei, it is considered that these two different pictures (shell and
cluster) coexist, and they compete with each other. Although the
$\alpha$–$\alpha$ cluster structure may persist in ${}^{8}\mathrm{Be}$, when
one more $\alpha$ cluster is added, in ${}^{12}\mathrm{C}$, the interaction
among $\alpha$ clusters gets stronger, and the system has a shorter
$\alpha$–$\alpha$ distance [7, 8]. In this case, the $\alpha$ clusters are
trapped in the interaction range of the spin-orbit interaction. Although the
traditional $\alpha$ cluster model (Brink model) [9] is incapable of treating
the spin-orbit interaction, its effect is significant if we allow the breaking
of the $\alpha$ clusters. The ground state of ${}^{12}\mathrm{C}$ is found to
have a mixed nature of shell and cluster components [10, 11, 12]. On the other
hand, the second $0^{+}$ state of 12C is well known $\alpha$ clustering state
called the Hoyle state. Since this state is nearby the three-$\alpha$ breakup
threshold, the wave function is dilute, and this state has a well-developed
$\alpha$ clustering structure.

It is interesting to investigate how clustering structure is changed when a
hyperon such as a $\Lambda$ particle is injected into 8Be and 12C. Here it
should be noted that there is no Pauli principle between nucleons and a
$\Lambda$, and the $\Lambda N$ interaction is attractive, but weaker than $NN$
interaction. Using this property, some authors studied the structure of
${}^{9}_{\Lambda}$Be and ${}^{13}_{\Lambda}$C from the viewpoint of dynamical
change of the core nuclei, 8Be and 12C, due to the addition of $\Lambda$
particle. For instance, Motoba et al. [13], pointed out that the
$\alpha$–$\alpha$ distance in ${}^{9}_{\Lambda}$Be was shrunk by about 20 % in
comparison with that in the 8Be core nucleus by $\Lambda$ injection. In the
Carbon isotope, one of the present authors (E. H.) pointed out that dynamical
change due to the addition of a $\Lambda$ particle is dependent on the states
in the core nucleus of 12C within the framework of $3\alpha$ and
$3\alpha+\Lambda$ three- and four-body OCM (orthogonal condition model) [14].
The ground state of 12C, $0^{+}_{1}$, is a mixture of shell and cluster
structure; the $\alpha$–$\alpha$ distance does not change due to the addition
of a $\Lambda$ particle. On the other hand, the $\alpha$–$\alpha$ distance is
dramatically contracted in the Hoyle state of ${}^{13}_{\Lambda}$C, which is
well-developed clustering state [14]. However, it should be noted that this
calculation was done without taking into account the breaking effect of
$\alpha$ clusters in 12C. In addition, in Ref. [15], they discussed the
similarity and difference in several states of 12C and ${}^{13}_{\Lambda}$C.
In this way, there are some discussions on the change of the $\alpha$–$\alpha$
distance w/o the $\Lambda$ particle and the change of the structure. However,
there remain never discussed effects of the clustering in such Be and C
isotopes due to addition of $\Lambda$ particles. The question is how the
clustering is broken when $\Lambda$ particles shrinks the $\alpha$–$\alpha$
distance. The traditional cluster model is incapable of describing such
breaking situation and we must extend the model space to incorporate the spin-
orbit contribution, which is the driving force of breaking clusters.

Thus, in this work, we focus on how the clustering is changed and broken due
to the addition of a $\Lambda$ particle(s) in 8Be, ${}^{9}_{\Lambda}$Be,
${}^{10}_{{\Lambda\Lambda}}$Be, 12C, ${}^{13}_{\Lambda}$C, and
${}^{14}_{{\Lambda\Lambda}}$C. In the case of Be isotopes, as mentioned, the
$\Lambda$ particle(s) shrinks the $\alpha$–$\alpha$ relative distance [16,
14], but the resultant distance might still be outside the range of the spin-
orbit interaction, and the $\alpha$ cluster structure could persist. On the
contrary, when $\Lambda$ particle(s) is added to ${}^{12}\mathrm{C}$, the
distances between clusters get even shorter. Since the spin-orbit interaction
works in the inner regions of the nuclear systems, the breaking of $\alpha$
clusters is expected to be enhanced. Therefore, the ground state would
approach more $jj$-coupling shell-model side. Indeed, as shown in the study of
antisymmetrized molecular dynamics [17], the slightly deformed ground state of
${}^{12}\mathrm{C}$ is changed into a spherical shape in ${}^{13}_{\Lambda}$C.
It is worthwhile to check this point in terms of the cluster-shell
competition.

In most of the conventional $\alpha$ cluster models, the contribution of the
non-central interactions (spin-orbit and tensor interactions) vanishes. To
include the spin-orbit effect, we have developed the antisymmetrized quasi-
cluster model (AQCM) [18, 19, 20, 21, 22, 23, 24, 25, 10, 26, 27, 28, 29, 30,
31]. This method allows us to smoothly transform $\alpha$-cluster model wave
functions to $jj$-coupling shell model ones, and we call the clusters that
feel the effect of the spin-orbit interaction quasi-clusters. We have
previously introduced AQCM to ${}^{12}\mathrm{C}$ and discussed the
competition between the cluster states and $jj$-coupling shell model state
[10]. The consistent description of ${}^{12}\mathrm{C}$ and
${}^{16}\mathrm{O}$, which has been a long-standing problem of microscopic
cluster models, has been achieved. Also, not only the competition between the
cluster states and the lowest shell-model configuration, the effect of single-
particle excitation was further included in the description of the ground
state [30].

This paper is organized as follows. The framework is described in Sec. II. The
results are shown in Sec. III. The conclusions are presented in Sec. IV.

##  II framework

The wave function is fully antisymmetrized, and different basis states are
superposed based on the generator coordinate method (GCM) after the angular
momentum projection, and the amplitude for each basis state is determined by
diagonalizing the norm and Hamiltonian matrices.

###  II.1 Single-particle wave function

In our framework, every single particle is described in a Gaussian form as in
many traditional cluster models, including the Brink model [9],

$\phi^{\tau,\sigma}\left(\bm{r}\right)=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\exp\left[-\nu\left(\bm{r}-\bm{\zeta}\right)^{2}\right]\chi^{\tau,\sigma},$
(1)

where the Gaussian center parameter $\bm{\zeta}$ is related to the expectation
value of the position of the nucleon, and $\chi^{\tau,\sigma}$ is the spin-
isospin part of the wave function. The $\alpha$ cluster is expressed by four
nucleons with different spin and isospin sharing the same $\bm{\zeta}$ value.
For the size parameter $\nu$, here we use $\nu=1/2b^{2}$ and $b=1.46$ fm. The
Slater determinant is constructed from these single-particle wave functions by
antisymmetrizing them. The $\Lambda$ particle is represented by the same local
Gaussian-type wave function.

This traditional $\alpha$ cluster wave function cannot take into account the
effect of non-central interactions including the spin-orbit interaction. We
can extend the model based on the AQCM, by which the contribution of the spin-
orbit interaction due to the breaking of $\alpha$ clusters is included. Here
the $\bm{\zeta}$ values in Eq. (1) are changed to complex numbers. When the
original value of the Gaussian center parameter $\bm{\zeta}$ is $\bm{R}$,
which is real and related to the spatial position of this nucleon, it is
transformed by adding the imaginary part as

$\bm{\zeta}=\bm{R}+i\lambda\bm{e}^{\text{spin}}\times\bm{R},$ (2)

where $\bm{e}^{\text{spin}}$ is a unit vector for the intrinsic-spin
orientation of this nucleon. The control parameter $\lambda$ is associated
with the breaking of the cluster. After this transformation, the $\alpha$
clusters are called quasi-clusters. The two nucleons in the same quasi-cluster
with opposite spin orientation have $\bm{\zeta}$ values that are complex
conjugate to each other. This situation corresponds to the time-reversal
motion of two nucleons.

In our previous analysis on ${}^{12}\mathrm{C}$ [10], we have introduced two
parameters representing the distances between quasi-clusters and their
breaking ($\lambda$). The subclosure configuration of
$\left(s_{1/2}\right)^{4}\,\left(p_{3/2}\right)^{8}$ of the $jj$-coupling
shell model can be obtained at the limit of small relative distances and
$\lambda=1$.

###  II.2 Angular momentum projection and GCM

Each AQCM Slater determinant is projected to the eigenstates of parity and
angular momentum by using the projection operator $P_{J^{\pi}}^{K}$,

$P_{J^{\pi}}^{K}=P^{\pi}\frac{2J+1}{8\pi^{2}}\int
d\Omega\,{D_{MK}^{J}}^{*}R\left(\Omega\right).$ (3)

Here ${D_{MK}^{J}}$ is the Wigner $D$-function and $R\left(\Omega\right)$ is
the rotation operator for the spatial and spin parts of the wave function.
This integration over the Euler angle $\Omega$ is numerically performed. The
operator $P^{\pi}$ is for the parity projection
($P^{\pi}=\left(1+P^{r}\right)/\sqrt{2}$ for the positive-parity states, where
$P^{r}$ is the parity-inversion operator), which is also performed
numerically.

The AQCM basis states with different distances between quasi-clusters and
$\lambda$ values are superposed based on GCM. We also generate Gaussian
centers for the $\Lambda$ particles using random numbers, and the basis states
with different positions are superposed. The coefficients
$\left\\{c^{K}_{i}\right\\}$ for the linear combination of the Slater
determinants are obtained together with the energy eigenvalue $E$ when we
diagonalize the norm and Hamiltonian matrices, namely by solving the Hill-
Wheeler equation.

$\sum_{j}(<\Phi_{i}|(P_{J^{\pi}}^{K})^{\dagger}HP_{J^{\pi}}^{K}|\Phi_{j}>-E<\Phi_{i}|(P_{J^{\pi}}^{K})^{\dagger}P_{J^{\pi}}^{K}|\Phi_{j}>)c^{K}_{j}=0.$
(4)

###  II.3 Hamiltonian

The Hamiltonian consists of kinetic energy and potential energy terms. For the
potential part, the interaction consists of the central, spin-orbit, and
Coulomb terms. The nucleon-nucleon interaction is Volkov No.2 [32] with the
Majorana exchange parameter of $M=0.6$, which has been known to reproduce the
scattering phase shift of 4He–4He [33]. For the spin-orbit part, we use the
spin-orbit term of the G3RS interaction [34], which is a realistic interaction
originally developed to reproduce the nucleon-nucleon scattering phase shifts.
The strength of the spin-orbit interactions [10] is set to
$V_{ls}^{1}=V_{ls}^{2}=1450\,\mathrm{MeV}$, which reproduces the binding
energy of 12C from the three-$\alpha$ threshold. For the nucleon-$\Lambda$
interaction, we employ only the central part; YNG-ND interaction [35]. The
$k_{F}$ value for ${}^{9}_{\Lambda}$Be and ${}^{10}_{\Lambda\Lambda}$Be is
0.962 fm-1 as in Ref. [14] and 1.17 fm-1 for ${}^{13}_{\Lambda}$C and
${}^{14}_{\Lambda\Lambda}$C as in Ref. [17]. For the $\Lambda$-$\Lambda$
interaction, we adopt the one called “NS” in Ref. [35], which allows the
reproduction of the binding energy of ${}^{6}_{\Lambda\Lambda}$He.

##  III Results

Figure 1:  (a): Energy curves of $0^{+}$ state of ${}^{8}\mathrm{Be}$ as a
function of the distance between two ${}^{4}\mathrm{He}$ clusters. Solid line
is for $\lambda=0$ (pure two $\alpha$’s) and dotted and dashed lines are for
two quasi-clusters with $\lambda=0.1$ and 0.2, respectively. (b): Same as (a)
but for the $1/2^{+}$ state of ${}^{9}_{\Lambda}$Be. (c): Same as (a) but for
the $0^{+}$ state of ${}^{10}_{\Lambda\Lambda}$Be.

###  III.1 Ground states of ${}^{8}\mathrm{Be}$, ${}^{9}_{\Lambda}$Be, and
${}^{10}_{\Lambda\Lambda}$Be

We start the discussion with ${}^{8}\mathrm{Be}$. Our Hamiltonian gives the
energy of $-27.57$ MeV for the $\alpha$ cluster, and thus, $-55.1$ MeV is the
two-$\alpha$ threshold energy (experimentally $-56.6$ MeV, to which our
theoretical value does not contradict). Figure 1 (a) shows the energy curves
of the $0^{+}$ state of ${}^{8}\mathrm{Be}$ as a function of the distance
between two ${}^{4}\mathrm{He}$ clusters. The solid line is for $\lambda=0$
(pure two $\alpha$’s), and the dotted and dashed line are for two quasi-
clusters with $\lambda=0.1$ and 0.2, respectively. The energy minimum point
appears around the relative distance of $\sim$3.5 fm. This distance is quite
large, and this is outside of the interaction range of the spin-orbit
interaction. Therefore, the $\lambda$ value that gives the minimum energy is
zero (solid line), which means that the $\alpha$ clusters are not broken. The
$\alpha$ breaking effect can be seen in more inner regions, where the energies
of dotted and dashed lines are lower than the solid line. The $\alpha$
clusters are surely broken there. However, at short relative distances, the
energy itself is high enough, and the spin-orbit interaction only plays a role
in reducing the increase of the excitation energy to some extent when two
clusters get closer.

The situation is slightly different in Figure 1 (b), which is for the
$1/2^{+}$ of ${}^{9}_{\Lambda}$Be, where one $\Lambda$ particle is added. We
superpose 50 Slater determinants with different positions for the $\Lambda$
particle and diagonalize the Hamiltonian based on the GCM for each cluster-
cluster distance and $\lambda$. Owing to the $\Lambda$ particle added, the
attractive effect is increased, and the optimal distance between the two
${}^{4}\mathrm{He}$ nuclei (lowest energy point) is around 3 fm, slightly
shorter than the ${}^{8}\mathrm{Be}$ case. Here, the solid line ($\lambda=0$)
and the dotted line ($\lambda=0.1$) almost degenerate, and thus, the $\alpha$
clusters are slightly broken due to the spin-orbit effect. The tendency is a
bit enhanced in ${}^{10}_{\Lambda\Lambda}$Be shown in Fig 1 (c). The optimal
cluster-cluster distance is less than 3 fm, where the dotted line
($\lambda=0.1$) is slightly lower than the solid line ($\lambda=0$). The
number of Slater determinants with different positions for the $\Lambda$
particles is increased to 100 for each 4He–4He distance and $\lambda$. In this
way, since the 4He–4He distances are large in ${}^{9}_{\Lambda}$Be and
${}^{10}_{\Lambda\Lambda}$Be, we find that the $\alpha$-cluster braking effect
is rather small.

###  III.2 Ground states of ${}^{12}\mathrm{C}$, ${}^{13}_{\Lambda}$C, and
${}^{14}_{\Lambda\Lambda}$C

Next we discuss ${}^{12}\mathrm{C}$ and ${}^{13}_{\Lambda}$C, and
${}^{14}_{\Lambda\Lambda}$C. The three-$\alpha$ threshold energy is $-82.7$
MeV in our calculation compared with the experimental value of $-84.9$ MeV.
Figure 2 (a) shows the energy curves of $0^{+}$ state of ${}^{12}\mathrm{C}$
with an equilateral triangular configuration as a function of the distance
between two ${}^{4}\mathrm{He}$ clusters. The solid line is for $\lambda=0$
(pure three $\alpha$’s). Since one ${}^{4}\mathrm{He}$ is added to
${}^{8}\mathrm{Be}$, the energy minimum point appears around the relative
distance of 2.5–3.0 fm, shorter by 1 fm than the previous ${}^{8}\mathrm{Be}$
case before allowing the breaking of $\alpha$ clusters. Therefore, it is
considered that the three $\alpha$ clusters step in the interaction range of
the spin-orbit interaction. The dotted line ($\lambda=0.1$) and dashed line
($\lambda=0.2$) almost degenerate at the region of the lowest energy (the
relative cluster-cluster distance shrinks to 2 fm there).

This tendency is enhanced in Fig. 2 (b), which is for the $1/2^{+}$ of
${}^{13}_{\Lambda}$C, where one $\Lambda$ particle is added. Owing to the
$\Lambda$ particle added, the attractive effect is increased, and the optimal
distance between the ${}^{4}\mathrm{He}$ nuclei is around 2.5 fm (solid line)
before breaking the $\alpha$ clusters. When we allow the breaking, the energy
curves become almost flat inside the relative 4He–4He distance of 2 fm. The
energy minimum points of the dotted ($\lambda=0.1$) and dashed ($\lambda=0.2$)
lines are lower than that of the solid line ($\lambda=0$).

The attractive effect of the $\Lambda$ particles is much more enhanced in Fig.
2 (c), which is for the $0^{+}$ state of ${}^{14}_{\Lambda\Lambda}$C. The
optimal distance between the ${}^{4}\mathrm{He}$ nuclei (energy minimum point)
is around 2.2 fm before breaking the $\alpha$ clusters (solid line). When we
allow the breaking, the energy minimum point appears at the relative
cluster–cluster distance of $\sim$1.4 fm, where the dashed line
($\lambda$=0.2) gives the lowest energy, and $\alpha$ clusters are
significantly broken. We can confirm that the optimal cluster distance gets
shorter, and the breaking of $\alpha$ clusters becomes larger with the
increasing number of $\Lambda$ particles added to the system.

Figure 2:  (a): Energy curves of $0^{+}$ state of ${}^{12}\mathrm{C}$ as a
function of the distance between three ${}^{4}\mathrm{He}$ clusters with
equilateral triangular configuration. Solid line is for $\lambda=0$ (pure
three $\alpha$’s) and dotted and dashed lines are for two quasi-clusters with
$\lambda=0.1$ and 0.2, respectively. (b): Same as (a) but for the $1/2^{+}$
state of ${}^{13}_{\Lambda}$C. (c) Same as (a) but for the $0^{+}$ state of
${}^{14}_{\Lambda\Lambda}$C

###  III.3 Superposition of states with different 4He$-^{4}$He distance and
breaking parameter $\lambda$

To demonstrate the relation between the effect of $\alpha$ breaking and spin-
orbit interaction, we calculate the ground state energies of 8Be,
${}^{9}_{\Lambda}$Be, ${}^{10}_{\Lambda\Lambda}$Be (Table 1) and those of 12C,
${}^{13}_{\Lambda}$C, ${}^{14}_{\Lambda\Lambda}$C (Table 2) with two models:
“AQCM”’ which explicitly takes account of the breaking effect of $\alpha$, and
“Brink model” which does not involve the $\alpha$ breaking effect
($\lambda=0$). We superpose Slater determinants with different positions of
the $\Lambda$ particle(s), 4He–4He cluster distances, and $\alpha$-breaking
parameter $\lambda$ and diagonalize the Hamiltonian based on the GCM.

For the Be case (Table 1), the energy difference between Brink and AQCM is
less than 0.2 MeV in ${}^{8}\mathrm{Be}$, which means that the spin-orbit
interaction does not break the $\alpha$ clusters since they are separated by a
certain distance. The situation is basically the same when $\Lambda$
particle(s) is added. The difference is about 0.5-0.6 MeV in
${}^{9}_{\Lambda}$Be and ${}^{10}_{\Lambda\Lambda}$Be. Concerning the ground
state energy of ${}^{10}_{\Lambda\Lambda}$Be, the binding energy
($B_{\Lambda\Lambda}$) of $17.5\pm 0.4$ MeV from 8Be has been reported in Ref.
[36], which has been revised to $14.7\pm 0.4$ MeV in Ref. [37] (see the
discussions in Refs. [38, 39]), and the present result (15.23 MeV) is almost
consistent with the latter case.

For the C case (Table 2), the energy difference between Brink and AQCM is
about 3.3 MeV in ${}^{12}\mathrm{C}$, and this is much enhanced with the
increasing number of the $\Lambda$ particles added. The difference increases
to 5.2 MeV in ${}^{14}_{\Lambda\Lambda}$C. This is because the spin-orbit
interaction works in the inner region of the nuclear systems; the glue-like
effect of $\Lambda$ particles shrinks the system and induces more contribution
of the spin-orbit interaction.

To clarify the mixing of the $jj$-coupling shell model components in each
state, we utilize the expectation value of the one-body spin-orbit operator,

$\hat{O}^{LS}=\sum_{i}\bm{l}_{i}\cdot\bm{s}_{i}/\hbar^{2},$ (5)

where $\bm{l}_{i}$ and $\bm{s}_{i}$ are the orbital angular momentum and the
spin operators for the $i$th nucleon. The sum runs over the nucleons. The
expectation value is zero for the pure $\alpha$ cluster state owing to the
antisymmetrization effect. Also, the $\bm{l}_{i}\cdot\bm{s}_{i}/\hbar^{2}$
value is $0.5$ for one nucleon in the $p_{3/2}$ orbit, and the eigen value is
$4$ for the subclosure configuration of the $jj$-coupling shell model
$(\left(s_{1/2}\right)^{4}\,\left(p_{3/2}\right)^{8})$ in 12C.

The expectation values of the one-body spin-orbit operator for the ground
states of ${}^{8}\mathrm{Be}$, ${}^{9}_{\Lambda}$Be, and
${}^{10}_{\Lambda\Lambda}$Be are listed in the column “one-body LS” in Table
1. Although the value increases with the number of $\Lambda$ particles added,
it is rather small and cluster structure is considered to be not broken.
However, this is completely different in the C case. The expectation values of
the one-body spin-orbit operator for the ground states of ${}^{12}\mathrm{C}$,
${}^{13}_{\Lambda}$C, and ${}^{14}_{\Lambda\Lambda}$C are listed in the column
“one-body LS” in Table 2. The value is $1.55$ for ${}^{12}\mathrm{C}$, and we
can reconfirm that the ground state has mixed configurations of shell and
cluster aspects. As the number of the $\Lambda$ particles added increases, we
can see that the ground states approach the $jj$-coupling shell model side.
The values for ${}^{13}_{\Lambda}$C and ${}^{14}_{\Lambda\Lambda}$C are 1.86
and 2.05, respectively.

Table 1:  Ground state energies of ${}^{8}\mathrm{Be}$, ${}^{9}_{\Lambda}$Be, and ${}^{10}_{\Lambda\Lambda}$Be (“energy ($J^{\pi}$)”) after performing the GCM calculations. “Brink” is for the Brink model ($\lambda=0$); two-$\alpha$ clusters without the breaking, and “AQCM” is for the AQCM calculation, where different $\lambda$ states are mixed. “one-body LS” is for the expectation values of the one-body spin-orbit operator. The values in the parenthesis show the experimental values. $B_{\Lambda}$, $B_{\Lambda\Lambda}$ are also shown. All energies are in MeV.  8Be | energy ($0^{+}$) |  | one-body LS
---|---|---|---
Brink | $-54.75$ |  | 0.00
AQCM | $-54.94$ | ($-56.50$) | 0.12
${}^{9}_{\Lambda}$Be | energy ($1/2^{+}$) | $B_{\Lambda}$ | one-body LS
Brink | $-60.97$ |  | 0.00
AQCM | $-61.53$ | 6.59 (6.71 [17]) | 0.29
${}^{10}_{\Lambda\Lambda}$Be | energy ($0^{+}$) | $B_{\Lambda\Lambda}$ | one-body LS
Brink | $-69.60$ |  | 0.00
AQCM | $-70.17$ | 15.23 ($14.7\pm 0.4$ [37]) | 0.44
Table 2:  Ground state energies of ${}^{12}\mathrm{C}$, ${}^{13}_{\Lambda}$C, and ${}^{14}_{\Lambda\Lambda}$C (“energy ($J^{\pi}$)”) after performing the GCM calculations. “Brink” is for the Brink model ($\lambda=0$); three-$\alpha$ clusters with equilateral triangular shapes without the breaking, and “AQCM” is for the AQCM calculation, where different $\lambda$ states are mixed. “one-body LS” is for the expectation values of the one-body spin-orbit operator. The values in the parenthesis show the experimental values. All energies are in MeV.  12C | energy ($0^{+}$) |  | one-body LS
---|---|---|---
Brink | $-86.84$ |  | 0.00
AQCM | $-90.12$ | ($-92.16$) | 1.55
${}^{13}_{\Lambda}$C | energy ($1/2^{+}$) | $B_{\Lambda}$ | one-body LS
Brink | $-97.77$ |  | 0.00
AQCM | $-102.00$ | 11.88 (11.69 [17]) | 1.86
${}^{14}_{\Lambda\Lambda}$C | energy ($0^{+}$) | $B_{\Lambda\Lambda}$ | one-body LS
Brink | $-110.58$ |  | 0.00
AQCM | $-115.74$ | 25.62 | 2.05

Figure 3:  Excited $0^{+}$ state comprised of pure three $\alpha$ clusters in
${}^{14}_{\Lambda\Lambda}$C as a function of distances between
$\alpha$–$\alpha$ (solid lines). Ground state is represented by the AQCM basis
state with the 4He–4He distance of $1.4$ fm and $\lambda=0.2$ (a) and 4He–4He
distance of $2.2$ fm and $\lambda=0.0$ (b), which are shown by the solid
circles.

###  III.4 pure $\alpha$ cluster state orthogonal to the ground state

We have discussed that the ground states shift to the $jj$-coupling shell
model side by adding $\Lambda$ particles, and the final question is where the
“pure” three-$\alpha$ cluster state appears in ${}^{14}_{\Lambda\Lambda}$C. We
can discuss it by preparing the pure three-$\alpha$ cluster states and
orthogonalizing them to the ground state. The shift of the ground state to the
$jj$-coupling shell-model-side after allowing the breaking of $\alpha$
clusters is found to play a crucial role.

The solid line in Fig. 3 (a) shows the excited $0^{+}$ state with equilateral
triangular configurations of pure three-$\alpha$ clusters as a function of the
relative distances between the $\alpha$ clusters. At each $\alpha$–$\alpha$
distance, the wave function is orthogonalized to the ground state. Here the
ground state is represented by the optimal AQCM basis state (4He–4He distance
of $1.4$ fm and $\Lambda=0.2$) shown by the solid circle. Therefore, the two-
by-two matrix is diagonalized at every point on the horizontal axis. It is
found that the pure cluster state appears around the excitation energy of
$E_{x}=15$ MeV with the relative $\alpha$–$\alpha$ distance of $\sim$2.5 fm.
To simplify the discussion, the positions for the Gaussian center parameters
for the $\Lambda$ particles are set to origin only in Figs. 3 (a) and (b).

This situation is quite different if the $\alpha$ cluster is assumed to be not
broken due to the spin-orbit interaction in the ground state. This is an
artificial calculation, but we can clearly see the influence of the cluster-
shell competition in the excited state; Fig. 3 (b) shows the result when the
ground state is represented by the Brink model, which is prepared by changing
the $\lambda$ value to zero and the 4He–4He distance to 2.2 fm. The excited
$0^{+}$ state is quite influenced by this change of the ground state. The
energy is pushed up by more than 10 MeV, and the optimal $\alpha$–$\alpha$
distance is increased to $\sim$3 fm. This is because if the ground state is a
pure three-$\alpha$ cluster state, the excited states need to be more
clusterized to satisfy the orthogonal condition. On the other hand, if the
ground state has different components other than the cluster structure, it is
easier for the pure cluster state to be orthogonal to the ground state. This
effect has been known in ${}^{12}\mathrm{C}$ and called the “shrink effect” of
the second $0^{+}$ state; when the $\alpha$ breaking component is mixed in the
ground state, the second $0^{+}$ state orthogonal to the ground state shrinks.
We found that this shrinking effect is much more enhanced in
${}^{14}_{\Lambda\Lambda}$C.

##  IV Conclusions

The effect of adding hyperon(s) in nuclear systems is a fundamental problem in
nuclear structure physics. We analyzed this effect in the context of cluster-
shell competition and discussed the difference between Be and C cases. The
antisymmetrized quasi-cluster model (AQCM) is a useful tool to treat the
cluster states and shell-model states on the same footing, and we added
$\Lambda$ particle(s) to 8Be and 12C.

The cluster breaking effect is negligibly small in ${}^{8}\mathrm{Be}$, where
$\alpha$–$\alpha$ cluster structure keeps enough distance; they stay out of
the interaction range of the spin-orbit interaction, which breaks the $\alpha$
clusters. The situation holds even after $\Lambda$ particle(s) is added. The
glue-like effect of $\Lambda$ particles surely shrinks the cluster-cluster
distance, but clusters are not yet broken.

The situation is completely different in the C case since the additional
$\alpha$ cluster shrinks the cluster-cluster distance, and clusters are in the
interaction range of the spin-orbit interaction. The ground state of
${}^{12}\mathrm{C}$ contains the component of the $jj$-coupling shell model.
The energy difference between the traditional Brink model and AQCM is about
3.3 MeV in ${}^{12}\mathrm{C}$, and this is much enhanced with the increasing
number of the $\Lambda$ particles added. The energy difference is about 5.2
MeV in ${}^{14}_{\Lambda\Lambda}$C. This is because the spin-orbit interaction
works in the inner region of the nuclear systems, and the glue-like effect of
$\Lambda$ particles shrinks the system and induces more contribution of the
spin-orbit interaction. In ${}^{14}_{\Lambda\Lambda}$C, the breaking of
$\alpha$ clusters in 12C is much enhanced by the addition of the $\Lambda$
particles. The energy and structure of the excited $0^{+}$ state with a pure
cluster structure are found to be drastically affected by the transition of
the ground state to the $jj$-coupling shell model side.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number 19J20543, 22K03618, and
JP18H05407. The numerical calculations have been performed using the computer
facility of Yukawa Institute for Theoretical Physics, Kyoto University
(Yukawa-21).

## References

  * Itagaki  _et al._ [2004] N. Itagaki, S. Aoyama, S. Okabe, and K. Ikeda, Cluster-shell competition in light nuclei, Phys. Rev. C 70, 054307 (2004).
  * Mayer and Jensen [1955] M. G. Mayer and H. G. Jensen, “Elementary theory of nuclear shell structure”, John Wiley, Sons, New York, Chapman, Hall, London (1955).
  * Itagaki and Okabe [2000] N. Itagaki and S. Okabe, Molecular orbital structures in ${}^{10}\mathrm{Be}$, Phys. Rev. C 61, 044306 (2000).
  * Itagaki  _et al._ [2000] N. Itagaki, S. Okabe, and K. Ikeda, Important role of the spin-orbit interaction in forming the ${1/2}^{+}$ orbital structure in Be isotopes, Phys. Rev. C 62, 034301 (2000).
  * Itagaki  _et al._ [2002] N. Itagaki, S. Hirose, T. Otsuka, S. Okabe, and K. Ikeda, Triaxial deformation in ${}^{10}\mathrm{Be}$, Phys. Rev. C 65, 044302 (2002).
  * Otsuka  _et al._ [2022] T. Otsuka, T. Abe, T. Yoshida, Y. Tsunoda, N. Shimizu, N. Itagaki, Y. Utsuno, J. Vary, P. Maris, and H. Ueno, $\alpha$-clustering in atomic nuclei from first principles with statistical learning and the hoyle state character, Nature Communications 13, 2234 (2022).
  * Kamimura [1981] M. Kamimura, Transition densities between the $0^{+}_{1}$, $2^{+}_{1}$, $4^{+}_{1}$, $0^{+}_{2}$, $2^{+}_{2}$, $1^{-}_{1}$ and $3^{-}_{1}$ states in ${}^{12}\mathrm{C}$ derived from the three-alpha resonating-group wave functions, Nuclear Physics A 351, 456 (1981).
  * Uegaki  _et al._ [1977] E. Uegaki, S. Okabe, Y. Abe, and H. Tanaka, Structure of the Excited States in 12C I, Prog. Theor. Phys. 57, 1262 (1977).
  * Brink [1966] D. M. Brink, The alpha-particle model of light nuclei, Proc. Int. School Phys.“Enrico Fermi” XXXVI, 247 (1966).
  * Itagaki [2016] N. Itagaki, Consistent description of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ using a finite-range three-body interaction, Phys. Rev. C 94, 064324 (2016).
  * Kanada-En’yo [2007] Y. Kanada-En’yo, The Structure of Ground and Excited States of 12C, Prog. Theor. Phys. 117, 655 (2007).
  * Chernykh  _et al._ [2007] M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-Cosel, and A. Richter, Structure of the Hoyle State in ${}^{12}\mathrm{C}$, Phys. Rev. Lett. 98, 032501 (2007).
  * Motoba  _et al._ [1985] T. Motoba, H. Bando, K. Ikeda, and T. Yamada, Chapter III. Production, Structure an Decay of Light $p$-Shell Lambda Hypernuclei, Progress of Theoretical Physics Supplement 81, 42 (1985).
  * Hiyama  _et al._ [1997] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Three- and Four-Body Cluster Models of Hypernuclei Using the G-Matrix $\Lambda N$ Interaction: ${}^{9}_{\Lambda}$Be, ${}^{13}_{\Lambda}$C, ${}^{6}_{\Lambda\Lambda}$He and ${}^{10}_{\Lambda\Lambda}$Be, Progress of Theoretical Physics 97, 881 (1997).
  * Funaki  _et al._ [2017] Y. Funaki, M. Isaka, E. Hiyama, T. Yamada, and K. Ikeda, Multi-cluster dynamics in ${}^{13}_{\Lambda}\mathrm{C}$ and analogy to clustering in ${}^{12}\mathrm{C}$, Physics Letters B 773, 336 (2017).
  * Motoba  _et al._ [1983] T. Motoba, H. Bando, and K. Ikeda, Light p-Shell Hypernuclei by the Microscopic Three-Cluster Model, Progress of Theoretical Physics 70, 189 (1983).
  * Isaka  _et al._ [2011] M. Isaka, M. Kimura, A. Dote, and A. Ohnishi, Deformation of hypernuclei studied with antisymmetrized molecular dynamics, Phys. Rev. C 83, 044323 (2011).
  * Itagaki  _et al._ [2005] N. Itagaki, H. Masui, M. Ito, and S. Aoyama, Simplified modeling of cluster-shell competition, Phys. Rev. C 71, 064307 (2005).
  * Masui and Itagaki [2007] H. Masui and N. Itagaki, Simplified modeling of cluster-shell competition in carbon isotopes, Phys. Rev. C 75, 054309 (2007).
  * Yoshida  _et al._ [2009] T. Yoshida, N. Itagaki, and T. Otsuka, Appearance of cluster states in ${}^{13}\mathrm{C}$, Phys. Rev. C 79, 034308 (2009).
  * Itagaki  _et al._ [2011] N. Itagaki, J. Cseh, and M. Płoszajczak, Simplified modeling of cluster-shell competition in ${}^{20}\mathrm{Ne}$ and ${}^{24}\mathrm{Mg}$, Phys. Rev. C 83, 014302 (2011).
  * Suhara  _et al._ [2013] T. Suhara, N. Itagaki, J. Cseh, and M. Płoszajczak, Novel and simple description for a smooth transition from $\alpha$-cluster wave functions to $jj$-coupling shell model wave functions, Phys. Rev. C 87, 054334 (2013).
  * Itagaki  _et al._ [2016] N. Itagaki, H. Matsuno, and T. Suhara, General transformation of $\alpha$ cluster model wave function to $jj$-coupling shell model in various $4N$ nuclei, Prog. Theor. Exp. Phys. 2016, 093D01 (2016).
  * Matsuno  _et al._ [2017] H. Matsuno, N. Itagaki, T. Ichikawa, Y. Yoshida, and Y. Kanada-En’yo, Effect of ${}^{12}\mathrm{C}+\alpha$ clustering on the $E0$ transition in ${}^{16}\mathrm{O}$, Prog. Theor. Exp. Phys. 2017, 063D01 (2017).
  * Matsuno and Itagaki [2017] H. Matsuno and N. Itagaki, Effects of cluster-shell competition and BCS-like pairing in 12C, Prog. Theor. Exp. Phys. 2017, 123D05 (2017).
  * Itagaki and Tohsaki [2018] N. Itagaki and A. Tohsaki, Nontrivial origin for the large nuclear radii of dripline oxygen isotopes, Phys. Rev. C 97, 014307 (2018).
  * Itagaki  _et al._ [2018] N. Itagaki, H. Matsuno, and A. Tohsaki, Explicit inclusion of the spin-orbit contribution in the Tohsaki-Horiuchi-Schuck-Röpke wave function, Phys. Rev. C 98, 044306 (2018).
  * Itagaki  _et al._ [2020a] N. Itagaki, A. V. Afanasjev, and D. Ray, Possibility of ${}^{14}\mathrm{C}$ cluster as a building block of medium-mass nuclei, Phys. Rev. C 101, 034304 (2020a).
  * Itagaki  _et al._ [2020b] N. Itagaki, T. Fukui, J. Tanaka, and Y. Kikuchi, ${}^{8}\mathrm{He}$ and ${}^{9}\mathrm{Li}$ cluster structures in light nuclei, Phys. Rev. C 102, 024332 (2020b).
  * Itagaki and Naito [2021] N. Itagaki and T. Naito, Consistent description for cluster dynamics and single-particle correlation, Phys. Rev. C 103, 044303 (2021).
  * Itagaki  _et al._ [2022] N. Itagaki, T. Naito, and Y. Hirata, Persistence of cluster structure in the ground state of ${}^{11}\mathrm{B}$, Phys. Rev. C 105, 024304 (2022).
  * Volkov [1965] A. Volkov, Equilibrium deformation calculations of the ground state energies of 1p shell nuclei, Nucl. Phys. 74, 33 (1965).
  * Okabe and Abe [1979] S. Okabe and Y. Abe, The Structure of 9Be by a Molecular Model. II, Prog. Theor. Phys. 61, 1049 (1979).
  * Tamagaki [1968] R. Tamagaki, Potential Models of Nuclear Forces at Small Distances, Prog. Theor. Phys. 39, 91 (1968).
  * Yamamoto  _et al._ [1994] Y. Yamamoto, T. Motoba, H. Himeno, K. Ikeda, and S. Nagata, Hyperon-Nucleon and Hyperon-Hyperon Interactions in Nuclei, Progress of Theoretical Physics Supplement 117, 361 (1994).
  * Danysz  _et al._ [1963] M. Danysz, K. Garbowska, J. Pniewski, T. Pniewski, J. Zakrzewski, E. R. Fletcher, J. Lemonne, P. Renard, J. Sacton, W. T. Toner, D. O’Sullivan, T. P. Shah, A. Thompson, P. Allen, M. Heeran, A. Montwill, J. E. Allen, M. J. Beniston, D. H. Davis, D. A. Garbutt, V. A. Bull, R. C. Kumar, and P. V. March, Observation of a double hyperfragment, Phys. Rev. Lett. 11, 29 (1963).
  * Dalitz  _et al._ [1989] R. H. Dalitz, D. H. Davis, P. H. Fowler, A. Montwill, J. Pniewski, and J. A. Zakrzewski, The identified $\Lambda\Lambda$ hypernuclei and the predicted $H$-particle, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 426, 1 (1989).
  * Hiyama  _et al._ [2002] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Four-body cluster structure of $a=7-10$ double-$\Lambda$ hypernuclei, Phys. Rev. C 66, 024007 (2002).
  * Hiyama and Nakazawa [2018] E. Hiyama and K. Nakazawa, Structure of $s=-2$ hypernuclei and hyperon-hyperon interactions, Annual Review of Nuclear and Particle Science 68, 131 (2018).