# Prospective Learning:
Principled Extrapolation to the Future

The Future Learning Collective

###### Abstract

Learning is a process which can update decision rules, based on past
experience, such that future performance improves. Traditionally, machine
learning is often evaluated under the assumption that the future will be
identical to the past in distribution or change adversarially. But these
assumptions can be either too optimistic or pessimistic for many problems in
the real world. Real world scenarios evolve over multiple spatiotemporal
scales with partially predictable dynamics. Here we reformulate the learning
problem to one that centers around this idea of dynamic futures that are
partially learnable. We conjecture that certain sequences of tasks are not
retrospectively learnable (in which the data distribution is fixed), but are
prospectively learnable (in which distributions may be dynamic), suggesting
that prospective learning is more difficult in kind than retrospective
learning. We argue that prospective learning more accurately characterizes
many real world problems that (1) currently stymie existing artificial
intelligence solutions and/or (2) lack adequate explanations for how natural
intelligences solve them. Thus, studying prospective learning will lead to
deeper insights and solutions to currently vexing challenges in both natural
and artificial intelligences.

##  1 Introduction

All learning is for the future. Learning involves updating decision rules or
policies, based on past experiences, to improve future performance. The dogma
of much of machine learning is that future scenarios faced by the learner are
similar to past ones, e.g., all data, including what the learner learns from
and what the learner gets tested on, are independent and sampled from the same
probability distribution (Glivenko, 1933; Cantelli, 1933; Vapnik &
Chervonenkis, 1971; Vapnik, 1991; Valiant, 1984). This is not specific to
supervised learning: algorithms for sub-fields ranging from robust learning
(White, 1982; Huber & Ronchetti, 2009) (test distribution can be slightly
different from the training distribution), online learning (Cesa-Bianchi &
Lugosi, 2006; Orabona, 2019) (data distribution can arbitrarily drift over
time) to adversarial learning (Madry et al., 2017) (test distribution can be
adversarial) and reinforcement learning (Bellman, 1954; Recht, 2019) (data
distribution changes as the learner procures more data), all use variants of
this assumption. We refer to these algorithms as retrospective learners
because they learn representations that are optimal for past data, not future
data. Retrospective learning is certainly useful: it is the basis for,
arguably all, real-world learning systems today. Modifications to the core
retrospective assumption, e.g., covariate shift detection, domain adaptation,
meta-learning and continual learning, help bridge the gap that arises from
using representations that are optimal on past data to make inferences on
future data. Each of these ideas makes a specific assumption about future
data, e.g., covariate shift assumes that only the marginal on the inputs
changes, meta-learning assumes that new tasks are similar to past tasks.
Learning systems can fail catastrophically when they are faced with scenarios
outside these assumptions (Giurfa et al., 2001; Vermaercke et al., 2014;
Fleuret et al., 2011; Kim et al., 2018; Yao et al., 2022; Lazaridou et al.,
2021).

This paper attempts to rethink the fundamental assumptions of learning to
address the above gap. We propose the concept of prospective learning, a
generalization of retrospective learning, that relaxes the assumption that
past (training) and future (test) distributions are identical and stationary.
The goal of prospective learning explicitly emphasizes the ability to predict
the future, or prospection. In comparison, adversarial learning or online
learning make no attempt at predicting the future. While the idea of
prospection is by no means new (Seligman et al., 2013), its role has largely
been neglected in machine learning. Prospective learning embodies the eventual
goal of machine learning which is to make predictions for the foreseeable
future.

##  2 A failure mode of retrospective learning

Figure 1: This figure depicts an infinite sequence of data distributions
formed by alternating between two distributions, $P_{A}$ and $P_{B}$. Both
distributions define binary classification problems, namely task A and task B,
with 2-dimensional inputs where the classes are separable using a single line.
A single hypothesis, that doesn’t change with time, cannot generalize
arbitrarily well to this sequence of distributions. However, the sequence of
distributions evolve according to a simple alternating pattern which can be
exploited to generalize to tasks in the future. A prospective learner should
ideally leverage this pattern by modeling the dynamics under which the tasks
evolve with time.

Consider an infinite sequence of distributions denoted by
$P=\\{P_{t}\\}_{t\in\mathbb{N}}$ where time $t$ is a natural number. We see
data from the first $t^{\prime}$ distributions and we seek a hypothesis that
generalizes to distributions that we will see in the future.

We illustrate the insufficiency of retrospective learning using a simple
example. Consider two distributions $P_{A}$ and $P_{B}$; both $P_{A}$ and
$P_{B}$ define binary classification tasks with 2-dimensional inputs. The
classes in both distributions can be perfectly separated by a linear
classifier. Consider the sequence of distributions $P$, formed by alternating
between $P_{A}$ and $P_{B}$ at regular intervals. We would like to design a
learner $L$ that learns using a finite amount of data and generalizes to the
infinite sequence of distributions depicted in Figure 1.

No single hypothesis can achieve an arbitrarily small error on both $P_{A}$
and $P_{B}$ simultaneously. As a result, if a retrospective learner picks a
hypothesis that is unchanged over time, then it cannot achieve an arbitrarily
small error on the sequence of distributions even if the learner has access to
a large number of samples.

We can also use methods from continual or meta-learning to tackle a sequence
of distributions. At every time step, the learner encounters an unknown
distribution and uses a few samples from this distribution in order to build a
hypothesis for it. Even if the learner has seen a large number of
distributions, it still requires at least 1 sample at every time step. Such a
learner lowers the sample complexity of learning the unknown task with the
help of a learnt inductive bias (Baxter, 2000). However, it still fails to
model one key aspect of the problem.

Many standard methods do not predict the future sequence of distributions;
this is helpful if the sequence of distributions follows some pattern like in
the example in Figure 1. For this example, if we know that the tasks alternate
at every $N$ time steps, then we can maintain two different hypotheses—one for
$P_{A}$ and another for $P_{B}$—and alternate between them. This helps us
generalize better but more importantly allows us to do so using fewer samples.
We do not need additional samples when the task switches to adapt to an
unknown distribution, since we can perfectly predict the future sequence of
distributions. In general, data distributions can change with time, and
approximating the underlying dynamics of distribution shift can help us build
a hypothesis that predicts the next distribution.

##  3 The theoretical foundations for prospective learning

###  3.1 Retrospective learning

To formalize prospective learning, we contrast it with traditional
(retrospective) learning (Mohri et al., 2018). Consider a distribution $P$
over inputs $x$ and outputs $y$, in which case we have a supervised learning
problem. A learning task is generally associated with a distribution $P$, a
hypothesis class $\mathcal{H}$, and a risk function $R(\cdot)$. For
simplicity, in this work, we represent a learning task by its associated
distribution $P$, assuming that $\mathcal{H}$ and $R(\cdot)$ are fixed.
Therefore, we use the terms “task” and “distribution” interchangeably. Using
the dataset $D$, drawn independently and identically from $P\in\mathcal{P}$,
we seek a hypothesis $h$ that minimizes the risk $R(\cdot)$. The hypothesis
$h$ is selected from the hypothesis class $\mathcal{H}$. The risk of a
hypothesis $h$ can be defined as $R(h)=\mathbb{E}_{(x,y)\sim
P}\left[\ell((x,y),h)\right]$ where $\ell$ is the loss function. For a
classification problem, we often consider the 0-1 loss,
$\ell((x,y),h)=\mathds{1}\\{h(x)\neq y\\}$, and for a regression problem, we
often consider the mean-squared error $\ell((x,y),h)=(h(x)-y)^{2}$.

Classical machine learning posits that a learning task is unchanged with
respect to time $t$, i.e., $P_{t}\equiv P;\ \forall t$ (Glivenko, 1933;
Cantelli, 1933). One may formalize learning for a class of problems using the
probably approximately correct (PAC) model (Valiant, 1984; Vapnik &
Chervonenkis, 1971). Under this model, we seek a learning algorithm that
achieves an arbitrarily small risk on any given task $P$ using samples from
that task.

######  Definition 1 (PAC-learnability)

(From Valiant (1984); Kearns & Vazirani (1994)) A family of tasks
$\mathcal{P}$ is PAC-learnable using hypothesis class $\mathcal{H}$ if there
exists a learner $L$ with the following property: For all $\epsilon,\delta>0$
and tasks $P\in\mathcal{P}$, if $L$ has access to dataset $D$ of at least $n$
independent and identically distributed samples from $P$, then it outputs a
hypothesis $\hat{h}\in\mathcal{H}$ which satisfies

$\mathbb{P}_{D}[|R(\hat{h})-R(h^{*})|<\epsilon]\geq 1-\delta,$ (1)

where $\mathbb{P}_{D}$ is the probability over draws of the dataset $D$ and
$h^{*}$ is the optimal hypothesis from the hypothesis class $\mathcal{H}$. The
learner $L$ is said to be efficiently PAC-learnable if the sample complexity
$n$ is a polynomial function of $\epsilon^{-1}$ and $\delta^{-1}$.

While Inequality 1 is relatively simple, it can be used to frame the
foundation of the most recent AI revolution, spanning computer vision
(Krizhevsky et al., 2012), natural language processing (Devlin et al., 2019),
diagnostics (McKinney et al., 2020) and protein folding (Senior et al., 2020).
Yet, its limitations have been widely acknowledged for decades in ML and
statistics (see Bozinovski & Fulgosi (1976); Hand (2006)). Specifically, real-
world data rarely exhibit such simple dynamics. Specifically, the assumption
that training (past) data and test/evaluation (future) data are sampled from
the exact same fixed distribution is woefully inadequate. Empirically, modern
deep learning approaches continue to fail in scenarios that are trivially easy
for humans (Madry et al., 2017; Alcorn et al., 2019), and which we believe
would benefit from prospective learning capabilities. Moreover, existing
retrospective learning theory is inadequate to characterize most of the above
generalized learning scenarios (Thrun, 1996) (though see Arjovsky (2021);
Baxter (2000) for some compelling results).

###  3.2 Prospective learning

Sub-fields like online (Cesa-Bianchi & Lugosi, 2006), transfer (Bozinovski &
Fulgosi, 1976), multitask (Bengio et al., 1992; Caruana, 1997) meta- (Baxter,
2000; Wang, 2020), continual (Ring, 1994), and lifelong learning (Thrun &
Mitchell, 1995) present different ways to generalize the retrospective learner
defined in Inequality 1. Most generalizations focus on adapting to new tasks;
given data from a set of tasks, the goal is to adapt to a new task using a
small number of samples from it. On the other hand, a prospective learner
attempts to model how the tasks evolve with time; the emphasis is on
predicting what new tasks will look like prospectively, as opposed to tackling
them retrospectively.

We propose a working formal definition of prospective learning which
generalizes the classical retrospective learning scenario described by
Inequality 1. Instead of learning a task where the distribution, risk, and
hypothesis are fixed across time, we assume that they can evolve over time.
For simplicity, hereafter we keep the risk and hypothesis class fixed, and
assume that only the data distribution changes over time. To formalize this
idea, we consider a fixed sequence of distributions $P:=\\{P_{t}\\}_{t\geq
0}$. Here, $P_{t}$ is the distribution at time $t$. Time $t$ can take values
from $[0,\infty)$ or be limited to the set of natural numbers, among other
choices.

Let $D_{t^{\prime}}$ denote a dataset drawn from $P$ before time $t^{\prime}$.
Using a prospective learner $L$ that takes $D_{t^{\prime}}$ as input, we wish
to estimate a time-indexed sequence of decision functions or hypotheses that
generalizes to future tasks encountered after time $t^{\prime}$. We denote
this estimated hypothesis sequence by $\hat{h}=\\{\hat{h}_{t}(\cdot)\\}_{t\geq
0}$ where $\hat{h}_{t}$ is the hypothesis at time $t$. We provide a visual
illustration of the objective of prospective learning in Figure 2.

Figure 2: A schematic diagram illustrating the objective of prospective
learning and the setup leading up to the Definition 2. $P=\\{P_{t}\\}_{t\geq
0}$ is the sequence of distributions and $D_{t^{\prime}}$ denotes a dataset
drawn from $P$, over the finite time interval $0\leq t\leq t^{\prime}$. Using
the prospective learner $L$ that takes $D_{t^{\prime}}$ as input, we wish to
estimate a time-indexed sequence of decision functions or hypotheses
$\hat{h}_{t^{\prime}}(t)$ that generalizes to future tasks encountered after
time $t^{\prime}$. Therefore, the focus of prospective learning is to learn a
sequence of hypotheses from data received up to $t^{\prime}$ such that it
generalizes to future tasks after time $t^{\prime}$.

Finally, let $\mathcal{H}$ denote a hypothesis class that contains sequences
of hypotheses. For a hypothesis sequence $h\in\mathcal{H}$, we can define the
risk at time $t$, assuming we have supervised learning problems, as
$R_{t}(h_{t})=\mathbb{E}_{(x,y)\sim P_{t}}\left[\ell((x,y),h_{t})\right]$
where $\ell$ is some choice of a loss function. We can now provide a formal
definition of prospective-learnability of the sequence $P$ using the
hypothesis class $\mathcal{H}$ as follows.

######  Definition 2 (Prospective-Learnability)

Consider a sequence of distributions $P:=\\{P_{t}\\}_{t\geq 0}$ and a
hypothesis class $\mathcal{H}$. We say that $\mathcal{H}$ is prospectively-
learnable with respect to $P$ and a chosen reference hypothesis sequence $h$,
with precision $\epsilon>0$ and confidence $\delta>0$ if there exists a
learner $L$ and $\bar{t}\geq 0$ that satisfies the following property: for any
$t^{\prime}>\bar{t}$, if $L$ has access to a dataset $D_{t^{\prime}}$ with at
least $n_{t^{\prime}}$ samples drawn from $P$ up to time $t^{\prime}$, then it
outputs a hypothesis sequence $\hat{h}\in\mathcal{H}$ such that it satisfies

$\lim_{T\to\infty}\frac{1}{T-t^{\prime}}\int_{t^{\prime}}^{T}\text{d}t\
\mathbb{P}_{D_{t^{\prime}}}[|R_{t}(\hat{h})-R_{t}(h)|<\epsilon]\geq 1-\delta.$
(2)

Here, $n_{t^{\prime}}$ and $\bar{t}$ would depend on the sequence $P$, the
complexity of the hypothesis class $\mathcal{H}$, the sequence of risks
$\\{R_{t}\\}_{t\geq 0}$, and the choice of $\epsilon$ and $\delta$.

In words, the sequence of hypotheses $\hat{h}$, learned by $L$ from all the
data points $D_{t^{\prime}}$ received up to time $t^{\prime}>\bar{t}$, makes
predictions, with precision no worse than $\epsilon$ and with a probability of
at least $1-\delta$ on future tasks beyond time $t^{\prime}$, with respect to
a reference hypothesis sequence $h$.

Definition 2 offers some flexibility when choosing the reference hypothesis
sequence $h$; the performance of the prospective learner $L$ is measured
against this reference. Some reasonable choices for $h$ are:

  1. 1.

Strong reference $h^{\ast}$, the optimal sequence of hypotheses given the
entire distribution sequence $P$ where $h^{\ast}_{t}$ is the optimal
hypothesis for $P_{t}$;

  2. 2.

Weak reference $h^{0}$, the sequence of hypotheses that performs at random
chance where $h^{0}_{t}$ is the hypothesis that performs at random chance with
respect to $P_{t}$.

Depending on the “strictness” of these choices, the class of sequences $P$
that are prospectively-learnable according to Definition 2 will vary.

In contrast to a retrospective learner, a prospective learner should respect
Inequality 2 for all $t^{\prime}>\bar{t}$ where $\bar{t}$ is finite. This is a
natural condition to impose because the prospective learner needs to see a few
tasks before it can build an accurate model of how the tasks evolve in the
future. Here, $\bar{t}$ is the minimum amount of time that we need to observe
data from the sequence $P$ in order to be prospectively learnable. Therefore,
$\bar{t}$ is a function of $P$, the hypothesis class $\mathcal{H}$, the risk,
$R$, in addition to $\epsilon,\delta$.

Equipped with this definition of prospective learnability, we can attempt to
understand the properties of prospectively learnable sequences of
distributions. Once we have defined the sequence of distributions that governs
a given prospective learning setting (that is, those that govern changes in
distributions, risks, and constraints), defining quantities that we use in
retrospective learning theory, such as VC-dimension (Vapnik & Chervonenkis,
2015), is analogous. In this way, the primary research question for
prospective learning is determining which distribution sequences are
intrinsically low-dimensional (e.g., have a dependence structure), because
otherwise learning cannot be achieved regardless of sample complexity.

When designing and implementing algorithms to perform prospective learning, it
is important to determine the desirable properties of these algorithms and
their output hypotheses. We state the following conjecture as an entry point
to the discussion about the hypothesis sequences learned by prospective-
learners.

######  Conjecture 1 (Continuum Hypothesis of Learning Hypotheses)

Suppose $P=\\{P_{t}\\}_{t\geq 0}$ is a sequence of distributions. We
conjecture that there are three distinct complexity classes for learning.

  1. 1.

$P$ is PAC-Learnable: With the introduction of potentially time-varying
$P_{t}$, $R_{t}$ and $\mathcal{H}_{t}$, PAC-learnability takes a different
form. Specifically, whenever any of the above change in such a way that the
relative risk, $|R_{t}(\hat{h})-R_{t}(h)|$ remains bounded above by $\epsilon$
with probability $1-\delta$, then it remains PAC-learnable. A simple example
of such a scenario is in a classification problem, where the posteriors can
change dramatically, but the discriminant boundaries remain fixed, so the
relative risk (assuming risk is expected $0-1$ loss) remains constant.

  2. 2.

$P$ is prospectively learnable: This is where the task evolves such that some
complexity measure of the sequence $P$ is low, in which case, we conjecture
that a prospective learner can learn a sequence of hypotheses
$\hat{h}\in\mathcal{H}$ after observing $n_{t^{\prime}}$ amount of data drawn
from $P$ for a duration $t^{\prime}>\bar{t}$ such that $\hat{h}$ can make
accurate predictions in the future. Note that $\mathcal{H},\bar{t}$ and
$n_{t^{\prime}}$ would depend on the complexity of $P$. For instance, when the
complexity of $P$ is high, we may need to search for $\hat{h}$ from a more
complex class of hypotheses and observe a larger amount of data from $P$ for a
longer time.

  3. 3.

$P$ is not prospectively learnable: Learning such a sequence of distributions
would be impossible and it requires us to observe the sequence $P$ for an
infinite period of time or collect an infinite amount of data from it. We deem
such sequences too complicated to learn prospectively.

While we do not propose a complexity measure for a distribution sequence in
this work, we note that quantities such as predictive information (Bialek et
al., 2001), which compute the mutual information between the past and the
future of a time series, can inspire the construction of an appropriate
measure. We leave this and the discovery of further desirable properties of
hypothesis sequences suitable for prospective learning as future work.

Conjecture 1 motivates the development of prospective learning algorithms that
can output dynamic sequences of hypotheses. Such learners must update
hypotheses as time evolves. Note that prospective learning as defined here is
equivalent to retrospective learning if the distribution sequence is a
constant. Thus traditional retrospective learning can be seen as a special
case of prospective learning. But this formulation of prospective learning is
also quite general, and it has relations with continual-, lifelong-,
transfer-, meta-, online-, and OOD-learning. However, we leave the task of
verifying whether the learnability of these paradigms is encapsulated by
Definition 2 as future work.

##  4 Related Work

#### Online Learning

Both online learning (Cesa-Bianchi & Lugosi, 2006; Hoi et al., 2021; Ying &
Pontil, 2008) and prospective learning deal with task sequences. The goal of
the online learner is to iteratively estimate the hypothesis such that it
minimizes regret, which is the difference between the learner’s risk and the
risk of the best model in hindsight. It is usually studied using a worst-case
analysis and we assume that the sequence of tasks can even be generated by an
adversary. Many algorithms use a follow-the-leader (FTL) strategy which
selects the best model averaged over the past, since it achieves an
asymptotically optimal regret. As a result, an online learner that uses FTL
doesn’t consider the temporal dynamics of the sequence and has no prospective
abilities. The same is true for other popular algorithms like online gradient
descent (OGD) and its variants.

In Figure 3, we evaluate FTL and OGD on two different alternating sequence of
binary classification tasks in the online setting. In Figure 3(a), we consider
two tasks with their labels flipped and observe that both OGD and FTL usually
have high test error when the task switches. Hence both learners fail to
exploit the dynamics of the task sequence. In Figure 3(b), we consider an
alternating sequence of tasks where the first and second task have different
class-conditional means. We observe that this sequence is weakly prospective
learnable by FTL since it converges to some fixed hypothesis that works better
than random on both tasks. We expect that a bona fide prospective learner
would successfully and optimally tackle both of these problems as it would
model the temporal dynamics of the task sequence.

(a)

(b)

Figure 3: We consider two different alternating sequences of tasks where the
learning task shifts between task A and task B periodically every 500 time-
steps. In this online setting, we see one new sample every time-step. We plot
the test error of the current task against the number of time-steps for FTL
and OGD.
(a) For task A, the samples from one class belong to a Gaussian with mean
$[1,1]^{\top}$ and samples from the other class belong to a Gaussian with mean
$[-1,-1]^{\top}$. Both Gaussians have isotropic covariance $I$. Task B is
similar to Task A, except that the distributions for the two labels are
interchanged. The OGD algorithm’s error peaks at each task switch and slowly
reduces afterwards as it learns the new task by unlearning the one before. On
the other hand, the FTL algorithm seems to learn task A successfully, but
completely fails at learning task B. We speculate that for FTL, the model
weights are biased to task A.
(b) For task A, the samples from one class belong to a Gaussian with mean
$[1,1]^{\top}$ and samples from the other class belong to a Gaussian with mean
$[-1,-1]^{\top}$. For task B, the samples from one class belong to a Gaussian
with mean $[1,-1]^{\top}$ and samples from the other class belong to a
Gaussian with mean $[-1,1]^{\top}$. All Gaussians have isotropic covariance
$I$. Unlike (a), the class conditional means are completely different making
the task sequence slightly easier. OGD learns each task much faster, but test
error of the current task peaks at each task switch. Interestingly, FTL is
able to progressively reduce its error as it observes more data from the
sequence. It manages to converge to a fixed hypothesis that acheives
reasonably low error on both task A and task B. However, FTL does not converge
to the Bayes optimal hypothesis for either task since that requires it to be
prospective.
In conclusion, neither OGD nor FTL attempt to model the temporal dynamics of
the task sequence. We expect a (reasonable) prospective learner would achieve
smaller error on both problems.

#### Online Meta-Learning

Online meta-learning (Finn et al., 2019; Denevi et al., 2019; Yao et al.,
2020) refers to a class of algorithms that implement meta-learning updates in
an online manner. These methods seek to minimize a variant of regret which
inspires the use of the follow-the-meta-leader strategy (Finn et al., 2019).
In practice, this translates to implementing model-agnostic meta-learning
(MAML) (Finn et al., 2017) but with the tasks sampled uniformly at random from
all previously seen tasks. While this minimizes the worst-case regret, this
choice is sub-optimal if we know that the task sequence evolves with time
according to a predictable pattern. Hence, methods like online-MAML will fail
to be strongly prospective learnable for some task sequences.

#### Continual Learning

Unlike PAC Learning and online learning, continual learning (Van de Ven &
Tolias, 2019; Zenke et al., 2017; Ramesh & Chaudhari, 2021; Vogelstein et al.,
2020; Geisa et al., 2021) lacks a universally agreed-upon definition. In the
current literature, it often refers to a wide gamut of scenarios that vary in
the goal (e.g., avoiding catastrophic forgetting, positive forward/backward
transfer), or storage and computational constraints. A lack of a universally
agreed-upon definition limits our ability to formally delineate differences.
That said, prospective learning resembles continual learning, albeit with some
key differences. Suppose we would like to tackle a sequence of tasks where the
task switches from A to B, and back, at regular intervals. Many existing
continual learning methods will eventually learn both task A and B, and will
therefore be able to switch back and forth with high accuracy—but shortly
after the task switches. In other words, current continual algorithms will
fail to predict when the task will switch. In contrast, a prospective learner,
that satisfies Definition 2, will learn how the sequence of tasks evolves, and
therefore predict when the task switches. Formally, this learner should
perform at least as well as the continual learner that does not model the
dynamics of the tasks. In simple terms, a continual learner would iteratively
“adapt” to future tasks while a prospective learner would “anticipate” them a
priori. We certainly expect continual learners to benefit from prospective
abilities and believe learners should ideally incorporate both adaptation and
anticipation to actively learn from and make inferences about the future.

#### Streaming Learning

The setting of streaming learning is also closely related to prospective
learning but the objective can often vary. For example, the goal of Gama et
al. (2013) is to minimize the error on samples from an underlying stationary
process. More recent works on streaming learning such as Hayes et al. (2020)
minimize the error on a fixed held-out dataset or on all the past data—neither
of which emphasizes prospection.

#### Out-of-distribution (OOD) Generalization

The OOD generalization problem (Shen et al., 2021; Geisa et al., 2021;
Arjovsky, 2020; Ye et al., 2022) seeks to minimize the generalization error on
an OOD task where the hypothesis is selected using data from a potentially
different set of domains. OOD is typically defined for batch data, not
sequential data. That said, we recover the OOD generalization setting if we
assume that the distribution shifts at time $t’$—which is the point up to
which we see data— and then never changes again.

#### Few-shot Learning

Few-shot learning (Triantafillou et al., 2019; Wang et al., 2020) refers to a
broad range of settings where the model adapts to a new task using limited
amounts of labeled data. Usually, the adaptation step has a limited
computational budget but this is difficult to write down formally. Prospective
learners are also expected to learn using very few samples with one key
difference: they generalize to future tasks using ‘time’ as another context
variable.

It is evident that when the stochastic process is a constant process,
prospective learning becomes equivalent to PAC learning. Thus PAC learning can
be seen as a special case of prospective learning. Furthermore, as discussed
above, prospective learning is quite general and it has relations with various
learning paradigms such as continual-, lifelong-, transfer-, meta-, online-,
and OOD-learning. However, learning paradigms besides PAC learning and online
learning lack formal and universally agreed-upon definitions of learnability.
Therefore, concrete comparison of these paradigms with prospective learning is
limited.

##  5 Discussion

Classical machine learning limits the ability of learning systems to
effectively learn in environments that (somewhat) predictably change. Here we
have proposed an alternative to the classical learning problem that works
around the assumption that training (past) and test (future) distributions are
not identical, but predictably related to some degree. This reframing of the
learning problem can produce flexible agents that effectively adapt in novel
contexts with invariance along key dimensions relevant to learning. As a
result, algorithms that follow a prospective learning framework can
effectively learn for the future, overcoming a fundamental limitation of most
modern AI.

We have provided an answer to the question “What is prospective learning?”,
yet we have largely ignored the broader question of “how is prospective
learning implemented?”. Assuming that natural intelligences (NI; including,
but not necessarily limited to, humans) implement prospective learning
Seligman et al. (2013), motivates synergistic research between NI and AI.
Carefully designed, ecologically appropriate experiments across species,
phyla, and substrates, will enable (i) quantifying the limitations and
capabilities of existing NIs with respect to prospective learning criteria,
and (ii) refining the mechanistic hypotheses generated by the AI theory.

Discovering how prospective learning is implemented in NIs may accelerate
incorporating prospective learning capabilities in AI. For example,
experiments in NI across taxa can establish milestones for experiments in AI
(e.g., (Turing, 1950)). Implementing and deploying AIs and observing NIs
exhibiting prospective learning ‘in the wild’ will provide real-world evidence
to deepen our understanding of learning for the future. These AI
implementations could be purely software, leverage specialized (neuromorphic)
hardware (Vogelstein et al., 2007; Davies, 2019), or even include wetware and
hybrids (Silva et al., 2020), as prospective learning is, by definition,
substrate independent.

In closing, there are many ways in which how we frame intelligence in machine
learning differs from the kinds of problems that we really want to (and
evolved to) solve in the real world, including both objective functions and
prior knowledge. Other discrepancies are hard to even formalize (Mitchell,
2019). Here we have argued that the problem of shifts in the world can be
formulated considerably better than to just appeal to out of distribution
learning. We highlight how many current developments in machine learning can
be seen as steps towards prospective learning. We believe that formulating the
learning problem as a prospective one will produce important synergies between
multiple branches of machine learning and cognitive sciences, providing a more
thorough foundation and new ways of providing logical precision.

## Acknowledgements

This paper was supported by an NSF AI Institute Planning award (#2020312),
NSF-Simons Research Collaborations on the Mathematical and Scientific
Foundations of Deep Learning (MoDL), NSF-Simons Research Collaborations on
Transferable, Hierarchical, Expressive, Optimal, Robust, Interpretable
Networks (THEORINET, #2031985), Microsoft Research, and DARPA. RR and PC were
supported by grants from the National Science Foundation (IIS-2145164,
CCF-2212519), the Office of Naval Research (N00014-22-1-2255), and cloud
computing credits from Amazon Web Services. The authors would like to
especially thank Kathryn Vogelstein for putting up with endless meetings at
the Vogelstein residence in order to make these ideas come to life.

## The Future Learning Collective

Ashwin De Silva1∗; Rahul Ramesh2∗; Lyle Ungar2; Marshall Hussain Shuler1; Noah
J. Cowan1; Michael Platt2; Chen Li1; Leyla Isik1; Seung-Eon Roh1; Adam
Charles1; Archana Venkataraman1; Brian Caffo1; Javier J. How1; Justus M
Kebschull1; John W. Krakauer1; Maxim Bichuch1; Kaleab Alemayehu Kinfu1; Eva
Yezerets1; Dinesh Jayaraman2; Jong M. Shin1; Soledad Villar1; Ian Phillips1;
Carey E. Priebe1; Thomas Hartung1; Michael I. Miller1; Jayanta Dey1; Ningyuan
(Teresa) Huang1; Eric Eaton2; Ralph Etienne-Cummings1; Elizabeth L. Ogburn1;
Randal Burns1; Onyema Osuagwu4; Brett Mensh5; Alysson R. Muotri6; Julia
Brown7; Chris White8; Weiwei Yang8; Andrei A. Rusu9 Timothy Verstynen3; Konrad
P. Kording2; Pratik Chaudhari2✉; Joshua T. Vogelstein1✉;

1 Johns Hopkins University; 2 University of Pennsylvania; 3 Carnegie Mellon
University; 4 Morgan State University 5 Howard Hughes Medical Institute; 6
University of California, San Diego 7 MindX; 8 Microsoft Research 9 DeepMind

∗ Equal contribution
✉ Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
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# First lattice QCD calculation of $J/\psi$ semileptonic decay containing $D$
and $D_{s}$ particles

Yu Meng<EMAIL_ADDRESS>School of Physics, Zhengzhou University, Zhengzhou
450001, China  Jin-Long Dang  School of Physics, Peking University, Beijing
100871, China  Chuan Liu<EMAIL_ADDRESS>School of Physics, Peking
University, Beijing 100871, China  Center for High Energy Physics, Peking
University, Beijing 100871, China  Collaborative Innovation Center of Quantum
Matter, Beijing 100871, China  Xin-Yu Tuo  Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USA  Haobo Yan  School of Physics,
Peking University, Beijing 100871, China  Helmholtz-Institut für Strahlen- und
Kernphysik, Universität Bonn, 53115 Bonn, Germany  Yi-Bo Yang  CAS Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing 100190, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, People’s Republic
of China  School of Fundamental Physics and Mathematical Sciences, Hangzhou
Institute for Advanced Study, UCAS, Hangzhou 310024, China  International
Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China  Ke-Long
Zhang<EMAIL_ADDRESS>Computer Network Information Center, Chinese Academy of
Sciences, Beijing,100083, China

(August 28, 2024)

###### Abstract

We perform the first lattice calculation on the semileptonic decay of $J/\psi$
using the (2+1)-flavor Wilson-clover gauge ensembles generated by CLQCD
collaboration. Three gauge ensembles with different lattice spacings, from
0.0519 fm to 0.1053 fm, and pion masses, $m_{\pi}\sim$ 300 MeV, are utilized.
After a naive continuum extrapolation using three lattice spacings, we obtain
$\operatorname{Br}(J/\psi\rightarrow D_{s}e\nu_{e})=1.90(6)(5)_{V_{cs}}\times
10^{-10}$ and $\operatorname{Br}(J/\psi\rightarrow
De\nu_{e})=1.21(6)(9)_{V_{cd}}\times 10^{-11}$, where the first errors are
statistical, and the second come from the uncertainties of CKM matrix element
$V_{cs(d)}$. The ratios of the branching fractions between lepton $\mu$ and
$e$ are also calculated as $R_{J/\psi}(D_{s})=0.97002(8)$ and
$R_{J/\psi}(D)=0.97423(15)$ after performing a continuum limit including only
$a^{2}$ term. The ratios provide necessary theoretical support for the future
experimental test of lepton flavor universality.

##  I Introduction

The high-precision frontier of flavor physics is one of the most important
goals in contemporary particle physics. It not only involves a precise test of
the standard model but also closely relates to the search for new physics
beyond the standard model. This type of test is typically categorized into two
approaches. The first involves directly comparing with experimental results to
identify any significant discrepancies, for example, the particle mass,
leptonic decay constant, and the decay rate of a particular physical process.
The second approach entails studying various flavor-changing weak decay
processes to accurately extract the Cabibbo-Kobayashi-Maskawa(CKM) matrix
elements, thereby assessing the unitarity of the CKM matrix elements in a row
or column [1]. In both cases, precise theoretical calculations and
experimental measurements are essential and indispensable. Historically,
however, the two are not always given at the same time, and the high-precision
result that appears first often provides an important reference for the test
of the other.

The discovery of the $J/\psi$ particle in 1974 has led to a revolution in
modern particle physics and serves as a solid experimental foundation for
establishing the standard model [2, 3]. The main decay channels of $J/\psi$
particle are strong decay and electromagnetic decay, and these properties have
been extensively studied for several decades both experimentally and
theoretically. For the weak decay, the first estimated branching ratio of
$J/\psi$ weak decay is at the order of $10^{-8}$ [4], which is far from the
experimental detection accuracy for a long time. With the BESIII experiment
already accumulating more than 10 billion $J/\psi$ events [5], and even larger
statistics for future Super Tau Charm Facility [6], these weak decays will be
marginally possible to be measured. Therefore, such rare decay has attracted
much attention in recent years from the experiments [7, 8, 9, 10] and various
phenomenological studies, for example, Bauer-Stech-Wirbel (BSW) model [11],
the QCD sum rules (QCDSR) [12], the Bethe-Salpeter (BS) method [13], covariant
light-front quark model (CLFQM) [14]. However, these phenomenological
calculations are rather precarious especially in the charm sector since the
$J/\psi$ particle is compactly bound by the nonperturbative strong
interaction. At the present stage, a genuine nonperturbative calculation can
not only provide a model-independent comparison with previous phenomenological
studies but also provide a precise and more reliable theoretical assist for
future experiments.

This work aims to present the first lattice calculation on $J/\psi$
semileptonic decay, and particularly determine the branching fraction
$\operatorname{Br}(J/\psi\rightarrow D/D_{s}l\nu_{l})$ with $l=e,\mu$. To that
end, we have developed a method capable of simultaneously extracting multiple
form factors, requiring only direct inputs of the lattice data. By
constructing several appropriate scalar functions, the related form factors
are calculated directly. Besides, we also calculate a large number of time
separations between the initial and final particles in our simulation. Only
when the ground states of initial and final particles dominate, a plateau
independent of the time separation will emerge, thus effectively avoiding the
mixing of excited states. To further control the discretization effect, three
gauge ensembles with different lattice spacings are utilized and a continuum
limit is finally performed.

The rest of this paper is organized as follows. In Sec. II, we introduce the
methodology utilized in this work for calculating the semileptonic decay rate.
This section is divided into three parts: in Sec. II.1 a relationship between
$J/\psi\rightarrow D/D_{s}l\nu_{l}$ decay width and relevant form factors are
given; in section. II.2, the scalar function method is introduce to obtain the
form factors on the lattice; in Sec. II.3 the hadronic function is extracted
from the lattice data. In Sec. III we give details of the simulations and show
the main results. This section is further divided into four parts: in Sec.
III.1 the numerical values of $D_{s}$ and $D$ masses, together with the
dispersion relation are presented; in Sec. III.2 the numerical results of four
form factors are presented; in section III.3 the decay width is obtained and a
continuum extrapolation under three lattice spacings is performed; in Sec.
III.4 the differential decay width of $J/\psi\rightarrow D/D_{s}l\nu_{l}$ are
presented for a better comparison with the future experiment; in section. IV,
a relationship with traditional form factors is established. Finally, we
conclude in Sec. V.

##  II Methodology

###  II.1 Decay width of $J/\psi\rightarrow D/D_{s}l\nu_{l}$

The amplitude of $J/\psi\rightarrow D/D_{s}l\nu_{l}$ in the lowest-order
standard model is expressed by

$i\mathcal{M}=-i\frac{G_{F}}{\sqrt{2}}V_{cs(d)}H_{\mu}(p,p^{\prime})g_{\mu\nu}\bar{u}_{l}\gamma_{\nu}(1-\gamma_{5})u_{\nu_{l}}$
(1)

where $l=e,\mu$. $p^{\prime}$ and $p$ are the four-momenta of $J/\psi$ and
$D/D_{s}$ particles, respectively. Here we introduce the
$H_{\mu}(p,p^{\prime})$ to denote the nonperturbative hadronic interaction
between the initial and final states,

$H_{\mu}(p,p^{\prime})\equiv\langle
D/D_{s}(p)|J_{\mu}^{W}|J/\psi(\epsilon,p^{\prime})\rangle$ (2)

Such hadronic function can be generally decomposed by four variant form
factors in the following way[15]

$\displaystyle H_{\mu}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha}(p^{\prime})H_{\mu\alpha}(p,p^{\prime})$
$\displaystyle H_{\mu\alpha}(p,p^{\prime})$ $\displaystyle=$ $\displaystyle
F_{1}(q^{2})g_{\mu\alpha}+\frac{F_{2}(q^{2})}{Mm}p^{\prime}_{\mu}p_{\alpha}+\frac{F_{3}(q^{2})}{m^{2}}p_{\mu}p_{\alpha}$
(3) $\displaystyle-$
$\displaystyle\frac{iF_{0}}{Mm}\epsilon_{\mu\alpha\rho\sigma}p^{\prime}_{\rho}p_{\sigma}$

where $M$ is the $J/\psi$ mass and $m$ the $D/D_{s}$ mass. The
$\epsilon_{\alpha}(p^{\prime})$ is the polarization vector of $J/\psi$
particle. Compared to the previous parameterization[15], the form factors in
Eq.(II.1) have been rescaled and have the same dimension.

A direct calculation on the decay width of $J/\psi\rightarrow D/D_{s}l\nu_{l}$
in the rest frame of $J/\psi$, by employing Eq. (1) and Eq. (II.1), leads to

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2M}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}2E}\int\frac{d^{3}\vec{q_{l}}}{(2\pi)^{3}2E_{l}}\int\frac{d^{3}\vec{q}_{\nu}}{(2\pi)^{3}2E_{\nu_{l}}}$
$\displaystyle\times$
$\displaystyle(2\pi)^{4}\delta^{4}(p^{\prime}-p-q_{l}-q_{\nu})\frac{1}{3}|\mathcal{M}|^{2}$
$\displaystyle=$
$\displaystyle\frac{G_{F}^{2}V_{cs(d)}^{2}}{12M^{2}}\frac{1}{32\pi^{3}}\int_{m_{l}^{2}}^{(M-m)^{2}}dq^{2}\times\big{[}c_{0}(E_{l}^{+}-E_{l}^{-})$
$\displaystyle+$
$\displaystyle\frac{c_{1}}{2}((E_{l}^{+})^{2}-(E_{l}^{-})^{2})+\frac{c_{2}}{3}((E_{l}^{+})^{3}-(E_{l}^{-})^{3})\big{]}$

where $(q^{2},E_{l})$ are two independent Dalitz variables. The $q^{2}$ is the
momentum transfer squared, $E$ the energy of $D/D_{s}$ particle, $m_{l}$ the
lepton’s mass, $q_{l}$ the lepton’s four-momentum, $E_{l}$ the lepton’s
energy, $q_{\nu_{l}}$ the neutrino’s four-momentum, and $E_{\nu_{l}}$ the
neutrino’s energy. Note that the $E$ is related to $q^{2}$ by
$E=(M^{2}+m^{2}-q^{2})/(2M)$. The coefficients $c_{i}(i=0,1,2)$ are
combinations of $M,m,m_{l},E,F_{i}(i=0,1,2,3)$ and are therefore
$q^{2}$-dependent. The expressions of $c_{i}$ are summarized in the Appendix
A.

The phase space boundary in the $(q^{2},E_{l})$-plane is constrained by[15]

$\displaystyle E_{l}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2M}\Big{[}q^{2}+m_{l}^{2}-\frac{1}{2q^{2}}\big{(}(q^{2}-M^{2}+m^{2})(q^{2}+m_{l}^{2})$
(5) $\displaystyle\mp$ $\displaystyle
2M|\vec{p}|(q^{2}-m_{l}^{2})\big{)}\Big{]}$

where $\vec{p}$ is the momentum of $D/D_{s}$ particle and
$2M|\vec{p}|=\sqrt{(M^{2}-m^{2})^{2}+q^{4}-2q^{2}(M^{2}+m^{2})}$.

###  II.2 Extraction of the form factor on the lattice

In this section, we proceed to extract the form factor $F_{i}(i=0,1,2,3)$
using the scalar function method, which has been widely applied to various
physical processes in recent years [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
and achieved great successes.

We start our discussion with a Euclidean hadronic function in the infinite
volume

$H_{\mu\nu}(\vec{x},t)=\langle
0|\phi_{h}(\vec{x},t)J_{\mu}^{W}(0)|J/\psi_{\nu}(\epsilon,p^{\prime})\rangle,t>0$
(6)

where $\phi_{h}$ is the interpolating operator of $D/D_{s}$ particle with
$h=D$ or $h=D_{s}$, respectively.
$J_{\mu}^{W}=\bar{s}\gamma_{\mu}(1-\gamma_{5})c$ for $h=D_{s}$ and
$J_{\mu}^{W}=\bar{l}\gamma_{\mu}(1-\gamma_{5})c$ for $h=D$, here $l,s,c$ are
the light, strange, and charm quarks. The
$|J/\psi_{\nu}(\epsilon,p^{\prime})\rangle$ is the $J/\psi$ state with
specific polarization direction $\nu$ and momentum $p^{\prime}=(iM,\vec{0})$.
At large time $t$, the hadronic function is saturated by the single $h=D$ or
$D_{s}$ state

$\displaystyle H_{\mu\nu}(x)\doteq
H_{\mu\nu}^{h}(x)=\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{1}{2E_{h}}e^{-E_{h}t+i\vec{p}\cdot\vec{x}}$
$\displaystyle\times\langle
0|\phi_{h}(0)|\phi_{h}(\vec{p})\rangle\langle\phi_{h}(\vec{p})|J_{\mu}^{W}(0)|J/\psi_{\nu}(p^{\prime})\rangle$
(7)

Considering the following parameterizations

$\displaystyle\langle 0|\phi_{h}(0)|\phi_{h}(\vec{p})\rangle$ $\displaystyle=$
$\displaystyle Z_{h}$
$\displaystyle\langle\phi_{h}(\vec{p})|J_{\mu}^{V}(0)|J/\psi_{\nu}(\epsilon,p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{F_{0}}{Mm}\epsilon_{\mu\alpha\rho\sigma}p^{\prime}_{\rho}p_{\sigma}$
$\displaystyle\langle\phi_{h}(\vec{p})|J_{\mu}^{A}(0)|J/\psi_{\nu}(\epsilon,p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle-
F_{1}(q^{2})\delta_{\mu\alpha}-\frac{F_{2}(q^{2})}{Mm}p^{\prime}_{\mu}p_{\alpha}$
(8) $\displaystyle-$
$\displaystyle\frac{F_{3}(q^{2})}{m^{2}}p_{\mu}p_{\alpha}$

where we have divided the weak current $J_{\mu}^{W}$ into two parts, i.e.
$J_{\mu}^{W}=J_{\mu}^{V}-J_{\mu}^{A}$ for convenience. The four-momentum of
the final state $\phi_{h}$ is denoted by $p=(iE_{h},\vec{p})$ and the transfer
momentum square is defined by $q^{2}=(M-E_{h})^{2}-|\vec{p}|^{2}$ as $J/\psi$
is at rest. Then, the spatial Fourier transform of
$H_{\mu\nu}(\vec{x},t)\equiv V_{\mu\nu}(\vec{x},t)-A_{\mu\nu}(\vec{x},t)$
yields

$\displaystyle\tilde{V}_{\mu\nu}(\vec{p},t)$ $\displaystyle\doteq$
$\displaystyle\tilde{V}_{\mu\nu}^{h}(\vec{p},t)=\frac{F_{0}(q^{2})}{Mm}\frac{Z_{h}}{2E_{h}}e^{-E_{h}t}\epsilon_{\mu\alpha\rho\sigma}p^{\prime}_{\rho}p_{\sigma}$
$\displaystyle\tilde{A}_{\mu\nu}(\vec{p},t)$ $\displaystyle\doteq$
$\displaystyle\tilde{A}_{\mu\nu}^{h}(\vec{p},t)=\frac{Z_{h}}{2E_{h}}e^{-E_{h}t}\big{(}-F_{1}(q^{2})\delta_{\mu\alpha}$
(9) $\displaystyle-$
$\displaystyle\frac{F_{2}(q^{2})}{Mm}p^{\prime}_{\mu}p_{\alpha}-\frac{F_{3}(q^{2})}{m^{2}}p_{\mu}p_{\alpha}\big{)}$

Similar to our previous studies [19, 25], the form factor $F_{0}$ can be
accessed to a scalar function

$\displaystyle\mathcal{I}_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{M|\vec{p}|^{2}}\epsilon_{\mu\nu\rho\sigma}p^{\prime}_{\rho}p_{\sigma}\tilde{V}_{\mu\nu}(\vec{p},t)$
(10)

and it is obtained by

$\displaystyle F_{0}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{mE_{h}}{Z_{h}}e^{E_{h}t}\int
d^{3}\vec{x}\frac{j_{1}(|\vec{p}||\vec{x}|)}{|\vec{p}||\vec{x}|}\epsilon_{\mu\nu\alpha
0}x_{\alpha}V_{\mu\nu}(\vec{x},t)$

where $j_{n}(x)$ are the spherical Bessel functions. To extract
$F_{i}(i=1,2,3)$, we construct the following scalar functions

$\displaystyle\mathcal{I}_{1}$ $\displaystyle\equiv$
$\displaystyle\delta_{\mu\nu}\tilde{A}_{\mu\nu}(\vec{p},t)$
$\displaystyle\mathcal{I}_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{E}{M}\frac{p^{\prime}_{\mu}p_{\nu}}{|\vec{p}|^{2}}\tilde{A}_{\mu\nu}(\vec{p},t)$
$\displaystyle\mathcal{I}_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{p_{\mu}p_{\nu}}{|\vec{p}|^{2}}\tilde{A}_{\mu\nu}(\vec{p},t)$
(12)

and it immediately leads to

$\displaystyle F_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{2E_{h}e^{E_{h}t}}{3m^{2}Z_{h}}\left[E_{h}^{2}I_{2}-E_{h}|\vec{p}|^{2}(I_{3}+I_{4})-m^{2}I_{1}-|\vec{p}|^{2}I_{5})\right]$
$\displaystyle F_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{2E_{h}e^{E_{h}t}}{mZ_{h}}\left[E_{h}I_{2}-E_{h}^{2}I_{4}-E_{h}I_{5}-|\vec{p}|^{2}I_{3}\right]$
$\displaystyle F_{3}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{2E_{h}e^{E_{h}t}}{3m^{2}Z_{h}}\big{[}E_{h}^{2}I_{2}+3m_{h}^{2}(E_{h}I_{4}+I_{5})-m^{2}I_{1}$
(13) $\displaystyle-$
$\displaystyle|\vec{p}|^{2}(E_{h}I_{3}+E_{h}I_{4}+I_{5})\big{]}$

where $I_{i}(i=1,2,3,4,5)$ are scalar functions that are directly related to
the correlation functions $A_{\mu\nu}(\vec{x},t)$. The expressions of these
$I_{i}$ are listed in the Appendix B.

Note that both $V_{\mu\nu}$ and $A_{\mu\nu}$ denote the correlation function
in the infinite volume. In real lattice simulations, we need to use the
hadronic function $V_{\mu\nu}^{L}(\vec{x},t)$ and $A_{\mu\nu}^{L}(\vec{x},t)$
which are calculated on a finite-volume lattice to replace the infinite-volume
$V_{\mu\nu}(\vec{x},t)$ and $A_{\mu\nu}(\vec{x},t)$. Such replacement only
amounts for exponentially suppressed finite-volume effects as they are
suppressed exponentially when $|\vec{x}|$ becomes large. One can introduce a
spatial integral truncation $R=|\vec{x}|$ and examine at large $R$ whether the
finite-volume effects are well under control or not, as we did in our previous
study [19]. In our latter work on the $D_{s}^{*}$ radiative decay [25], such
inspections have also been performed, and the finite-volume effects are
confirmed to be well-controlled (this part of the analysis was not included in
the article). In this work, we also present the examination and the results
are summarized in the Appendix C. It is also observed that the finite-volume
effects can be negligible. Therefore, we can empirically conclude that for the
case where the intermediate state is a charm or heavier meson, we can make the
substitution directly without any additional operations. However, when the
intermediate state is a light meson, e.g. $\pi$ or $K$, the exponentially
suppressed finite-volume effects may not be ignored. In that case, one can
use, for example, the infinite volume reconstruction(IVR) method to deal with
these problems [18].

###  II.3 Hadronic function on the lattice

The hadronic function $H_{\mu\nu}(\vec{x},t)$ can be extracted from a three-
point function

$\displaystyle
C_{\mu\nu}^{h}(\vec{x},t;t_{s})=\langle\mathcal{O}_{h}(t)J_{\mu}^{W}(0)\mathcal{O}_{J/\psi,\nu}^{\dagger}(-t_{s})\rangle$
(14)

where interpolating operators are chosen as
$\mathcal{O}_{J/\psi,\nu}^{\dagger}=-\bar{c}\gamma_{\mu}c$,
$\mathcal{O}_{h=D}=\bar{l}\gamma_{5}c$, and
$\mathcal{O}_{h=D_{s}}=\bar{s}\gamma_{5}c$ . In this work, we only consider
the connected contribution. Then, it has the following contractions

$\displaystyle C_{\mu\nu}^{D}(\vec{x},t;t_{s})$ $\displaystyle=$
$\displaystyle\langle\gamma_{\nu}\gamma_{5}S_{c}^{\dagger}(t,-t_{s})\gamma_{5}\gamma_{5}S_{l}(t,0)\gamma_{\mu}(1-\gamma_{5})S_{c}(0,-t_{s})\rangle$
$\displaystyle C_{\mu\nu}^{D_{s}}(\vec{x},t;t_{s})$ $\displaystyle=$
$\displaystyle\langle\gamma_{\nu}\gamma_{5}S_{c}^{\dagger}(t,-t_{s})\gamma_{5}\gamma_{5}S_{s}(t,0)\gamma_{\mu}(1-\gamma_{5})S_{c}(0,-t_{s})\rangle$

Then, the hadroinc function $H_{\mu\nu}(\vec{x},t)$ is determined directly
through

$\displaystyle
H_{\mu\nu}(\vec{x},t)=\frac{2M}{Z_{M}}e^{Mt}C_{\mu\nu}^{h}(\vec{x},t)$ (16)

In our calculations, $M$, $m$, $Z_{M}$, and $Z_{h}$ are extracted from the
two-point function
$C^{(2)}(\vec{p},t)=\sum\limits_{\vec{x}}\cos(\vec{p}\cdot\vec{x})\langle\mathcal{O}_{h}(\vec{x},t)\mathcal{O}_{h}^{\dagger}(0)\rangle$
by a single-state fit

$\displaystyle
C^{(2)}(\vec{p},t)=\frac{Z_{h}^{2}}{2E_{h}}\left(e^{-E_{h}t}+e^{-E_{h}(T-t)}\right)$
(17)

with $m_{h}=E_{h}(\vec{p}=0)$ the ground-state energy and $Z_{h}=\langle
h|\mathcal{O}^{\dagger}_{h}|0\rangle$ is the overlap amplitude for the ground
state. The calculation of three-point function $C_{\mu\nu}^{h}(\vec{x},t)$ is
similar to our previous studies [19, 25], so we won’t go into these details in
this paper.

##  III Simulations and results

Ensemble | C24P29 | F32P30 | H48P32
---|---|---|---
$a(\textrm{fm})$ | 0.10530(18) | 0.07746(18) | 0.05187(26)
$a\mu_{s}$ | -0.2400 | -0.2050 | -0.1700
$a\mu_{c}$ | 0.4479 | 0.2079 | 0.0581
$L^{3}\times T$ | $24^{3}\times 72$ | $32^{3}\times 96$ | $48^{3}\times 144$
$N_{\textrm{cfg}}\times N_{\textrm{src}}$ | $450\times 72$ | $719\times 96$ | $100\times 72$
$m_{\pi}(\textrm{MeV})$ | 292.7(1.2) | 303.2(1.3) | 317.2(0.9)
$t$ | 6-17 | 6-20 | 8-30
$Z_{V}$ | 0.79814(23) | 0.83548(12) | 0.86855(04)
$Z_{A}$ | 0.85442(85) | 0.88161(64) | 0.90113(36)
Table 1:  Parameters of gauge ensembles used in this work. From top to bottom,
we list the ensemble name, the lattice spacing $a$, the bare quark mass
including the strange quark $a\mu_{s}$ and charm quark $a{\mu}_{c}$, the
spatial and temporal lattice size $L$ and $T$, the number of the measurements
of the correlation function for each ensemble $N_{\textrm{cfg}}\times
N_{\textrm{src}}$, the pion mass $m_{\pi}$, the range of the time separation
$t$ between the initial hadron and the electromagnetic current, vector
normalization constant $Z_{V}$, and axial vector normalization constant
$Z_{A}$. Here, $L$, $T$ and $t$ are given in lattice units.

We employ three (2+1)-flavor Wilson-clover gauge ensembles generated by the
CLQCD collaboration with lattice spacings $a\approx 0.1053,0.0775,0.0519$ fm
[26], the parameters of which are shown in Table. 1. The dynamical quarks of
the ensembles use the tadpole-improved tree-level Symanzik gauge action and
the tadpole-improved tree-level clover fermions. In recent years, plenty of
studies have been carried out based on these configurations [27, 28, 29, 30,
25]. The same setup has been utilized in our previous lattice study on
$D_{s}^{*}$ radiative decay. For more details of the ensembles, we refer to
Ref. [25]. Since the ensembles have similar volumes and pion masses in
physical units and are therefore expected to provide a fully well-controlled
continuous extrapolation.

###  III.1 Mass spectrum and dispersion relation

Figure 1:  For the ensembles of C24P29, F32P30, and H48P32, the energy levels of $D$ particle with different momentum $\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$ extracted from two-point functions calculated using point source propagators. The horizontal gray bands denote the fitting ranges. Ensemble | C24P29 | F32P30 | H48P32
---|---|---|---
$M$(MeV) | 3098.6(3) | 3094.0(2) | 3096.6(6)
$Z_{M}$ | 0.2868(3) | 0.1718(2) | 0.0847(3)
$aE_{D}(|\vec{n}|^{2}=0)$ | 1.0257(8) | 0.7505(4) | 0.5010(7)
$aE_{D}(|\vec{n}|^{2}=1)$ | 1.0537(10) | 0.7747(5) | 0.5174(9)
$aE_{D}(|\vec{n}|^{2}=2)$ | 1.0807(14) | 0.7971(7) | 0.5334(14)
$aE_{D}(|\vec{n}|^{2}=3)$ | 1.1068(19) | 0.8195(10) | 0.5493(20)
$aE_{D}(|\vec{n}|^{2}=4)$ | 1.1290(27) | 0.8411(15) | 0.5645(30)
$\mathcal{Z}_{\textrm{latt}}^{D}$ | 1.012(18) | 1.040(12) | 1.030(33)
$Z_{D}$ | 0.1028(8) | 0.0604(12) | 0.0274(9)
$aE_{D_{s}}(|\vec{n}|^{2}=0)$ | 1.0658(3) | 0.7801(2) | 0.5211(3)
$aE_{D_{s}}(|\vec{n}|^{2}=1)$ | 1.0928(4) | 0.8026(2) | 0.5369(5)
$aE_{D_{s}}(|\vec{n}|^{2}=2)$ | 1.1191(5) | 0.8244(3) | 0.5525(8)
$aE_{D_{s}}(|\vec{n}|^{2}=3)$ | 1.1446(6) | 0.8456(5) | 0.5680(12)
$aE_{D_{s}}(|\vec{n}|^{2}=4)$ | 1.1674(8) | 0.8652(8) | 0.5842(20)
$\mathcal{Z}_{\textrm{latt}}^{D_{s}}$ | 1.035(6) | 1.026(6) | 1.043(21)
$Z_{D_{s}}$ | 0.2165(5) | 0.1376(3) | 0.0721(6)
Table 2: Mass spectra of the $J/\psi$ particle, energy levels $E_{D}$,
$E_{D_{s}}$ with $\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$, overlap
function $Z_{M/D/D_{s}}$, and the coefficients
$\mathcal{Z}_{\textrm{latt}}^{D/D_{s}}$ defined by the discrete dispersion
relation.

The energy levels of the particle $D/D_{s}$ are extracted from the two-point
functions which are calculated by the point source propagators. A single-state
correlated fit with the formula Eq. (17) is utilized and the numerical fitting
results of the spectra are summarized in Table 2. The effective levels of the
particle $D$ are shown in Fig. 1 for all the ensembles and the horizontal gray
bands therein denote the fitting center values and statistical errors
estimated by the jackknife method. The dispersion relation of $D$ particle is
also checked by using its energy levels. It is found that the discrete
dispersion relation

$4\sinh^{2}\frac{E_{h}}{2}=4\sinh^{2}\frac{m_{h}}{2}+\mathcal{Z}_{\textrm{latt}}^{h}\cdot
4\sum\limits_{i}\sin^{2}\frac{\vec{p}_{i}}{2}$ (18)

describes the energies and momenta well and there is a nice linear behavior
between $4\sinh^{2}(E_{h}/2)$ and $4\sum\limits_{i}\sin^{2}(\vec{p}_{i}/2)$ as
illustrated in Fig. 1. For the results of $D_{s}$ particle, one can refer to
our previous paper [25].

###  III.2 Form factor

Ensemble | C24P29 | F32P30 | H48P32
---|---|---|---
$F_{0}^{D_{s}}(|\vec{n}|^{2}=0)$ | 1.932(10) | 1.164(6) | 1.246(13)
$F_{0}^{D_{s}}(|\vec{n}|^{2}=1)$ | 1.731(8) | 1.459(4) | 1.111(10)
$F_{0}^{D_{s}}(|\vec{n}|^{2}=2)$ | 1.529(7) | 1.315(4) | 0.978(9)
$F_{0}^{D_{s}}(|\vec{n}|^{2}=3)$ | 1.340(6) | 1.450(4) | 0.857(9)
$F_{0}^{D_{s}}(|\vec{n}|^{2}=4)$ | 1.151(6) | 0.986(4) | 0.738(8)
$d_{0}^{D_{s},(0)}$ | 1.229(7) | 1.084(4) | 0.812(9)
$d_{0}^{D_{s},(1)}$ | 2.042(26) | 3.077(33) | 4.793(159)
$d_{0}^{D_{s},(2)}$ | -0.021(81) | 0.303(169) | 3.16(2.02)
$F_{1}^{D_{s}}(|\vec{n}|^{2}=0)$ | 1.350(5) | 1.161(3) | 0.870(8)
$F_{1}^{D_{s}}(|\vec{n}|^{2}=1)$ | 1.200(5) | 1.024(3) | 0.771(6)
$F_{1}^{D_{s}}(|\vec{n}|^{2}=2)$ | 1.740(5) | 0.908(3) | 0.691(6)
$F_{1}^{D_{s}}(|\vec{n}|^{2}=3)$ | 0.973(5) | 0.817(3) | 0.628(6)
$F_{1}^{D_{s}}(|\vec{n}|^{2}=4)$ | 0.876(5) | 0.739(3) | 0.571(6)
$d_{1}^{D_{s},(0)}$ | 0.914(5) | 0.784(3) | 0.604(6)
$d_{1}^{D_{s},(1)}$ | 0.965(20) | 1.464(19) | 2.208(114)
$d_{1}^{D_{s},(2)}$ | 0.857(67) | 2.813(108) | 10.29(1.54)
$F_{2}^{D_{s}}(|\vec{n}|^{2}=0)$ | 0.183(12) | 0.125(7) | 0.048(14)
$F_{2}^{D_{s}}(|\vec{n}|^{2}=1)$ | 0.102(9) | 0.048(6) | 0.020(10)
$F_{2}^{D_{s}}(|\vec{n}|^{2}=2)$ | 0.056(9) | 0.005(5) | 0.005(9)
$F_{2}^{D_{s}}(|\vec{n}|^{2}=3)$ | 0.040(8) | -0.006(5) | 0.003(10)
$F_{2}^{D_{s}}(|\vec{n}|^{2}=4)$ | 0.047(7) | 0.006(5) | 0.010(10)
$d_{2}^{D_{s},(0)}$ | 0.041(7) | -0.005(5) | 0.004(10)
$d_{2}^{D_{s},(1)}$ | -0.134(30) | -0.223(30) | -0.276(140)
$d_{2}^{D_{s},(2)}$ | 1.574(101) | 4.838(168) | 9.12(1.85)
$F_{3}^{D_{s}}(|\vec{n}|^{2}=0)$ | 1.166(12) | 1.034(7) | 0.921(17)
$F_{3}^{D_{s}}(|\vec{n}|^{2}=1)$ | 1.099(9) | 0.977(5) | 0.787(12)
$F_{3}^{D_{s}}(|\vec{n}|^{2}=2)$ | 1.016(8) | 0.902(5) | 0.688(12)
$F_{3}^{D_{s}}(|\vec{n}|^{2}=3)$ | 0.929(7) | 0.818(4) | 0.624(11)
$F_{3}^{D_{s}}(|\vec{n}|^{2}=4)$ | 0.826(7) | 0.729(4) | 0.576(11)
$d_{3}^{D_{s},(0)}$ | 0.870(7) | 0.785(4) | 0.601(11)
$d_{3}^{D_{s},(1)}$ | 1.109(28) | 1.723(36) | 1.842(140)
$d_{3}^{D_{s},(2)}$ | -0.730(91) | -2.107(205) | 21.78(2.40)
Table 3: For the ensembles of C24P29, F32P30, and H48P32, the numerical results of $F_{i}^{D_{s}}(q^{2})$ and the polynomial fitting coefficient $d_{i}^{D_{s},(j)}$ for $\vec{p}=2\pi\vec{n}/L(|\vec{n}|^{2}=0,1,2,3,4),i=0,1,2,3$, and $j=0,1,2$. Ensemble | C24P29 | F32P30 | H48P32
---|---|---|---
$F_{0}^{D}(|\vec{n}|^{2}=0)$ | 1.870(25) | 1.637(11) | 1.237(31)
$F_{0}^{D}(|\vec{n}|^{2}=1)$ | 1.668(17) | 1.436(8) | 1.105(21)
$F_{0}^{D}(|\vec{n}|^{2}=2)$ | 1.456(14) | 1.241(6) | 0.956(18)
$F_{0}^{D}(|\vec{n}|^{2}=3)$ | 1.259(13) | 1.067(6) | 0.800(17)
$F_{0}^{D}(|\vec{n}|^{2}=4)$ | 1.069(13) | 0.914(6) | 0.665(17)
$d_{0}^{D,(0)}$ | 1.066(17) | 0.953(8) | 0.699(22)
$d_{0}^{D,(1)}$ | 2.070(85) | 2.806(68) | 5.961(524)
$d_{0}^{D,(2)}$ | -0.069(231) | 1.750(349) | -4.39(5.83)
$F_{1}^{D}(|\vec{n}|^{2}=0)$ | 1.234(10) | 1.069(5) | 0.820(11)
$F_{1}^{D}(|\vec{n}|^{2}=1)$ | 1.084(8) | 0.931(4) | 0.713(8)
$F_{1}^{D}(|\vec{n}|^{2}=2)$ | 0.960(8) | 0.818(4) | 0.629(8)
$F_{1}^{D}(|\vec{n}|^{2}=3)$ | 0.859(8) | 0.730(4) | 0.568(8)
$F_{1}^{D}(|\vec{n}|^{2}=4)$ | 0.764(9) | 0.660(4) | 0.515(8)
$d_{1}^{D,(0)}$ | 0.766(10) | 0.677(5) | 0.527(9)
$d_{1}^{D,(1)}$ | 0.869(61) | 1.164(47) | 1.780(227)
$d_{1}^{D,(2)}$ | 0.797(160) | 3.048(231) | 12.18(2.64)
$F_{2}^{D}(|\vec{n}|^{2}=0)$ | 0.032(29) | -0.007(14) | -0.108(48)
$F_{2}^{D}(|\vec{n}|^{2}=1)$ | -0.047(18) | -0.074(9) | -0.097(28)
$F_{2}^{D}(|\vec{n}|^{2}=2)$ | -0.084(15) | -0.101(8) | -0.068(20)
$F_{2}^{D}(|\vec{n}|^{2}=3)$ | -0.089(14) | -0.097(7) | -0.028(19)
$F_{2}^{D}(|\vec{n}|^{2}=4)$ | -0.066(14) | -0.073(7) | 0.0043(20)
$d_{2}^{D,(0)}$ | -0.0647(14) | -0.821(7) | -0.004(20)
$d_{2}^{D,(1)}$ | -0.433(100) | -0.635(75) | -1.695(517)
$d_{2}^{D,(2)}$ | 1.719(315) | 4.530(441) | 6.22(6.73)
$F_{3}^{D}(|\vec{n}|^{2}=0)$ | 1.195(29) | 1.075(14) | 0.818(25)
$F_{3}^{D}(|\vec{n}|^{2}=1)$ | 1.123(18) | 1.005(9) | 0.672(14)
$F_{3}^{D}(|\vec{n}|^{2}=2)$ | 1.029(15) | 0.914(7) | 0.578(11)
$F_{3}^{D}(|\vec{n}|^{2}=3)$ | 0.931(14) | 0.817(7) | 0.518(10)
$F_{3}^{D}(|\vec{n}|^{2}=4)$ | 0.814(15) | 0.720(7) | 0.469(9)
$d_{3}^{D,(0)}$ | 0.813(17) | 0.746(8) | 0.480(10)
$d_{3}^{D,(1)}$ | 1.311(90) | 1.896(76) | 1.229(299)
$d_{3}^{D,(2)}$ | -0.868(277) | -1.677(420) | 22.37(4.57)
Table 4: For the ensembles of C24P29, F32P30, and H48P32, the numerical
results of $F_{i}^{D}(q^{2})$ and the polynomial fitting coefficient
$d_{i}^{D,(j)}$ for
$\vec{p}=2\pi\vec{n}/L(|\vec{n}|^{2}=0,1,2,3,4),i=0,1,2,3$, and $j=0,1,2$.

Figure 2:  For the ensembles of C24P29, F32P30, and H48P32, the form factors
$F_{i}^{D_{s}}(q^{2})$(top) and $F_{i}^{D}(q^{2})$(bottom) with
$\vec{p}=2\pi\vec{n}/L(|\vec{n}|^{2}=0,1,2,3,4)$ and $i=0,1,2,3$, where
$F_{i}^{D_{s}}(q^{2})$ denotes the form factor for the process
$J/\psi\rightarrow D_{s}l\nu_{l}$, and $F_{i}^{D}(q^{2})$ for
$J/\psi\rightarrow Dl\nu_{l}$.

Figure 3:  For the ensembles of C24P29, F32P30, and H48P32, the polynomial
fits for the form factors $F_{i}^{D_{s}}(q^{2})$(left) and
$F_{i}^{D}(q^{2})$(right) with
$\vec{p}=2\pi\vec{n}/L(|\vec{n}|^{2}=0,1,2,3,4)$ and $i=0,1,2,3$, where
$F_{i}^{D_{s}}(q^{2})$ denotes the form factor for the process
$J/\psi\rightarrow D_{s}l\nu_{l}$, and $F_{i}^{D}(q^{2})$ for
$J/\psi\rightarrow Dl\nu_{l}$. The black boxes indicate the results at
$q^{2}=0$.

The lattice results of $F_{i}^{D/D_{s}}(q^{2})(i=0,1,2,3)$ as a function of
the time separation $t$ are shown in Fig. 2, together with a series of momenta
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$. Here we present all the
results of the three ensembles, i.e. C24P29, F32P30, and H48P32 from left to
right, respectively. It shows that both $F_{i}^{D}$ and $F_{i}^{D_{s}}$ have
obvious $t$ dependence as $t$ increases, indicating sizable excited-state
effects associated with the initial and final states. With enough time
intervals utilized in this work, we could observe obvious plateaus in a large
enough time region. It is therefore natural to perform a correlated fit for
these lattice data to a constant at a suitable time region. All results of
$F_{i}^{D/D_{s}}(q^{2})$ with different momenta $\vec{p}$ are obtained in this
way and are denoted by the gray bands in the figure.

With the five momentum modes of the form factor $F_{i}^{D/D_{s}}(q^{2})$ taken
into account, a momentum extrapolation can be performed by a polynomial form

$F_{i}^{h}(q^{2})=d_{i}^{h,(0)}+d_{i}^{h,(1)}\cdot q^{2}+d_{i}^{h,(2)}\cdot
q^{4}$ (19)

where the coefficients $d_{i}^{h,(j)}$ with $j=0,1,2$ are introduced. As shown
in Fig. 3, such a polynomial formula in Eq. (19) can describe the lattice data
very well. In the following, we can see that the $q^{2}$ with five momentum
modes $\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$ have covered the full
phase space for all gauge ensembles in our calculation. The numerical results
of the coefficients $d_{i}^{D,(j)}$ and $d_{i}^{D_{s},(j)}$ are summarized in
Table 3 and Table 4.

Ensemble | C24P29 | F32P30 | H48P32 | Cont.Limit
---|---|---|---|---
$\operatorname{Br}(J/\psi\rightarrow D_{s}e\nu_{e})\times 10^{10}$ | $0.837(7)_{\textrm{stat}}(10)_{V_{cs}}$ | $1.168(6)_{\textrm{stat}}(14)_{V_{cs}}$ | $1.526(26)_{\textrm{stat}}(19)_{V_{cs}}$ | $1.90(6)_{\textrm{stat}}(5)_{V_{cs}}$
$\operatorname{Br}(J/\psi\rightarrow D_{s}\mu\nu_{\mu})\times 10^{10}$ | $0.813(7)_{\textrm{stat}}(10)_{V_{cs}}$ | $1.133(6)_{\textrm{stat}}(14)_{V_{cs}}$ | $1.480(25)_{\textrm{stat}}(18)_{V_{cs}}$ | $1.84(6)_{\textrm{stat}}(5)_{V_{cs}}$
$\operatorname{Br}(J/\psi\rightarrow De\nu_{e})\times 10^{11}$ | $0.459(7)_{\textrm{stat}}(17)_{V_{cd}}$ | $0.651(6)_{\textrm{stat}}(24)_{V_{cd}}$ | $0.915(24)_{\textrm{stat}}(33)_{V_{cd}}$ | $1.21(6)_{\textrm{stat}}(9)_{V_{cd}}$
$\operatorname{Br}(J/\psi\rightarrow D\mu\nu_{\mu})\times 10^{11}$ | $0.447(7)_{\textrm{stat}}(16)_{V_{cd}}$ | $0.634(6)_{\textrm{stat}}(23)_{V_{cd}}$ | $0.892(23)_{\textrm{stat}}(32)_{V_{cd}}$ | $1.18(6)_{\textrm{stat}}(9)_{V_{cd}}$
Table 5: Numerical results of the branching fraction
$\operatorname{Br}(J/\psi\rightarrow D/D_{s}l\nu_{l})$ with $l=e,\mu$ for all
the gauge ensembles. The first errors are the statistical errors from the
lattice simulation, and the second ones come from the uncertainties of the CKM
matrix $V_{cs(d)}$.

Figure 4:  The lattice results of $\operatorname{Br}(J/\psi\rightarrow
D/D_{s}e\nu_{e})$(left) and $R_{J/\psi}(D/D_{s})$(right) as a function of the
lattice spacings. On the left, a continuum extrapolation including $a^{4}$
term is performed. In the right, only a linear $a^{2}$ behavior is considered
since the discretization effects are greatly canceled for the ratios
$R_{J/\psi}(D)$ and $R_{J/\psi}(D_{s})$ . The errors of lattice spacing have
been included in the continuous limit, which are presented by the horizontal
error bars.

###  III.3 Decay width

Using the numerical values of the coefficients $d_{i}^{D/D_{s},(j)}$ as
inputs, and combining with the Eq. (II.1), the differential decay rate
$d\Gamma/(dq^{2}V_{cs(d)}^{2})$ can be calculated directly with the continuous
$q^{2}$ covering the full phase space. After integrating the $q^{2}$ in the
full phase space and taking into account the values of $V_{cs}=0.975(6)$ and
$V_{cd}=0.221(4)$ [31], we immediately obtain the total branching fraction of
the corresponding decay channels $\operatorname{Br}(J/\psi\rightarrow
D/D_{s}l\nu_{l})$, numerical values of which are listed in the Table. 5.

The lattice results for the branching fraction $J/\psi\rightarrow
D/D_{s}e\nu_{e}$ at different lattice spacings are shown in Fig. 4, together
with the ratio of between lepton $\mu$ and $e$ denoted by
$R_{J/\psi}(D/D_{s})=\operatorname{Br}(J/\psi\rightarrow
D/D_{s}\mu\nu_{\mu})/\operatorname{Br}(J/\psi\rightarrow D/D_{s}e\nu_{e})$. It
is observed the branching fraction can be described well by a continuous limit
extrapolation that contains the $a^{4}$-order term, whereas the ratios only
the $a^{2}$ term. This is because the ratio can greatly eliminate the
additional discretization effects. The leading $a^{2}$ term is expected since
the ensembles used in the work have adopted the tadpole-improved tree-level
Symanzik gauge action and the tadpole-improved tree-level clover fermion
action. The next-leading order $a^{4}$ term has a sizeable effect on the
charmonium system especially for a coarse lattice spacing that is larger than
0.1 fm, for instance. The continuous extrapolatin including $a^{4}$ term has
also been used in previous lattice studies on charmonium system [32, 33].

Since we only have three lattice spacing values, we cannot make the continuum
extrapolation that includes $a^{4}$ term in a controlled fashion. Instead, we
can only ensure a well-controlled continuum limit in a linear $a^{2}$ behavior
given three lattice values. Because of this, we will call the former a naive
continuum extrapolation in this paper. Finally, we report the first lattice
prediction as

$\displaystyle\operatorname{Br}(J/\psi\rightarrow D_{s}e\nu_{e})$
$\displaystyle=$ $\displaystyle 1.90(6)_{\textrm{stat}}(5)_{V_{cs}}\times
10^{-10}$ $\displaystyle\operatorname{Br}(J/\psi\rightarrow De\nu_{e})$
$\displaystyle=$ $\displaystyle 1.21(6)_{\textrm{stat}}(9)_{V_{cd}}\times
10^{-11}$ $\displaystyle R_{J/\psi}(D_{s})$ $\displaystyle=$ $\displaystyle
0.97002(8)_{\textrm{stat}}$ $\displaystyle R_{J/\psi}(D)$ $\displaystyle=$
$\displaystyle 0.97423(15)_{\textrm{stat}}$ (20)

where the first errors are the statistical errors, and the seconds result from
the uncertainties of CKM matrix element $V_{cs(d)}$. The errors of the
branching fraction $\operatorname{Br}(J/\psi\rightarrow D/D_{s}l\nu_{l})$ with
the different lepton $l=\mu$ and $l=e$ almost completely cancel, eventually
leading to an extremely high precision value.

Note that the light quark masses in our calculation are nonphysical, it may
have an influence on the process $J/\psi\rightarrow Dl\nu_{l}$, where the
light quarks appear as valence quarks. For the process $J/\psi\rightarrow
D_{s}l\nu_{l}$ , the nonphysical light quark only occurs in the sea, so its
effect is expected to be small [1]. Other systematic errors that have not been
seriously considered include the effects from the neglected disconnected
diagrams and the quenching of the charm quark. All these effects could be
studied in future systematic lattice studies using e.g. the gauge ensembles
with physical pion mass, with charm sea quarks, and with more lattice
spacings.

###  III.4 Differential decay width

Figure 5:  The lattice results of the $q^{2}$-expansion for $J/\psi\rightarrow D_{s}$ and $J/\psi\rightarrow D$ form factors. The red-shaded regions correspond to the final results in a continuum limit $a\rightarrow 0$. Figure 6:  Predictions for the differential decay widths for $J/\psi\rightarrow D_{s}e\nu_{e}$ and $J/\psi\rightarrow De\nu_{e}$, divided by $V_{cs}$ and $V_{cd}$, respectively. $J/\psi\rightarrow D_{s}/D$ | $c_{0}$ | $c_{1}$ | $c_{2}$
---|---|---|---
$F_{0}^{D_{s}}$ | 3.38(5) | 1.29(20) | 0.10(19)
$F_{1}^{D_{s}}$ | 2.62(4) | 0.57(11) | 0.19(10)
$F_{2}^{D_{s}}$ | 0.09(6) | -0.05(23) | -0.03(23)
$F_{3}^{D_{s}}$ | 2.55(7) | 0.17(24) | 1.14(24)
$F_{0}^{D}$ | 2.78(15) | 2.37(58) | -0.48(46)
$F_{1}^{D}$ | 2.29(6) | 0.46(13) | 0.27(9)
$F_{2}^{D}$ | 0.24(18) | -0.51(83) | -0.25(69)
$F_{3}^{D}$ | 1.58(7) | -0.25(30) | 1.06(26)
Table 6: Numerical results of the $q^{2}$-expansion coefficients
$c_{i},i=0,1,2$ for both processes $J/\psi\rightarrow D_{s}l\nu_{l}$ and
$J/\psi\rightarrow Dl\nu_{l}$.

To further compare with future experimental and phenomenological studies, in
this section, we will give the results of the differential $q^{2}$
distribution of the decay width, i.e. $d\Gamma/dq^{2}$. For this purpose, we
utilize a general $q^{2}$-expansion parametrization of form factors. In the
above section, it has been found the polynomial $q^{2}$-expansion describes
the lattice data very well, as shown in Fig. 3. Therefore, the formula is
given by

$F(a^{2},q^{2})=\sum\limits_{n=0}^{n_{\textrm{max}}}(c_{i}+d_{i}a^{2}+f_{i}a^{4})q^{2n}$
(21)

where the terms $d_{i}$ and $f_{i}$ describe the discretization effects. The
$a^{4}$ term is found to be necessary in the above discussions due to the
coarse lattice spacing of the ensemble C24P29, thus we include it here. The
$q^{2}$ distribution of these form factors are shown in Fig. 5 and the
numerical values of coefficients $c_{0},c_{1}$ and $c_{2}$ are collected in
the Table 6.

Taking into account the PDG values: $m_{J/\psi}=3.09690(1)\textrm{GeV}$,
$m_{D_{s}}=1.96834(7)\textrm{GeV}$, and $m_{D}=1.86966(5)\textrm{GeV}$, one
can obtain the differential decay width $d\Gamma/dq^{2}$, as shown in Fig. 6.
Integrating the $q^{2}$, it immediately leads to the branching fraction
$\operatorname{Br}(J/\psi\rightarrow D_{s}e\nu_{e})=1.99(14)\times 10^{-10}$
and $\operatorname{Br}(J/\psi\rightarrow De\nu_{e})=1.44(17)\times 10^{-11}$,
where the errors are total errors including the uncertainties both from the
statistics and CKM matrix element $V_{cs(d)}$. It is seen the results obtained
here are almost compatible with Eq. (III.3), but with larger statistical
errors. Therefore, we will take the values in Eq. (III.3) as our final
reports.

##  IV Discussion

The hadronic matrix element of $J/\psi\rightarrow D/D_{s}$ should have the
similar lorentz structure with other semileptonic decay $P\rightarrow V$, for
example, $D_{s}\rightarrow\phi$ [34], $B\rightarrow D^{*}$ [35], and
$B\rightarrow J/\psi$ [36]. Therefore, the method proposed in this work can be
directly extended to the above processes. Previous lattice studies on these
channels have utilized another parameterization scheme [37]

$\displaystyle
H_{\mu}(p^{\prime},q)=\epsilon_{\mu\alpha\beta\delta}\epsilon_{\alpha}(p^{\prime},\lambda)p^{\prime\beta}p^{\delta}\frac{2V(q^{2})}{M+m}$
$\displaystyle+$ $\displaystyle
2MA_{0}(q^{2})\frac{\epsilon(p^{\prime},\lambda)\cdot q}{q^{2}}q_{\mu}$
$\displaystyle+$
$\displaystyle(M+m)A_{1}(q^{2})\left[\epsilon_{\mu}(p^{\prime},\lambda)-\frac{\epsilon(p^{\prime},\lambda)\cdot
q}{q^{2}}q_{\mu}\right]$ $\displaystyle-$ $\displaystyle
A_{2}(q^{2})\frac{\epsilon(p^{\prime},\lambda)\cdot
q}{M+m}\left[p^{\prime}_{\mu}+p_{\mu}-\frac{M^{2}-m^{2}}{q^{2}}q^{\mu}\right]$

where form factors $V$ and $A_{i}$ are introduced, rather than $F_{i}$ in this
work. The kinematic constraint gives

$A_{0}(0)=\frac{m+M}{2M}A_{1}(0)-\frac{M-m}{2M}A_{2}(0)$ (23)

Note that the form factors with a factor of $q_{\mu}$ in Eq. (IV) have no
significant contribution to the experimental rate, due to its combination with
the leptonic current tends to zero as the lepton mass vanishes. The form
factors $F_{i}$ introduced in this work can be related to the conventional
form factors $V$ and $A_{i}$ by the following way

$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{(m+M)}{2mM}F_{0}$
$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2M}\left(F_{1}-\frac{M^{2}-m^{2}+q^{2}}{2mM}F_{2}-\frac{M^{2}-m^{2}-q^{2}}{2m^{2}}F_{3}\right)$
$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle\frac{F_{1}}{m+M},\quad
A_{2}=\frac{m+M}{2m^{2}M}(mF_{2}+MF_{3})$ (24)

In our simulations, we find $F_{2}(0)\simeq 0$, and it naturally leads to the
well-known relationship given in Eq. (23). Given the $q^{2}$-expansion
coefficients $c_{i}$, the results of $V(0)$ and $A_{i}(0)$ are determined
directly, numerical values of which are summarized in Table. 8.

| $V(0)$ | $A_{0}(0)$ | $A_{1}(0)$ | $A_{2}(0)$
---|---|---|---|---
$J/\psi\rightarrow D_{s}$ | 1.40(2) | 0.11(1) | 0.52(1) | 1.71(5)
$J/\psi\rightarrow D$ | 1.19(6) | 0.13(2) | 0.46(1) | 1.23(9)
Table 7: Numerical results of the conventional form factors $V(0),A_{i}(0)$.

The experimental search for $J/\psi$ semileptonic decay has been ongoing for
many years, but it has not been detected to date. The highest upper limit
comes from BESIII experiment on $J/\psi\rightarrow De\nu_{e}$, where the full
$1.01\times 10^{10}$ $J/\psi$ samples are utilized. However, there are still
three orders of magnitude away from the theoretical prediction for the process
$J/\psi\rightarrow De\nu_{e}$. Since there exist no measurements using the
full $J/\psi$ samples on $J/\psi\rightarrow D_{s}l\nu_{l}$ so far, we can
deduce from previous measurements that the upper limit of the channel
$J/\psi\rightarrow D_{s}e\nu_{e}$ might be $10^{-8}$ if all $J/\psi$ samples
are used. The experimental results of $J/\psi$ semileptonic decay are
summarized in Table. 7, together with the lattice results of this work for
comparisons. Nowadays, the future experiment, such as Super Tau Charm Facility
[6], has the great potential to improve the upper limit to a level of
$10^{-10}$. By combining the certain experimental measurement of the branching
fraction and the form factors given by lattice QCD, one can extract the CKM
matrix element $V_{cs(d)}$, therefore providing an alternative precision test
for the standard model.

channels | Upper limit/Br |  $J/\psi$ number | Refs
---|---|---|---
$J/\psi\rightarrow D_{s}e\nu_{e}$ | $4.9\times 10^{-5}$ | $5.8\times 10^{7}$ | [7]
$J/\psi\rightarrow D_{s}e\nu_{e}$ | $1.3\times 10^{-6}$ | $2.3\times 10^{8}$ | [8]
$J/\psi\rightarrow De\nu_{e}$ | $7.1\times 10^{-8}$ | $1.01\times 10^{10}$ | [9]
$J/\psi\rightarrow D\mu\nu_{\mu}$ | $5.6\times 10^{-7}$ | $1.01\times 10^{10}$ | [10]
$J/\psi\rightarrow D_{s}e\nu_{e}$ | $1.90(8)\times 10^{-10}$ |  | this work
$J/\psi\rightarrow D_{s}\mu\nu_{\mu}$ | $1.84(8)\times 10^{-10}$ |  | this work
$J/\psi\rightarrow De\nu_{e}$ | $1.21(11)\times 10^{-11}$ |  | this work
$J/\psi\rightarrow D\mu\nu_{\mu}$ | $1.18(11)\times 10^{-11}$ |  | this work
Table 8:  Upper limits of the branching fraction of $J/\psi\rightarrow
D/D_{s}l\nu_{l}$ from BESIII experiments and the branching fractions(Br)
predicted by lattice QCD.

##  V Conclusion

In this work, we present the first lattice QCD calculation on the semileptonic
decay of $J/\psi$ particle. The weak decay $J/\psi\rightarrow D_{s}l\nu_{l}$
and $J/\psi\rightarrow Dl\nu_{l}$ are studied respectively. The (2+1)-flavor
Wilson-clover gauge ensembles with $m_{\pi}\sim 300$ MeV are utilized. After a
native continuum limit under three lattice spacings, we finally obtain the
lattice prediction $\operatorname{Br}(J/\psi\rightarrow
D_{s}e\nu_{e})=1.90(6)(5)_{V_{cs}}\times 10^{-10}$ and
$\operatorname{Br}(J/\psi\rightarrow De\nu_{e})=1.21(6)(9)_{V_{cd}}\times
10^{-11}$ , where the first errors are statistical errors, and the seconds
come from the uncertainties of CKM matrix element $V_{cs(d)}$. Given the form
factors of $J/\psi\rightarrow D$ and $J/\psi\rightarrow D_{s}$, the ratios of
the branching fractions between lepton $\mu$ and $e$ are computed immediately.
The values of the ratios are given by $R_{J/\psi}(D_{s})=0.97002(8)$ and
$R_{J/\psi}(D)=0.97423(15)$ with only the statistical errors included.

Combining with the exact experimental measurement in the future, the lepton
flavor universality can be checked carefully and the CKM matrix element
$V_{cs(d)}$ can also be extracted, thus providing an alternative precision
test for the standard model. The lattice scheme adopted in this work can be
directly applied to other $P\rightarrow V$ semileptonic decays, for example,
$D\rightarrow K^{*}$ [38], $B\rightarrow K^{*}$ [39], and $B\rightarrow D^{*}$
[40].

###### Acknowledgements.

We thank the CLQCD collaborations for providing us their gauge configurations
with dynamical fermions [26], which are generated on HPC Cluster of ITP-CAS,
the Southern Nuclear Science Computing Center(SNSC), the Siyuan-1 cluster
supported by the Center for High Performance Computing at Shanghai Jiao Tong
University and the Dongjiang Yuan Intelligent Computing Center. The
calculations were performed using the Chroma software suite [41] with QUDA
[42, 43, 44] through HIP programming model [45]. The authors are supported by
NSFC of China under Grant No.12293060, No.12293063, No.12293062, No. 12305094,
No.11935017, and the National Key R&D Program of China No.2021YFB0300203. The
numerical calculations are supported by the SongShan supercomputer at the
National Supercomputing Center in Zhengzhou.

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##  Appendix A Coefficients of $c_{0},c_{1}$, and $c_{2}$

$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{m^{4}}\Big{[}-8E^{3}F_{0}^{2}m^{2}M+4E^{3}F_{3}m_{l}^{2}(2F_{2}m+F_{3}M)+4E^{3}M(F_{2}m+F_{3}M)^{2}+4E^{2}F_{0}^{2}m^{2}(m^{2}+3M^{2}+m_{l}^{2})$
$\displaystyle+$ $\displaystyle
2E^{2}m_{l}^{2}(m^{2}(F_{2}^{2}+F_{3}^{2}+4F_{1}F_{3})-2F_{3}F_{2}mM-2F_{3}^{2}M^{2})-2E^{2}(F_{2}m+F_{3}M)(-4F_{1}m^{2}M+(F_{2}m+F_{3}M)(m^{2}+M^{2}))$
$\displaystyle-$ $\displaystyle
2E^{2}F_{3}^{2}m_{l}^{4}-4Em^{2}m_{l}^{2}(2F_{0}^{2}M+F_{2}m(F_{1}+2F_{3})+F_{3}M(3F_{1}+F_{3}))-8EF_{0}^{2}m^{2}M^{3}-4EF_{1}^{2}m^{4}M$
$\displaystyle-$ $\displaystyle
4EF_{1}F_{2}m^{5}-12EF_{1}F_{2}m^{3}M^{2}-4EF_{1}F_{3}m^{4}M-12EF_{1}F_{3}m^{2}M^{3}-4EF_{2}^{2}m^{4}M-8EF_{2}F_{3}m^{3}M^{2}$
$\displaystyle-$ $\displaystyle
4EF_{3}^{2}m^{2}M^{3}+2F_{0}^{2}m^{2}((M^{2}+m_{l}^{2})^{2}-m^{4})+2m^{2}(m^{2}+M^{2})(F_{1}^{2}m^{2}+(2F_{1}M+F_{2}m+F_{3}M)(F_{2}m+F_{3}M))$
$\displaystyle-$ $\displaystyle
2m^{2}m_{l}^{2}(m^{2}((F_{1}+F_{3})^{2}+F_{2}^{2})-2F_{2}mM(F_{1}+F_{3})-2F_{3}M^{2}(2F_{1}+F_{3}))+2F_{3}m^{2}m_{l}^{4}(2F_{1}+F_{3})\Big{]}$
$\displaystyle+$
$\displaystyle\frac{8}{m}F_{0}F_{1}\left((E-M)(2ME-M^{2}-m^{2}-m_{l}^{2})\right)$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{m^{4}}\Big{[}-8E^{3}(F_{2}m+F_{3}M)^{2}-8E^{2}\big{(}2F_{0}^{2}m^{2}M+2F_{1}m^{2}(F_{2}m+F_{3}M)-(F_{2}m+F_{3}M)(F_{2}mM+F_{3}(M^{2}+m_{l}^{2}))\big{)}$
$\displaystyle+$ $\displaystyle
8Em^{2}\big{(}F_{0}^{2}(m^{2}+3M^{2}+m_{l}^{2})-F_{1}^{2}m^{2}+4F_{1}M(F_{2}m+F_{3}M)+F_{1}F_{3}m_{l}^{2}+(F_{2}m+F_{3}M)^{2}\big{)}$
$\displaystyle+$ $\displaystyle
8m^{2}M\big{(}-F_{0}^{2}(m^{2}+M^{2})+F_{1}^{2}m^{2}-2F_{1}M(F_{2}m+F_{3}M)-(F_{2}m+F_{3}M)^{2}\big{)}$
$\displaystyle-$ $\displaystyle
8m^{2}m_{l}^{2}(M(F_{0}^{2}+F_{3}(2F_{1}+F_{3}))+F_{2}m(F_{1}+F_{3}))\Big{]}+\frac{16}{m}F_{0}F_{1}(2ME-M^{2}-m^{2})$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{m^{4}}\Big{[}-8E^{2}(F_{2}m+F_{3}M)^{2}-16Em^{2}(F_{0}^{2}M+F_{1}F_{2}m+F_{1}F_{3}M)$
$\displaystyle+$ $\displaystyle
8m^{2}\big{(}F_{0}^{2}(m^{2}+M^{2})-F_{1}^{2}m^{2}+2F_{1}M(F_{2}m+F_{3}M)+(F_{2}m+F_{3}M)^{2}\big{)}\Big{]}$

##  Appendix B Scalar function $I_{i}$

##  Appendix C Finite-volume effects

Figure 7: For ensemble F32P30, $\tilde{I}_{i}^{D_{s}}(\vec{x},t)$(left) and
$\tilde{I}_{i}^{D}(\vec{x},t)$(right) as a function of the spatial range
truncation $|\vec{x}|$ for t=1.16 fm, here $i=0,1,2,3,4,5$. The p=1 denotes
the projected momentum $|\vec{p}|=2\pi/L$.

The hadronic function $H_{\mu\nu}(x)$ is dominated by the $D_{s}$ or $D$ state
at large distance, therefore the size of the integrand for those scalar
functions is exponentially suppressed when $|\vec{x}|$ becomes large. We can
introduce a spatial integral truncation parameter $|\vec{x}|$ in this space
integral, and study its dependence. Three volumes of the ensemble are 2.5272
fm, 2.4787 fm, 2.4898 fm for L=24, 32 and 48, respectively. The smallest is
L=32. We therefore take this ensemble as an illustration. There are several
structures that appear in scalar functions, as given below

$\displaystyle\tilde{I}_{0}$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu\alpha 0}x_{\alpha}V_{\mu\nu}(\vec{x},t)$
$\displaystyle\tilde{I}_{1}$ $\displaystyle=$
$\displaystyle\delta_{ij}A_{ij}(\vec{x},t)$ $\displaystyle\tilde{I}_{2}$
$\displaystyle=$ $\displaystyle A_{00}(\vec{x},t)$
$\displaystyle\tilde{I}_{3}$ $\displaystyle=$ $\displaystyle
x_{i}A_{0i}(\vec{x},t)$ $\displaystyle\tilde{I}_{4}$ $\displaystyle=$
$\displaystyle x_{i}A_{i0}(\vec{x},t)$ $\displaystyle\tilde{I}_{5}$
$\displaystyle=$
$\displaystyle\frac{j_{2}(|\vec{p}||\vec{x}|)}{(|\vec{p}||\vec{x}|)^{2}}x_{i}x_{j}A_{ij}(\vec{x},t)$
(27)

The space integrals of these structures at a truncation parameter $|\vec{x}|$
are shown in Fig. 7. As far as the scalar structures in this work are
concerned, there all exist plateaus for $|\vec{x}|\gtrsim 1.5$ fm, indicating
that the hadronic function has negligible contribution beyond this range.
Hence, the finite-volume effects should be well-controlled in our
calculations.