# Explicit String bundles††thanks: orcid.org/0000-0002-3478-0522
This document is released under a CC0 license:
creativecommons.org/publicdomain/zero/1.0/.

David Michael Roberts

(1 July 2014)

These notes are the written version of a talk delivered at Herriot-Watt
University on 26 June 2014 at the Workshop on Higher Gauge Theory and Higher
Quantization. DMR is supported by ARC grant number DP120100106.
Thanks to David Baraglia, Michael Murray, Christian Saemann and Raymond Vozzo
for helpful converstations. If we look back at the historical development of
bundles, the notion of a principal $H$-bundle for $H$ a Lie group arose via
considerations of homogeneous spaces $G/H$, and the defining bundle $G\to
G/H$. Here $H$ is a closed subgroup of $G$, and this will be a general
assumption for the talk. For instance, we can conside Stiefel manifolds,
Grassmann manifolds, projective spaces, Minkowski space, spheres,…. I am going
to leverage this in order to address the

######  Challenge.

Write down a (nontrivial) 2-bundle. Equivalently, write down a Čech cocycle
with values in an interesting crossed module $(K\xrightarrow{t}H,H\times
K\xrightarrow{a}K)$.

Recall [Bre94] that the cocycle equations are

$\displaystyle h_{ij}^{\alpha}h_{jk}^{\beta}=$ $\displaystyle
t(k_{ijk}^{\alpha\beta\gamma})h_{ik}^{\gamma}$ $\displaystyle
t(h_{ij}^{a},k_{jkl}^{\beta\lambda\varepsilon})k_{ijl}^{\alpha\varepsilon\delta}=$
$\displaystyle k_{ijk}^{\alpha\beta\gamma}k_{ikl}^{\gamma\lambda\delta}.$

where the $h_{ij}^{\alpha}$ are $H$-valued functions, the
$k_{ijk}^{\alpha\beta\gamma}$ are $K$-valued functions and the two sorts of
indices label open sets of the base space. We shall return to this
momentarily. Note that at this point we haven’t even started to consider
connections, which are necessary for gauge theory (and in fact we won’t even
go so far today).

######  Note.

I am _not_ going to use good open covers (that is, those such that non-empty
finite intersections are contractible), since in many geometric situations
there are naturally arising open covers that are not good. Instead, I will be
using _truncated globular hypercovers_ (these are open covers with particular
properties), and I will define these in a moment. For now, suffice it to say,
this is why there are two different sorts of indices on the cocycle.

Christian Saemann asked (Feb 2013): I want a 2-bundle on (conformally
compactified)111recall that this is diffeomorphic to $S^{5}\times S^{1}$
$\mathbb{R}^{5,1}$. So let’s try lifting the frame bundle of $S^{5}\times
S^{1}$ to a _String_ bundle. Note that the $S^{1}$ factor contributes nothing
(its frame bundle is trivial) so just work over $S^{5}$. Note that the frame
bundle of $S^{5}$ is most definitely not trivial.

The frame bundle $FS^{5}\to S^{5}$ is classified by a map $S^{5}\supset
S^{4}\to SO(5)$, called the _clutching_ or _transition function_. Since
$S^{5}$ is 4-connected, the first Stiefel-Whitney class $w_{1}$ necessarily
vanishes, as does the characteristic class $p_{1}/2\in
H^{4}(S^{5},\mathbb{Z})$ that is the obstruction to lifting to a String
bundle. Thus we can be assured that the lift we are after does exist. From the
vanishing of $w_{1}$ we know the transition function lifts to a function
$S^{4}\to\operatorname{Spin}(5)$, and so defines a class in
$\pi_{4}(\operatorname{Spin}(5))$, which is the group $\mathbb{Z}/2\mathbb{Z}$
[MT64]. Since $FS^{5}$ is not trivial, the transition function needs to
represent the non-trivial homotopy class. We want to write down an explicit
function in coordinates, rather than use some abstract representative.

To approach this, we first use the exceptional isomorphism
$\operatorname{Spin}(5)\simeq Sp(2)$, where $Sp(2)$ is the group of $2\times
2$ unitary quaternionic matrices. The non-trivial class in $\pi_{4}(Sp(2))$ is
represented by a map $S^{4}\to S^{3}\simeq Sp(1)\hookrightarrow Sp(2)$ and
here $Sp(1)$ is the group of unit quaternions. The map between spheres is (up
to homotopy) the suspension of the Hopf map $S^{3}\to S^{2}$, which is not a
priori a smooth map, and the inclusion is
$q\mapsto\left(\begin{array}[]{cc}q&0\\\ 0&1\end{array}\right)$. Note that
this implies that $FS^{5}$ lifts to an $Sp(1)$-bundle, and this is what we
shall assue without further comment.

The first task is then to write down a smooth, non-null-homotopic smooth map
$S^{4}\to Sp(1)$. We shall use quaternionic coordinates on
$S^{4}=\mathbb{HP}^{1}$, that is, homogeneous coordinates $[p;q]$ where at
least one of $p$, $q$ is non-zero.

######  Proposition.

The smooth function

$T[p;q]=\frac{2p\bar{q}i\bar{p}q-|p|^{4}+|q|^{4}}{|p|^{4}+|q|^{4}}\quad(\in
Sp(1))$

represents the non-trivial class of $\pi_{4}(Sp(1))$, and hence is the
transition function for $FS^{5}$.

Now we want to shift perspective a little bit, and note that the function $T$
gives rise to a smooth _functor_ from the Čech groupoid
$U\times_{S^{5}}U\rightrightarrows U$ over $S^{5}$ coming from the open cover
by two discs222One should take these as open discs, and so the intersection
would be $S^{4}\times(-\varepsilon,\varepsilon)$; we extend $T$ to this
slightly larger subspace by taking it constant in the direction of the
interval. $U:=D_{+}\coprod D_{-}\to S^{5}$. For future notational convenience,
write $U^{[2]}=U\times_{S^{5}}U$.

Since we now have an explicit Čech cocycle (this is precisely what the above
functor is) for $FS^{5}$, we can talk about lifting this to a Čech cocycle for
the 2-group $\operatorname{String}_{Sp(1)}$. But what _is_ this? There are
many models for String 2-groups, and we shall take the crossed module
$(\widehat{\Omega Sp(1)}\to PSp(1))$, where $PSp(1)$ is the group of smooth
paths $[0,1]\to Sp(1)$ based at $1\in Sp(1)$, and $\widehat{\Omega Sp(1)}$ is
the universal central extension of the subgroup $\Omega Sp(1)\subset PSp(1)$
of loops [Bae+07]. Note that the abstract details of what I’m considering
doesn’t rely on this choice of model. Notice that $(\widehat{\Omega Sp(1)}\to
PSp(1))$ comes with a map to the crossed module $(1\to Sp(1))$, and that the
former gives rise to a groupoid (which I shall call
$\operatorname{String}(3)$, as $Sp(1)\simeq\operatorname{Spin}(3)$), namely
the action groupoid for $\widehat{\Omega Sp(1)}$ acting on $PSp(1)$ via the
given homomorphism, and a 2-groupoid $\mathbf{B}\operatorname{String}(3)$ with
a single object (using the 2-group structure). More generally, we can repeat
these constructions with any compact, simple, simply connected Lie group $G$
to get a 2-group $\operatorname{String}_{G}$. Also, given an inclusion of Lie
groups333that induces an isomorphism $H^{3}(G,\mathbb{Z})\to
H^{3}(H,\mathbb{Z})$; the examples listed below all satisfy this, as can be
calculated via the long-exact sequence in homotopy. $H\to G$ gives an
inclusion of Lie 2-groups
$\operatorname{String}_{H}\to\operatorname{String}_{G}$.

In the Čech groupoid $U^{[2]}\rightrightarrows U$ we don’t have $U^{[2]}$ a
disjoint union of contractible opens, so we take an open cover $V\to U^{[2]}$
where $V$ _is_ such a disjoint union (or, at least, acyclic enough). Since the
non-trivial part of $U^{[2]}$ is $D_{+}\cap D_{-}\sim\mathbb{HP}^{1}$, we will
take $V$ to be the two $\mathbb{H}$ charts $\mathbb{H}_{+}$ and
$\mathbb{H}_{-}$ given by non-vanishing of each of the two homogenous
coordinates. Then if we take the fibred product $V^{[2]}=V\times_{U^{[2]}}V$
we get a Lie 2-groupoid $V^{[2]}\rightrightarrows V\rightrightarrows U$, which
I call a _truncated globular hypercover_.444This may look familiar if you’ve
seen bundle 2-gerbes [Ste04] before. The nontrivial component of $V^{[2]}$ (it
contains boring bits like $D_{+}$) is the intersection
$\mathbb{H}_{+}\cap\mathbb{H}_{-}=\mathbb{H}^{\times}$. Notice that if we
wanted to use a good open cover then $U$ would necessarily have had more open
sets, and so more overlaps. In some sense we have made a trade-off in the
number of open sets and the slight increase in complexity of the description.
Also, we can finally see where the two sorts of indices in the cocycle
equation above come from: the indices $i,j,\ldots$ label open sets appearing
in $U$, and the indices $\alpha,\beta,\ldots$ label the open sets appearing in
$V$.

So, finally, a Čech cocycle on $S^{5}$ with values in
$\operatorname{String}(3)$ is ‘just’ a 2-functor

$(V^{[2]}\rightrightarrows V\rightrightarrows
U)\to\mathbf{B}\operatorname{String}(3).$

If we break this down, it is determined by components

$\displaystyle V\to PSp(1)$ $\displaystyle V^{[2]}\to\widehat{\Omega PSp(1)}$

and since we have so few open sets in the globular hypercover, functoriality
follows automatically. In our particular case, we want the first map to lift
the given $V\to U^{[2]}\to Sp(1)$.

Recall that $V$ is (essentially) $\mathbb{H}_{+}\coprod\mathbb{H}_{-}$, we
define the lift in two parts:

$\displaystyle T_{+}(q)$
$\displaystyle=\left(s\mapsto\frac{|q|^{4}-s^{2}+2\bar{q}iq}{|q|^{4}+s^{2}}\right)$
$\displaystyle T_{-}(p)$
$\displaystyle=\left(s\mapsto\frac{|p|^{4}s^{2}-1+2\bar{p}ip}{|p|^{4}s^{2}+1}\cdot\left(\frac{s-i}{s+i}\right)^{2}\right)$

To define the remaining component of the 2-functor, we first take the
difference of these two maps to get a function $\mathbb{H}^{\times}\to\Omega
Sp(1)$

$T_{\Omega}(q)=\left(s\mapsto\frac{(s+Q)(sQ-1)}{(s-Q)(sQ+1)}\cdot\left(\frac{s-i}{s+i}\right)^{2}\right),\quad\text{where
}Q=\bar{q}iq.$

Now we need to lift this map through the projection $\widehat{\Omega
Sp(1)}\to\Omega Sp(1)$ (this is not a priori possible, but one calculates the
possible obstructions and they vanish). To do this, we need a workable
description of what $\widehat{\Omega Sp(1)}$ is. There are multiple papers
constructing this e.g. [Mic87, Mur88, MS03]. We shall use the description of
it as the quotient group

$\frac{P\Omega Sp(1)\rtimes U(1)}{\widetilde{\Omega^{2}Sp(1)}}$

The precise embedding of the simply-connected covering group
$\widetilde{\Omega^{2}Sp(1)}$ is not important, just that we can represent
elements as equivalence classes of pairs consisting of paths in $\Omega Sp(1)$
and elements of $U(1)$.

One calculates the final answer to be as follows. For any
$q\in\mathbb{H}^{\times}$, let $q_{t}$ be any path (in $\mathbb{H}^{\times}$)
$1\rightsquigarrow q$, and the lift to the central extension is

$T_{\widehat{\Omega}}(q)=[T_{\Omega}(q_{t}),1].$

This is independent of the choice of path and is smooth. This function,
together with $T_{\pm}$, defines the Čech cocycle we are interested in. We
know that this cocycle is not a coboundary, since geometrically realising
everything we get a map $S^{5}\to B\operatorname{String}(3)$ that picks out
the nontrivial class in
$\pi_{5}(B\operatorname{String}(3))\simeq\pi_{5}(B\operatorname{Spin}(3))\simeq\pi_{4}(\operatorname{Spin}(3))\simeq\pi_{4}(Sp(1))=\mathbb{Z}/2\mathbb{Z}$.
One can also check (easily, as there are so few open sets involved in the open
covers), that these functions satisfy the cocycle equations displayed at the
beginning of the notes.

Now this is just one example, and a pretty exceptional example at that, as the
dimensions involved are right on the boundary of where the obstructions
vanish, not to mention the use of quaternions. One can take a more global
approach that leads to many more examples as follows. The total space of the
frame bundle $FS^{5}$, as an $Sp(1)$-bundle, is nothing other than the
homogenous bundle $SU(3)\to SU(3)/Sp(1)=S^{5}$, using the embedding
$Sp(1)\simeq SU(2)\to SU(3)$ as a block matrix. One can calculate that
$\operatorname{String}_{SU(3)}/\operatorname{String}(3)\simeq SU(3)/Sp(1)$, so
that the underlying groupoid of $\operatorname{String}_{SU(3)}$ is the ‘total
space’ of the $\operatorname{String}(3)$ bundle. Another way to view this is
to consider the transitive $\operatorname{String}_{SU(3)}$ action on $S^{5}$
via the projection to $SU(3)$; then $\operatorname{String}(3)$ is the
stabiliser of the basepoint.

This picture generalises to any $\operatorname{String}_{G}$ acting on $G/H$
for $H<G$, and at this point we can use any model of
$\operatorname{String}_{G}$, including non-strict models, and even 2-groups in
differentiable stacks, which have underlying Lie groupoids. There are a number
of interesting exceptional examples which should be amenable to the same
treatment as above, for instance:

  * •

$\operatorname{String}_{G_{2}}\to G_{2}/SU(3)=S^{6}$

  * •

$\operatorname{String}_{\operatorname{Spin}(7)}\to\operatorname{Spin}(7)/G_{2}=S^{7}$

  * •

$\operatorname{String}_{Sp(2)}\to Sp(2)/Sp(1)=S^{7}$

  * •

$\operatorname{String}_{F_{4}}\to
F_{4}/\operatorname{Spin}(9)=\mathbb{OP}^{2}$

The first three of these have explicit transition functions written down by
Püttmann in [Püt11]. $\mathbb{OP}^{2}$ admits a cover by three
$\mathbb{R}^{16}$ charts, and is 7-connected.

######  Exercise.

Write down transition functions for the $\operatorname{Spin}(9)$ bundle on
$\mathbb{OP}^{2}$, and lift them to
$\operatorname{String}(9)=\operatorname{String}_{\operatorname{Spin}(9)}$-valued
transition functions using a globular hypercover.

The astute reader will have realised that this method only gives a single
example on each homogeneous space with that particular structure group, which
in the case of $S^{5}$ is ok as there is only one nontrivial
$\operatorname{String}(3)$ bundle. But, for instance,
$\operatorname{String}_{SU(3)}$ bundles on $S^{6}$ are classified by an
integer (and in fact the example above is a generator). However, using the
Eckmann-Hilton argument, one can show that over a sphere $S^{k+1}$, given a
$G$-bundle with transition function $t\colon S^{k}\to G$ representing a
generator of $g\in\pi_{k}(G)$, we can obtain the transition functions for the
bundles corresponding to elements $g^{n}$ by taking the pointwise power
$t^{n}\colon S^{k}\to G$ for any $n\in\mathbb{Z}$. The same will be true for
the lifted 2-bundles, where we take pointwise powers of the 2-group-valued
functor $(V^{[2]}\rightrightarrows V)\to\operatorname{String}_{H}$. Thus, for
spheres at least, we can in principle give Čech cocycle descriptions for all
String bundles.

As a final note, the abstract picture in the penultimate paragraph is not
restricted to smooth geometry: one can equally well take holomorphic 2-groups,
assuming one has them. However, in current work with Raymond Vozzo we have
found that the basic gerbe on a simple, simply-connected complex reductive Lie
group, which is holomorphic [Bry94, Bry00], is also multiplicative, so defines
a weak 2-group in complex analytic stacks. This means we can define
holomorphic String bundles on complex homogeneous spaces, which can be plugged
into the higher twistor correspondence of Saemann-Wolf (eg [SW14]).

## References

  * [Bae+07] John C. Baez, Danny Stevenson, Alissa S. Crans and Urs Schreiber  “From loop groups to 2-groups”  In _Homology, Homotopy and Applications_ 9.2, 2007, pp. 101–135  URL: https://doi.org/10.4310/HHA.2007.v9.n2.a4
  * [Bre94] Lawrence Breen  “On the classification of 2-gerbes and 2-stacks”, Astérisque 225  Société mathématique de France, 1994  URL: http://www.numdam.org/item/AST_1994__225__1_0/
  * [Bry00] Jean-Luc Brylinski  “Gerbes on complex reductive Lie groups”, 2000  URL: https://arXiv.org/abs/math/0002158
  * [Bry94] Jean-Luc Brylinski  “Holomorphic gerbes and the Beilinson regulator”  In _$K$ -theory - Strasbourg, 1992_, Astérisque 226  Société mathématique de France, 1994  URL: http://www.numdam.org/item/AST_1994__226__145_0/
  * [Mic87] Jouko Mickelsson  “Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization”  In _Communications in Mathematical Physics_ 110.2, 1987, pp. 173–183  URL: https://doi.org/10.1007/BF01207361
  * [MS03] M.. Murray and D. Stevenson  “Higgs fields, bundle gerbes and string structures”  In _Comm. Math. Phys._ 243.3, 2003, pp. 541–555  URL: https://doi.org/10.1007/s00220-003-0984-4
  * [MT64] Mamoru Mimura and Hirosi Toda  “Homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$”  In _J. Math. Kyoto Univ_ 3.2, 1964, pp. 217–250  URL: https://doi.org/10.1215/kjm/1250524818
  * [Mur88] M.. Murray  “Another construction of the central extension of the loop group”  In _Comm. Math. Phys._ 116.1, 1988, pp. 73–80  URL: https://doi.org/10.1007/BF01239026
  * [Püt11] Thomas Püttmann  “On the commutator of unit quaternions and the numbers 12 and 24”, 2011  URL: https://arXiv.org/abs/1101.5147
  * [Ste04] D. Stevenson  “Bundle 2-gerbes”  In _Proc. Lond. Math. Soc._ 88.3, 2004, pp. 405–435  URL: https://doi.org/10.1112/S0024611503014357
  * [SW14] Christian Sämann and Martin Wolf  “Non-Abelian Tensor Multiplet Equations from Twistor Space”  In _Communications in Mathematical Physics_ 328.2, 2014, pp. 527–544  URL: https://doi.org/10.1007/s00220-014-2022-0


# Few-Shot Domain Adaptation for Grammatical Error Correction
via Meta-Learning

Shengsheng Zhang1,2, Yaping Huang1, Yun Chen3, Liner Yang2,
Chencheng Wang4, Erhong Yang2
1Beijing Jiaotong University, Beijing, China
2Beijing Language and Culture University, Beijing, China
3Shanghai University of Finance and Economics, Shanghai, China
4Beijing University of Technology, Beijing, China

###### Abstract

Most existing Grammatical Error Correction (GEC) methods based on sequence-to-
sequence mainly focus on how to generate more pseudo data to obtain better
performance. Few work addresses few-shot GEC domain adaptation. In this paper,
we treat different GEC domains as different GEC tasks and propose to extend
meta-learning to few-shot GEC domain adaptation without using any pseudo data.
We exploit a set of data-rich source domains to learn the initialization of
model parameters that facilitates fast adaptation on new resource-poor target
domains. We adapt GEC model to the first language (L1) of the second language
learner. To evaluate the proposed method, we use nine L1s as source domains
and five L1s as target domains. Experiment results on the L1 GEC domain
adaptation dataset demonstrate that the proposed approach outperforms the
multi-task transfer learning baseline by 0.50 $F_{0.5}$ score on average and
enables us to effectively adapt to a new L1 domain with only 200 parallel
sentences.

##  1 Introduction

Grammatical Error Correction (GEC) aims to correct errors in text. For
example, “He notice the picture.” can be corrected to “He notices the
picture.”. A GEC system takes an incorrect sentence as input and outputs the
corresponding correct sentence. With the development of deep learning, GEC has
drawn the attention of many researchers during the last few years.

Most existing methods (Chollampatt and Ng, 2018; Junczys-Dowmunt et al., 2018;
Zhao et al., 2019) frame GEC as a sequence-to-sequence (seq2seq) task and have
obtained high performance on the general domain while using a large number of
training examples. However, these seq2seq-based models cannot gain
satisfactory performance in special GEC domains due to domain shift and the
limited in-domain data. For instance, Nadejde and Tetreault (2019) use the GEC
model trained on the general domain to test on specific domains and find that
the performance drops dramatically. One way to tackle this issue is transfer
learning (Nadejde and Tetreault, 2019), in which a GEC model is pretrained on
the high-resource general domain and then fine-tuned on a low-resource target
domain. Although leading to empirical improvements in the target domain, this
method suffers from model over-fitting and catastrophic forgetting when the
in-domain data is insufficient Sharaf et al. (2020).

Figure 1: Difference between multi-task transfer learning and meta learning.
Solid lines denote the learning of initial parameters and dashed lines are the
path of fine-tuning. Pink and gray represent the source task and target task,
respectively.

In this paper, we frame GEC system for different domains as different tasks
and propose a meta-learning method for few-shot GEC domain adaptation.
Specifically, we use model-agnostic meta-learning algorithm (MAML; Finn et al.
(2017)) to learn the initialization of model parameters from high-resource
domains, which can quickly adapt to a new target domain with a minimal amount
of data. Fig.1 shows the difference between our method and the multi-task
transfer learning method in Nadejde and Tetreault (2019). Their method first
trains GEC model on multi-domain data and then fine-tunes it on a target
domain.

To evaluate the proposed method, we adapt GEC model to Chinese as a Second
Language (CSL) learner’s first language (L1). We construct a few-shot GEC
domain adaptation dataset by making use of 4 resource-poor L1s as the test
domains and the rest 10 L1s as the source and valid domains. Our experiments
on the constructed dataset show that our method can effectively adapt to a new
domain using only 200 parallel sentences and outperform the multi-task
transfer learning method by 0.50 $F_{0.5}$ score on average. To our best
knowledge, we are the first to apply meta-learning to GEC.

##  2 Method

###  2.1 GEC Domain Adaptation

Given an erroneous sentence $X=\\{x_{1},...,x_{M}\\}$ and a learner’s domain
$d$, a Neural Machine Translation (NMT)-based model for domain-aware GEC
models the conditional probability of the output sentence
$Y=\\{y_{1},...,y_{N}\\}$ with neural networks as follows:

$p(Y|X,d;\theta)=\prod_{t=1}^{N}p(y_{t}|y_{1:t-1},x_{1:M},d;\theta),$ (1)

where $\theta$ is a set of model parameters. Following Madotto et al. (2019),
we first adapt $\theta$ to the learner’s domain $d$ and then model the output
sentence conditional on the erroneous input sentence with:

$p(Y|X;\theta_{d})=\prod_{t=1}^{N}p(y_{t}|y_{1:t-1},x_{1:M};\theta_{d}),$ (2)

where $\theta_{d}$ is the set of domain-aware model parameters. A learner’s
domain can be defined with different criterion, such as the L1 and the
proficiency level. In this paper, we use the L1 as the criterion and adapt a
GEC system to the learner’s L1. Since our method is agnostic to the definition
of domains, it can be easily extended to other type of domain-aware GEC
systems.

###  2.2 Few-Shot GEC Domain Adaptation via Meta Learning

We propose to apply the model-agnostic meta-learning (MAML; Finn et al. (2017)
in few-shot GEC domain adaptation. We use MAML to learn a good initialization
of model parameters $\theta^{0}$, which can quickly adapt to new domains using
few training examples. We call the proposed meta-learning method for GEC
domain adaptation as MetaGEC.

We define a set of source tasks
$\mathscr{T}=\\{\mathcal{T}_{d_{1}},...,\mathcal{T}_{d_{k}}\\}$, where each
task $\mathcal{T}_{d_{i}}$ is a GEC system of a specific domain $d_{i}$ and
$k$ is the number of learner’s domains. For each meta-learning episode, we
randomly sample a task $\mathcal{T}_{d_{i}}$ from $\mathscr{T}$. Then we
sample two batches independently from task $\mathcal{T}_{d_{i}}$’s data, a
support batch $D_{d_{i}}^{s}$ and a query batch $D_{d_{i}}^{q}$. We first use
$D_{d_{i}}^{s}$ to update the GEC model parameters $\theta$ as follows:

$\theta^{{}^{\prime}}_{d_{i}}=\theta-\alpha\nabla_{\theta}\mathcal{L}_{D_{d_{i}}^{s}}(\theta),$
(3)

where $\alpha$ is the learning rate and $\mathcal{L}$ is the cross-entropy
loss function:

$\mathcal{L}_{D_{d_{i}}^{s}}(\theta)=-\sum_{D_{d_{i}}^{s}}\log p(Y|X,\theta).$
(4)

After that, we evaluate the updated parameters $\theta_{d_{i}}^{{}^{\prime}}$
on $D_{d_{i}}^{q}$ and update the original model parameters $\theta$ with
gradient computed from this evaluation. It is possible to aggregate multiple
episodes of source tasks before updating $\theta$. Therefore the original
model parameters $\theta$ are updated as follows:

$\theta=\theta-\beta\sum_{d_{i}}\nabla_{\theta}\mathcal{L}_{D_{d_{i}}^{q}}(\theta_{d_{i}}^{{}^{\prime}}),$
(5)

where $\beta$ is the meta learning rate. The full algorithm is shown in
Algorithm 1.

Algorithm 1 Meta learning for few-shot GEC domain adaptation

Require: $\mathscr{T}$: set of source tasks
Require: $\alpha,\beta$: step size hyperparameters

1:Randomly initialize $\theta$

2:while not done do

3: Sample batch of tasks $\mathcal{T}_{d_{i}}\sim\mathscr{T}$

4: for all $\mathcal{T}_{d_{i}}$ do

5: $(D_{d_{i}}^{s},D_{d_{i}}^{q})\sim D_{d_{i}}$

6: Evaluate$\nabla_{\theta}\mathcal{L}_{D_{d_{i}}^{s}}(\theta)$ using
$D_{d_{i}}^{s}$

7: Compute adapted parameters with gra-

8: dient descent:
$\theta_{d_{i}}^{{}^{\prime}}=\theta-\alpha\nabla_{\theta}\mathcal{L}_{D_{d_{i}}^{s}}(\theta)$

9: end for

10: Update meta $\theta:$

11:
$\theta=\theta-\beta\sum_{d_{i}}\nabla_{\theta}\mathcal{L}_{D_{d_{i}}^{q}}(\theta_{d_{i}}^{\prime})$

12:end while

The update of meta parameters involves second-order partial derivatives, which
is computationally expensive. In our experiments, we use a first-order
approximation to save memory consumption following previous work Gu et al.
(2018).

Corpus | #Sentence | #SrcToken | #TgtToken
---|---|---|---
Lang-8 | 1.09M | 14M | 15M
HSK | 88K | 1.78M | 1.76M
Table 1: Data statistics for the Lang-8 and HSK datasets

After the meta-training phrase, task-specific learning is done on a small
amount of examples from a new target task $\mathcal{T}_{d}$, in order to
obtain a task-specific model $\theta_{d}$.

##  3 Experiments

###  3.1 Settings

Dataset We use two datasets in our experiments: Lang-8111https://lang-8.com
and HSK222http://hsk.blcu.edu.cn/. Both dataset are written by CSL learners
and corrected by Chinese native speakers. We tokenize the datasets by
jieba333https://github.com/fxsjy/jieba and apply Byte Pair Encoding Sennrich
et al. (2016) to limit vocabulary
size.444https://github.com/rsennrich/subword-nmt We first pretrain our model
on Lang-8 and then study GEC domain adaptation on the HSK dataset with the
pretrained model. Table 1 shows the statistics of both datasets. HSK consists
of examination essays written by CSL learners with fourteen different L1s.
First, we choose four domains with the least data as the test domains,
including German (De), Russian (Ru), French (Fr) and Mongolian (Mo). Then, we
randomly sample one domain from the rest domains as the valid domain, while
the other domains serve as the source domains. Specifically, we use Indonesian
(In) as the valid domain, and Korean (Ko), Traditional Chinese (Zh-tw),
Japanese (Ja), Singapore English (En-Sg), Malay (Ma), Burmese (Bu), Thai (Th),
Vietnamese (Vi) and English (En) as the source domains. For each source
domain, we sample 1000 parallel sentences as the in-domain dataset. For valid
domain, we sample 200, 800, and 400 parallel sentences as the in-domain
training set, development set, and test set respectively. For each test
domain, we sample 200 parallel sentences as the in-domain training set, and
divide the rest data in HSK into development set and test set according to a
two-to-one ratio. The valid and test domains are also called target domains.
We use the ERRANT555https://github.com/chrisjbryant/errant to make the gold
edits of grammatical errors in sentences of each test set.

GEC System

Target Task | No Fine-tuning | Fine-tuning | MTL+Fine-tuning | MetaGEC
---|---|---|---|---
In | 19.18 | 25.40 | 37.09 | 37.46
De | 24.43 | 30.03 | 37.76 | 39.43
Ru | 21.44 | 33.98 | 40.15 | 39.14
Fr | 29.10 | 35.48 | 43.19 | 43.49
Mo | 29.30 | 36.72 | 48.07 | 49.21
Average | 24.69 | 32.32 | 41.25 | 41.75
Table 2:  $F_{0.5}$ score on the test set of the target tasks, where In is the
valid task and all other L1s are the test tasks.

We utilize the Transformer (Vaswani et al., 2017) implemented by
fairseq666https://github.com/pytorch/fairseq as our GEC model. We follow the
model configure in transformer_wmt_en_de and set batch size to 4000 tokens.
For pretraining on Lang-8, we follow the training instructions in Ott et al.
(2018).777https://github.com/pytorch/fairseq/blob/v0.9.0/examples/scaling_nmt/README.md
For meta training, we use the same Adam optimizer except that we set lr=1e-5
for the outer loop and lr=1e-7 for the inner loop. At test time, we fine-tune
the model on the target task’s training set with lr=5e-4. For all models, we
translate with beam search using beam_size=12.

Baselines We compare MetaGEC with three baselines: (1) No Fine-tuning Sharaf
et al. (2020): the method that evaluates the pretrained GEC model on the
target task’s test set; (2) Fine-tuning Sharaf et al. (2020): the method that
fine-tunes the pretrained GEC model on the target task’s training data
directly; (3) MTL+Fine-tuning Nadejde and Tetreault (2019): the multi-task
transfer learning method we discussed in Section 1. It first fine-tunes the
pretrained GEC model on all data of the source tasks in a multi-task learning
framework, and then fine-tunes the resulting model on the target task’s
training data. As an evaluation metric, we use $F_{0.5}$ score computed by
applying the MaxMatch888https://www.comp.nus.edu.sg/~nlp/conll14st.html
($M^{2}$) scorer (Dahlmeier and Ng, 2012). We repeat the baselines and our
method three times with different seeds and report the averaged score.

###  3.2 Results

Table 2 shows the evaluation results of MetaGEC and the baselines. Overall,
MetaGEC outperforms all the baselines, improving No Fine-tuning, Fine-tuning
and MTL+Fine-tuning by 17.06, 9.43 and 0.50 $F_{0.5}$ on average. This
indicates that MetaGEC has successfully found a good initialization of model
parameters for fast domain adaptation. We also observe that for Ru, MetaGEC
performs worse than the baseline MTL+Fine-tuning. Ru benefits the most when
fine-tuning with in-domain data (No Fine-tuning to Fine-tuning) among all five
target tasks. In contrast, it benefits the least from multi-task learning
(Fine-tuning to MTL+Fine-tuning). We hypothesize that for Ru, fine-tuning with
in-domain data is more important than the way we choose to utilize the data
contained in source tasks. Since MetaGEC is different from MTL+Fine-tuning in
the way of utilizing data from the source tasks, our hypothesis also partially
explains the degraded performance of MetaGEC on Ru.

Figure 2: Impact of the number of source tasks. We report the averaged
$F_{0.5}$ score on the test sets of the five target tasks (In, De, Ru, Fr,
Mo).

To study the impact of the number of source tasks, we experiment with
different number of source tasks and report the averaged $F_{0.5}$ score on
the test sets of the five target tasks, as shown in Fig. 2. Note that we only
ran the experiments once here. We use Ko, Zh-tw, Ja, Ma and Bu as the source
tasks when the number of source tasks is 5, and gradually add Th, En-Sg, En
and Vi when increasing the number of source tasks from 5 to 9. We observe that
when including more source tasks at the meta training phase, we can obtain
better performance on the target tasks, demonstrating that better
initialization model can be learned with more source tasks.

##  4 Related Work

Grammatical Error Correction The traditional GEC approaches include two
categories: specific rule-based methods (Heidorn et al., 1982; Bustamante and
León, 1996) and statistical machine translation (SMT)-based approaches
(Brockett et al., 2006; Junczys-Dowmunt and Grundkiewicz, 2014). Specific
ruled-based methods only correct certain types of errors in the text. SMT-
based approaches greatly improve the performance of GEC. But they are
surpassed by deep learning-based methods. Junczys-Dowmunt et al. (2018) cast
GEC as a low-resource NMT task. Due to the limited public data, many works
(Lichtarge et al., 2019; Kiyono et al., 2019; Wang et al., 2019; Kaneko et
al., 2020) pay attention to how to generate more pseudo data to improve the
performance of neural GEC models.
GEC Domain Adaptation Rozovskaya and Roth (2011) use Naive Bayes classifier to
adapt a model to the L1 of the learner. Chollampatt et al. (2016) first train
a neural network joint model on the data labeled by L1 of the learner and then
integrate it into a SMT based GEC system. Nadejde and Tetreault (2019) utilize
transfer learning method to adapt a model to different domains.
Meta Learning Recently, meta-learning (Lake et al., 2015; Andrychowicz et al.,
2016; Finn et al., 2017) has attracted lots of attention. Meta-learning aims
at solving how to achieve fast adaption on new data. Current meta-learning
methods can be classified into two categories: 1) Learning strategies and
policies (Andrychowicz et al., 2016). 2) Learning good initial parameters of
model (Finn et al., 2017). Many works have applied meta-learning to Natural
Language Processing tasks, such as low-resource NMT (Gu et al., 2018),
personalizing dialogue agents (Madotto et al., 2019) and few-shot NMT
adaptation (Sharaf et al., 2020).

##  5 Conclusion

In this paper, we introduce MetaGEC, a model-agnostic meta-learning algorithm
for few-shot GEC domain adaptation. MetaGEC exploits a set of data-rich source
domains to learn the initialization of model parameters that facilitates fast
adaptation for a new target domain with a minimal amount of training examples.
Experiment results demonstrate the effectiveness of the proposed method. In
the future, we will apply different meta-learning methods in the GEC task.

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# Form factor for Dalitz decays from $J/\psi$ to light pseudoscalars

Chunjiang Shi<EMAIL_ADDRESS>Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, People’s Republic of China  Ying Chen<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China  School of Physical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, People’s Republic of China  Xiangyu Jiang
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China  CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
People’s Republic of China  Ming Gong Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, People’s Republic of China  Zhaofeng Liu Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, People’s Republic
of China  Center for High Energy Physics, Peking University, Beijing 100871,
People’s Republic of China  Wei Sun Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, People’s Republic of China

###### Abstract

We calculate the form factor $M(q^{2})$ for the Dalitz decay
$J/\psi\to\gamma^{*}(q^{2})\eta_{(N_{f}=1)}$ with $\eta_{(N_{f})}$ being the
SU($N_{f}$) flavor singlet pseudoscalar meson. The difference among the
partial widths $\Gamma(J/\psi\to\gamma\eta_{(N_{f})})$ at different $N_{f}$
can be attributed in part to the $\mathbf{U}_{A}(1)$ anomaly that induces a
$N_{f}$ scaling. $M(q^{2})$’s in $N_{f}=1,2$ are both well described by the
single pole model $M(q^{2})=M(0)/(1-q^{2}/\Lambda^{2})$. Combined with the
known experimental results of the Dalitz decays $J/\psi\to Pe^{+}e^{-}$, the
pseudoscalar mass $m_{P}$ dependence of the pole parameter $\Lambda$ is
approximated by $\Lambda(m_{P}^{2})=\Lambda_{1}(1-m_{P}^{2}/\Lambda_{2}^{2})$
with $\Lambda_{1}=2.64(4)~{}\mathrm{GeV}$ and
$\Lambda_{2}=2.97(33)~{}\mathrm{GeV}$. These results provide inputs for future
theoretical and experimental studies on the Dalitz decays $J/\psi\to
Pe^{+}e^{-}$.

##  I Introduction

The electromagnetic (EM) Dalitz decay of a hadron $A$, namely, $A\to
B\gamma^{*}\to Bl^{+}l^{-}$, refers to the decay process that $A$ decays into
$B$ by emitting a time-like photon which then converts to a lepton pair
$l^{+}l^{-}$. The differential partial decay width with respect to the
invariant mass $q^{2}\equiv m^{2}_{l^{+}l^{-}}$ of the lepton pair can be
expressed by $d\Gamma(q^{2})/dq^{2}=(d\Gamma(q^{2})/dq^{2})_{\mathrm{point-
like}}|f_{AB}(q^{2})|^{2}$ Landsberg (1985), where
$(d\Gamma(q^{2})/dq^{2})_{\mathrm{point-like}}$ can be calculated exactly in
QED for point-like particle $A$ and $B$ and the $f_{AB}(q^{2})$ is called the
transition form factor (TFF) of the transition $A\to B$, and is an important
probe to the EM structure of the $AB\gamma$ vertex and also the internal
structure of the hadron $A$ (and $B$ if it is also structured).
Experimentally, the TFF $f_{AB}(q^{2})$ can be derived by taking the ratio
$[d\Gamma(q^{2})/dq^{2}]/\Gamma(A\to
B\gamma)\propto|f_{AB}(q^{2})|^{2}/|f_{AB}(0)|^{2}\equiv|F_{AB}(q^{2})|^{2}$
with the normalization $F_{AB}(0)=1$, where many systematic uncertainties
cancel. The Dalitz decays of light hadrons, such as $\phi\to\pi^{0}e^{+}e^{-}$
Achasov  _et al._ (2002); Anastasi  _et al._ (2016), $\phi\to\eta e^{+}e^{-}$
Babusci  _et al._ (2015), $\omega\to\pi^{0}e^{+}e^{-}$ Akhmetshin  _et al._
(2005), $\omega\to\pi^{0}\mu^{+}\mu^{-}$ Dzhelyadin  _et al._ (1979), have
been widely studied in experiments. It should be noted that the experimental
studies on Dalitz decays usually require large statistics, since they are rare
decays for a hadron.

The BESIII Collaboration (BESIII) has accumulated more than $10^{10}$ $J/\psi$
events Ablikim  _et al._ (2022a), based on which the Dalitz decays of $J/\psi$
to light hadrons can be researched. On the other hand, the light pseudoscalars
(P), such as $\eta,\eta^{\prime},\eta(1405/1475)$ and $X(1835)$ etc., are
observed to have large production rates on the $J/\psi$ radiative decays
Workman  _et al._ (2022). So it is expected the Daltiz decays $J/\psi\to
Pl^{+}l^{-}$ can be investigated to a high precision. Actually, BESIII has
performed the experimental studies on the processes $J/\psi\to
e^{+}e^{-}(\eta,\eta^{\prime},\pi^{0})$ Ablikim  _et al._ (2014), $J/\psi\to
e^{+}e^{-}\eta$ Ablikim  _et al._ (2019a), $J/\psi\to e^{+}e^{-}\eta^{\prime}$
Ablikim  _et al._ (2018, 2019b), $J/\psi\to e^{+}e^{-}\eta(1405)$ Ablikim  _et
al._ (2023) and $J/\psi\to e^{+}e^{-}(X(1835),X(2120),X(2370)$ Ablikim  _et
al._ (2022b). With the large $\psi(3686)$ data ensemble, BESIII also studies
the process $\psi(3686)\to e^{+}e^{-}\eta_{c}$. The TTFs $F_{\psi p}(q^{2})$
are extracted for the processes
$J/\psi\to\eta,\eta^{\prime},\eta(1405),X(1835)$ and $\psi(3686)\to\eta_{c}$,
and the $q^{2}$-dependence can be described by the single-pole model

$F_{\psi P}=\frac{1}{1-q^{2}/\Lambda^{2}},$ (1)

based on the vector meson dominance (VMD) Landsberg (1985); Fu  _et al._
(2012); Gu  _et al._ (2019), and the pole parameter $\Lambda$ varies in the
range from 1.7 to 3.8 GeV.

Intuitively, the Dalitz decay and the radiative decay of charmonium into light
pseudoscalars happen through the annihilation of the charm quark and
antiquark. According to the OZI rule, the dominant contribution comes from the
initial state radiation of the virtual and real photons from the charm
(anti)quark. In this sense, the TFF of $J/\psi$ to pseudoscalars should
reflect the electromagnetic properties of $J/\psi$. Therefore, For a same
initial vector charmonium, Dalitz decays are insensitive to the properties of
the final state light hadrons. A theoretical derivation of $F_{\psi P}(q^{2})$
from QCD is desirable but is still challenging since $F_{\psi P}(q^{2})$ is
obviously in the non-perturbative regime of QCD. The phenomenological studies
of $F_{\psi\eta^{(^{\prime})}}(q^{2})$ can be found in Ref. He and Fan (2022)
where the analysis is carried out in the full kinematic region based on QCD
models and in Ref. Chen  _et al._ (2015) where the $J/\psi\to\gamma^{*}P$ is
discussed within the framework of the effective Lagrangian approach and the
$\eta_{c}-\eta-\eta^{\prime}$ mixing is considered.

Lattice QCD may take the mission to give reliable predictions of TFF of
$J/\psi$ to light hadrons. A recent $N_{f}=2$ lattice QCD calculation confirms
the large production rate of the flavor singlet pseudoscalar meson
$\eta_{(2)}$ in the $J/\psi$ radiative decay Jiang  _et al._ (2023). In that
work, the EM form factor is obtained at quite a few values of time-like
$q^{2}$, from which the on-shell form factor at $q^{2}=0$ is obtained through
a polynomial interpolation. By assuming the $\mathrm{U}_{A}(1)$ anomaly
dominance and using the $\eta-\eta^{\prime}$ mixing angle, this on-shell form
factor results in the branching fractions of $J/\psi\to\gamma\eta$ and
$J/\psi\to\gamma\eta^{\prime}$ that are close to the experimental values.
Actually, the $q^{2}$ dependence of this decay form factor is described better
by the single-pole model in Eq. (1) (see below).

Recently, we generated a large gauge ensemble with $N_{f}=1$ strange sea
quarks. The $N_{f}=1$ QCD is a well defined theory and a simplified version of
QCD. It has no chiral symmetry breaking but the $\mathrm{U}_{A}(1)$ anomaly
that has a close relation with the unique light pseudoscalar meson
$\eta_{(1)}$. So we will revisit the production rate of $\eta_{(1)}$ in the
$J/\psi$ radiative decays. We will test the $\mathrm{U}_{A}(1)$ anomaly
dominance in this process by looking at the $N_{f}$ dependence of the partial
decay width, since the $\mathrm{U}_{A}(1)$ anomaly is proportional to $N_{f}$.
In the meantime, we will explore the $q^{2}$-dependence of the related TFF and
its sensitivity to the light pseudoscalar mass since our sea quark is much
heavier than that in Ref. Jiang  _et al._ (2023). The related calculations
involve necessarily the annihilation effect of strange quarks which are dealt
with using the distillation method Peardon  _et al._ (2009).

This work is organized as follows. The numerical procedures and results are
presented in Sec. II. Section III is devoted to the discussions and the
physical implications of our results. Sect. IV is the summary of this work.

##  II Numerical details

###  II.1 $N_{f}=1$ Gauge Ensemble

We generate gauge configurations with $N_{f}=1$ dynamical strange quarks on an
$L^{3}\times T=16^{3}\times 128$ anisotropic lattice. We use the tadpole-
improved Symanzik’s gauge action for anisotropic lattices Morningstar and
Peardon (1997); Chen  _et al._ (2006) and the tadpole-improved anisotropic
clover fermion action Zhang and Liu (2001); Su  _et al._ (2006). The RHMC
algorithm implemented in Chroma software Edwards and Joo (2005) is used to
generate the $N_{f}=1$ gauge configurations. The parameters in the action are
tuned to give the anisotropy $\xi=a_{s}/a_{t}\approx 5$, where $a_{t}$ and
$a_{s}$ are the temporal and spatial lattice spacings, respectively. The scale
setting takes the following procedure. Experimentally, there is an interesting
relation between pseudoscalar meson masses $m_{PS}$ and the vector meson
masses $m_{V}$ of the quark configuration $q_{l}\bar{q}$,

$\Delta m^{2}\equiv m_{V}^{2}-m_{PS}^{2}\approx
0.56-0.58~{}~{}\mathrm{GeV}^{2}$ (2)

where $q_{l}$ stands for the $u,d,s$ quarks and $q$ stands for $u,d,s,c$
quarks. The masses of these vector and pseudoscalar mesons from PDG Workman
_et al._ (2022) are listed in Table 1 along with their mass squared
differences. So we assume the relation of Eq. (2) is somewhat general for
light mesons and use it to set the scale parameter $a_{t}$. We make the least
squares fitting to the mass squared differences over the $n\bar{n}$,
$n\bar{s}$, $n\bar{c}$ and $s\bar{c}$ systems where $n$ refers to the $u,d$
quarks, and get the mean value $\overline{\Delta m^{2}}=0.568(8)$
$\mathrm{GeV}^{2}$, which serves as an input to give the lattice scale
parameter $a_{t}^{-1}=6.66(5)$ GeV. Since the HPQCD collaboration determines
the $s\bar{s}$ pseudoscalar meson mass to be $m_{\eta_{s}}=0.686(4)$ GeV from
the connected quark diagram Davies  _et al._ (2010), we use the ratio
$m_{\phi}/m_{\eta_{s}}=1.487(9)$ to set the bare mass parameters of strange
quarks. Even though $\eta_{s}$ is not a physical state, the mass squared
difference $m_{\phi}^{2}-m_{\eta_{s}}^{2}\approx 0.570~{}\mathrm{GeV}^{2}$
also satisfies the empirical relation of Eq. (2). Finally, we obtain
$m_{\eta_{s}}=693.1(3)(6.0)$ MeV, $m_{\phi}=1027.2(5)(7.7)$ MeV and
$m_{\phi}^{2}-m_{\eta_{s}}^{2}=0.570~{}\mathrm{GeV}^{2}$ on our gauge
ensemble. This serves as a self-consistent check of our lattice setup. The
details of the gauge ensemble are given in Table 2. For the valence charm
quark, we use the same fermion action as the strange sea quarks and the charm
quark mass parameters are tuned to give $(m_{\eta_{c}}+3m_{J/\psi})/4=3069$
MeV.

The quark propagators are calculated in the framework of the distillation
method Peardon  _et al._ (2009). Let $\\{V_{i},i=1,2,\ldots,70\\}$ be the set
of the $N_{V}=70$ eigenvectors (with smallest eigenvalues) of the gauge
covariant Laplacian operator on the lattice. We use these eigenvectors to
calculate the perambulators of strange and charm quarks, which are encoded
with the all-to-all quark propagators and facilitate the treatment of quark
disconnected diagrams. In the meantime, these eigenvectors provide a smearing
scheme for quark fields, namely, $\psi^{(s)}=V^{\dagger}V\psi$, where
$\psi^{(s)}$ is the smeared quark field of $\psi$, $V$ is a matrix with each
column being an eigenvector. All the meson interpolation operators in this
work are built from the smeared charm and strange quark fields.

Table 1:  Experimental values of the masses of the pseudoscalar (P) and vector mesons (V) of quark configurations $n\bar{q}$, $n\bar{s}$, $n\bar{c}$, $s\bar{c}$, $n\bar{b}$ and $s\bar{b}$ Workman  _et al._ (2022). Here $n$ refers to the light $u,d$ quarks. The right most column lists the $m_{V}^{2}-m_{PS}^{2}~{}(\mathrm{GeV}^{2})$. In the row of $s\bar{s}$ states, the mass of the $s\bar{s}$ pseudoscalar $\eta_{s}$ is determined by the HPQCD collaboration from lattice QCD calculations Davies  _et al._ (2010). $q_{l}\bar{q}$ |  $m_{V}$ (GeV) |  $m_{PS}$ (GeV) | $m_{V}^{2}-m_{PS}^{2}~{}(\mathrm{GeV}^{2})$
---|---|---|---
$n\bar{n}$ | 0.775 | 0.140 | 0.581
$n\bar{s}$ | 0.896 | 0.494 | 0.559
$s\bar{s}$ | 1.020 | 0.686 Davies  _et al._ (2010) | 0.570
$n\bar{c}$ | 2.010 | 1.870 | 0.543
$s\bar{c}$ | 2.112 | 1.968 | 0.588
$n\bar{b}$ | 5.325 | 5.279 | 0.481
$s\bar{b}$ | 5.415 | 5.367 | 0.523
Table 2: Parameters of the gauge ensemble. $L^{3}\times T$ | $\beta$ |  $a_{t}^{-1}$(GeV) | $\xi$ |  $m_{\eta_{s}}$(MeV) |  $m_{\phi}$ (MeV) | $N_{\mathrm{cfg}}$
---|---|---|---|---|---|---
$16^{3}\times 128$ | 2.0 | $6.66(5)$ | $\sim 5.0$ | $693(5)$ | $1027(8)$ | $1547$

###  II.2 Pseudoscalar meson $\eta_{(1)}$

Figure 1: Lattice results of the energies of $\eta_{(1)}$. Left panel: the
effective mass of two-point correlation functions of $\eta_{(1)}$ and
$\eta_{s}$, where $\eta_{(1)}$ using the operators $\mathcal{O}_{\gamma_{5}}$
and $\mathcal{O}_{\gamma_{4}\gamma_{5}}$. The green points are from the
original two-point function of $\mathcal{O}_{\gamma_{5}}$, the blue points are
from the subtracted two-point function in Eq. (3) with $t_{0}=3a_{t}$, and the
red points are from the two-point function of
$\mathcal{O}_{\gamma_{4}\gamma_{5}}$. The plateau regions of red and blue
points merge together in large $t/a_{t}$ range. The purple points stand for
unphysical $\eta_{s}$, which only include connected diagram. Middle panel: the
effective energies $E_{\eta_{(1)}}^{(\vec{q})}(t)$ with the momentum mode
$\vec{n}$ of $\vec{q}$ up to $|\vec{n}|^{2}=9$, where the data points are from
the correlation functions of $\mathcal{O}_{\gamma_{4}\gamma_{5}}(\vec{q})$.
The fitted $E_{\eta_{(1)}}^{(\vec{q})}$ values are given in Table 3. Right
panel: the dispersion relation of $\eta_{(1)}$. The grey band illustrate the
dispersion relation in Eq. (9) with the fitted $\xi=4.88(1)$ and the
$\chi^{2}$ per degree of freedom $\chi^{2}/\mathrm{d.o.f}=0.32$.

We use two interpolation operators for $\eta_{(1)}$, namely
$\mathcal{O}_{\gamma_{5}}=\bar{s}^{(s)}\gamma_{5}s^{(s)}$ and
$\mathcal{O}_{\gamma_{4}\gamma_{5}}=\bar{s}^{(s)}\gamma_{4}\gamma_{5}s^{(s)}$
to calculate the correlation functions $C_{\gamma_{5}\gamma_{5}}(t)$ and
$C_{(\gamma_{4}\gamma_{5})(\gamma_{4}\gamma_{5})}(t)$.
$C_{\gamma_{5}\gamma_{5}}(t)$ has a finite volume artifact that it approaches
to a nonzero constant when $t$ is large, as shown in Fig. 1. This artifact
comes from the topology of QCD vacuum and can be approximately expressed as
$a^{5}(\chi_{\mathrm{top}}+Q^{2}/V)/T$ where $a$ is the lattice spacing (in
the isotropic case), $\chi_{\mathrm{top}}$ is the topological susceptibility,
$Q$ is the topological charge, $V$ is the spatial volume and $T$ is the
temporal extension of the lattice Aoki  _et al._ (2007); Bali  _et al._
(2015); Dimopoulos  _et al._ (2019). In contrast,
$C_{(\gamma_{4}\gamma_{5})(\gamma_{4}\gamma_{5})}(t)$ damps to zero for large
$t$, which is the normal large $t$ behavior. The constant term of
$C_{\gamma_{5}\gamma_{5}}(t)$ can be subtracted by taking the difference

$C_{\gamma_{5}\gamma_{5}}^{\prime}(t)=C_{\gamma_{5}\gamma_{5}}(t)-C_{\gamma_{5}\gamma_{5}}(t+t_{0}),$
(3)

and we take $t_{0}=3a_{t}$ in practice. The effective mass functions
$m_{\mathrm{eff}}(t)=\ln\frac{C_{\Gamma\Gamma}^{(^{\prime})}(t)}{C_{\Gamma\Gamma}^{(^{\prime})}(t+1)}$
of the two correlations are shown in Fig. 1, where one can see that decent
mass plateaus show up when $t/a_{t}>15$ and agree with each other. The
effective masses of the connected parts of the two correlation functions are
also shown for comparison. Their plateaus correspond to the mass
$m_{\eta_{s}}$ of $\eta_{s}$. The data analysis gives the results

${m}_{\eta_{s}}=693.1(3)~{}\text{MeV},~{}~{}~{}m_{\eta_{(1)}}=783.0(5.5)~{}\text{MeV}.$
(4)

Here $m_{\eta_{s}}$ is the mass parameter from the connected diagram and is
consistent with the value $m_{\eta_{s}}=686(4)$ obtained by HPQCD at the
physical strange quark mass Davies  _et al._ (2010). This indicates that our
sea quark mass parameter is tuned to be almost at the strange quark mass.
$m_{\eta_{(1)}}$ is determined from the correlation function that includes the
connected diagram and quark annihilation diagram, and is therefore the mass of
the physical state $\eta_{(1)}$.

###  II.3 Form Factor for $J/\psi\to\gamma^{*}\eta_{(1)}$

The transition matrix element $\mathcal{M}$ for the process
$J/\psi\to\gamma^{*}\eta_{(1)}$ can be expressed in terms of one form factor
$M(q^{2})$, namely,

$\displaystyle\mathcal{M}_{\psi\eta_{(1)}\gamma^{*}}^{\mu}$
$\displaystyle\equiv$
$\displaystyle\langle\eta_{(1)}(p_{\eta})|j_{\mathrm{em}}^{\mu}(0)|\psi(p_{\psi},\lambda)\rangle$
(5) $\displaystyle=$ $\displaystyle
M(q^{2})\epsilon^{\mu\nu\rho\sigma}p_{\psi,\nu}p_{\eta,\rho}\epsilon_{\sigma}(p_{\psi},\lambda),$

where $q^{2}=(p_{\psi}-p_{\eta})^{2}$ is the virtuality of the photon,
$\epsilon_{\sigma}(p_{J/\psi},\lambda)$ is the polarization vector of $J/\psi$
and $j_{\mathrm{em}}^{\mu}=\bar{c}\gamma^{\mu}c$ is the electromagnetic
current of charm quark (we only consider the initial state radiation and
ignore photon emissions from sea quarks and the final state). The matrix
element $\mathcal{M}$ is encoded in the following three-point functions

$C^{\mu
i}_{(3)}(\vec{q};t,t^{\prime})=\sum\limits_{\vec{y}}e^{i\vec{q}\cdot\vec{y}}\langle\mathcal{O}_{\eta}(\vec{p^{\prime}},t)j_{\mathrm{em}}^{\mu}(\vec{y},t^{\prime})\mathcal{O}_{\psi}^{i,\dagger}(\vec{p},0)\rangle$
(6)

with $\vec{q}=\vec{p^{\prime}}-\vec{p}$, where
$\mathcal{O}_{\eta}(\vec{p^{\prime}},t)$ and
$\mathcal{O}_{\psi}^{i}(\vec{p},t)$ are the interpolating field operators for
$\eta_{(1)}$ and $J/\psi$ with spatial momenta $\vec{p^{\prime}}$ and
$\vec{p}$, respectively. For $t\gg t^{\prime}$, $t^{\prime}\gg 0$ and in the
rest frame of $J/\psi$ ($\vec{p}=0$), the explicit spectral expression of
$C^{\mu i}_{(3)}(\vec{q};t,t^{\prime})$ reads

$\displaystyle C^{\mu i}_{(3)}(\vec{q};t,t^{\prime})$ $\displaystyle\approx$
$\displaystyle\frac{Z_{\eta}(\vec{q})Z_{\psi}^{*}}{4V_{3}E_{\eta}(\vec{q})m_{\psi}}e^{-E_{\eta}(\vec{q})(t-t^{\prime})}e^{-m_{\psi}t^{\prime}}$
(7) $\displaystyle\times$
$\displaystyle\sum\limits_{\lambda}\langle{\eta_{(1)}}(\vec{q})|j_{\mathrm{em}}^{\mu}|J/\psi(\vec{0},\lambda)\rangle\epsilon^{*,i}(\vec{0},\lambda),$

where $V_{3}$ is the spatial volume,
$Z_{\eta}(\vec{q})=\langle\Omega|\mathcal{O}_{\eta_{(1)}}(\vec{q})|{\eta_{(1)}}(\vec{q})\rangle$,
and
$Z_{\psi}\epsilon^{i}(\vec{0},\lambda)=\langle\Omega|\mathcal{O}_{\psi}^{i}(\vec{0})|J/\psi(\vec{0},\lambda)\rangle$.
Note that $Z_{\eta}$ has a $\vec{q}$ dependence due to the smeared operator
$\mathcal{O}_{\eta}$ Bali  _et al._ (2016). The parameters $m_{\psi}$,
$E_{\eta}(\vec{q})$, $Z_{\psi}$ and $Z_{\eta}(\vec{q})$ can be derived from
the two-point correlation functions

$\displaystyle C_{(2),\eta}(\vec{q},t)$ $\displaystyle\approx$
$\displaystyle\frac{1}{2E_{\eta}(\vec{q})V_{3}}|Z_{\eta}(\vec{q})|^{2}e^{-E_{\eta}(\vec{q})t},$
$\displaystyle C_{(2),\psi}^{ii}(t)$ $\displaystyle\approx$
$\displaystyle\frac{1}{2m_{\psi}V_{3}}|Z_{J/\psi}|^{2}e^{-m_{\psi}t}.$ (8)

Thus we can extract the matrix element
$\langle\eta_{(1)}|j_{\mathrm{em}}^{\mu}|J/\psi\rangle$ through Eqs. (7) and
(II.3).

Therefore, the major numerical task is the calculation of $C_{(3)}^{\mu
i}(\vec{q};t,t^{\prime})$. The local EM current
$j_{\mathrm{em}}^{\mu}(x)=[\bar{c}\gamma^{\mu}c](x)$ mentioned above (the
charm quark field $c$ and $\bar{c}$ are the original field, which are not
smeared) is not conserved anymore on the finite lattice and should be
renormalized. We determine the renormalization factor $Z_{V}^{t}=1.147(1)$ and
$Z_{V}^{s}=1.191(2)$ for the temporal and spatial components of
$j_{\mathrm{em}}^{\mu}(x)$, respectively, by calculating the relevant
electromagnetic form factors of $\eta_{c}$ Dudek  _et al._ (2006); Yang  _et
al._ (2013). In practice, only $Z_{V}^{s}$ is involved and is incorporated
implicitly in $j_{\mathrm{em}}^{\mu}(x)$. We use the operator
$\mathcal{O}_{\gamma_{4}\gamma_{5}}=\bar{s}^{(s)}\gamma_{4}\gamma_{5}s^{(s)}$
for $\mathcal{O}_{\eta}$ and $\mathcal{O}_{\psi}^{i}$ takes the form
$\bar{c}^{(s)}\gamma^{i}c^{(s)}$ in Eq. (6). The three-point function
$C_{(3)}^{\mu i}(\vec{q};t,t^{\prime})$ is calculated in the rest frame of
$J/\psi$ ($\vec{p}=0$), such that $\eta_{(1)}$ moves in a spatial momentum
$\vec{p}^{\prime}=\vec{q}$. The right panel of Fig. 1 shows the dispersion
relation of $\eta_{(1)}$

$a_{t}^{2}E_{\eta}^{2}(\vec{q})=a_{t}^{2}m_{\eta}^{2}+\frac{1}{\xi^{2}}\left(\frac{2\pi}{L}\right)^{2}|\vec{n}|^{2},$
(9)

where $\vec{n}$ stands for the momentum mode of
$\vec{q}=\frac{2\pi}{La_{s}}\vec{n}$. It is seen that $E_{\eta}^{2}(\vec{q})$
exhibits a perfect linear behavior in $|\vec{q}|^{2}$ up to $|\vec{n}|^{2}=9$
and the fitted slope gives $\xi=4.88(1)$ which deviates from the renormalized
anisotropy $\xi\approx 5.0$ by less than 3%.

After the Wick’s contractions, the three-point function $C^{\mu i}_{(3)}$ is
expressed in terms of quark propagators, and the schematic quark diagram is
illustrated in Fig. 2. There are two separated quark loops connected by
gluons. The strange quark loop on the right-hand side can be calculated in the
framework of the distillation method. The left part $G^{\mu i}$ comes from the
contraction of $\mathcal{O}_{\psi}^{i}$ and the current
$j^{\mu}_{\mathrm{em}}$, namely,

$G^{\mu
i}(\vec{p},\vec{q};t^{\prime}+\tau,\tau)=\sum\limits_{\vec{y}}e^{i\vec{q}\cdot\vec{y}}j_{\mathrm{em}}^{\mu}(\vec{y},t^{\prime}+\tau)O_{\psi}^{i\dagger}(\vec{p},\tau),$
(10)

and is dealt with the distillation method Chen  _et al._ (2023). Considering
$\mathcal{O}_{\psi}^{i}(\vec{p},t)=\sum\limits_{\vec{y}}e^{-i\vec{p}\cdot\vec{y}}[\bar{c}^{(s)}\gamma_{i}c^{(s)}](\vec{y},t)$,
the explicit expression of $G^{\mu i}$ at the source time slice $\tau=0$ is

$\displaystyle G^{\mu i}(\vec{p},\vec{q};t,0)$ $\displaystyle=$
$\displaystyle\sum\limits_{\vec{x}}e^{i\vec{q}\cdot\vec{x}}\mathrm{Tr}\left\\{\gamma_{5}[S_{c}V(0)]^{\dagger}(\vec{x},t)\gamma_{5}\gamma^{\mu}\right.$
$\displaystyle\times$
$\displaystyle\left.\left[S_{c}V(0)\right](\vec{x},t)[V^{\dagger}(0)D(\vec{p})\gamma^{i}V(0)]\right\\},$

where $S_{c}=\langle c\bar{c}\rangle_{U}$ is the all-to-all propagator of
charm quark for a given gauge configuration $U$ and $D(\vec{p})$ is a
$3L^{3}\times 3L^{3}$ diagonal matrix with the diagonal elements being
$\delta_{ab}e^{i\vec{p}\cdot\vec{y}}$ ($\vec{y}$ labels the column or row
indices and $a,b=1,2,3$ refer to the color indices). The
$\gamma_{5}$-hermiticity $S_{c}=\gamma_{5}S_{c}^{\dagger}\gamma_{5}$ implies
$[V^{\dagger}(0)S_{c}](\vec{x},t)=\gamma_{5}[S_{c}V(0)]^{\dagger}(\vec{x},t)\gamma_{5}$,
such that only $S_{c}V(0)$ is required, while $S_{c}V(0)$ can be obtained by
solving the system of linear equations

$M[U;m_{c}][S_{c}V(0)]=V(0),$ (12)

where $M[U;m_{c}]$ is the fermion matrix in the charm quark action (the linear
system solver defined by $M[U;m_{c}]x=b$ is applied $4N_{V}$ times for Dirac
indices $\alpha=1,2,3,4$ and all the columns of $V(0)$). In order to increase
the statistics, the above procedure runs over all the time range, say,
$\tau\in[0,T-1]$. Averaging over $\tau\in[0,T-1]$ improves the precision of
the calculated $C^{\mu i}_{(3)}$ drastically.

Figure 2: Schematic diagram for the process $J/\psi\to\gamma^{*}\eta_{(1)}$.
Figure 3: Fit of form factor $M(q^{2})$ for $J/\psi\to\gamma^{*}\eta_{(1)}$. The lattice data are plotted as data points and the grey bands show the fit by constants to the plateau regions. The fitted $M(q^{2})$ values are given in Table 3. mode $\vec{n}$ of $\vec{q}$ | $(1,2,2)$ | $(0,2,2)$ | $(1,1,2)$ | $(0,1,2)$ | $(1,1,1)$ | $(0,1,1)$ | $(0,0,1)$
---|---|---|---|---|---|---|---
$q^{2}/\mathrm{GeV}^{2}$ |  $-$0.6800(66) |  $-$0.1869(73) | 0.8777(91) | 1.459(10) | 2.756(14) | 3.499(16) | 4.337(20)
$E_{\eta_{(1)}}^{(\vec{q})}/\mathrm{GeV}$ | 1.803(14) | 1.710(17) | 1.5279(29) | 1.4291(24) | 1.2119(29) | 1.0886(20) | 0.9466(18)
$M(q^{2})/\mathrm{GeV}^{-1}$ | 0.00400(38) | 0.00503(41) | 0.00654(26) | 0.00714(30) | 0.00865(36) | 0.01377(28) | 0.01757(57)
Table 3: Fit values of $\eta_{(1)}$ form factor $M(q^{2})$. The momentum modes
$\vec{n}$ represent the relation $\vec{q}=\frac{2\pi}{L}\vec{n}$. The two-
point function $C_{\Gamma\Gamma}(t)$ and the three-point ratio function
$R^{\mu i}(\vec{q};t,t^{\prime})$ corresponding to the same momentum mode
$\vec{n}$ has been averaged for increasing signal of the energy
$E_{\eta_{(1)}}^{(\vec{q})}$ and form factor $M(q^{2})$, respectively.

It is observed that $J/\psi$ contribution dominates $C^{\mu
i}_{(3)}(\vec{q};t,t^{\prime})$ when $t^{\prime}>40$. Combining Eqs.
(5,7,II.3), we have the following expression

$\displaystyle R^{\mu i}(\vec{q};t,t^{\prime})$ $\displaystyle\equiv$
$\displaystyle\frac{Z_{\psi}Z_{\eta}(\vec{q})C^{\mu
i}_{(3)}(\vec{q};t,t^{\prime})}{V_{3}C_{(2),\eta}(-\vec{q},t-t^{\prime})C_{(2),\psi}(t^{\prime})}$
(13) $\displaystyle\approx$ $\displaystyle M(q^{2};t-t^{\prime})\epsilon^{\mu
ij}q_{j}$

for the fixed $t^{\prime}/a_{t}=40$, from which we obtain
$M(q^{2},t-t^{\prime})$ for each $q^{2}$. Fig. 3 shows the $t-t^{\prime}$
dependence of $M(q^{2},t-t^{\prime})$ at several $q^{2}$ close to $q^{2}=0$.
It is seen that a plateau region appears beyond $t-t^{\prime}>10$ for each
$q^{2}$, where $M(q^{2})$ is obtained through a constant fit. The grey bands
illustrate the fitted values and fitting time ranges, along with the jackknife
errors. We also test the fit function form
$M(q^{2},t-t^{\prime})=M(q^{2})+c(q^{2})e^{-\delta E(t-t^{\prime})}$ with the
exponential term being introduced to account for the higher state
contamination. The fitted values of $M(q^{2})$ in this way are consistent with
those in the constant fit but have much larger errors. Therefore, we use the
results from the constant fit for the values of $M(q^{2})$. The derived
$M(q^{2})$ up to $q^{2}=4.3~{}\mathrm{GeV}^{2}$ data points are list in Table.
3.

Instead of a polynomial functions form used by Ref. Jiang  _et al._ (2023), we
use the single pole model to describe the $q^{2}$ dependence of $M(q^{2})$

$M(q^{2})=\frac{M(0)}{1-q^{2}/\Lambda^{2}}\equiv M(0)F_{\psi\eta}(q^{2}).$
(14)

As indicated by the red band in Fig. 4, the model fits the overall behaviors
of $M(q^{2})$ very well with the parameters

$\displaystyle M(0)$ $\displaystyle=$ $\displaystyle
0.00541(13)~{}\mathrm{GeV}^{-1},$ $\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle 2.465(22)~{}\mathrm{GeV}.$ (15)

##  III Discussions

###  III.1 The partial decay width of $J/\psi\to\gamma\eta_{(1)}$

The partial decay width $\Gamma(J/\psi\to\gamma\eta_{(1)})$ is dictated by the
on-shell form factor $M(q^{2}=0)$ through the relation

$\Gamma(J/\psi\to\gamma\eta_{(1)})=\frac{4\alpha}{27}|M(0)|^{2}|\vec{p}_{\gamma}|^{3},$
(16)

where the electric charge of charm quark $Q=+2e/3$ has been incorporated,
$\alpha\equiv\frac{e^{2}}{4\pi}=1/134$ is the fine structure constant at the
charm quark mass scale, and
$|\vec{p}_{\gamma}|=(m_{\psi}^{2}-m_{\eta_{(1)}}^{2})/2m_{\psi}$ is the on-
shell momentum of the photon. Using the value of $M(0)$ in Eq. (II.3), the
partial decay width and the corresponding branching fraction are predicted as

$\displaystyle\Gamma(J/\psi\to\gamma\eta_{(1)})$ $\displaystyle=$
$\displaystyle 0.097(5)~{}\mathrm{keV}$
$\displaystyle\mathrm{Br}(J/\psi\to\gamma\eta_{(1)})$ $\displaystyle=$
$\displaystyle 1.06(5)\times 10^{-3},$ (17)

where the experimental value $\Gamma_{J/\psi}=92.6~{}\mathrm{keV}$ is used.
Since an $\eta$ state in the process $J/\psi\to\gamma\eta$ is produced by
gluons through its flavor singlet component, the results in Eq. (III.1) should
be compared with the experimental result
$\mathrm{Br}(J/\psi\to\gamma\eta^{\prime})=5.25(7)\times 10^{-3}$ Workman  _et
al._ (2022) ($\eta^{\prime}$ is mainly a flavor singlet) and
$\mathrm{Br}(J/\psi\to\gamma\eta_{(2)})=4.16(49)\times 10^{-3}$ in the
$N_{f}=2$ case at $m_{\pi}\approx 350~{}\mathrm{MeV}$ Jiang  _et al._ (2023).
Obviously $\mathrm{Br}(J/\psi\to\gamma\eta_{(1)})$ is four or five times
smaller that in $N_{f}=2$ and $N_{f}=2+1$ cases.

This large difference can be understood as follows. The decay process
$J/\psi\to\gamma\eta_{(N_{f})}$ takes place in the procedure that the
$c\bar{c}$ pair annihilates into gluons (after a photon radiation), which then
convert into $\eta_{(N_{f})}$. There are two mechanisms for gluons to couple
to $\eta_{(N_{f})}$. The first is the $\mathrm{U}_{A}(1)$ anomaly manifested
by the anomalous axial vector current relation (in the chiral limit)

$\partial_{\mu}j_{5}^{\mu}(x)=\sqrt{N_{f}}\frac{g^{2}}{32\pi}G_{\mu\nu}^{a}(x)\tilde{G}^{a,\mu\nu}(x)\equiv\sqrt{N_{f}}q(x),$
(18)

where
$j_{5}^{\mu}=\frac{1}{\sqrt{N_{f}}}\sum\limits_{k=1}^{N_{f}}\bar{q}_{k}\gamma_{5}\gamma^{\mu}q_{k}$
is the flavor singlet axial vector current for $N_{f}$ flavor quarks, and
$q(x)$ is the topological charge density. The $\mathrm{U}_{A}(1)$ anomaly
induces the anomalous gluon-$\eta$ coupling with the strength observed by the
matrix element $\langle 0|q(0)|\eta_{(N_{f})}\rangle$. With the matrix element
$\langle
0|\partial_{\mu}j_{5}^{\mu}(0)|\eta_{(N_{f})}\rangle=f_{\eta_{(N_{f})}}m_{\eta_{(N_{f})}}^{2}$,
from Eq. (18) one has the relation

$\langle
0|q(0)|\eta_{(N_{f})}\rangle=\frac{1}{\sqrt{N_{f}}}f_{\eta_{(N_{f})}}m_{\eta_{(N_{f})}}^{2}$
(19)

in the chiral limit. According to the Witten and Veneziano mechanism Witten
(1979); Veneziano (1979) for the mass of $\eta_{(N_{f})}$,
$m_{\eta_{(N_{f})}}^{2}=\frac{4N_{f}}{f_{\pi}^{2}}\chi_{\mathrm{top}}$, where
$\chi_{\mathrm{top}}$ is the topological susceptibility of the SU(3) pure
Yang-Mills theory, one has $\langle
0|q(0)|\eta_{(N_{f})}\rangle\propto\sqrt{N_{f}}$ in the chiral limit.

For massless quarks, the $\mathrm{U}_{A}(1)$ anomaly dominates the production
of $\eta_{(N_{f})}$ in the process $J/\psi\to\gamma\eta_{(N_{f})}$, then one
expects the $N_{f}$ scaling for the partial decay width

$\Gamma(J/\psi\to\gamma\eta_{(N_{f})})\propto|\langle
0|q(0)|\eta_{(N_{f})}\rangle|^{2}\propto N_{f},$ (20)

since $f_{\eta_{(N_{f})}}\approx f_{\pi}$ is independent of $N_{f}$ to the
lowest order in $1/N_{c}$. In Ref. Jiang  _et al._ (2023), this scaling
relation is used to predict the production rates of $\eta$ and $\eta^{\prime}$
from the form factor $M(0)$ of the $N_{f}=2$ case at $m_{\pi}\approx
350~{}\mathrm{MeV}$ along with the mixing $\eta-\eta^{\prime}$ mixing angle
$\theta_{\mathrm{lin}}=-24.5^{\circ}$ Workman  _et al._ (2022). The results
$\mathrm{Br}(J/\psi\to\gamma\eta)=1.15(14)\times 10^{-3}$ and
$\mathrm{Br}(J/\psi\to\gamma\eta^{\prime})=4.49(53)\times 10^{-3}$ are in
excellent agreement with the experimental values $1.11(3)\times 10^{-3}$ and
$5.25(7)\times 10^{-3}$, respectively. For the case of this study of $N_{f}=1$
strange quarks, the scaling relation implies
$\mathrm{Br}(J/\psi\to\gamma\eta_{(N_{f}=3)})=3.19(15)\times 10^{-3}$ which is
closer to the experiment value of $J/\psi\to\gamma\eta^{\prime}$. This is in
the right trend but still has a large discrepancy that can be attributed to
the quark mass dependence. The quark mass dependence appears in three places.
First, there is an additional term $2m_{s}j_{5}\equiv
2m_{s}(i\bar{s}\gamma_{5}s)$ in the right-hand side of Eq. (18) for $N_{f}=1$
strange quarks, which gives a correction to Eq. (19) as

$\langle
0|q(0)|\eta_{(1)}\rangle=f_{\eta_{(N_{f})}}m_{\eta_{(N_{f})}}^{2}-2m_{s}\langle
0|j_{5}(0)|\eta_{(1)}\rangle$ (21)

with $m_{s}\langle 0|j_{5}(0)|\eta_{(1)}\rangle>0$ Novikov  _et al._ (1980);
Feldmann (2000); Beneke and Neubert (2003); Cheng  _et al._ (2009); Singh
(2013); Qin  _et al._ (2018); Ding  _et al._ (2019); Bali  _et al._ (2021).
Secondly, the right-hand side of Eq. (19) has quark mass dependence itself
through $f_{\eta_{(N_{f})}}$ and $m_{\eta_{(N_{f})}}$. Since the strange quark
is not so light as $u,d$ quarks, this kind of the quark mass dependence may
not be negligible. The third place is the $\eta_{(N_{f})}$ production
procedure that the gluons couple perturbatively to $q\bar{q}$ pairs which then
couple to $\eta_{(N_{f})}$. Because of the vector-like $q\bar{q}$-gluon
coupling in QCD, the produced quark and antiquark have opposite chiralities.
So in the chiral limit, the production of $\eta_{(N_{f})}$ is prohibited to
all orders of the perturbative QCD due to the conservation of angular momentum
(the $q\bar{q}$ pair has chirality two and has a total spin $S=0$ and the
orbital angular momentum $L=0$). For massive quarks, the amplitude of this
process is proportional to the quark mass Chanowitz (2005); Chao  _et al._
(2007), which measures the mixing of quarks with different chiralities. Thus
the production of $s\bar{s}$ is drastically enhanced over that of
$(u\bar{u}+d\bar{d})/\sqrt{2}$. If the amplitude of this mechanism for the
$\eta_{(N_{f})}$ production has an opposite sign to that of the
$\mathrm{U}_{A}(1)$ anomaly, then the seemingly smaller value in Eq. (III.1)
can be understood. This possibility actually exists, as is manifested in a
previous lattice QCD study on the semileptonic decay from $D_{s}$ to
$\eta^{\prime}$ Bali  _et al._ (2015), where it is observed that, the
contribution of the direct coupling of $s\bar{s}$ to $\eta^{\prime}$ has an
opposite sign to that through the disconnected diagram.

Figure 4: The form factor $M(q^{2})$ for $J/\psi\to\gamma^{*}\eta_{(1,2)}$.
The data points are the lattice QCD results, and the shaded bands illustrate
the fit model $M(q^{2})=\frac{M(0)}{1-q^{2}/\Lambda^{2}}$ with the best fit
parameters $M(0)=0.01066(36)~{}\text{GeV}^{-1}$ for $N_{f}=1$ and
$\Lambda=2.442(36)~{}\text{GeV}$ for $N_{f}=2$. The $M(q^{2})$ data of
$J/\psi\to\gamma^{*}\eta_{(2)}$ are the same as those in Table II of Ref.
Jiang  _et al._ (2023) and the fit is performed using the jackknife method on
the original data sample.

###  III.2 The Dalitz decay form factors $J/\psi\to Pl^{+}l^{-}$

The form factor $M(q^{2})$ in Eq. (14) is actually the TFF for the Dalitz
decay $J/\psi\to\eta_{(1)}l^{+}l^{-}$ when $q^{2}>4m_{l}^{2}$, which is seen
to be well described by the single pole model with
$\Lambda=2.465(22)~{}\mathrm{GeV}$. In Ref. Jiang  _et al._ (2023), the Dalitz
TFF $M(q^{2})$ is also obtained in the $N_{f}=2$ lattice QCD at
$m_{\pi}\approx 350~{}\mathrm{MeV}$, and the value $M(0)$ is interpolated
using a polynomial function form. We refit the $q^{2}$-dependence of
$M(q^{2})$ using the same single pole model, as shown in Fig. 4. It is
observed that the single pole model describes the data better than the
polynomial model in the whole $q^{2}$ range, and the pole parameter
$\Lambda=2.442(36)~{}\mathrm{GeV}$ agrees well with the value for $N_{f}=1$.
This signals the single pole model may be universal for the Dalitz decays from
$J/\psi$ to light pseudoscalar mesons $P$ and the pole parameter $\Lambda$ is
insensitive to the number of light flavors $N_{f}$ and the mass of $P$.

In experiments, the TFF $F_{\psi P}$ can be extracted from the ratio

$\frac{d\Gamma(\psi\to Pl^{+}l^{-})/dq^{2}}{\Gamma(\psi\to
P\gamma)}=A(q^{2})|F_{\psi P}(q^{2})|^{2},$ (22)

where $A(q^{2})$ is a known kinematic factor Landsberg (1985); Fu  _et al._
(2012); Gu  _et al._ (2019)

$\displaystyle A(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{3\pi}\frac{1}{q^{2}}\left(1-\frac{4m_{l}^{2}}{q^{2}}\right)^{1/2}\left(1+\frac{2m_{l}^{2}}{q^{2}}\right)$
(23) $\displaystyle\times$
$\displaystyle\left[\left(1+\frac{q^{2}}{m_{\psi}^{2}-m_{P}^{2}}\right)^{2}-\frac{4m_{\psi}^{2}q^{2}}{(m_{\psi}^{2}-m_{P}^{2})^{2}}\right]^{3/2}$

derived from the QED calculation. BESIII has measured many Dalitz decay
processes of $J/\psi\to Pe^{+}e^{-}$ with $P=\eta$ Ablikim  _et al._ (2014,
2019a), $\eta^{\prime}$ Ablikim  _et al._ (2014, 2018, 2019b), $\eta(1405)$
Ablikim  _et al._ (2023), and $(X(1835),X(2120),X(2370))$ Ablikim  _et al._
(2022b). For some of these processes, the TFF are obtained and fitted through
the single pole model (along with resonance terms if experimental data are
precise enough Ablikim  _et al._ (2019a)) in Eq. (1) and the fitted values of
$\Lambda$ are collected in Table 4, where the values of $\Lambda$ derived from
lattice QCD are also presented in the last two rows for comparison. Although
the values of $\Lambda$ for the $J/\psi\to\eta,\eta^{\prime}$ Dalitz decays
are compatible with the lattice values, the values of $\Lambda$ for
$J/\psi\to\eta(1405),X(1835)$ are substantially smaller. So it is possible
that $\Lambda$ depends on the mass of the final state pseudoscalar meson.

Table 4: The values of the pole parameter $\Lambda$ of the TFF for different Dalitz decays $J/\psi\to Pe^{+}e^{-}$. The $N_{f}=1,2$ lattice QCD results of $\Lambda$ are also shown in the bottom two rows for comparison.  $V\to e^{+}e^{-}P$ | $\Lambda~{}(\text{GeV})$ | Ref.
---|---|---
$J/\psi\to e^{+}e^{-}\eta$ | $2.56\pm 0.04\pm 0.03$ |  Ablikim  _et al._ (2019a)
$J/\psi\to e^{+}e^{-}\eta^{\prime}$ | $3.1~{}\pm 1.0$ |  Ablikim  _et al._ (2014)
$J/\psi\to e^{+}e^{-}\eta(1405)$ | $1.96\pm 0.24\pm 0.06$ |  Ablikim  _et al._ (2023)
$J/\psi\to e^{+}e^{-}X(1835)$ | $1.75\pm 0.29\pm 0.05$ |  Ablikim  _et al._ (2022b)
$J/\psi\to\gamma^{*}\eta_{(2)}(718)$ | $2.44\pm 0.04$ |  Jiang  _et al._ (2023)
$J/\psi\to\gamma^{*}\eta_{(1)}(783)$ | $2.47\pm 0.02$ | this work
Figure 5: The $s$ dependence of the pole parameter $\Lambda$. The data points
indicate the experimental and lattice result of $\Lambda$ at different values
of $s=m_{P}^{2}$ (listed in Table 4), where $m_{P}$ is the mass of the
psuedoscalar meson in the process $J/\psi\to P\gamma^{*}$. The grey band shows
the model $\Lambda(s)=\Lambda_{1}(1-\frac{s}{\Lambda_{2}^{2}})$ with the
fitted parameters $\Lambda_{1}=2.64(4)~{}\mathrm{GeV}$ and
$\Lambda_{2}=2.97(33)~{}\mathrm{GeV}$. The $\chi^{2}$ per degree of freedom is
$\chi^{2}/\mathrm{d.o.f}=0.45$.

In principle, the production of each light pseudoscalar $P$ in the $J/\psi$
radiative decay or the Dalitz decay undergoes the same procedure that the
$c\bar{c}$ pair emits a photon of the virtuality $q^{2}$ and then annihilates
into gluons, whose invariant mass squared is labelled as $s$. Since the single
pole model describes $M(q^{2})$ very well while the $q^{2}$ and $s$ in the
$J/\psi-\gamma^{*}(q^{2})-(gg\cdots)^{*}(s)$ vertex are correlated, one
expects the the $s$-dependence of $\Lambda$. We assume a linear function form
for $\Lambda(s)$

$\Lambda(s)=\Lambda_{1}\left(1-\frac{s}{\Lambda_{2}^{2}}\right).$ (24)

Then using the values of $\Lambda$ in Table 4 that are measured from
experiments and lattice QCD studies at different $s=m_{P}^{2}$, the parameters
$\Lambda_{1}$ and $\Lambda_{2}$ can be fitted through the above equation.
Finally, we get

$\Lambda_{1}=2.64(4)~{}\mathrm{GeV},~{}~{}~{}\Lambda_{2}=2.97(33)~{}\mathrm{GeV}$
(25)

with $\chi^{2}$ per degree of freedom $\chi^{2}/\mathrm{d.o.f}=0.45$. The
values of $\Lambda_{1,2}$ in Eq. (25) can give inputs for theoretical and
experimental studies. Taking the process $J/\psi\to\eta^{\prime}e^{+}e^{-}$
for instance, the experimental value of $\Lambda$ has huge uncertainties, but
the model in Eq. (24) with the parameters in Eq. (25) gives a more precise
prediction

$\Lambda(s=m_{\eta^{\prime}}^{2})=2.36(3)~{}\mathrm{GeV}.$ (26)

Then according to Eq. (22) and using the experimental result of
$\mathrm{Br}(J/\psi\to\gamma\eta^{\prime})=5.25(7)\times 10^{-3}$, the
branching fraction of $J/\psi\to\eta^{\prime}e^{+}e^{-}$ is estimated to be
$6.05(3)(8)\times 10^{-5}$, which is compatible with the BESIII result
$6.59(7)(17)\times 10^{-5}$ Ablikim  _et al._ (2019b). When the $\rho$
resonance contribution is included, as did by BESIII for $J/\psi\to\eta
e^{+}e^{-}$ in Ref. Ablikim  _et al._ (2019a), the
$|F_{\psi\eta^{\prime}}(q^{2})|^{2}$ reads

$\displaystyle|F_{\psi\eta^{\prime}}(q^{2})|^{2}$ $\displaystyle=$
$\displaystyle|A_{\rho}|^{2}\left(\frac{m_{\rho}^{4}}{(q^{2}-m_{\rho}^{2})^{2}+m_{\rho}^{2}\Gamma_{\rho}^{2}}\right)$
(27) $\displaystyle+$
$\displaystyle|A_{\Lambda}|^{2}\left(\frac{1}{1-q^{2}/\Lambda^{2}}\right)^{2},$

where $A_{\rho}$ is the coupling constant of the $\rho$ meson and
$A_{\Lambda}$ is the coupling constant of the non-resonant contribution. For
$J/\psi\to\eta e^{+}e^{-}$, BESIII determines $A_{\rho}=0.23(4)$ and
$A_{\Lambda}=1.05(3)$ Ablikim  _et al._ (2019a), which give
$|F_{\psi\eta}(q^{2}\approx 0)|^{2}=1.11\pm 0.07\pm 0.07$. If we take the same
value for $A_{\rho}=0.23(4)$ and assume $A_{\Lambda}=1$ for the case of
$\eta^{\prime}$ (The $|F_{\psi\eta^{\prime}}(q^{2})|^{2}$ at $q^{2}\approx 0$
in Ref. Ablikim  _et al._ (2014) is consist with one within errors), then
using the PDG values of $m_{\rho}$ and $\Gamma_{\rho}$ Colquhoun  _et al._
(2023) we get

$\mathrm{Br}(J/\psi\to\eta^{\prime}e^{+}e^{-})=6.58_{-17}^{+21}(2)(9)\times
10^{-5},$ (28)

where the first error is is due to the uncertainty of $A_{\rho}$, the second
is from that of $\Lambda$, and the third is from that of the experimental
value of $\mathrm{Br}(J/\psi\to\gamma\eta^{\prime})$. This value agrees with
the experimental value better.

##  IV Summary

We generate a large gauge ensemble with $N_{f}=1$ dynamical strange quarks on
an anisotropic lattice with the anisotropy $a_{s}/a_{t}\approx 5.0$. The
pseudoscalar mass is measured to be $m_{\eta_{s}}=693.1(3)~{}\mathrm{MeV}$
without considering the quark annihilation effect, and
$m_{\eta_{(1)}}=783.0(5.5)~{}\mathrm{MeV}$ with the inclusion the quark
annihilation diagrams. We calculate the EM form factor $M(q^{2})$ for the
decay process $J/\psi\to\gamma^{*}(q^{2})\eta_{(1)}$ with $q^{2}$ being the
virtuality of the photon. By interpolating $M(q^{2})$ to the value at
$q^{2}=0$ through the VMD inspired single pole model in Eq. (14), the decay
width and the branching fraction of $J/\psi\to\gamma\eta_{(1)}$ is predicted
to be $\Gamma(J/\psi\to\gamma\eta_{(1)})=0.097(5)~{}\mathrm{keV}$ and
$\mathrm{Br}(J/\psi\to\gamma\eta_{(1)})=1.06(5)\times 10^{-3}$, respectively,
which are much smaller than those in the $N_{f}=2$ case and those in the
physical $N_{f}=2+1$ case. The difference among the partial widths
$\Gamma(J/\psi\to\gamma\eta_{(N_{f})})$ at different $N_{f}$ can be attributed
in part to the $\mathbf{U}_{A}(1)$ anomaly that induces a $N_{f}$ scaling.

It is interesting to see that $M(q^{2})$’s in $N_{f}=1,2$ are both well
described by the single pole model $M(q^{2})=M(0)/(1-q^{2}/\Lambda^{2})$.
Combined together with the known experimental results of the Dalitz decays
$J/\psi\to Pe^{+}e^{-}$ with $P$ being light pseudoscalar mesons, the $m_{P}$
dependence of the pole parameter $\Lambda$ is observed and can be expressed
approximately as $\Lambda(m_{P}^{2})=\Lambda_{1}(1-m_{P}^{2}/\Lambda_{2}^{2})$
with $\Lambda_{1}=2.64(4)~{}\mathrm{GeV}$ and
$\Lambda_{2}=2.97(33)~{}\mathrm{GeV}$. This result provide meaningful inputs
for future theoretical and experimental studied on Dalitz decays $J/\psi\to
Pe^{+}e^{-}$. As a direct application, this $m_{P}$ dependence expects a pole
parameter $\Lambda(s=m_{\eta^{\prime}}^{2})=2.36(3)~{}\mathrm{GeV}$, which is
more precise than the value $3.1(1.0)~{}\mathrm{GeV}$ measured by BESIII
Ablikim  _et al._ (2014) and whose prediction on
$\mathrm{Br}(J/\psi\to\eta^{\prime}e^{+}e^{-})$ agrees better with the
experimental value Ablikim  _et al._ (2019b).

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NNSFC) under Grants No. 11935017, No. 12293060, No. 12293065, No. 12293061,
No. 12293062, No. 12293063, No. 12075253, No. 12192264, No. 12175063, No.
12205311, No. 12070131001 (CRC 110 by DFG and NNSFC)), and the National Key
Research and Development Program of China (No. 2020YFA0406400) and the
Strategic Priority Research Program of Chinese Academy of Sciences (No.
XDB34030302). The Chroma software system Edwards and Joo (2005) and QUDA
library Clark  _et al._ (2010); Babich  _et al._ (2011) are acknowledged. The
computations were performed on the HPC clusters at Institute of High Energy
Physics (Beijing) and China Spallation Neutron Source (Dongguan), and the
ORISE computing environment.

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