# Homotopy types of gauge groups over Riemann surfaces

Masaki Kameko  Department of Mathematical Sciences, Shibaura Institute of
Technology, 307 Minuma-ku Fukasaku, Saitama-City 337-8570, Japan
<EMAIL_ADDRESS>,  Daisuke Kishimoto  Department of Mathematics,
Kyoto University, Kyoto, 606-8502, Japan<EMAIL_ADDRESS>and
Masahiro Takeda  Department of Mathematics, Kyoto University, Kyoto, 606-8502,
Japan<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$. We
study the homotopy types of gauge groups of principal $G$-bundles over Riemann
surfaces. This can be applied to an explicit computation of the homotopy
groups of the moduli spaces of stable vector bundles over Riemann surfaces.

###### Key words and phrases:

gauge group, Riemann surface, stable vector bundle, Samelson product

###### 2010 Mathematics Subject Classification:

57S05, 55Q15

##  1\. Introduction

Let $G$ be a compact connected Lie group, and let $P$ be a principal
$G$-bundle over a finite complex $X$. The _gauge group_ of $P$ is defined to
be the topological group of $G$-equivariant self-maps of $P$ which fix $X$.
There may be infinitely many distinct principal $G$-bundles over $X$. For
example, there are infinitely many bundles when $X$ is an orientable
4-manifold. Each bundle has a gauge group, so there may be potentially
certainly infinitely many gauge groups. However, Crabb and Sutherland [7]
showed that these gauge groups have only finitely many homotopy types. Then it
has been intensely studied the precise number of homotopy types of gauge
groups for specific $G$ and $X$. The study began with simply-connected Lie
groups [6, 10, 12, 15, 16, 18, 20, 28, 29], and recently, non-simply-connected
cases are also studied as in [11, 14, 17, 24].

In this paper, we study the homotopy types of gauge groups of principal
$G$-bundles over a compact connected Riemann surface, where
$\pi_{1}(G)\cong\mathbb{Z}$. This includes an important case, gauge groups of
principal $U(n)$-bundles over a Riemann surface, whose topology was first
studied by Atiyah and Bott [2]. To state the results, we need to define an
intrinsic structure of $G$. Suppose $\pi_{1}(G)\cong\mathbb{Z}$. Then there
are a compact connected simply-connected Lie group $H$ and a subgroup $C$ of
the center of $S^{1}\times H$ such that

(1.1) $G\cong(S^{1}\times H)/C$

Note that $H$ is uniquely determined by $G$, but $C$ is not. For example, if
$G=S^{1}\times H$, then $C$ can be any finite subgroup of $S^{1}\times
1\subset S^{1}\times H$. We define

$s(G)=|p_{2}(C)|$

where $p_{2}\colon S^{1}\times H\to H$. By Theorem 1.4 below, we can see that
$s(G)$ is independent from the choice of $C$.

######  Example 1.1.

Since $U(n)$ is the quotient of $S^{1}\times SU(n)$ by the diagonal central
subgroup isomorphic to $\mathbb{Z}/n$, we have $s(U(n))=n$.

Let $X$ be a compact connected Riemann surface. Then there is a one-to-one
correspondence between principal $G$-bundles over $X$ and
$\pi_{2}(BG)\cong\mathbb{Z}$. Let $\mathcal{G}_{k}(X,G)$ denote the gauge
group of a principal $G$-bundle over $X$ corresponding to $k\in\mathbb{Z}$.
Now we state our results.

######  Theorem 1.2.

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$, and
let $X$ be a compact connected Riemann surface. If $(k,s(G))=(l,s(G))$, then
$\mathcal{G}_{k}(X,G)$ and $\mathcal{G}_{l}(X,G)$ are homotopy equivalent
after localizing at any prime or zero.

We remark that the $p$-localization of a disconnected space will mean the
disjoint union of the $p$-localization of path-connected components. For a
prime $p$, Theriault [27] gave a $p$-local homotopy decomposition of
$\mathcal{G}_{k}(X,U(p))$, which implies the converse implication of Theorem
1.2 holds for $G=U(p)$. We will prove the converse implication of Theorem 1.2
holds for other Lie groups.

######  Theorem 1.3.

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$, and
let $X$ be a compact connected Riemann surface. If $G$ is locally isomorphic
to $S^{1}\times SU(n)^{r}$ or $S^{1}\times SU(4n-2)^{s}\times Sp(2n-1)^{t}$,
then the following statements are equivalent:

  1. (1)

$(k,s(G))=(l,s(G))$;

  2. (2)

$\mathcal{G}_{k}(X,G)$ and $\mathcal{G}_{l}(X,G)$ are homotopy equivalent
after localizing at any prime or zero.

The homotopy type of a gauge group $\mathcal{G}_{k}(X,G)$ is closely related
with a Samelson product in $G$, as we will see in Section 2. In our context,
the Samelson product of a generator of $\pi_{1}(G)\cong\mathbb{Z}$ and the
identity map of $G$ is of particular importance. Then we will prove the
following theorem, which is of independent interest.

######  Theorem 1.4.

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$, and
let $\epsilon$ denote a generator of $\pi_{1}(G)$. Then the Samelson product
$\langle\epsilon,1_{G}\rangle$ in $G$ is of order $s(G)$.

Now we consider an application. Gauge groups over a Riemann surface are
closely related to the moduli spaces of stable vector bundles over a Riemann
surface as follows. Let $X$ be a Riemann surface of genus $g$, and let
$M(n,k)$ denote the moduli space of stable vector bundles over $X$ of rank $n$
and degree $k$. Daskalopoulos and Uhlenbeck [8] showed that there is an
isomorphism

$\pi_{i}(M(n,k))\cong\pi_{i-1}(\mathcal{G}_{k}(X,U(n)))$

for $2<i\leq 2(g-1)(n-1)-2$ and $(n,k)\neq(2,2)$. There is a polystable Higgs
bundle analog due to Bradlow, Garcia-Prada and Gothen [5]. Then we can compute
the homotopy groups of these moduli spaces through the following homotopy
decomposition.

######  Theorem 1.5.

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$, and
let $X$ be a compact connected Riemann surface of genus $g$. If $s(G)$ divides
$k$, then

$\mathcal{G}_{k}(X,G)\simeq G\times(\Omega G)^{2g}\times\Omega^{2}G.$

Moreover, the above homotopy equivalence also holds after localizing at $p$
whenever $p$ does not divide $s(G)$.

The paper is structured as follows Section 2 recalls a connection of gauge
groups to Samelson products, and then proves Theorems 1.2 and 1.5 by assuming
Theorem 1.4 holds. Section 3 shows some general results on Samelson products
in a Lie group, which will be used for a practical computation. Sections 4 and
5 compute the Samelson products in $G$ when $H$ is simple. Finally, Section 6
collects all results so far together to prove Theorems 1.4 and Theorem 1.3.

### Acknowledgement

The authors were partly supported by JSPS KAKENHI No. 17K05263 (Kameko), No.
17K05248 (Kishimoto) and No. 21J10117 (Takeda).

##  2\. Gauge groups and Samelson products

This section recalls a connection of gauge groups to Samelson products, and
then Theorems 1.2 and 1.5 are proved by assuming Theorem 1.4 holds. First, we
recall a connection of gauge groups to mapping spaces. Let $G$ be a
topological group, and let $P$ be a principal $G$-bundle over a base $X$,
which is classified by a map $\alpha\colon X\to BG$. Recall that the gauge
group of $P$, denoted by $\mathcal{G}(P)$, is a topological group of
$G$-equivariant self-maps of $P$ which fix $X$. Gottlieb [9] proved that there
is a natural homotopy equivalence

$B\mathcal{G}(P)\simeq\operatorname{map}(X,BG;\alpha)$

where $\operatorname{map}(A,B;f)$ denotes the path component of the space of
maps $\operatorname{map}(A,B)$ containing a map $f\colon A\to B$. Then
evaluating at the basepoint of $X$ yields a homotopy fibration

(2.1) $\operatorname{map}_{*}(X,BG;\alpha)\to B\mathcal{G}(P)\to BG$

where $\operatorname{map}_{*}(X,BG;\alpha)$ is a subspace of
$\operatorname{map}(X,BG;\alpha)$ consisting of basepoint preserving maps. So
the gauge group $\mathcal{G}(P)$ is homotopy equivalent to the homotopy fiber
of the connecting map

$\partial_{\alpha}\colon G\to\operatorname{map}_{*}(X,BG;\alpha)$

of the above homotopy fibration.

Next, we assume $X=S^{n}$ for $n\geq 1$ and describe the connecting map
$\partial_{\alpha}$. Clearly, there is a homotopy equivalence
$\operatorname{map}_{*}(S^{n},BG;\alpha)\simeq\Omega^{n-1}_{0}G$, where
$\Omega^{n-1}_{0}G$ denotes the path component of $\Omega^{n-1}G$ containing
the constant map. Then by adjointing, the connecting map $\partial_{\alpha}$
corresponds to a map

$d_{\alpha}\colon S^{n-1}\wedge G\to G$

The original definition of Whitehead products in [30] and adjointness of
Whitehead products and Samelson products prove the following.

######  Lemma 2.1.

The map $d_{\alpha}$ is the Samelson product
$\langle\bar{\alpha},1_{G}\rangle$ in $G$, where $\bar{\alpha}\colon
S^{n-1}\to G$ is the adjoint of $\alpha\colon S^{n}\to BG$.

The following lemma due to Theriault [26] shows how to identify the homotopy
type of a gauge group $\mathcal{G}(P)$ from the order of a Samelson product
$\langle\bar{\alpha},1_{G}\rangle$.

######  Lemma 2.2.

Suppose that a map $f\colon X\to Y$ into an H-space $Y$ is of order
$n<\infty$. Then $(n,k)=(n,l)$ implies $F_{k}{}_{(p)}\simeq F_{l}{}_{(p)}$ for
any prime $p$, where $F_{k}$ denotes the homotopy fiber of a map $k\circ
f\colon X\to Y$

Finally, we recall a homotopy decomposition of a gauge group. Theriault [25]
showed a homotopy decomposition of a gauge group over principal $U(n)$-bundle
over a Riemann surface. We can easily see that his proof works in verbatim for
any compact connected Lie group $G$ with $\pi_{1}(G)\cong\mathbb{Z}$. Then we
get:

######  Proposition 2.3.

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$, and
let $X$ be a compact connected Riemann surface of genus $g$. Then there is a
homotopy equivalence

$\mathcal{G}_{k}(X,G)\simeq(\Omega G)^{2g}\times\mathcal{G}_{k}(S^{2},G).$

Now we prove Theorems 1.2 and 1.5 by assuming Theorem 1.4 holds.

###### Proof of Theorem 1.2.

Combine Lemmas 2.1, 2.2, Proposition 2.3 and Theorem 1.4. ∎

###### Proof of Theorem 1.5.

By Lemma 2.1 and Theorem 1.4, if $k$ is divisible by $s(G)$, then
$\mathcal{G}_{k}(S^{2},G)$ is homotopy equivalent to the homotopy fiber of the
constant map $G\to\Omega_{0}G$. So since $\pi_{2}(G)=0$,
$\mathcal{G}_{k}(S^{2},G)\simeq G\times\Omega^{2}G$. Thus by Proposition 2.3,
the proof is done. ∎

##  3\. Samelson products in Lie groups

This section shows some criteria for computing Samelson products in a Lie
group. For the rest of the paper, we will use the following notation.

  * •

Let $G$ be a compact connected Lie group with $\pi_{1}(G)\cong\mathbb{Z}$.

  * •

Let $\epsilon_{G}$ denote a generator of $\pi_{1}(G)\cong\mathbb{Z}$

  * •

Let $H$ and $C$ be as in the decomposition (1.1).

  * •

Let $j_{H}\colon\Sigma H\to BH$ denote the natural map.

  * •

Let $p_{G}\colon S^{1}\times H\to G$ denote the projection.

  * •

Let $p_{1}\colon S^{1}\times H\to S^{1}$ and $p_{2}\colon S^{1}\times H\to H$
denote projections.

  * •

Let $K=H/p_{2}(C)$.

  * •

Let $q_{G}\colon G\to K$ and $\bar{q}_{K}\colon H\to K$ denote projections.

We will abbreviate $\epsilon_{G},j_{H},p_{G},q_{G},\bar{q}_{K}$ to
$\epsilon,j,p,q,\bar{q}$, respectively, if $G,H,K$ are clear from the context.
First, we show two properties of the group $C$.

######  Lemma 3.1.

The abelian group $p_{2}(C)$ is cyclic.

###### Proof.

There is a fibration

(3.1) $S^{1}\to G\xrightarrow{q}K$

and so by the homotopy exact sequence, we can see that$\pi_{1}(K)\cong
p_{2}(C)$ is a quotient of $\pi_{1}(G)\cong\mathbb{Z}$. Then $p_{2}(C)$ is a
cyclic group, as stated. ∎

######  Lemma 3.2.

We may choose a group $C$ such that $|p_{1}(C)|=s(G)$.

###### Proof.

Note that $p_{2}(C)$ is a cyclic group by Lemma 3.1. We prove an inequality
$|p_{1}(C)|\geq s(G)$ always holds. If $|p_{1}(C)|<s(G)$, then
$C_{1}=|p_{1}(C)|C$ is a non-trivial subgroup of the center of $1\times
H\subset S^{1}\times H$. In particular, there is a covering

$C/C_{1}\to(S^{1}\times H)/C_{1}\to G.$

Then $\pi_{1}(G)\cong\mathbb{Z}$ includes a non-trivial finite abelian group
$C_{1}$, which is a contradiction. Thus $|p_{1}(C)|\geq s(G)$.

Suppose that $|p_{1}(C)|>s(G)$. Then $C_{2}=s(G)C$ is a finite subgroup of
$S^{1}\times 1\subset S^{1}\times H$. Then $(S^{1}\times H)/C_{2}\cong
S^{1}\times H$, implying

$G\cong(S^{1}\times H)/C\cong((S^{1}\times
H)/C_{2})/(C/C_{2})\cong(S^{1}\times H)/(C/C_{2}).$

By definition, $|p_{1}(C/C_{2})|=|p_{2}(C/C_{1})|=|p_{2}(C)|=s(G)$, and thus
the proof is finished. ∎

By Lemma 3.1, $\pi_{1}(K)\cong p_{2}(C)$ is a cyclic group of order $s(G)$.
For the rest of this section, we will also use the following notation.

  * •

Let $\bar{\epsilon}_{K}$ denote a generator of $\pi_{1}(K)$.

We will abbreviate it by $\bar{\epsilon}$ if $K$ is clear from the context.

Next, we show an upper bound and a lower bound for the order of
$\langle\epsilon,1_{G}\rangle$.

######  Lemma 3.3.

The order of $\langle\epsilon,1_{G}\rangle$, hence $\langle\epsilon,p\rangle$,
divides $s(G)$.

###### Proof.

The proof of Lemma 3.1 implies $q\circ\epsilon=\bar{\epsilon}$. Then since $q$
is a homomorphism, we get

$q_{*}(s(G)\langle\epsilon,1_{G}\rangle)=s(G)\langle
q\circ\epsilon,q\rangle=\langle s(G)\bar{\epsilon},q\rangle=0.$

So since there is a fibration (3.1), $s(G)\langle\epsilon,1_{G}\rangle$ lifts
to a map $S^{1}\wedge G\to S^{1}$. Since $S^{1}\wedge G$ is simply-connected,
this lift is trivial, and thus $s(G)\langle\epsilon,1_{G}\rangle$ itself is
trivial, completing the proof. ∎

######  Lemma 3.4.

The order of $\langle\bar{\epsilon},\bar{q}\rangle$ divides the order of
$\langle\epsilon,p\rangle$.

###### Proof.

Let $i\colon H\to S^{1}\times H$ denote the inclusion. By definition, $q\circ
p\circ i=\bar{q}$, and the proof of Lemma 3.2 implies that
$q\circ\epsilon=\bar{\epsilon}$. Then

$(1\wedge i)^{*}\circ
q_{*}(\langle\epsilon,p\rangle)=q_{*}(\langle\epsilon,p\circ i\rangle)=\langle
q\circ\epsilon,q\circ p\circ i\rangle=\langle\bar{\epsilon},\bar{q}\rangle$

and so the proof is done. ∎

Finally, we give a cohomological criterion for the Samelson product
$\langle\bar{\epsilon},\bar{q}\rangle$ being non-trivial. For an algebra $A$,
let $QA$ denote the module of indecomposables.

######  Lemma 3.5.

Suppose there are $x,y,z\in QH^{*}(BK;\mathbb{Z}/p)$ and a Steenrod operation
$\theta$ satisfying the following conditions:

  1. (1)

$|y|=2$ and $QH^{n}(BK;\mathbb{Z}/p)=\langle z\rangle$ for $n>2$;

  2. (2)

$\theta(x)$ is decomposable and includes the term $y\otimes z$;

  3. (3)

$(\bar{q}\circ j)^{*}(z)$ is non-trivial and not included in any element of
$\theta(H^{*}(\Sigma H;\mathbb{Z}/p))$.

Then the Samelson product $\langle\bar{\epsilon},\bar{q}\rangle$ is non-
trivial.

###### Proof.

Suppose that $\langle\bar{\epsilon},\bar{q}\rangle$ is trivial. Let
$\hat{\epsilon}\colon S^{2}\to BK$ and $\hat{q}\colon\Sigma H\to BK$ denote
the adjoint of $\bar{\epsilon}$ and $\bar{q}$, respectively. Then by
adjointness of Samelson products and Whitehead products, the Whitehead product
$[\hat{\epsilon},\hat{q}]$ is trivial, so that there is a homotopy commutative
diagram

$\textstyle{S^{2}\vee\Sigma
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\epsilon}\vee\hat{q}}$$\textstyle{BK\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{2}\times\Sigma
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{BK.}$

By the first condition, $\hat{\epsilon}^{*}(y)=u$, where $u$ is a generator of
$H^{2}(S^{2};\mathbb{Z}/p)\cong\mathbb{Z}/p$. Then by the first and the second
conditions, $\mu^{*}(\theta(x))$ includes the term $u\otimes\hat{q}^{*}(z)$.
Since $\hat{q}=\bar{q}\circ j$, the third condition implies
$u\otimes\hat{q}^{*}(z)\neq 0$. On the other hand, by the third condition,
$\theta(\mu^{*}(x))$ cannot include the term $u\otimes\hat{q}^{*}(z)$. Thus
since $\mu^{*}(\theta(x))=\theta(\mu^{*}(x))$, we obtain a contradiction.
Therefore $\langle\bar{\epsilon},\bar{q}\rangle$ is non-trivial, completing
the proof. ∎

Recall that compact simply-connected simple Lie groups with non-trivial center
are

$SU(n),\quad Sp(n),\quad Spin(n)\quad(n\geq 7),\quad E_{6},\quad E_{7}.$

Then in the following two sections, we will compute the Samelson product
$\langle\epsilon,p\rangle$ for $H$ being one of the above Lie groups.

##  4\. Classical case

This section determines the order of the Samelson product
$\langle\epsilon,p\rangle$ for $H=SU(n),Sp(n),Spin(n)$.

###  4.1. The case $H=SU(n)$

First we consider the case $H=SU(n)$.

######  Proposition 4.1.

If $H=SU(n)$, then $\langle\epsilon,p\rangle$ is of order $s(G)$.

###### Proof.

By Lemma 3.3, it suffices to show that the order of $\langle\epsilon,p\rangle$
is a non-zero multiple of $s(G)$. The center of $SU(n)$ is isomorphic to
$\mathbb{Z}/n$. Then since $U(n)=S^{1}\times_{\mathbb{Z}/n}SU(n)$, it follows
from Lemma 3.2 that there is a homomorphism $\rho\colon G\to U(n)$ which is a
$\frac{n}{s(G)}$ sheeted covering. Let $\alpha_{2i-1}$ denote a generator of
$\pi_{2i-1}(U(n))\cong\mathbb{Z}$ for $i=1,2,\ldots,n$. Then

$\rho_{*}(\epsilon)=\frac{n}{s(G)}\alpha_{1}.$

On the other hand, it is shown in [4] that the order of
$\langle\alpha_{1},\alpha_{2n-1}\rangle$ is a non-zero multiple of $n$. Since
$\rho_{*}\colon\pi_{2n-1}(G)\to\pi_{2n-1}(U(n))$ is an isomorphism, there is
$\tilde{\alpha}\in\pi_{2n-1}(G)$ such that
$\rho_{*}(\tilde{\alpha})=\alpha_{2n-1}$. Then since

$\rho_{*}(\langle\epsilon,\tilde{\alpha}\rangle)=\langle\rho_{*}(\epsilon),\rho_{*}(\tilde{\alpha})\rangle=\langle\frac{n}{s(G)}\alpha_{1},\alpha_{2n-1}\rangle=\frac{n}{s(G)}\langle\alpha_{1},\alpha_{2n-1}\rangle,$

the order of $\rho_{*}(\langle\epsilon,\tilde{\alpha}\rangle)$ is a non-zero
multiple of $s(G)$. Thus since the map
$\rho_{*}\colon\pi_{2n}(G)\to\pi_{2n}(U(n))$ is an isomorphism, the order of
$\langle\epsilon,\tilde{\alpha}\rangle$ is a non-zero multiple of $s(G)$ too.
Since $p_{*}\colon\pi_{2n-1}(S^{1}\times SU(n))\to\pi_{2n-1}(G)$ is an
isomorphism, there is $\beta\in\pi_{2n-1}(S^{1}\times SU(n))$ such that
$p\circ\beta=\tilde{\alpha}$. Thus since
$(1\wedge\beta)^{*}(\langle\epsilon,p\rangle)=\langle\epsilon,\tilde{\alpha}\rangle$,
the order of $\langle\epsilon,p\rangle$ is a non-zero multiple of $s(G)$,
completing the proof. ∎

###  4.2. The case $H=Sp(n)$

Next, we consider the case $H=Sp(n)$. Recall that the center of $Sp(n)$ is
isomorphic to $\mathbb{Z}/2$, and the quotient of $Sp(n)$ by its center is
denoted by $PSp(n)$. We apply Lemma 3.5 to the case $H=Sp(n)$. To this end, we
compute the mod 2 cohomology of $BPSp(2n)$ in low dimensions.

######  Lemma 4.2.

Let $\Delta=\\{\pm(1,\ldots,1)\in Sp(2)^{n}\\}$. Then for $*\leq 7$

$H^{*}(B(Sp(2)^{n}/\Delta);\mathbb{Z}/2)=\mathbb{Z}/2[x_{2},x_{3},x_{5}]\otimes\bigotimes_{k=1}^{n}\mathbb{Z}/2[x_{4,k}],\quad
Sq^{2}x_{4,k}=x_{2}x_{4,k}$

where $|x_{i}|=i$ and $|x_{4,k}|=4$.

###### Proof.

Consider the Serre spectral sequence for a homotopy fibration
$\mathbb{R}P^{\infty}\to BSp(2)^{n}\to B(Sp(2)^{n}/\Delta)$. Since
$H^{*}(\mathbb{R}P^{\infty};\mathbb{Z}/2)=\mathbb{Z}/2[w]$ with $|w|=1$,

$H^{*}(\mathbb{R}P^{\infty};\mathbb{Z}/2)=\Delta(w,Sq^{1}w,Sq^{2}Sq^{1}w).$

for $*\leq 7$, where $\Delta(a_{1},\ldots,a_{k})$ denotes the simple system of
generators in $a_{1},\ldots,a_{k}$. Clearly, $\tau(w)=x_{2}$ for a generator
$x_{2}$ of $H^{2}(B(Sp(2)^{n}/\Delta);\mathbb{Z}/2)\cong\mathbb{Z}/2$, where
$\tau$ denotes the transgression. Then we get
$H^{*}(B(Sp(2)^{n}/\Delta);\mathbb{Z}/2)$ for $*\leq 7$ as stated. It remains
to show $Sq^{2}x_{4,k}=x_{2}x_{4,k}$. Recall that

(4.1) $H^{*}(BSO(n);\mathbb{Z}/2)=\mathbb{Z}/2[w_{2},w_{3},\ldots,w_{n}],\quad
Sq^{i}w_{j}=\sum_{k=0}^{i}\binom{j+k-i-1}{k}w_{i-k}w_{j+k}$

where $w_{i}$ is the $i$-th Stiefel-Whitney class. Then since $PSp(2)\cong
SO(5)$,

$H^{*}(BPSp(2);\mathbb{Z}/2)=\mathbb{Z}/2[y_{2},y_{3},y_{4},y_{5}],\quad
Sq^{2}y_{4}=y_{2}y_{4}$

where $|y_{i}|=i$. Let $q_{k}\colon Sp(2)^{n}\to Sp(2)$ denote the $k$-th
projection for $k=1,2,\ldots,n$. Then $q_{k}^{*}(y_{2})=x_{2}$ and
$q_{k}^{*}(y_{4})=x_{4,k}$. Thus we obtain $Sq^{2}x_{4,k}=x_{2}x_{4,k}$,
completing the proof. ∎

######  Proposition 4.3.

For $*\leq 7$,

$H^{*}(BPSp(n);\mathbb{Z}/2)=\mathbb{Z}/2[x_{2},x_{3},x_{4},x_{5}],\quad
Sq^{2}x_{4}=x_{4}x_{2},\quad|x_{i}|=i.$

###### Proof.

We can compute the mod 2 cohomology of $BPSp(2n)$ in the same way as in the
proof of Lemma 4.2 by considering a homotopy fibration
$\mathbb{R}P^{\infty}\to BSp(2n)\to BPSp(2n)$. Then it remains to show
$Sq^{2}x_{4}=x_{4}x_{2}$. Let $\Delta$ be as in Lemma 4.2. Then there is an
inclusion $i\colon Sp(2)^{n}/\Delta\to PSp(2n)$. Clearly, $i^{*}(x_{2})=x_{2}$
and $i^{*}(x_{4})=x_{4,1}+\cdots+x_{4,n}$. Then we obtain
$Sq^{2}x_{4}=x_{4}x_{2}$ by Lemma 4.2. ∎

Now we prove:

######  Proposition 4.4.

If $H=Sp(n)$, then $\langle\epsilon,p\rangle$ is of order $s(G)$.

###### Proof.

Since the center of $Sp(n)$ is isomorphic to $\mathbb{Z}/2$, we only consider
$G=S^{1}\times_{\mathbb{Z}/2}Sp(n)$. In this case, $s(G)=2$, so by Lemmas 3.3,
it suffices to show $\langle\epsilon,p\rangle$ is non-trivial. First, we
consider the case $G=S^{1}\times_{\mathbb{Z}/2}Sp(2n-1)$. The natural
inclusion $Sp(2n-1)\to SU(4n-2)$ sends the center of $Sp(2n-1)$ injectively
into the center of $SU(4n-2)$. Then we get a homomorphism $G\to
S^{1}\times_{\mathbb{Z}/2}SU(4n-2)$ which is an isomorphism in $\pi_{1}$. It
is well known that the induced map
$\pi_{8n-5}(Sp(2n-1))\to\pi_{8n-5}(SU(4n-2))$ is an isomorphism, hence so is
$\pi_{8n-5}(G)\to\pi_{8n-5}(S^{1}\times_{\mathbb{Z}/2}SU(4n-2))$. Then the
proof of Proposition 4.1 implies that the Samelson product
$\langle\epsilon,p\rangle$ is non-trivial.

Next, we consider $G=S^{1}\times_{\mathbb{Z}/2}Sp(2n)$. We apply Lemma 3.5 to
$K=PSp(2n)$ by setting $x=z=x_{4},\,y=x_{2}$ and $\theta=Sq^{2}$. By
Proposition 4.3, the first and the second conditions of Lemma 3.5 are
satisfied. The proof of Proposition 4.3 implies $\bar{q}^{*}(x_{4})$ is non-
trivial, where $H^{4}(BSp(2n);\mathbb{Z}/2)\cong
QH^{4}(BSp(2n);\mathbb{Z}/2)\cong\mathbb{Z}/2$. Since the map

$j^{*}\colon QH^{4}(BSp(2n);\mathbb{Z}/2)\to\Sigma
QH^{3}(Sp(2n);\mathbb{Z}/2)$

is an isomorphism, we have $(\bar{q}\circ j)^{*}(x_{4})\neq 0$. Moreover, for
degree reasons, $(\bar{q}\circ j)^{*}(x_{4})$ is not included in any element
of $\theta(H^{*}(\Sigma Sp(2n);\mathbb{Z}/2))$. Then the third condition of
Lemma 3.5 is also satisfied. Thus $\langle\bar{\epsilon},\bar{q}\rangle$ is
non-trivial, and so by Lemma 3.4, $\langle\epsilon,p\rangle$ is non-trivial
too. ∎

###  4.3. The case $H=Spin(n)$

Finally, we consider the case $H=Spin(n)$. We show some properties of the mod
2 cohomology of $BSpin(n)$ that we are going to use. Recall that the mod 2
cohomology of $BSO(n)$ is given as in (4.1).

######  Lemma 4.5.

  1. (1)

The mod 2 cohomology of $BSpin(n)$ is given by

$H^{*}(BSpin(n);\mathbb{Z}/2)=\mathbb{Z}/2[u_{2},u_{3},\ldots,u_{n},z]/(u_{2},Sq^{2^{k}}Sq^{2^{k-1}}\cdots
Sq^{1}u_{2}\mid k\geq 0)$

where $\bar{q}_{SO(n)}^{*}(w_{j})=u_{j}$, $|z|=2^{h}$ for some $h>0$ and
$Sq^{i}u_{j}$ is computed by replacing $w_{j}$ with $u_{j}$ in the formula
(4.1).

  2. (2)

For $2\leq i\leq n$ with $i\neq 2^{k}+1$, $j_{Spin(n)}^{*}(u_{i})\neq 0$.

###### Proof.

(1) is a result of Quillen [23]. We prove the statement (2). It is well known
that $(j^{\prime})^{*}(w_{i})\neq 0$ for $i=2,3,\ldots,n$, where
$j^{\prime}\colon\Sigma SO(n)\to BSO(n)$ is the natural map. On the other
hand, it is shown in [13] that
$(\Sigma\bar{q}_{SO(n)})^{*}\circ(j^{\prime})^{*}(w_{i})\neq 0$. Then for
$2\leq i\leq n$ with $i\neq 2^{k}+1$,

$0\neq(\Sigma\bar{q}_{SO(n)})^{*}\circ(j^{\prime})^{*}(w_{i})=j^{*}\circ\bar{q}_{SO(n)}(w_{i})=j^{*}(u_{i}).$

Thus the statement (2) is proved. ∎

The following lemma is easily deduced from the formula (4.1).

######  Lemma 4.6.

In $H^{*}(BSO(n);\mathbb{Z}/2)$, we have:

  1. (1)

If $n\equiv 0,1\mod 4$, then $Sq^{2}w_{i}$ for $i=n-3,n-1$ are decomposable
and $Sq^{2}w_{n-1}$ includes the term $w_{2}w_{n-1}$;

  2. (2)

if $n\equiv 2\mod 8$, then $Sq^{5}w_{i}$ for $i=n-4,n-9$ are decomposable and
$Sq^{5}w_{n-4}$ includes the term $w_{2}w_{n-1}$;

  3. (3)

if $n\equiv 6\mod 8$, then $Sq^{3}w_{i}$ for $i=n-2,n-4$ are decomposable and
$Sq^{3}w_{n-2}$ includes the term $w_{2}w_{n-1}$;

  4. (4)

if $n\equiv 3\mod 4$, then $Sq^{2}w_{i}$ for $i=n-2,n$ are decomposable and
$Sq^{2}w_{n}$ includes the term $w_{2}w_{n}$.

Let $C_{n}$ denote the center of $Spin(n)$. Then we have:

  1. (1)

$C_{2n+1}\cong\mathbb{Z}/2$ and $Spin(2n+1)/C_{2n+1}\cong SO(2n+1)$.

  2. (2)

$C_{4n+2}\cong\mathbb{Z}/4$ and $Spin(4n+2)/(\mathbb{Z}/2)\cong SO(4n+2)$.

  3. (3)

$C_{4n}\cong\mathbb{Z}/2\times\mathbb{Z}/2$, $Spin(4n)/(\mathbb{Z}/2\times
1)\cong SO(4n)$ and $Spin(4n)/(1\times\mathbb{Z}/2)\cong Ss(4n)$.

######  Proposition 4.7.

If $H=Spin(n)$ and $K=SO(n)$, then $\langle\epsilon,p\rangle$ is of order
$s(G)$.

###### Proof.

We only give a proof for $n$ odd because the case $n$ even is quite similarly
proved. We apply Lemma 3.5 by setting $x=z=w_{n-1},\,y=w_{2}$ and
$\theta=Sq^{2}$. By Lemma 4.6, the first and the second conditions of Lemma
3.5 are satisfied. By Lemmas 4.5 and 4.6, $(\bar{q}\circ j)^{*}(w_{n-1})$ is
non-trivial and not included in any element of $Sq^{2}(H^{*}(\Sigma
Spin(n);\mathbb{Z}/2))$. Then the third condition of Lemma 3.5 is also
satisfied, so $\langle\bar{\epsilon},\bar{q}\rangle\neq 0$. Thus since
$s(G)=2$, Lemmas 3.3 and 3.4 complete the proof. ∎

Let $PO(n)=Spin(n)/C_{n}$. Then we have:

######  Corollary 4.8.

If $H=Spin(4n+2)$ and $K=PO(4n+2)$, then $\langle\epsilon,p\rangle$ is of
order $s(G)$.

###### Proof.

Let $\bar{\rho}\colon SO(4n+2)\to PO(4n+2)$ denote the projection. Then
$\bar{\rho}_{*}(\bar{\epsilon}_{SO(4n+2)})=2\bar{\epsilon}_{PO(4n+2)}$. Since
$S^{1}\wedge Spin(4n+2)$ is simply-connected, the map

$\bar{\rho}_{*}\colon[S^{1}\wedge Spin(4n+2),SO(4n+2)]\to[S^{1}\wedge
Spin(4n+2),PO(4n+2)]$

is an isomorphism. By definition,
$\bar{q}_{PO(4n+2)}=\bar{\rho}\circ\bar{q}_{SO(4n+2)}$. So by Proposition 4.7,

$2\langle\bar{\epsilon}_{PO(4n+2)},\bar{q}_{PO(4n+2)}\rangle=\bar{\rho}_{*}(\langle\bar{\epsilon}_{SO(4n+2)},\bar{q}_{SO(4n+2)}\rangle)\neq
0.$

Then by Lemma 3.3, the order of
$\langle\bar{\epsilon}_{PO(4n+2)},\bar{q}_{PO(4n+2)}\rangle$ is a non-zero
multiple of $s(G)=4$. Thus the proof is complete by Lemmas 3.3 and 3.4. ∎

Let $\Delta$ denote the diagonal subgroup of $\mathbb{Z}/2\times\mathbb{Z}/2$.

######  Proposition 4.9.

If $H=Spin(4n)$ and $p_{2}(C)=1\times\mathbb{Z}/2,\Delta$, then
$\langle\epsilon,p\rangle$ is of order $s(G)$.

###### Proof.

By triality of $Spin(8)$, the case $H=Spin(8)$ is proved by Proposition 4.7.
Then we assume $n>2$. The mod 2 cohomology of $PO(4n)$ was determined by Baum
and Browder [3] such that

$H^{*}(PO(4n);\mathbb{Z}/2)=\mathbb{Z}/2[v]/(v^{2^{r}})\otimes\Delta(u_{1},\ldots,\widehat{u_{2^{r}-1}},\ldots,u_{n-1}),\quad\bar{\rho}^{*}(u_{i})=w_{i}$

where $4n=2^{r}(2m+1)$, $|v|=1$ and $|u_{i}|=i$. The elements $v$ and $u_{1}$
correspond respectively to generators of subgroups $1\times\mathbb{Z}/2$ and
$\mathbb{Z}/2\times 1$ of $C_{4n}\cong\mathbb{Z}/2\times\mathbb{Z}/2$. The
Hopf algebra structure of $H^{*}(PO(4n);\mathbb{Z}/2)$ was also determined
such that

$\bar{\phi}(v)=0\quad\text{and}\quad\bar{\phi}(u_{i})=\sum_{j=1}^{i-1}\binom{i}{j}u_{j}\otimes
v^{i-j}$

where $\bar{\phi}$ is the reduced diagonal map. Let $\gamma\colon
PO(4n)^{2}\to PO(4n)$ denote the commutator map. Since $\bar{\epsilon}(v)\neq
0$, it suffices to show $\gamma^{*}(x)$ includes the term $v\otimes y$ such
that $\rho^{*}(y)\neq 0$, where $\rho\colon Spin(4n)\to PO(4n)$ denotes the
projection. Let $\mu\colon PO(4n)^{2}\to PO(4n)$ and $\Delta\colon PO(4n)\to
PO(4n)^{2}$ denote the multiplication and the diagonal map, respectively. Let
$\iota\colon PO(4n)\to PO(4n)$ be a map given by $\iota(x)=x^{-1}$, and let
$T\colon PO(4n)^{2}\to PO(4n)^{2}$ be the switching map. Then

$\gamma=\mu\circ(\mu\times\mu)\circ(1\times
1\times\iota\times\iota)\circ(1\times T\times 1)\circ(\Delta\times\Delta).$

Let $I_{k}=\widetilde{H}^{*}(PO(n)^{k};\mathbb{Z}/2)$. Now we compute
$\gamma^{*}(u_{i})$:

$\displaystyle u_{i}$ $\displaystyle\xmapsto{\mu^{*}}$ $\displaystyle
u_{i}\otimes 1+1\otimes u_{i}+iu_{i-1}\otimes v\mod I_{2}^{3}$
$\displaystyle\xmapsto{(\mu\times\mu)^{*}}$ $\displaystyle i(u_{i-1}\otimes
v\otimes 1\otimes 1+1\otimes 1\otimes u_{i-1}\otimes v+u_{i-1}\otimes 1\otimes
1\otimes v+1\otimes u_{i-1}\otimes v\otimes 1)$ $\displaystyle\mod
I_{1}\otimes 1\otimes I_{1}\otimes 1+1\otimes I_{1}\otimes 1\otimes
I_{1}+I_{4}^{3}$ $\displaystyle\xmapsto{(1\times
1\times\iota\times\iota)^{*}}$ $\displaystyle i(u_{i-1}\otimes v\otimes
1\otimes 1+1\otimes 1\otimes u_{i-1}\otimes v-u_{i-1}\otimes 1\otimes 1\otimes
v-1\otimes u_{i-1}\otimes v\otimes 1)$ $\displaystyle\mod I_{1}\otimes
1\otimes I_{1}\otimes 1+1\otimes I_{1}\otimes 1\otimes I_{1}+I_{4}^{3}$
$\displaystyle\xmapsto{(1\times T\times 1)^{*}}$ $\displaystyle
i(u_{i-1}\otimes 1\otimes v\otimes 1+1\otimes u_{i-1}\otimes 1\otimes
v-u_{i-1}\otimes 1\otimes 1\otimes v-1\otimes v\otimes u_{i-1}\otimes 1)$
$\displaystyle\mod I_{1}\otimes I_{1}\otimes 1\otimes 1+1\otimes 1\otimes
I_{1}\otimes I_{1}+I_{4}^{3}$
$\displaystyle\xmapsto{(\Delta\times\Delta)^{*}}$ $\displaystyle
i(u_{i-1}\otimes y-y\otimes u_{i-1})\mod I_{1}\otimes 1+1\otimes
I_{1}+I_{2}^{3}.$

Then for $n$ odd, $\gamma^{*}(u_{7})$ includes the term $v\otimes u_{6}$,
where $\rho^{*}(u_{6})\neq 0$ by Lemma 4.5, and for $n$ even,
$\gamma^{*}(u_{11})$ includes the term $v\otimes u_{10}$, where
$\rho^{*}(u_{10})\neq 0$ by Lemma 4.5. Thus the Samelson product
$\langle\bar{\epsilon},\bar{q}\rangle$ is non-trivial, completing the proof by
Lemmas 3.3 and 3.4 because $s(G)=2$. ∎

##  5\. Exceptional case

First, we consider the case $H=E_{6}$.

######  Proposition 5.1.

If $H=E_{6}$, then $\langle\epsilon,p\rangle$ is of order $s(G)$.

###### Proof.

Since the center of $E_{6}$ is isomorphic to $\mathbb{Z}/3$, we only need to
consider the case $G=S^{1}\times_{\mathbb{Z}/3}E_{6}$. The mod 3 cohomology of
$Ad(E_{6})$, which is the quotient of $E_{6}$ by its center, was determined by
Kono [19] such that

$H^{*}(Ad(E_{6});\mathbb{Z}/3)=\mathbb{Z}/3[x_{2},x_{8}]/(x_{2}^{9},x_{8}^{3})\otimes\Lambda(x_{1},x_{3},x_{7},x_{9},x_{11},x_{16})$

such that

$\bar{\phi}(x_{9})=x_{8}\otimes x_{1}+x_{2}\otimes x_{7}-x_{2}^{3}\otimes
x_{3}+x_{2}^{4}\otimes x_{1}\quad\text{and}\quad\bar{q}^{*}(x_{8})\neq 0$

where $|x_{i}|=i$. Then by the same computation as in the proof of Proposition
4.9, we can see that $\langle\bar{\epsilon},\bar{q}\rangle$ is non-trivial.
Thus by Lemmas 3.3 and 3.4, $\langle\epsilon,1_{G}\rangle$ is of order
$s(G)=3$. ∎

Next, we consider the case $H=E_{7}$. Because the center of $E_{7}$ is
isomorphic to $\mathbb{Z}/2$, we only need to consider the case
$G=S^{1}\times_{\mathbb{Z}/2}E_{7}$. The Hopf algebra structure of
$H^{*}(Ad(E_{7});\mathbb{Z}/2)$ was determined by Ishitoya, Kono and Toda
[13], from which we can see that the same computation as $Ad(E_{6})$ does not
apply to $Ad(E_{7})$. So we apply Lemma 3.5. Kono and Mimura [21] showed that
the mod 2 cohomology of $BAd(E_{7})$ is generated by elements $x_{i}$ for
$i\in\\{2,3,6,7,10,11,18,19,34,35,64,66,67,96,112\\}$, where $|x_{i}|=i$. We
determine $Sq^{2}x_{6}$.

Let $e_{1},e_{2},\ldots,e_{n}$ be the standard basis of $\mathbb{R}^{n}$.
Elements of the spin group $Spin(n)$ are expressed by using
$e_{1},e_{2},\ldots,e_{n}$. See [1, Chapter 3]. Recall from [1, Proposition
4.2] that there are two representations

$\Delta_{2n}^{+},\,\Delta^{-}_{2n}\colon Spin(2n)\to SU(2^{n-1})$

such that $\Delta_{n}^{+}$ has weights $\frac{1}{2}(\pm x_{1}\pm
x_{2}\pm\cdots\pm x_{n})$ with even numbers of minus signs and
$\Delta_{n}^{-}$ has weights $\frac{1}{2}(\pm x_{1}\pm x_{2}\pm\cdots\pm
x_{n})$ with odd numbers of minus signs.

######  Proposition 5.2.

There is a natural isomorphism

$Spin(4)\cong\operatorname{Ker}\Delta_{4}^{+}\times\operatorname{Ker}\Delta_{4}^{-}.$

###### Proof.

There is a product decomposition $Spin(4)\cong SU(2)\times SU(2)$ such that
$\Delta_{4}^{\pm}\colon Spin(4)\to SU(2)$ are identified with projections
$SU(2)\times SU(2)\to SU(2)$. Then the statement is proved. ∎

As in [1, Theorem 6.1], there is a homomorphism

$\theta\colon Spin(16)\to E_{8}$

whose kernel is $\\{1,e_{1}e_{2}\cdots e_{16}\\}$. Let $\mu\colon
Spin(4)\times Spin(12)\to Spin(16)$ denote the homomorphism covering the
inclusion

$SO(4)\times SO(12)\to SO(16),\quad(A,B)\mapsto\begin{pmatrix}A&O\\\
O&B\end{pmatrix}.$

Define $\bar{\mu}=\theta\circ\mu\colon Spin(4)\times Spin(12)\to E_{8}$. Then

$\operatorname{Ker}\bar{\mu}=\\{(1,1),\,(-1,-1),\,(e_{1}e_{2}e_{3}e_{4},e_{5}e_{6}\cdots
e_{16}),\,(-e_{1}e_{2}e_{3}e_{4},-e_{5}e_{6}\cdots e_{16})\\}.$

Recall from [1, Chapter 8] that $E_{7}$ is defined as the centralizer of
$\bar{\mu}(\operatorname{Ker}\Delta_{4}^{+}\times 1)$ in $E_{8}$. Then by
Lemma 5.2, there is a homomorphism

$\hat{\mu}\colon\operatorname{Ker}\Delta_{4}^{-}\times Spin(12)\to E_{7}.$

Since $-e_{1}e_{2}e_{3}e_{4}\in\operatorname{Ker}\Delta_{4}^{+}$,
$\bar{\mu}(-e_{1}e_{2}e_{3}e_{4},1)$ commutes with every element of $E_{7}$ in
$E_{8}$. Moreover,
$\bar{\mu}(-e_{1}e_{2}e_{3}e_{4},1)=\bar{\mu}(e_{1}e_{2}e_{3}e_{4},-1)=\hat{\mu}(e_{1}e_{2}e_{3}e_{4},-1)$,
which belongs to $E_{7}$ and is not the unit of $E_{7}$. Then we obtain:

######  Proposition 5.3.

The center of $E_{7}$ is $\\{1,\hat{\mu}(e_{1}e_{2}e_{3}e_{4},-1)\\}$.

Let $L=(\operatorname{Ker}\Delta_{4}^{-}\times
Spin(12))/\\{(1,1),(e_{1}e_{2}e_{3}e_{4},-1)\\}$. Then by Proposition 5.3,
there is a map

$\rho\colon L\to Ad(E_{7})$

which is an isomorphism in the second mod 2 cohomology.

######  Lemma 5.4.

In $H^{*}(BAd(E_{7});\mathbb{Z}/2)$, $Sq^{2}x_{6}$ is decomposable and
includes the term $x_{2}x_{6}$.

###### Proof.

By [21, 22], $(\bar{\mu}\circ(1\times\bar{q}))^{*}(x_{6})$ includes the term
$1\otimes u_{6}$, where $u_{i}$ is as in Lemma 4.5. Note that the composition

$Spin(12)\to\operatorname{Ker}\Delta_{4}^{-}\times Spin(12)\to
L\xrightarrow{q_{2}}SO(12)$

is the natural projection, where $q_{2}$ is the second projection. Then by
degree reasons,

$\rho^{*}(x_{6})+a\rho^{*}(x_{2})^{3}+b\rho^{*}(x_{3})^{2}=q_{2}^{*}(w_{6})$

for some $a,b\in\mathbb{Z}/2$. On the other hand, $q_{2}^{*}\colon
H^{2}(BSO(12);\mathbb{Z}/2)\to H^{2}(BL;\mathbb{Z}/2)$ is an isomorphism,
implying $\rho^{*}(x_{2})=q_{2}^{*}(w_{2})$. Then since
$Sq^{2}w_{6}=w_{2}w_{6}$ by (4.1) and $Sq^{2}x_{6}$ is decomposable by degree
reasons, $Sq^{2}x_{6}$ is decomposable and includes the term $x_{2}x_{6}$, as
stated. ∎

We are ready to prove:

######  Proposition 5.5.

If $H=E_{7}$, then $\langle\epsilon,p\rangle$ is of order $s(G)$.

###### Proof.

As mentioned above, we only need to consider
$G=S^{1}\times_{\mathbb{Z}/2}E_{7}$. We apply Lemma 3.5 by setting
$x=z=x_{6},\,y=x_{2}$ and $\theta=Sq^{2}$. By Lemma 5.4, the first and the
second conditions of Lemma 3.5 are satisfied. As in [22], $\bar{q}^{*}(x_{6})$
is a generator of $H^{6}(BE_{7};\mathbb{Z}/2)$ such that $(\bar{q}\circ
j)^{*}(x_{6})$ is non-trivial. Then by degree reasons, the third condition of
Lemma 3.5 is also satisfied, implying $\langle\bar{\epsilon},\bar{q}\rangle$
is non-trivial. Since $s(G)=2$, the proof is complete by Lemmas 3.3 and 3.4. ∎

##  6\. Proofs of Theorems 1.3 and 1.4

This section proves Theorems 1.3 and 1.4. First, we prove Theorem 1.4.

###### Proof of Theorem 1.4.

Suppose $H\cong H_{1}\times\cdots\times H_{k}$, where each $H_{i}$ is a simple
Lie group. Let $r_{i}\colon S^{1}\times H\to S^{1}\times H_{i}$ be the
projection, and let $G_{i}=(S^{1}\times H_{i})/(r_{i}(C))$ for
$i=1,2,\ldots,k$. By definition, $s(G)$ is the least common multiple of
$s(G_{1}),\ldots,s(G_{k})$.

Let $\bar{r}_{i}\colon G\to G_{i}$ and $\iota_{i}\colon S^{1}\times H_{i}\to
S^{1}\times H$ denote the projection and the inclusion, respectively. Then
$\bar{r}_{i}\circ\epsilon_{G}=\epsilon_{G_{i}}$ and $\bar{r}_{i}\circ
p_{G}\circ\iota_{i}=p_{G_{i}}$, so that

$(1\wedge\iota_{i})^{*}\circ(\bar{r}_{i})_{*}(\langle\epsilon_{G},p_{G}\rangle)=\langle\bar{r}_{i}\circ\epsilon_{G},\bar{r}_{i}\circ
p_{G}\circ\iota_{i}\rangle=\langle\epsilon_{G_{i}},p_{G_{i}}\rangle.$

Thus the order of $\langle\epsilon_{G},p_{G}\rangle$ is a non-zero multiple of
the order of $\langle\epsilon_{G_{i}},p_{G_{i}}\rangle$. So by Propositions
4.1, 4.4, 4.7, 5.1 and 5.5, the order of $\langle\epsilon_{G},p_{G}\rangle$ is
a non-zero multiple of $s(G_{i})$ for $i=1,2,\ldots,k$, hence so is
$\langle\epsilon_{G},1_{G}\rangle$. Therefore by Lemma 3.3, the proof is
complete. ∎

Next, we prove Theorem 1.3.

###### Proof of Theorem 1.3.

We only prove the case $H=SU(n)^{r}$ because the case $H=SU(4n-2)^{s}\times
Sp(2n-1)^{t}$ is proved analogously. The implication (1) $\Rightarrow$ (2)
follows from Theorem 1.2. We prove the implication (2) $\Rightarrow$ (1). Let
$\partial_{k}\colon G\to\operatorname{map}_{*}(S^{2},BG;k)\simeq\Omega_{0}G$
be as in Section 2, and let $q_{i}\colon H\to SU(n)$ be the projection onto
the $i$-th $SU(n)$. Then by Lemma 2.1, the proof of Proposition 4.1 implies
that the image of the map

$(\partial_{k})_{*}\colon\pi_{2n-1}(G)\to\pi_{2n-1}(\Omega_{0}G)$

is isomorphic to $\prod_{i=1}^{r}\mathbb{Z}/\frac{n!}{(k,|q_{i}(C)|)}$, where
$\pi_{2n-1}(\Omega_{0}G)\cong(\mathbb{Z}/n!)^{r}$. By (2.1), there is an exact
sequence

$0\to\prod_{i=1}^{r}\mathbb{Z}/\tfrac{n!}{(k,|q_{i}(C)|)}\to(\mathbb{Z}/n!)^{r}\to\pi_{2n-1}(B\mathcal{G}_{k}(S^{2},G))\to\pi_{2n-1}(BG)\cong\pi_{2n-1}(BSU(n)^{r})=0.$

Then since
$\pi_{2n-1}(B\mathcal{G}_{k}(S^{2},G))\cong\pi_{2n-2}(\mathcal{G}_{k}(S^{2},G))$,
$\pi_{2n-2}(\mathcal{G}_{k}(S^{2},G))\cong\prod_{i=1}^{r}\mathbb{Z}/(k,|q_{i}(C)|)$.
So if $\mathcal{G}_{k}(X,G)\simeq\mathcal{G}_{l}(X,G)$, then
$\pi_{2n-2}(\mathcal{G}_{k}(S^{2},G))\simeq\pi_{2n-2}(\mathcal{G}_{l}(S^{2},G))$,
implying

$(k,|q_{1}(C)|)\cdots(k,|q_{r}(C)|)=(l,|q_{1}(C)|)\cdots(l,|q_{r}(C)|).$

As in the proof of Theorem 1.3, $s(G)$ is the least common multiple of
$|q_{1}(C)|,\ldots,|q_{r}(C)|$. Then it is easy to see that the above equality
implies $(k,s(G))=(l,s(G))$, completing the proof. ∎

## References

  * [1] J.F. Adams, Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. With a foreword by J. Peter May; Edited by Zafer Mahmud and Mamoru Mimura.
  * [2] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523-615.
  * [3] P.F. Baum and W. Browder,The cohomology of quotients of classical groups, Topology 3 (1965), 305-336.
  * [4] R. Bott, A note on the Samelson products in the classical groups, Comment. Math. Helv. 34 (1960), 249-256.
  * [5] S. B. Bradlow, O. Garcia-Prada, and P. B. Gothen, Homotopy groups of moduli spaces of representations, Topology 47 (2008) 203-224.
  * [6] T. Cutler, The homotopy types of $Sp(3)$-gauge groups, Topology Appl. 236 (2018), 44-58.
  * [7] M.C. Crabb and W.A. Sutherland, Counting homotopy types of gauge groups, Proc. London Math. Soc. 83 (2000), 747-768.
  * [8] G.D. Daskalopoulos and K.K. Uhlenbeck, An application of transversality to the topology of the moduli space of stable bundles, Topology 34 (1995) 203-215.
  * [9] D.H. Gottlieb, Applications of bundle map theory, Trans. Amer. Math. Soc. 171 (1972) 23-50.
  * [10] H. Hamanaka and A. Kono, Unstable $K^{1}$-group and homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 149-155.
  * [11] S. Hasui, D. Kishimoto, A. Kono, and T. Sato, The homotopy types of $\mathrm{PU}(3)$\- and $\mathrm{PSp}(2)$-gauge groups, Algebr. Geom. Topol. 16 (2016), no. 3, 1813-1825.
  * [12] S. Hasui, D. Kishimoto, T. So, and S. Theriault, Odd primary homotopy types of the gauge groups of exceptional Lie groups, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1751-1762.
  * [13] K. Ishitoya, A. Kono, and H. Toda, Hopf algebra structure of mod 2 cohomology of simple Lie groups, Publ. RIMS 12 (1976), 141-167.
  * [14] Y. Kamiyama, D. Kishimoto, A. Kono, and S. Tsukuda, Samelson products of $SO(3)$ and applications, Glasg. Math. J. 49 (2007), no. 2, 405-409.
  * [15] D. Kishimoto and A. Kono, On the homotopy types of $\mathrm{Sp}(n)$ gauge groups, Algebr. Geom. Topol. 19 (2019), no. 1, 491-502.
  * [16] D. Kishimoto, A. Kono, and M. Tsutaya, On $p$-local homotopy types of gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 1, 149-160.
  * [17] D. Kishimoto, I. Membrillo-Solis, and S. Theriault, The homotopy types of $\mathrm{SO}(4)$-gauge groups, accepted by Eur. J. Math.
  * [18] D. Kishimoto, S. Theriault, and M. Tsutaya, The homotopy types of $G_{2}$-gauge groups, Topology Appl. 228 (2017), 92-107.
  * [19] A. Kono, Hopf algebra structure of simple Lie groups, J. Math. Kyoto Univ. 17 (1977), no. 2, 259-298.
  * [20] A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 295-297.
  * [21] A. Kono and M. Mimura, On the cohomology mod 2 of the classifying space of $AdE_{7}$, J. Math. Kyoto Univ. 18 (1978), no. 3, 535-541.
  * [22] A. Kono, M. Mimura, and N. Shimada, On the cohomology mod 2 of the classifying space of the 1-connected exceptional Lie group $E_{7}$, J. Pure Appl. Alg. 8 (1976), 267-283.
  * [23] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971) 197-212.
  * [24] S. Rea, Homotopy types of gauge groups of $\mathrm{PU}(p)$-bundles over spheres, J. Homotopy Rel. Str. 16 (2021), 61-74.
  * [25] S.D. Theriault, Odd primary homotopy decompositions of gauge groups, Algebr. Geom. Topol. 10 (2010) 535-564.
  * [26] S.D. Theriault, The homotopy types of $\mathrm{Sp}(2)$-gauge groups, Kyoto J. Math. 50 (2010) 591-605.
  * [27] S.D. Theriault, Homotopy decompositions of gauge groups over Riemann surfaces and applications to moduli spaces, Int. J. Math. 22 (2011), no. 12, 1711-1719.
  * [28] S.D. Theriault, The homotopy types of $SU(5)$-gauge groups, Osaka J. Math. 52 (2015), 15-29.
  * [29] S. Theriault, Odd primary homotopy types of $SU(n)$-gauge groups, Algebr. Geom. Topol. 17 (2017), no. 2, 1131-1150.
  * [30] G.W. Whitehead, On products in homotopy groups, Ann. Math. 47 (1946), no. 3, 460-475.


# Connection between near the $D_{s}^{+}D_{s}^{-}$ threshold enhancement in
$B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ and conventional charmonium $\chi_{c0}(2P)$

Dan Guo1,3<EMAIL_ADDRESS>Jun-Zhang Wang2,1<EMAIL_ADDRESS>Dian-
Yong Chen4,5<EMAIL_ADDRESS>Xiang Liu1,3,5<EMAIL_ADDRESS>1School of
Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
2School of Physics and Center of High Energy Physics, Peking University,
Beijing 100871, China
3Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
4School of Physics, Southeast University, Nanjing 210094, China
5Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province and Frontier Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China

###### Abstract

Focusing on recent measurement of $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ process
given by the LHCb Collaboration, we propose that this newly observed near the
$D_{s}^{+}D_{s}^{-}$ threshold enhancement can be due to the contribution of
the $\chi_{c0}(2P)$, which is a $P$-wave charmonium below the
$D_{s}^{+}D_{s}^{-}$ threshold. By performing a combined fit to the measured
$D_{s}^{+}D_{s}^{-}$, $D_{s}^{+}K^{+}$, and $D_{s}^{-}K^{+}$ invariant mass
spectra, introducing the $\chi_{c0}(2P)$ can well reproduce the near threshold
enhancement in the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum. When
depicting the whole $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum, the
contributions from higher charmonium $\psi(4230)$ and charmoniumlike state
$X_{0}(4140)$ are obvious. In addition, a charmed meson $D^{*}(3^{3}S_{1})$
with mass around 3015 MeV is found to be important to depict the
$D_{s}^{-}K^{+}$ invariant mass spectrum well. Especially, the importance of
node effect of the spatial wave function of the $\chi_{c0}(2P)$ is revealed,
by which the anomaly of the ratio $\Gamma(X\to D^{+}D^{-})/\Gamma(X\to
D_{s}^{+}D_{s}^{-})$ indicated by LHCb can be explained well. Finally, our
scenario of the newly observed enhancement structure resulted from the
$\chi_{c0}(2P)$ is enforced.

##  I Introduction

Very recently, the LHCb Collaboration found a novel phenomenon of near
threshold enhancement in the $D_{s}^{+}D_{s}^{-}$ invariant mass distribution
of the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ process LHCb:2022vsv . If adopting
the Breit-Wigner formula to depict this enhancement structure, the extracted
resonant parameters are $M=3956\pm 5({\rm stat.})\pm 10({\rm syst.})$ MeV and
$\Gamma=43\pm 13({\rm stat.})\pm 8({\rm syst.})$ MeV LHCb:2022vsv . Thus, this
enhancement structure is refereed to be the $X(3960)$ before decoding its
nature. In addition, the spin-parity test from LHCb suggests that the
$X(3960)$ favors $J^{PC}=0^{++}$. When checking the data of the
$D_{s}^{+}D_{s}^{-}$ invariant mass distribution, a dip around 4.14 GeV can be
found, which was due to another $J^{PC}=0^{++}$ state $X_{0}(4140)$ by LHCb
LHCb:2022vsv .

For understanding this novel phenomenon, some theoretical explanations were
proposed. The $X(3960)$ was explained as a threshold structure due to the
$D_{s}\bar{D}_{s}$ interaction, where the interpretation of a virtual or bound
state below the $D_{s}^{+}D_{s}^{-}$ threshold Ji:2022uie ; Xie:2022lyw , and
the $D_{s}^{+}D_{s}^{-}$ molecule description from the QCD sum rules method
Xin:2022bzt  and the one-boson-exchange model Chen:2022dad  were proposed. In
Ref. Bayar:2022dqa , the near threshold enhancement signal of the $X(3960)$
was interpreted to be the off-shell contribution of the $\chi_{c0}(3930)$,
which is a $P$-wave charmonium state found in the $B^{+}\to D^{+}D^{-}K^{+}$
decay LHCb:2020pxc .

A main reason why most of theoretical groups tried to categorize the newly
observed $X(3960)$ to be an exotic hadronic state is due to the measured ratio

$\displaystyle\frac{\Gamma(X\to D^{+}D^{-})}{\Gamma(X\to D_{s}^{+}D_{s}^{-})}$
$\displaystyle=$ $\displaystyle\frac{\mathcal{B}[B^{+}\to
D^{+}D^{-}K^{+}]\mathcal{FF}^{X}_{B^{+}\to
D^{+}D^{-}K^{+}}}{\mathcal{B}[B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}]\mathcal{FF}^{X}_{B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}}}$
(1) $\displaystyle=$ $\displaystyle 0.29\pm 0.09\pm 0.10\pm 0.08,$

given by LHCb when treating the $X(3960)$ in the $B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}$ process LHCb:2022vsv  and the $\chi_{c0}(3930)$ in
the $B^{+}\to D^{+}D^{-}K^{+}$ decay LHCb:2020pxc  as the same state (see Ref.
LHCb:2022vsv  for more details). Usually, the $s\bar{s}$ quark pair excited
from vacuum is harder than the $u\bar{u}$ and $d\bar{d}$ pairs. The ratio is
anomalous for a higher charmonium as indicated by LHCb LHCb:2022vsv . It seems
to provide enough motivation to assign the newly observed $X(3960)$ as an
exotic hadronic state. However, we must face the fact that the coupled-channel
effect is obvious for these $2P$ states of charmonium Duan:2020tsx . We still
should be cautious when definitely making a conclusion of the $X(3960)$ to be
an exotic state.

Thus, we briefly introduce how the coupled-channel effect plays the role in
the spectroscopy behavior of these $2P$ states of charmonium. When solving the
low mass puzzle of the $X(3872)$ Godfrey:1985xj ; Barnes:2005pb , the
importance of the coupled-channel was realized Barnes:2003vb ;
Kalashnikova:2005ui ; Ortega:2009hj ; Duan:2020tsx , where the bare mass of
the $\chi_{c1}(2P)$ can be decorated by the nearby $D\bar{D}^{\ast}$ channel.
After observing the charmoniumlike state $X(3915)$ Belle:2009and  in the
$\gamma\gamma\to J/\psi\omega$ fusion process, the Lanzhou group indicated
that the $X(3915)$ must have $0^{++}$ quantum number, which is a good
candidate of the charmonium $\chi_{c0}(2P)$ Liu:2009fe . Later, this
prediction was confirmed by the BaBar Collaboration BaBar:2012nxg . Thus, the
Particle Data Group (PDG) once collected the $X(3915)$ as the charmonium
$\chi_{c0}(3915)$ into The Review of Particle Physics (2012)
ParticleDataGroup:2012pjm  in a short time. Under this assignment to the
$X(3915)$, we have to explain why the mass gap between the $X(3915)$ and
$Z(3930)\equiv\chi_{c2}(2P)$ is smaller than that between the $\chi_{c0}(1P)$
and $\chi_{c2}(1P)$, or that between the $\chi_{b0}(2P)$ and $\chi_{b2}(2P)$
Godfrey:1985xj ; Barnes:2005pb ; Guo:2012tv . Finally, the Lanzhou group
clarified this issue by introducing the coupled-channel effect, and found that
the node effect of the spatial wave functions of the discussed $2P$ charmonoia
is crucial to exactly reproduce the narrow mass gap between the $X(3915)$ and
$Z(3930)$ Duan:2020tsx , which supports the assignment of the $X(3915)$ as a
$\chi_{c0}(2P)$ charmonium. In 2020, LHCb observed the $\chi_{c0}(3930)$ and
$\chi_{c2}(3930)$ simultaneously in the $B^{+}\to D^{+}D^{-}K^{+}$ decay
LHCb:2020pxc , which is the first signal of the $\chi_{c0}(2P)\to D\bar{D}$
decay mode. This observation confirms that the $\chi_{c0}(2P)$ state has
narrow width Liu:2009fe , and also checks the prediction from the Lanzhou
group that two $P$-wave charmonia $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ are
close to each other Chen:2012wy ; Duan:2020tsx . By these efforts, the name
$\chi_{c0}(3915)$ appeared in The Review of Particle Physics (2022)
Workman:2022ynf  again.

Coming back to the LHCb’s observation of the near threshold enhancement in the
$D_{s}^{+}D_{s}^{-}$ invariant mass spectrum LHCb:2022vsv , we should
carefully check whether this near threshold enhancement can be resulted by the
$\chi_{c0}(2P)$ under considering the unquenched effect. In this work, we
carry out a serious study on this issue. According to the measured mass of the
$\chi_{c0}(3930)$, we conclude that the $\chi_{c0}(2P)$ is below and near the
$D_{s}^{+}D_{s}^{-}$ threshold. However, the $\chi_{c0}(2P)$ can still
contribute to $B\to D_{s}^{+}D_{s}^{-}K^{+}$ and should have obvious effect to
the near threshold enhancement in the $D_{s}^{+}D_{s}^{-}$ invariant mass
spectrum if the width effect of the $\chi_{c0}(2P)$ is considered. For
depicting the dip at 4.14 GeV in the $D_{s}^{+}D_{s}^{-}$ invariant mass
spectrum, we still need to introduce a scalar $X_{0}(4140)$ state similar to
the treatment of LHCb LHCb:2022vsv . Besides, we notice event accumulation
around 4.2 GeV, which also inspires us to test the contribution of the higher
vector charmonium $\psi(4230)$ BESIII:2014rja ; BESIII:2016bnd ;
BESIII:2016adj ; Wang:2019mhs ; Wang:2017sxq ; Qian:2021neg ; Chen:2014sra ;
Chen:2017uof  to $B\to D_{s}^{+}D_{s}^{-}K^{+}$. In fact, these intermediate
states like the $\chi_{c0}(2P)$, $X_{0}(4140)$, and $\psi(4230)$ have the
reflections in the $D_{s}^{-}K^{+}$, and $D_{s}^{+}K^{+}$ invariant mass
spectra. However, if checking the $D_{s}^{-}K^{+}$ invariant mass spectrum,
one may find the event cluster around 3 GeV compared with the $D_{s}^{+}K^{+}$
invariant mass distributions, which indicates the contributions from the
higher charmed mesons Song:2015fha ; Wang:2016krl . Thus, in the present work,
we perform a combined fit to these measured three invariant mass distributions
of the $B\to D_{s}^{+}D_{s}^{-}K^{+}$ decay by including the
charmonia/charmonium-like intermediate states and charmed mesons intermediate
states simultaneously to test our proposal. Moreover, if making a connection
between the near threshold enhancement and the $\chi_{c0}(2P)$, we still need
to suitably explain the anomaly of the ratio shown in Eq. (1). Inspired by the
node effect to solve the small mass gap between the $X(3915)$ and $Z(3930)$
Duan:2020tsx , we introduce the node effect, and reproduce the anomalous ratio
in Eq. (1). In this work, we want to emphasize that this anomaly of the ratio
of $\Gamma(X\to D^{+}D^{-})/\Gamma(X\to D_{s}^{+}D_{s}^{-})$ does not conflict
with the $P$-wave charmonium assignment to depict the near threshold
enhancement in the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum. Instead,
reproducing this anomalous ratio in the present work may further enforce our
scenario.

This work is organized as follows. In the following Section, we focus on the
interpretation of three invariant mass distributions, and in Sec. III, we
present an explanation to the abnormal coupling ratio. Finally, this work ends
with a short summary in Sec. IV.

##  II Decoding the enhancement structures in the invariant mass spectra of
$B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$

Figure 1: The schematic diagrams of different intermediate resonant
contributions to the $B^{+}(P)\to
K^{+}(p_{1})D_{s}^{+}(p_{2})D_{s}^{-}(p_{3})$ decay. Diagram (a) presents the
intermediate charmonium contribution, while diagram (b) stands for the charmed
meson contribution.

As discussed in the Introduction, there are two different kinds of
intermediate state contributions to the process $B^{+}\to
K^{+}D_{s}^{-}D_{s}^{+}$ as shown in Fig. 1. For the processes corresponding
to Fig. 1 (a), we consider the contributions from charmonia $\chi_{c0}(2P)$,
$\psi(4230)$, and charmoniumlike $X(4140)$. For the process shown in Fig. 1
(b), all the possible charmed mesons above the $D_{s}^{+}K^{-}$ threshold with
proper $J^{P}$ quantum numbers like $0^{+}$, $1^{-}$ and $2^{+}$ could be as
the intermediate state. According to the spectroscopy behavior of the higher
charmed mesons Song:2015fha ; Wang:2016krl , there exist several candidates
like the $D_{1}^{\ast}(2600)$, $D_{2}^{\ast}(2460)$, $D^{\ast}(2^{3}P_{0})$,
$D^{\ast}(1^{3}D_{1})$, $D^{\ast}(3^{3}S_{1})$, $D^{\ast}(2^{3}P_{2})$ and
$D^{\ast}(1^{3}F_{2})$ Song:2015fha ; Workman:2022ynf . For depicting the
event cluster around 3 GeV shown in the $D_{s}^{-}K^{+}$ invariant mass
spectrum, finally we consider the charmed mesons with mass around 3 GeV for
simplicity. Since the widths of the $D^{\ast}(2^{3}P_{0})$,
$D^{\ast}(1^{3}D_{1})$ and $D^{\ast}(1^{3}F_{2})$ are all larger than 200 MeV
Song:2015fha ; Workman:2022ynf , only the $D^{\ast}(3^{3}S_{1})$ with the
width to be 80.36 MeV is promising when refining the $D_{s}^{-}K^{+}$,
$D_{s}^{+}K^{+}$ and $D_{s}^{+}D_{s}^{-}$ mass spectra simultaneously.

As discussed above, two kinds of intermediate resonant contributions are
considered for the $B^{+}\to K^{+}D_{s}^{+}D_{s}^{-}$ process, which are
$B^{+}\to K^{+}(c\bar{c})\to K^{+}[(c\bar{c})\to D_{s}^{+}D_{s}^{-}]$ and
$B\to\bar{D}^{\ast 0}(3^{3}S_{1})D_{s}^{+}\to(\bar{D}^{\ast 0}(3^{3}S_{1})\to
D_{s}^{-}K^{+})D_{s}^{+}$. In this work, the effective Lagrangian approach is
employed to describe the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ decay, which has
been successfully applied to describe other similar $B$ decay processes
Duan:2021bna ; Wang:2021crr . Since the primary concern of this work is to
explore the role of different intermediate states in describing the line shape
of invariant mass distribution of $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$, the
effective Lagrangians relevant to the weak decay of $B$ meson can be
constructed by considering the invariance under the isospin and parity
transformation, which can be written as

$\displaystyle\mathcal{L}_{BXK}$ $\displaystyle=$ $\displaystyle
ig_{BXK}XB\bar{K},$ $\displaystyle\mathcal{L}_{B\psi K}$ $\displaystyle=$
$\displaystyle ig_{B\psi
K}\psi_{\mu}(\partial^{\mu}B^{{\dagger}}K-B\partial^{\mu}K^{{\dagger}}),$ (2)
$\displaystyle\mathcal{L}_{BD^{\ast}D_{s}}$ $\displaystyle=$ $\displaystyle
ig_{BD^{\ast}D_{s}}D^{\ast}_{\mu}(\partial^{\mu}D_{s}^{\dagger}B-D_{s}\partial^{\mu}B^{\dagger}).$

Here, $X$ refers to the scalar $\chi_{c0}(2P)$ and $X_{0}(4140)$, while the
$\psi(4230)$ and $D^{\ast}(3^{3}S_{1})$ are abbreviated to be $\psi$ and
$D^{\ast}$, respectively.

For the $(c\bar{c})\to D_{s}^{+}D_{s}^{-}$ and $D^{\ast}(3^{3}S_{1})\to DK$
interactions, the relevant effective Lagrangians based on the heavy quark
symmetry and chiral symmetry are Duan:2021bna ; Huang:2021kfm ; Chen:2012nva ;
Wang:2020kej

$\displaystyle\mathcal{L}_{XD_{s}\bar{D}_{s}}$ $\displaystyle=$ $\displaystyle
ig_{XD_{s}\bar{D}_{s}}XD_{s}\bar{D}_{s},$ $\displaystyle\mathcal{L}_{\psi
D_{s}D_{s}}$ $\displaystyle=$ $\displaystyle ig_{\psi
D_{s}D_{s}}\psi_{\mu}(\partial^{\mu}D_{s}^{{\dagger}}D_{s}-D_{s}\partial^{\mu}D_{s}^{{\dagger}}),$
(3) $\displaystyle\mathcal{L}_{D^{\ast}D_{s}K}$ $\displaystyle=$
$\displaystyle
ig_{D^{\ast}D_{s}K}D^{\ast}_{\mu}(\partial^{\mu}D_{s}^{\dagger}K-D_{s}\partial^{\mu}K^{\dagger}).$

With the above preparation, we obtain the decay amplitude of the resonant
contribution to the process $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$, i.e.,

$\displaystyle\mathcal{M}_{X}$ $\displaystyle=$
$\displaystyle\frac{g_{X}}{(p_{2}+p_{3})^{2}-m_{X}^{2}+im_{X}\Gamma_{X}},$
$\displaystyle\mathcal{M}_{\psi}$ $\displaystyle=$
$\displaystyle\frac{g_{\psi}(P^{\mu}+p_{1}^{\mu})(p_{2}^{\nu}-p_{3}^{\nu})\tilde{g}_{\mu\nu}(\psi)}{(p_{2}+p_{3})^{2}-m_{\psi}^{2}+im_{\psi}\Gamma_{\psi}},$
(4) $\displaystyle\mathcal{M}_{D^{\ast}}$ $\displaystyle=$
$\displaystyle\frac{g_{D^{\ast}}(P^{\mu}+p_{2}^{\mu})(p_{1}^{\nu}-p_{3}^{\nu})\tilde{g}_{\mu\nu}(D^{\ast})}{(p_{1}+p_{3})^{2}-m_{D^{\ast}}^{2}+im_{D^{\ast}}\Gamma_{D^{\ast}}}$

with
$\tilde{g}_{\mu\nu}(\psi)=-g_{\mu\nu}+(p_{2\mu}+p_{3\mu})(p_{2\nu}+p_{3\nu})/m_{\psi}^{2}$
and
$\tilde{g}_{\mu\nu}(D^{\ast})=-g_{\mu\nu}+(p_{1\mu}+p_{3\mu})(p_{1\nu}+p_{3\nu})/m_{D^{\ast}}^{2}$,
where the coupling constants are defined as
$g_{X}=g_{XD_{s}\bar{D}_{s}}g_{BXK}$, $g_{\psi}=g_{\psi D_{s}D_{s}}g_{B\psi
K}$ and $g_{D^{\ast}}=g_{D^{\ast}D_{s}K}g_{BD^{\ast}D_{s}}$.

In addition to the resonant contribution, there is the background contribution
for the discussed process. In the realistic calculation, we consider a
constant amplitude to simulate the background term, which is,

$\displaystyle\mathcal{M}_{bkg}=g_{bkg}.$ (5)

And then, the total decay amplitude is the coherent sum of each resonant
amplitudes and nonresonant background amplitude, i.e.,

$\displaystyle\mathcal{M}_{Total}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{bkg}+e^{i\phi_{1}}\mathcal{M}_{\chi_{c0}(2P)}+e^{i\phi_{2}}\mathcal{M}_{X_{0}(4140)}$
(6)
$\displaystyle+e^{i\phi_{3}}\mathcal{M}_{\psi(4230)}+e^{i\phi_{4}}\mathcal{M}_{D^{\ast}(3^{3}S_{1})},$

where $\phi_{i}$ denotes the phase angle between the resonant amplitudes and
the nonresonant background amplitude, which are considered as free parameters.
Moreover, the coupling constants $g_{bkg}$, $g_{\chi_{c0}(2P)}$,
$g_{X_{0}(4140)}$, $g_{\psi(4230)}$ and $g_{D^{\ast}(3^{3}S_{1})}$ are also
treated as parameters, which can be determined by fitting the experimental
data. While, the masses and widths of the involved intermediate states are
fixed, and the concrete values of the resonant parameters are collected in
Table 1.

The differential decay width of $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ relative to
the invariant mass distribution of ${D_{s}^{+}D_{s}^{-}}$ is,

$\frac{\mathrm{d}\Gamma}{\mathrm{d}m_{D_{s}^{+}D_{s}^{-}}}=\frac{1}{(2\pi)^{5}}\frac{1}{16m_{B}^{2}}\left|\mathcal{M}_{Total}\right|^{2}\left|\vec{p}_{1}\right|\left|\vec{p}_{2}^{\ast}\right|\mathrm{d}\Omega_{1}\mathrm{d}\Omega_{23}^{\ast},$
(7)

where $\vec{p}_{2}^{\ast}$ and $\Omega_{23}^{\ast}$ are the three momentum and
solid angle of the $D_{s}^{+}$ meson in the center of the $D_{s}^{+}D_{s}^{-}$
system mass frame, respectively. Similarly, one can obtain the differential
decay width relative to the invariant mass distributions of $D_{s}^{+}K^{+}$
and $D_{s}^{-}K^{+}$.

|  |
---|---|---
(a) | (b) | (c)
Figure 2: (Color online.) Our best fit to the invariant mass spectra of (a) $D_{s}^{+}D_{s}^{-}$, (b) $D_{s}^{+}K^{+}$, and (c) $D_{s}^{-}K^{+}$ of the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ process. Here, the red solid step line indicates the total fitting results, while the individual contributions from background and the intermediate states are also given. Table 1:  The resonant parameters of the involved intermediate states. State |  Mass (MeV) |  Width (MeV)
---|---|---
$\chi_{c0}(2P)]$ LHCb:2020pxc  |  3923.8 |  17.4
$X_{0}(4140)$ LHCb:2022vsv  |  4133 |  67
$\psi(4230)$ LHCb:2022vsv  |  4230 |  55
$D^{\ast}(3^{3}S_{1})$ Song:2015fha  |  3015 |  80.36
Table 2:  The values of the parameters obtained by fitting the LHCb data LHCb:2022vsv .  Parameters |  Values ($\times 10^{-7}$) |  Parameters |  Values
---|---|---|---
$g_{bkg}$ |  $108.0\pm 9.1$ |  – – |  – –
$g_{\chi_{c0}(2P)}$ |  $(73.8\pm 2.9)\ \mathrm{GeV}^{2}$ |  $\phi_{1}$ |  1.01$\pm$0.01
$g_{X_{0}(4140)}$ |  $(34.6\pm 9.2)\ \mathrm{GeV}^{2}$ |  $\phi_{2}$ |  4.67$\pm$0.01
$g_{\psi(4230)}$ |  8.0$\pm 1.5$ |  $\phi_{3}$ |  1.12$\pm$0.03
$g_{D^{\ast}(3^{3}S_{1})}$ |  10.5$\pm 1.0$ |  $\phi_{4}$ |  4.27$\pm$0.06

In the following, we can simultaneously reproduce the invariant mass
distributions of $D_{s}^{+}D_{s}^{-}$, $D_{s}^{+}K^{+}$ and $D_{s}^{-}K^{+}$
of the process $B^{+}\to K^{+}D_{s}^{+}D_{s}^{-}$. Our best fit to the
experimental data is presented in Fig. 2 with $\chi^{2}/\mathrm{dof}=1.73$ and
the corresponding parameter values are listed in Table 2. Here, the candidates
in Fig. 2 can be directly associated to the differential width of $B^{+}\to
K^{+}D_{s}^{+}D_{s}^{-}$. Thus, our fitting parameters have been calibrated to
reproduce the decay width of the process $B^{+}\to K^{+}D_{s}^{+}D_{s}^{-}$ by
using the experimental ratio $\mathcal{B}(B^{+}\to
K^{+}D_{s}^{+}D_{s}^{-})/\mathcal{B}(B^{+}\to K^{+}D^{+}D^{-})=0.525\pm 0.033$
LHCb2022Qian  and branching ratio $\mathcal{B}(B^{+}\to
K^{+}D^{+}D^{-})=2.2\pm 0.5\pm 0.5$ Workman:2022ynf . As shown in Fig. 2 (a),
these two structures reported by LHCb, i.e., the enhancement near the
$D_{s}^{+}D_{s}^{-}$ threshold and the dip near 4.14 GeV, can be well
reproduced by introducing the below-threshold $\chi_{c0}(2P)$ and the
$X_{0}(4140)$, respectively. In addition, the event accumulation near 4.2 GeV
can also be roughly described by the introduced $\psi(4230)$ contribution. It
is obvious that the present work provides an alternative approach to explain
the reported near threshold structure in the $D_{s}^{+}D_{s}^{-}$ invariant
mass spectrum LHCb:2022vsv . In our scenario, it is not necessary to introduce
a new charmoniumlike state $X(3960)$ as treated by LHCb LHCb:2022vsv .

In the present scheme, our fitting results also show a good consistency with
the invariant mass distributions of $D_{s}^{+}K^{+}$ and $D_{s}^{-}K^{+}$.
From Fig. 2 (b) and (c) one can notice that the reflection from the
$\chi_{c0}(2P)$ plays the dominant role for resulting in the broad structure
around 3.2 GeV in both $D_{s}^{+}K^{+}$ and $D_{s}^{-}K^{+}$ invariant mass
spectra. However, as discussed in the last section, there exists an apparent
difference between the $D_{s}^{+}K^{+}$ and $D_{s}^{-}K^{+}$ invariant mass
distributions. Thus, for the discussed $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$
decay, we introduce an additional higher charmed meson besides the
intermediate charmonia. After introducing the $D^{\ast}(3^{3}S_{1})$ resonant
contribution, the shoulder-like shape in the $D_{s}^{-}K^{+}$ invariant mass
distribution from 3.0 to 3.2 GeV can be well depicted. Moreover, the
interference between the background term and the reflection contribution of
the $D^{\ast}(3^{3}S_{1})$ in the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum
can naturally explain the bump near 4.5 GeV in the $D_{s}^{+}D_{s}^{-}$
invariant mass spectrum, which can also be an important hint of the
intermediate $D^{\ast}(3^{3}S_{1})$ contributing to the process $B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}$. The refined properties of the unestablished charmed
meson $D^{\ast}(3^{3}S_{1})$ can be revealed by the more precise experimental
measurement of the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ decay at LHCb and Belle
II.

##  III Explanation to the abnormal ratio involved in the
$\chi_{c0}(2P)D^{+}D^{-}$ and $\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}$ interactions

After reproducing the invariant mass distributions of $D_{s}^{+}D_{s}^{-}$,
$D_{s}^{+}K^{+}$ and $D_{s}^{-}K^{+}$ in our scheme, we can further extract
the fit fraction of $B^{+}\to K^{+}(\chi_{c0}(2P)\to D_{s}^{+}D_{s}^{-})$ to
be $(20.8\pm 11.0)\%$, where the systematical and statistical uncertainties of
the experiments have been included. Then, one gets the ratio111In this work,
we do not adopt the LHCb’s value as given in Eq. (1). In our scheme, we
extract the ratio of ${\mathcal{B}[B^{+}\to
D^{+}D^{-}K^{+}]\mathcal{FF}^{\chi_{c0}(2P)}_{B^{+}\to D^{+}D^{-}K^{+}}}$ and
${\mathcal{B}[B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}]\mathcal{FF}^{\chi_{c0}(2P)}_{B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}}}$ by the present experimental data.

$\displaystyle\frac{\mathcal{B}[B^{+}\to
D^{+}D^{-}K^{+}]\mathcal{FF}^{\chi_{c0}(2P)}_{B^{+}\to
D^{+}D^{-}K^{+}}}{\mathcal{B}[B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}]\mathcal{FF}^{\chi_{c0}(2P)}_{B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}}}=0.34\pm 0.20.$ (8)

In fact, this value in Eq. (8) is the ratio of $\Gamma(\chi_{c0}(2P)\to
D^{+}D^{-})$ and $\Gamma(\chi_{c0}(2P)\to D_{s}^{+}D_{s}^{-})$, which
indicates the relative magnitude between the coupling of the $\chi_{c0}(2P)$
with the $D^{+}D^{-}$ channel and that of the $\chi_{c0}(2P)$ with the
$D_{s}^{+}D_{s}^{-}$ channel. Considering the phase space integral and taking
the mass distributions of the $\chi_{c0}(2P)$ into account, the ratio of the
coupling constants is obtained to be

$\displaystyle\frac{g_{\chi_{c0}(2P)D^{+}D^{-}}}{g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}}=0.15\pm
0.05.$ (9)

If making a naive theoretical estimate and considering the SU(3) flavor
symmetry, the ratio
$g_{\chi_{c0}(2P)D^{+}D^{-}}/g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}=1$ can be
got. In fact, the $s$ quark has larger mass than the $u(q)$ quark, and the
creation possibility of the $s\bar{s}$ quark pair from vacuum is less than
that of the $u\bar{u}(d\bar{d})$ quark pair. Thus, this ratio is expected to
be larger than 1. Obviously, the extracted ratio shown in Eq. (9) does not
satisfy the above estimate. To some extent, there exists an unexpected
reversal between coupling strengths $g_{\chi_{c0}(2P)D^{+}D^{-}}$ and
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$, which should be explained
quantitatively.

As a main task of this work, we explain why the anomaly of this ratio can
happen by introducing the nontrivial node effect. Generally, the transition
matrix of the open-charm process $\chi_{c0}(2P)\to D_{{(s)}}\bar{D}_{{(s)}}$
can be expressed as

$\displaystyle\mathcal{M}=\langle
D_{{(s)}}\bar{D}_{{(s)}}|\mathcal{T}|\chi_{c0}(2P)\rangle,$ (10)

where $\mathcal{T}$ is the transition operator. In the quark pair creation
(QPC) model Micu:1968mk ; LeYaouanc:1972vsx ; LeYaouanc:1973ldf ;
LeYaouanc:1974cvx ; LeYaouanc:1977fsz ; LeYaouanc:1977gm ; Guo:2019wpx ;
Guo:2022xqu ; Song:2015nia , which was extensively applied to study the Okubo-
Zweig-Iizuka (OZI) allowed strong decays of conventional hadron in the past
decades, the transition operator $\mathcal{T}$ is defined as

$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle-3\gamma\sum_{m}\langle 1m;1-m|00\rangle\int
d{\mathbf{k}}_{3}d{\mathbf{k}}_{4}\delta^{3}({\mathbf{k}}_{3}+{\mathbf{k}}_{4})$
(11)
$\displaystyle\times\mathcal{Y}_{1m}\left(\frac{{\mathbf{k}}_{3}-{\mathbf{k}}_{4}}{2}\right)\chi_{1,-m}^{34}\phi_{0}^{34}\omega_{0}^{34}d_{3i}^{\dagger}(\mathbf{k}_{3})b_{4j}^{\dagger}(\mathbf{k}_{4}).$

where $\mathcal{Y}_{1m}(\mathbf{k})=|\mathbf{k}|Y_{1m}(\theta,\phi)$,
$\chi_{1,-m}^{34}$, $\varphi_{0}^{34}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$
and $\omega_{0}^{34}=\delta_{\alpha_{3}\alpha_{4}}$ are the spatial, spin,
flavor and color parts of the wave functions, respectively. $\alpha_{3}$ and
$\alpha_{4}$ are the color indexes of the created quark pair from the vacuum.
In the QPC model, the parameter $\gamma$ is introduced to represent the
strength of the quark-antiquark pair creation from the vacuum, which satisfies
the relation $\gamma_{u\bar{u}/d\bar{d}}=\sqrt{3}\gamma_{s\bar{s}}$ and is a
universal value for the specific initial state system.

In the center-of-mass frame of charmonium $\chi_{c0}(2P)$, the transition
matrix element in Eq. (10) is proportional to the overlap integral of the wave
functions in the momentum space, which reads as

$\displaystyle I(\mathbf{P},m_{c},m_{\bar{c}},m_{q})=\int
d^{3}\mathbf{p}\;\Psi_{n_{A}L_{A}M_{L_{A}}}(\mathbf{P}+\mathbf{p})\;\mathcal{Y}_{lm}(\mathbf{p})$
$\displaystyle\quad\quad\times\Psi^{\ast}_{n_{B}L_{B}M_{L_{B}}}(\frac{m_{q}}{m_{c}+m_{q}}\mathbf{P}+\mathbf{p})\Psi^{\ast}_{n_{C}L_{C}M_{L_{C}}}(\frac{m_{q}}{m_{\bar{c}}+m_{q}}\mathbf{P}+\mathbf{p}),\
\ $ (12)

where $\mathbf{P}$ denotes the momentum of either outgoing meson, and
$m_{c(\bar{c})}$ and $m_{q}$ are masses of charm(anticharm) quark and the
light quark, respectively. $\Psi_{nLM_{L}}(\mathbf{p})$ stands for the spatial
part of the meson wave function, while the notation $A$, $B$ and $C$ refer to
the $\chi_{c0}(2P)$, $D_{(s)}$ and $\bar{D}_{(s)}$ states, respectively. In
the present study, the wave functions obtained in the unquenched potential
model Wang:2019mhs  are employed. For the $n$th radial excitation of a meson
system, its radial part of the spatial wave function $R_{nL}(p)$ has $(n-1)$
nodes. For the involved $\chi_{c0}(2P)$ state, its radial spatial wave
function only has one node, which makes us to easily distinguish positive and
negative values of the radial wave function $R_{nL}^{\chi_{c0}(2P)}(p)$. Thus,
the value of the integral in Eq. (12) corresponding to some interval of
integration can be either negative or positive, where these two parts can be
partly cancelled with each other. In fact, such kind of cancellation is
resulted from the node of radial wave function. The sensitivity of
$I(\mathbf{P},m_{c},m_{\bar{c}},m_{q})$ to the position of the node of radial
wave functions of the involved hadrons is called as node effect.

By the effective Lagrangian of the ${\chi_{c0}(2P)D\bar{D}}$ coupling, one can
obtain the decay amplitude of $\chi_{c0}(3930)\to D\bar{D}$, which is
$\mathcal{M}_{\chi_{c0}(2P)\to D\bar{D}}=g_{\chi_{c0}(2P)D\bar{D}}$. This
decay amplitude can be calculated by the QPC model. By connecting the decay
widths deduced from these two amplitudes, the coupling constants
$g_{\chi_{c0}(2P)D\bar{D}}$ can be obtained. In the present work, only the
parameter $\gamma$ in the QPC model should be fixed by reproducing the width
of the $\chi_{c2}(2P)$, which is $34.2\pm 6.6\pm 1.1\;\mathrm{MeV}$
LHCb:2020pxc . By the calculation, one gets $\gamma^{2}=40.9\pm 8.2$, which
will be applied to the following investigation.

Figure 3: (Color online.) The decay width of $\chi_{c0}(2P)\to D\bar{D}$
dependent on the parameter $\beta$. Here, the blue band indicates the
uncertainty resulted from the value of $\gamma$. The horizontal gray band with
red dashed line stands for the measured width of the $\chi_{c0}(3930)$
LHCb:2020pxc . The vertical dashed refers to $\beta=0.824$ GeV, where the
estimated width of $\chi_{c0}(2P)\to D\bar{D}$ reaches up to the lower limit
of the measured width of the $\chi_{c0}(3930)$.

In order to show the node effect to the decay width and discuss possible
theoretical uncertainty from the wave function, we adopt a simple harmonic
oscillator (SHO) form to depict the radial wave function of the
$\chi_{c0}(2P)$. Explicitly, it reads,

$\displaystyle\Psi_{nlm}(\mathbf{p})=R_{nl}(p,\beta)Y_{lm}(\Omega_{p}),$ (13)
$\displaystyle
R_{nl}(p,\beta)=\frac{(-1)^{n}(-i)^{l}}{\beta^{3/2}}e^{-\frac{p^{2}}{2\beta^{2}}}\sqrt{\frac{2n!}{\Gamma(n+l+3/2)}}{\left(\frac{p}{\beta}\right)}^{l}$
$\displaystyle\quad\quad\quad\quad\times
L_{n}^{l+1/2}\left(\frac{p^{2}}{\beta^{2}}\right),$ (14)

where the node position of the radial wave function is determined by the
oscillator parameter $\beta$. The value of $\beta$ for the $\chi_{c0}(3930)$
is 0.78 GeV, which is determined by the unquenched quark model Wang:2019mhs .
Thus, in our calculation, we select a $\beta$ range to be $0.68\sim 0.88$ GeV
when taking the uncertainty of the theoretical model into account.

For the $\chi_{c0}(2P)$, $\chi_{c0}(2P)\to D\bar{D}$ is an OZI-allowed decay
channel. Its decay width dependent on the parameter $\beta$ is presented in
Fig. 3, where the node effect is obvious since the calculated decay width
becomes smaller when the $\beta$ value is close to 0.746 GeV near the
estimated value $\beta=0.78$ GeV from the unquenched quark model Wang:2019mhs
. This fact show that the $\chi_{c0}(2P)\to D\bar{D}$ decay can be suppressed
by the node effect. Thus, it is possible to get weak coupling for the
$\chi_{c0}(2P)D\bar{D}$ interaction when the node effect is considered. In
Fig. 3, we also list the experimental width of the $\chi_{c0}(3930)$
determined by LHCb LHCb:2020pxc . If reproducing this width value, we should
take $\beta=(0.824\sim 0.853)$ GeV, which is not obviously deviated from the
value 0.78 GeV mentioned above. We have reason to believe that the node effect
should be seriously considered when discussing the coupling of
$\chi_{c0}(2P)D\bar{D}$.

Figure 4: The mass dependence of the coupling
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ on different $\beta$ values. Here, six
typical values of the mass of $\chi_{c0}(2P)$ are adopted, which are 3.937,
3.939, 3.941, 3.943, 3.945 and 3.947 GeV, respectively. The uncertainty of
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ results from the uncertainty of
parameter $\gamma$, and the red dashed horizontal lines stands for the average
values for different $\beta$.

In the following, we discuss how to estimate coupling constant
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$. If only taking central mass of the
$\chi_{c0}(2P)$ as input, the process $\chi_{c0}(2P)\to D_{s}^{+}D_{s}^{-}$ is
kinematically forbidden. Although the $\chi_{c0}(2P)$ is below the threshold
of $D_{s}^{+}D_{s}^{-}$, the decay $\chi_{c0}(2P)\to D_{s}^{+}D_{s}^{-}$ still
can happen when considering the mass distribution of the $\chi_{c0}(2P)$222The
mass gap between the $\chi_{c0}(3930)\equiv\chi_{c0}(2P)$ and the
$D_{s}^{+}D_{s}^{-}$ threshold is $12\pm 2$ MeV and the measured width of
$\chi_{c0}(3930)$ is $17.4\pm 5.1\pm 0.8$ MeV LHCb:2020pxc ., which is similar
to the case of the $a_{0}(980)\to K\bar{K}$ decay. By the effective Lagrangian
approach, the decay amplitude of the $\chi_{c0}\to D_{s}^{+}D_{s}^{-}$ is
$\mathcal{M}_{\chi_{c0}(2P)\to
D_{s}^{+}{D}_{s}^{-}}=g_{\chi_{c0}(2P)D_{s}^{+}{D}_{s}^{-}}$, by which the
expression of this decay width can be obtained. For $\chi_{c0}\to
D_{s}^{+}D_{s}^{-}$, its decay width can be also deduced by the QPC model. And
then, the expression of $g_{\chi_{c0}(2P)D_{s}^{+}{D}_{s}^{-}}$ can be given
by the above preparation with the connection of the decay widths from the
effective Lagrangian approach and the QPC model.

Figure 5: (Color online.) The $\beta$ dependence of the coupling constant
ratio. The horizontal red dashed line with gray band indicates the coupling
constant ratio extracted from LHCb data. The vertical dashed line corresponds
to $\beta=0.824$ GeV.

We first check the $g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ value when taking six
typical values of the mass distribution of the $\chi_{c0}(2P)$, all of which
are above the $D_{s}^{+}D_{s}^{-}$ threshold. By this way, these
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ value corresponding to the above typical
mass values are calculable. In Fig. 4, the results of
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ dependent on several typical mass values
of the $\chi_{c0}(2P)$ are collected, where four typical $\beta$ values are
taken. We may conclude that the obtained $g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$
values are almost stable when changing the mass of the $\chi_{c0}(2P)$ as
shown in Fig. 4. Thus, we may adopt an extrapolation to estimate the realistic
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ value, which is considered to be
consistent with the obtained stable value of
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ corresponding to the mass range above
the $D_{s}^{+}D_{s}^{-}$ threshold. We also check the node effect of
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$. Our result shows that the coupling
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ is not sensitive to these selected
$\beta$ values. For the given $\beta$ range333The node effect of the coupling
$g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ is obvious if taking $\beta<0.5$ GeV,
which is deviated from the $\beta$ value fixed by the unquenched quark model
Wang:2019mhs ., the node effect to $g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ is
not apparent, which is completely different from the behavior of the node
effect to $g_{\chi_{c0}(2P)D^{+}D^{-}}$.

With the above preparation, we further present the ratio of
$g_{\chi(c0)(2P)D_{s}^{+}D_{s}^{-}}$ and $g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$
in Fig. 5, which is dependent on the parameter $\beta$. For a comparison, we
also list the corresponding ratio extracted from LHCb data (see the value in
Eq. (9)). Fig. 5 illustrates how the node effect plays the crucial role to be
responsible for explaining the coupling reversal phenomenon. Here, in an
extreme case, the coupling constants ratio
$g_{\chi_{c0}(2P)D^{+}D^{-}}/g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ could be
zero when $\beta=0.746$ GeV. In particular, we find the obtained ratio can
overlap with the one extracted from LHCb data if the $\beta$ range is
$(0.702-0.722)\ \mathrm{GeV}$ or $(0.773-0.807)\ \mathrm{GeV}$, where the
later one is a litter bit smaller than the $\beta$ parameter range
($(0.824-0.853)\ \mathrm{GeV}$) determined by the width of the
$\chi_{c0}(3930)$. This small discrepancy of the $\beta$ range can be
understood. Since the width of the $\chi_{c0}(3930)$ is composed of its
partial decay widths from open-charm decay channel, two gluon $gg$ process,
the hidden-charm channel like $\omega J/\psi$ Belle:2009and , and even
electromagnetic transition, the realistic $\beta$ range becomes smaller if
adopting the $\chi_{c0}(2P)\to D\bar{D}$ partial decay width to determine it.
Finally, the anomaly of the ratio
$g_{\chi_{c0}(2P)D^{+}D^{-}}/g_{\chi_{c0}(2P)D_{s}^{+}D_{s}^{-}}$ can be
explained well, which not only reflects the importance of the node effect, but
also enforces our scenario that the near $D_{s}^{+}D_{s}^{-}$ threshold
enhancement can be due to the $\chi_{c0}(2P)$ contribution.

##  IV Summary

Very recently, the LHCb Collaboration announced a near threshold enhancement,
referred to be the $X(3960)$ before deciphering its nature, in the
$D_{s}^{+}D_{s}^{-}$ invariant mass distribution of the process $B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}$ LHCb:2022vsv . The $X(3960)$ favors $J^{PC}=0^{++}$
suggested by LHCb LHCb:2022vsv . If relating the $X(3960)$ to the
$\chi_{c0}(3930)$ reported in the $B^{+}\to D^{+}D^{-}K^{+}$ decay
LHCb:2020pxc , LHCb obtained the ratio $\Gamma(X\to D^{+}D^{-})/\Gamma(X\to
D_{s}^{+}D_{s}^{-})$ to be $0.29\pm 0.09\pm 0.10\pm 0.08$ LHCb:2022vsv , which
is not consistent with of the naive expectation of assigning the $X(3960)$ as
a charmonium $\chi_{c0}(2P)$. Thus, LHCb claimed that the observed $X(3960)$
should be a candidate of the $c\bar{c}s\bar{s}$ tetraquark LHCb:2022vsv .

In this work, we propose that this enhancement phenomenon near the
$D_{s}^{+}D_{s}^{-}$ threshold is resulted from a conventional $P$-wave
charmonium $\chi_{c0}(2P)$ below the $D_{s}^{+}D_{s}^{-}$ threshold
Duan:2020tsx ; Duan:2021bna ; Liu:2009fe ; Belle:2009and ; BaBar:2012nxg ;
LHCb:2020pxc ; Chen:2012wy . For testing such scenario, a combined fit to the
measured $D_{s}^{+}D_{s}^{-}$, $D_{s}^{+}K^{+}$, and $D_{s}^{-}K^{+}$
invariant mass spectra is performed. Here, by introducing the $\chi_{c0}(2P)$
contribution, indeed we reproduce the near threshold enhancement in the
$D_{s}^{+}D_{s}^{-}$ invariant mass spectrum well. For depicting the whole
$D_{s}^{+}D_{s}^{-}$ invariant mass spectrum given by LHCb, we also introduce
higher charmonium $\psi(4230)$ BESIII:2014rja ; BESIII:2016bnd ;
BESIII:2016adj ; Wang:2019mhs ; Wang:2017sxq ; Qian:2021neg ; Chen:2014sra ;
Chen:2017uof  and a scalar charmoniumlike state $X_{0}(4140)$, which are as
the intermediate states in the discussed $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$
decay. Additionally, we also reveal that the predicted charmed meson
$D^{*}(3^{3}S_{1})$ with mass around 3015 MeV Song:2015fha ; Wang:2016krl  may
play crucial role when describing the $D_{s}^{-}K^{+}$ invariant mass spectrum
well.

In this work, another important issue is to explain the anomaly of the ratio
of $\Gamma(X\to D^{+}D^{-})/\Gamma(X\to D_{s}^{+}D_{s}^{-})$ indicated by
LHCb. Our study reveals that the node effect of the spatial wave function of
the $\chi_{c0}(2P)$ is the main reason to result in this anomaly of the ratio.
Finally, explaining the anomaly of the ratio of $\Gamma(X\to
D^{+}D^{-})/\Gamma(X\to D_{s}^{+}D_{s}^{-})$ enforces our scenario again,
i.e., the newly observed enhancement structure near the $D_{s}^{+}D_{s}^{-}$
threshold can be due to the $\chi_{c0}(2P)$ contribution.

In a summary, after observing the enhancement phenomenon near the
$D_{s}^{+}D_{s}^{-}$ threshold in $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$
LHCb:2022vsv  inspired interest of the community in deciphering its underlying
mechanism. LHCb tried to assign this observed enhancement structure as a
tetraquark LHCb:2022vsv . In fact, there is no hurry to introduce exotic state
assignment before exhausting possible conventional explanations. In this work,
an alternative explanation is proposed, where the observed near threshold
enhancement structure and the corresponding anomaly of the ratio of
$\Gamma(X\to D^{+}D^{-})/\Gamma(X\to D_{s}^{+}D_{s}^{-})$ indicate the
existence of a $\chi_{c0}(2P)$ charmonium below the $D_{s}^{+}D_{s}^{-}$
threshold, which is a crucial step to establish the $\chi_{c0}(2P)$
charmonium.

## Acknowledgements

This work is supported by the China National Funds for Distinguished Young
Scientists under Grant No. 11825503 and No. 12175037, National Key Research
and Development Program of China under Contract No. 2020YFA0406400, and the
111 Project under Grant No. B20063, the National Natural Science Foundation of
China under Grant No. 12047501. J.-Z.W. is also supported by the National
Postdoctoral Program for Innovative Talent.

## References

  * (1) [LHCb], Observation of a resonant structure near the $D_{s}^{+}D_{s}^{-}$ threshold in the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ decay, [arXiv:2210.15153 [hep-ex]].
  * (2) T. Ji, X. K. Dong, M. Albaladejo, M. L. Du, F. K. Guo and J. Nieves, Establishing the heavy quark spin and light flavor molecular multiplets of the $X(3872)$, $Z_{c}(3900)$ and $X(3960)$, [arXiv:2207.08563 [hep-ph]].
  * (3) J. M. Xie, M. Z. Liu and L. S. Geng, Production rates of $D_{s}^{+}D_{s}^{-}$ and $D\bar{D}$ molecules in $B$ decays, [arXiv:2207.12178 [hep-ph]].
  * (4) Q. Xin, Z. G. Wang and X. S. Yang, Analysis of the $X(3960)$ and related tetraquark molecular states via the QCD sum rules, [arXiv:2207.09910 [hep-ph]].
  * (5) R. Chen and Q. Huang, Charmoniumlike resonant explanation on the newly observed $X(3960)$, [arXiv:2209.05180 [hep-ph]].
  * (6) M. Bayar, A. Feijoo and E. Oset, The $X(3960)$ seen in $D_{s}^{+}D_{s}^{-}$ as the $X(3930)$ state seen in $D^{+}D^{-}$, [arXiv:2207.08490 [hep-ph]].
  * (7) R. Aaij et al. [LHCb], Amplitude analysis of the $B^{+}\to D^{+}D^{-}K^{+}$ decay, Phys. Rev. D 102, 112003 (2020).
  * (8) M. X. Duan, S. Q. Luo, X. Liu and T. Matsuki, Possibility of charmoniumlike state $X(3915)$ as $\chi_{c0}(2P)$ state, Phys. Rev. D 101, no.5, 054029 (2020).
  * (9) S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32, 189-231 (1985).
  * (10) T. Barnes, S. Godfrey and E. S. Swanson, Higher charmonia, Phys. Rev. D 72, 054026 (2005).
  * (11) T. Barnes and S. Godfrey, Charmonium options for the $X(3872)$, Phys. Rev. D 69, 054008 (2004).
  * (12) Y. S. Kalashnikova, Coupled-channel model for charmonium levels and an option for $X(3872)$, Phys. Rev. D 72, 034010 (2005).
  * (13) P. G. Ortega, J. Segovia, D. R. Entem and F. Fernandez, Coupled channel approach to the structure of the $X(3872)$, Phys. Rev. D 81, 054023 (2010).
  * (14) S. Uehara et al. [Belle], Observation of a charmonium-like enhancement in the $\gamma\gamma\to\omega J/\psi$ process, Phys. Rev. Lett. 104, 092001 (2010).
  * (15) X. Liu, Z. G. Luo and Z. F. Sun, $X(3915)$ and $X(4350)$ as new members in $P$-wave charmonium family, Phys. Rev. Lett. 104, 122001 (2010).
  * (16) J. P. Lees et al. [BaBar], Study of $X(3915)\to J/\psi\omega$ in two-photon collisions, Phys. Rev. D 86, 072002 (2012).
  * (17) J. Beringer et al. [Particle Data Group], Review of Particle Physics (RPP), Phys. Rev. D 86, 010001 (2012).
  * (18) F. K. Guo and U. G. Meissner, Where is the $\chi_{c0}(2P)$?, Phys. Rev. D 86, 091501 (2012).
  * (19) D. Y. Chen, J. He, X. Liu, T. Matsuki and T. Matsuki, Does the enhancement observed in $\gamma\gamma\to D\bar{D}$ contain two $P$-wave higher charmonia?, Eur. Phys. J. C 72, 2226 (2012).
  * (20) R. L. Workman [Particle Data Group], Review of Particle Physics, PTEP 2022, 083C01 (2022).
  * (21) M. Ablikim et al. [BESIII], Study of $e^{+}e^{-}\to\omega\chi_{cJ}$ at center-of-mass energies from 4.21 to 4.42 GeV, Phys. Rev. Lett. 114, no.9, 092003 (2015).
  * (22) M. Ablikim et al. [BESIII], Precise measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ cross section at center-of-mass energies from 3.77 to 4.60 GeV, Phys. Rev. Lett. 118, no.9, 092001 (2017).
  * (23) M. Ablikim et al. [BESIII], Evidence of Two Resonant Structures in $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$, Phys. Rev. Lett. 118, no.9, 092002 (2017).
  * (24) J. Z. Wang, D. Y. Chen, X. Liu and T. Matsuki, Constructing $J/\psi$ family with updated data of charmoniumlike $Y$ states, Phys. Rev. D 99, no.11, 114003 (2019).
  * (25) J. Z. Wang, H. Xu, J. J. Xie and X. Liu, Production of the charmoniumlike state Y(4220) through the $p\bar{p}\to Y(4220)\pi^{0}$ reaction, Phys. Rev. D 96, no.9, 094004 (2017).
  * (26) R. Q. Qian, Q. Huang and X. Liu, Predicted $\Lambda\bar{\Lambda}$ and $\Xi^{-}\bar{\Xi}^{+}$ decay modes of the charmoniumlike $Y(4230)$, Phys. Lett. B 833, 137292 (2022).
  * (27) D. Y. Chen, X. Liu and T. Matsuki, Observation of $e^{+}e^{-}\to\chi_{c0}\omega$ and missing higher charmonium $\psi(4S)$, Phys. Rev. D 91, no.9, 094023 (2015).
  * (28) D. Y. Chen, X. Liu and T. Matsuki, Interference effect as resonance killer of newly observed charmoniumlike states $Y(4320)$ and $Y(4390)$, Eur. Phys. J. C 78, no.2, 136 (2018).
  * (29) Q. T. Song, D. Y. Chen, X. Liu and T. Matsuki, Higher radial and orbital excitations in the charmed meson family, Phys. Rev. D 92, no.7, 074011 (2015).
  * (30) J. Z. Wang, D. Y. Chen, Q. T. Song, X. Liu and T. Matsuki, Revealing the inner structure of the newly observed $D_{2}^{\ast}(3000)$, Phys. Rev. D 94, no.9, 094044 (2016).
  * (31) M. X. Duan, J. Z. Wang, Y. S. Li and X. Liu, Role of the newly measured $B\to KD\bar{D}$ process to establish $\chi_{c0}(2P)$ state, Phys. Rev. D 104, no.3, 034035 (2021).
  * (32) J. Z. Wang, X. Liu and T. Matsuki, Evidence supporting the existence of $P_{c}(4380)^{\pm}$ from the recent measurements of $B_{s}\to J/\psi p\bar{p}$, Phys. Rev. D 104, no.11, 114020 (2021).
  * (33) Q. Huang, J. Z. Wang, R. G. Ping and X. Liu, Detecting the polarization in $\chi_{cJ}\to\phi\phi$ decays to probe hadronic loop effect, Phys. Rev. D 103, no.9, 096006 (2021).
  * (34) D. Y. Chen, X. Liu and T. Matsuki, $\eta$ transitions between charmonia with meson loop contributions, Phys. Rev. D 87, no.5, 054006 (2013).
  * (35) J. Z. Wang, Q. S. Zhou, X. Liu and T. Matsuki, Toward charged $Z_{cs}(3985)$ structure under a reflection mechanism, Eur. Phys. J. C 81, no.1, 51 (2021).
  * (36) Wenbin Qian, Observation of several new tetra-quark states at the LHCb experiment.
  * (37) L. Micu, Decay rates of meson resonances in a quark model, Nucl. Phys. B 10, 521-526 (1969).
  * (38) A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Naive quark pair creation model of strong interaction vertices, Phys. Rev. D 8, 2223-2234 (1973).
  * (39) A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Naive quark pair creation model and baryon decays, Phys. Rev. D 9, 1415-1419 (1974).
  * (40) A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Resonant Partial Wave Amplitudes in $\pi N\to\pi\pi N$ According to the Naive Quark Pair Creation Model, Phys. Rev. D 11, 1272 (1975).
  * (41) A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Strong Decays of $\psi^{\prime\prime}$ (4.028) as a Radial Excitation of Charmonium, Phys. Lett. B 71, 397-399 (1977).
  * (42) A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Why Is $\psi$(4.414) SO Narrow?, Phys. Lett. B 72, 57-61 (1977).
  * (43) D. Guo, C. Q. Pang, Z. W. Liu and X. Liu, Study of unflavored light mesons with $J^{PC}=2^{--}$, Phys. Rev. D 99, no.5, 056001 (2019).
  * (44) D. Guo, W. Chen, H. X. Chen, X. Liu and S. L. Zhu, Newly observed $a_{0}(1817)$ as the scaling point of constructing the scalar meson spectroscopy, Phys. Rev. D 105, no.11, 114014 (2022).
  * (45) Q. T. Song, D. Y. Chen, X. Liu and T. Matsuki, Charmed-strange mesons revisited: mass spectra and strong decays, Phys. Rev. D 91, 054031 (2015).