# The nonabelian Hodge correspondence for balanced Hermitian metrics of Hodge-
Riemann type

Xuemiao Chen  and  Richard A. Wentworth  Department of Mathematics, University
of Maryland, College Park, MD 20742, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper extends the nonabelian Hodge correspondence for Kähler manifolds to
a larger class of hermitian metrics on complex manifolds called balanced of
Hodge-Riemann type. Essentially, it grows out of a few key observations so
that the known results, especially the Donaldson-Uhlenbeck-Yau theorem and
Corlette’s theorem, can be applied in our setting. Though not necessarily
Kähler, we show that the Sampson-Siu Theorem proving that harmonic maps are
pluriharmonic remains valid for a slightly smaller class by using the known
argument. Special important examples include those balanced metrics arising
from multipolarizations.

###### Key words and phrases:

Hermitian-Einstein metric, Higgs bundle, harmonic map

###### 2020 Mathematics Subject Classification:

Primary: 53C07, 14J60; Secondary: 58E20

R.W.’s research is supported in part by NSF grant DMS-1906403.

##  1\. Introduction

Let $X$ be a compact, complex manifold of dimension $n$. Recall that a
hermitian metric on $X$ is called _balanced_ if $d\omega^{n-1}=0$, where
$\omega$ is the fundamental (Kähler) $(1,1)$-form of the metric. The balanced
metrics are a more restrictive class than the _Gauduchon_ metrics, which
satisfy $\partial\bar{\partial}\omega^{n-1}=0$. Nevertheless, there are many
examples of balanced, non-Kähler, metrics (cf. [18, p. 292]).

In this paper we consider a further condition. We say that a balanced metric
is of _Hodge-Riemann type_ , if it admits an expression:

(1.1) $\displaystyle\frac{\omega^{n-1}}{(n-1)!}=\omega_{0}\wedge\Omega_{0}$

where $\omega_{0}$ (resp. $\Omega_{0}$) is a real $(1,1)$ (resp. $(n-2,n-2)$),
and $\Omega_{0}$ satisfies the Hodge-Riemann bilinear relations (see
Definition 2.1 for the precise definition).

The condition of being balanced of Hodge-Riemann type seems very restrictive.
However, many examples of non-Kähler metrics satisfying this property come
from _multipolarizations_. Namely, let
$\omega_{0},\omega_{1},\ldots,\omega_{n-2}$ be positive $(1,1)$-forms on $X$,
and suppose

(1.2) $\frac{\omega^{n-1}}{(n-1)!}=\omega_{0}\wedge\cdots\wedge\omega_{n-2}$

such that

  * •

$d\omega^{n-1}=0$;

  * •

$d(\omega_{1}\wedge\cdots\wedge\omega_{n-2})=0$

(e.g. both conditions are automatic if the $\omega_{i}$ are Kähler). Then by a
result of Timorin [22], $\omega$ is of Hodge-Riemann type. Even when the
$\omega_{i}$ are Kähler metrics, $\omega$ is not so in general.

In this note we show that certain properties of Hermitian-Einstein metrics and
equivariant harmonic maps familiar for Kähler manifolds continue to hold for
balanced metrics of Hodge-Riemann type. Namely, we prove

  1. (1)

a generalized Bogomolov-Miyaoka-Yau inequality for $\omega$-polystable
holomorphic (and Higgs) bundles (Corollary 3.4);

  2. (2)

a version of the Sampson-Siu pluriharmonicity theorem for harmonic maps to
targets with nonpositive complexified sectional curvature (Theorem 4.1 and
Corollary 4.3);

  3. (3)

the nonabelian Hodge correspondence relating $\omega$-stable Higgs bundles
with vanishing Chern classes to irreducible representations of the fundamental
group (Theorem 5.1).

Let us remark that for the class of Gauduchon metrics, items (2) and (3) do
_not_ hold in general (see [2]), and the statement of item (1) cannot even be
formulated.

The simple idea behind these generalizations is well explained in [21, Lemma
1.1]. Let us focus on item (3) above. Suppose $D$ be a complex connection on a
vector bundle $E\to X$. A hermitian metric $h$ on $E$ gives a decomposition
$D=D^{\prime\prime}+D^{\prime}$ (see (5.2)). Conversely, a Higgs bundle
defines an operator $D^{\prime\prime}$, and a metric allows one to complete it
to a complex connection $D$ by setting $D^{\prime}=(D^{\prime\prime})^{\ast}$.
Let $F_{D}=D^{2}$ be the curvature of $D$, and $G_{D}=(D^{\prime\prime})^{2}$
the _pseudo-curvature_. Flatness of $D$ is the equation $F_{D}=0$, whereas $D$
arises from a Higgs bundle iff $G_{D}=0$.

Now suppose $\omega$ is a balanced metric on $X$, so that the degree and slope
stability of holomorphic bundles can be defined. Given an $\omega$-slope
stable Higgs bundle with $\operatorname{ch}_{1}(E)=0$, one can find a metric
$h$ so that the associated connection $D$ satisfies

(1.3) $F_{D}\wedge\omega_{0}\wedge\Omega_{0}=0$

Similarly, given a flat connection $D$ one can find a _harmonic metric_ ,
meaning that

(1.4) $G_{D}\wedge\omega_{0}\wedge\Omega_{0}=0$

Thus, under the assumptions, the forms $F_{D}$ and $G_{D}$ are “primitive”, in
the sense of (2.1) below.

The nonabelian Hodge correspondence follows by showing that if in addition
$\operatorname{ch}_{2}(E)=0$, then (1.3) implies $F_{D}=0$, and on the other
hand, (1.4) always implies $G_{D}=0$ (as pointed out in [21, p. 17], the
“pseudo-Chern class” defined by $G_{D}$ automatically vanishes by the flatness
of $D$). Now, if we assume $\omega$ is of Hodge-Riemann type, then these
conclusions hold by integrating $\operatorname{tr}(F_{D}\wedge F_{D})$ or
$\operatorname{tr}(G_{D}\wedge G_{D})$ against $\Omega_{0}$, and using the
vanishing of the Chern classes and the Hodge-Riemann bilinear relations. Thus,
we see that the Kähler condition may be relaxed.

Acknowledgements. The authors warmly thank Yang Li for comments on an earlier
version of this paper.

##  2\. Hodge-Riemann forms

In this section, we recall the notion of a Hodge-Riemann form on a polarized
complex vector space, i.e. a complex space $V$ with a constant Kähler form
$\omega_{0}$. Denote $\Lambda^{p,q}$ to be the space of constant $(p,q)$ forms
over $V$. We fix $\Omega_{0}$ to be any real $(n-p-q,n-p-q)$ form. On
$\Lambda^{p,q}$, we can define a Hermitian form as

$Q(\alpha,\beta):=(\sqrt{-1})^{p-q}(-1)^{\frac{(p+q)(p+q-1)}{2}}*(\alpha\wedge\overline{\beta}\wedge\Omega_{0})$

The space of _primitive_ forms of degree $(p,q)$ associated to
$(\Omega_{0},\omega_{0})$ is defined as

(2.1)
$P^{p,q}=\\{\alpha\in\Lambda^{p,q}:\alpha\wedge\omega_{0}\wedge\Omega_{0}=0\\}.$

######  Definition 2.1.

We call $\Omega_{0}$ a Hodge-Riemann form for degree $(p,q)$ with respect to
$\omega_{0}$ if

  1. (1)

there exists a $Q$-orthogonal decomposition

$\Lambda^{p,q}=\mathbb{C}\omega_{0}\oplus P^{p,q};$

  2. (2)

$Q$ is positive definite on $P^{p,q}$.

######  Remark 2.2.

It follows from the classical Hodge-Riemann relation that
$\Omega_{0}=\omega_{0}^{n-p-q}$ is a Hodge-Riemann form (cf. [24, Thm. 6.32]).

In general, the Hodge-Riemann property of a form is difficult to verify.
However, we have the following result, which has been used to get the mixed
Hodge-Riemann relation (see [8]).

######  Proposition 2.3 ([22, Main Theorem], see also [12]).

For any constant positive $(1,1)$ forms $\omega_{1},\cdots,\omega_{k}$ on
$(V,\omega_{0})$, $\omega_{1}\wedge\cdots\wedge\omega_{n-k}$ is a Hodge-
Riemann form with respect to $\omega_{0}$ for any degrees $(p,q)$ satisfying
$p+q=k$.

As a special case, this gives

######  Corollary 2.4.

For any constant positive $(1,1)$ forms $\omega_{1},\cdots,\omega_{n-2}$ on
$(V,\omega_{0})$

  1. (1)

$\omega_{0}\wedge\cdots\wedge\omega_{n-2}$ is a strictly positive $(n-1,n-1)$
form. In particular, there exists a positive $(1,1)$ form $\omega$ so that

$\frac{\omega^{n-1}}{(n-1)!}=\omega_{0}\wedge\cdots\wedge\omega_{n-2}$

(cf. [18, p. 279], and also [23]).

  2. (2)

$\omega_{1}\wedge\cdots\wedge\omega_{n-2}$ is a Hodge-Riemann form for degrees
$(p,q)$ satisfying $p+q=2$ with respect to $\omega_{0}$.

This combined with [20, Cor. 8.5] implies the following

######  Proposition 2.5.

For any constant Kähler forms $\omega_{1},\omega_{2}$ on $(V,\omega_{0})$, the
form

$\omega_{1}^{n-2}+\omega_{1}\wedge\omega_{2}^{n-3}+\cdots\omega_{2}^{n-2}$

is a Hodge-Riemann form for degree $(p,q)$ with $p+q=2$ with respect to
$\omega_{0}$.

######  Remark 2.6.

It is known that the Hodge-Riemann property is not invariant under convex
linear combinations. For example, fix any two Kähler forms $\omega_{1}$ and
$\omega_{2}$ on $\mathbb{C}^{4}$, $\omega_{1}^{2}+a\omega_{2}^{2}$ is not a
Hodge-Riemann form for degree $(1,1)$ for certain positive values of $a$ (see
[20, Rem. 9.3] and also [22, Rem. 3] for other examples).

Timorin’s result motivates the following:

######  Definition 2.7.

A hermitian metric $\omega$ on a complex manifold $X$ is said to be _balanced
of Hodge-Riemann type_ if the following hold:

  1. (1)

we have an expression

$\frac{\omega^{n-1}}{(n-1)!}=\omega_{0}\wedge\Omega_{0}$

where $\omega_{0}$ is a strongly positive real $(1,1)$ form on $X$ and
$\Omega_{0}$ is a real $(n-2,n-2)$;

  2. (2)

at every point $\Omega_{0}$ is a Hodge-Riemann form for $(p,q)$, $p+q=2$;

  3. (3)

$\Omega_{0}$ and $\omega_{0}\wedge\Omega_{0}$ are closed.

Note that (3) is equivalent to $\omega$ being balanced and $\Omega_{0}$ being
closed.

##  3\. Bogomolov-Miyaoka-Yau inequality

Below we show how the Donaldson-Uhlenbeck-Yau (resp. Hitchin-Simpson) theorem
relating stability of holomorphic (resp. Higgs) bundles to the existence of
Hermitian-Einstein type metrics results in a Chern class inequality. The main
result is Corollary 3.4. In this section, assume $(X,\omega)$ is a compact
complex Hermitian manifold that satisfies items (1) and (3) of Definition 2.7,
as well as the Hodge-Riemann condition (2) for the case $(p,q)=(1,1)$.

Recall that associated to every coherent analytic sheaf $\mathcal{E}\to X$ is
a holomorphic line bundle $\det\mathcal{E}$. The first Chern class of
$\mathcal{E}$ is by definition $c_{1}(\mathcal{E}):=c_{1}(\det\mathcal{E})\in
H^{2}(X,\mathbb{Z})\cap H^{1,1}_{\bar{\partial}}(X)$. We define the
$\omega$-degree of $\mathcal{E}$ by:

$\deg\mathcal{E}:=\int_{X}c_{1}(\mathcal{E})\wedge\frac{\omega^{n-1}}{(n-1)!}=\int_{X}c_{1}(\mathcal{E})\wedge\omega_{0}\wedge\Omega_{0}$

Because of the balanced condition, this is well-defined on the class of
$c_{1}(\mathcal{E})$. The _slope_ of a (nonzero) torsion-free sheaf is

$\mu(\mathcal{E})=\frac{\deg\mathcal{E}}{\operatorname{rank}\mathcal{E}}$

Then we say a holomorphic bundle $\mathcal{E}\to X$ is _$\omega$ -stable_ if
$\mu(\mathcal{S})<\mu(\mathcal{E})$ for every coherent subsheaf
$\mathcal{S}\subset\mathcal{E}$ with
$0<\operatorname{rank}\mathcal{S}<\operatorname{rank}\mathcal{E}$.

A _Higgs bundle_ on $X$ is a pair $(\mathcal{E},\theta)$, where
$\mathcal{E}\to X$ is a holomorphic bundle, $\theta$ is a holomorphic $1$-form
with values in $\operatorname{End}\mathcal{E}$, and $\theta\wedge\theta=0$. We
say that a Higgs bundle is $\omega$-stable if
$\mu(\mathcal{S})<\mu(\mathcal{E})$ for every coherent subsheaf
$\mathcal{S}\subset\mathcal{E}$ with
$0<\operatorname{rank}\mathcal{S}<\operatorname{rank}\mathcal{E}$ _and_
$\theta(\mathcal{S})\subset\mathcal{S}\otimes\Omega^{1}_{X}$, where
$\Omega^{1}_{X}$ is the holomorphic cotangent sheaf of $X$. Thus, stable
vector bundles are a special case of stable Higgs bundles, where $\theta\equiv
0$. Finally, we say that $(\mathcal{E},\theta)$ is _$\omega$ -polystable_ if
$(\mathcal{E},\theta)$ splits as a direct sum of Higgs subbundles, all with
the same slope.

Given a hermitian metric $h$ on $\mathcal{E}$, let
$\bar{\partial}_{E}+\partial_{E}$ denote the Chern connection of
$(\mathcal{E},h)$, $\theta^{\ast}$ the hermitian adjoint of $\theta$ with
respect to $h$. Thus, $\theta^{\ast}$ is a $(0,1)$-form with values in
$\operatorname{End}E$, satisfying $\partial_{E}\theta^{\ast}=0$. We will
consider the _complex_ connection

$D=\bar{\partial}_{E}+\partial_{E}+\theta+\theta^{\ast}$

and its curvature $F_{D}$. To be explicit, we will write:
$D=(\mathcal{E},\theta,h)$.

######  Definition 3.1.

Fix a Higgs bundle $(\mathcal{E},\theta)$ on $X$. A hermitian metric $h$ on
$\mathcal{E}$ is called Hermitian-Einstein (HE) if

(3.1)
$\sqrt{-1}F_{(\mathcal{E},\theta,h)}\wedge\omega_{0}\wedge\Omega_{0}=\lambda\cdot\operatorname{Id}\cdot\omega_{0}^{2}\wedge\Omega_{0}$

for some constant $\lambda$.

Now we have the following generalized Donaldson-Uhlenbeck-Yau, Hitchin-Simpson
theorem.

######  Theorem 3.2.

$(\mathcal{E},\theta)$ is $\omega$-polystable if and only if it admits a HE
metric. Moreover, if $(\mathcal{E},\theta)$ is $\omega$-stable, such a metric
is unique up to scaling.

We will use the key fact that for balanced metrics, the Kähler identities hold
for $(1,0)$ and $(0,1)$ forms ([10, Prop. 1]; see also [16, Lemma 7.1.1]).

######  Lemma 3.3.

Given an $n$-dimensional hermitian manifold $(X,\omega)$ with
$d\omega^{n-1}=0$, the following hold:

$\bar{\partial}^{*}\alpha^{0,1}=-\sqrt{-1}\Lambda\partial\alpha^{0,1}\quad,\quad\partial^{*}\alpha^{1,0}=\sqrt{-1}\Lambda\bar{\partial}\alpha^{1,0}$

for any $(0,1)$-form $\alpha^{0,1}$, and any $(1,0)$-form $\alpha^{1,0}$.

###### Proof of Theorem 3.2.

That the existence of a HE metric implies polystability is well-known. For the
converse, it suffices to assume $(\mathcal{E},\theta)$ is $\omega$-stable.
Since $\omega$ is balanced, it is in particular Gauduchon, and so by the
result of Li-Yau [15], generalized to Higgs bundles by Lübke-Teleman [17],
there is a metric $\widetilde{h}$ such that

$\sqrt{-1}F_{(\mathcal{E},\theta,\widetilde{h})}\wedge\frac{\omega^{n-1}}{(n-1)!}=\widetilde{\lambda}\cdot\operatorname{Id}\cdot\frac{\omega^{n}}{n!}$

where $\widetilde{\lambda}=2\pi\mu(\mathcal{E})/\operatorname{vol}(X,\omega)$.
Now there is a positive function $f$ such that

$\omega_{0}\wedge\frac{\omega^{n-1}}{(n-1)!}=f\cdot\frac{\omega^{n}}{n!}$

Choose $\lambda$ such that

(3.2)
$\lambda\int_{X}f\cdot\frac{\omega^{n}}{n!}=\widetilde{\lambda}\operatorname{vol}(X,\omega)=2\pi\mu(\mathcal{E})$

Then we can find a function $\varphi$ satisfying:
$\Delta_{\omega}\varphi=2(\lambda f-\widetilde{\lambda})$. Let
$h=e^{\varphi}\widetilde{h}$. Then

$F_{(\mathcal{E},\theta,h)}=F_{(\mathcal{E},\theta,\widetilde{h})}-\partial\bar{\partial}\varphi\cdot\operatorname{Id}$

By Lemma 3.3, the Hodge and Dolbeault laplacians on functions are related:
$\Delta_{\omega}=2\Delta_{\bar{\partial}}$. Hence,

$-i\partial\bar{\partial}\varphi\wedge\frac{\omega^{n-1}}{(n-1)!}=\frac{1}{2}\Delta\varphi\frac{\omega^{n}}{n!}=\lambda\omega_{0}\wedge\frac{\omega^{n-1}}{(n-1)!}-\widetilde{\lambda}\frac{\omega^{n}}{n!}$

The result follows. ∎

As a direct corollary of this, we have the following generalized Bogomolov-
Miyaoka-Yau inequality

######  Corollary 3.4.

For any $\omega$-polystable rank $r$ Higgs bundle $(\mathcal{E},\theta)$, the
following inequality holds:

$\int_{X}(2rc_{2}(\mathcal{E})-(r-1)c_{1}(\mathcal{E})^{2})\wedge\Omega_{0}\geq
0$

where the equality holds if and only if $\mathcal{E}$ is projectively flat.

###### Proof.

Eq. (3.1) implies

$F_{(\mathcal{E},\theta,h)}-\frac{1}{r}\operatorname{tr}(F_{(\mathcal{E},\theta,h)})\cdot\operatorname{Id}\cdot\,\omega_{0}$

is primitive. Now use the Hodge-Riemann property of $\Omega_{0}$.

∎

For emphasis, we state the following version of the Donaldson-Uhlenbeck-Yau
theorem for the slope stability condition defined by multipolarizations (cf.
[11]).

######  Theorem 3.5.

Suppose $X$ is a compact Kähler manifold with $(n-1)$ Kähler forms
$\omega_{0},\cdots,\omega_{n-2}$. Given a holomorphic vector bundle
$\mathcal{E}$ that is slope stable with respect to
$[\omega_{0}]\cup\cdots\cup[\omega_{n-2}]$, there exists a Hermitian-Einstein
metric $h$ on $\mathcal{E}$, i.e.

$\sqrt{-1}F_{(\mathcal{E},h)}\wedge\omega_{0}\wedge\cdots\wedge\omega_{n-2}=\lambda\cdot\operatorname{Id}\cdot\,\omega^{2}_{0}\wedge\omega_{1}\wedge\cdots\wedge\omega_{n-2}$

for some constant $\lambda$. Moreover, such a metric is unique up to constant
rescalings.

######  Remark 3.6.

We emphasize here that the Chern connection of $(\mathcal{E},h)$ is _not_ a
Yang-Mills connection in general.

By Corollary 3.4 and Proposition 2.3, we have the following generalization of
the Bogomolov-Gieseker inequality for multipolarizations, proven in the
projective case by Miyaoka [19, Cor. 4.7].

######  Corollary 3.7.

Suppose $X$ is a compact Kähler manifold with $(n-1)$ Kähler forms
$\omega_{0},\cdots,\omega_{n-2}$, and $\mathcal{E}$ is a slope stable
holomorphic vector bundle over $X$ with respect to
$[\omega_{0}]\cup\cdots\cup[\omega_{n-2}]$. Then the following holds:

$\int_{X}(2rc_{2}(\mathcal{E})-(r-1)c_{1}^{2}(\mathcal{E}))\wedge\Omega_{j}\geq
0$

for any $j=0,\ldots,n-2$. Here,
$\Omega_{j}=\omega_{0}\wedge\cdots\wedge\omega_{j-1}\wedge\omega_{j+1}\cdots\wedge\omega_{n-2}$.
Moreover, the equality holds for some $j$ if and only if $\mathcal{E}$ is
projectively flat.

######  Remark 3.8.

The gauge theoretic side of the HE connections defined via multipolarizations
is studied in [4].

##  4\. The Sampson-Siu theorem

In this section, we prove

######  Theorem 4.1.

Let $X$ be a compact complex manifold with a balanced metric of Hodge-Riemann
type, and assume $\Omega_{0}$ is strongly positive. If $N$ is a Riemannian
manifold with nonpositive complexified sectional curvature, then every
harmonic map $u:X\to N$ is pluriharmonic.

In the statement of the theorem, $\omega$ satisfies the condition of
Definition 2.7, but we make the _additional assumption_ that $\Omega_{0}$ is a
_strongly positive_ $(n-2,n-2)$-form in the sense of [7, Ch. III, Def. 1.1].

###### Proof of Theorem 4.1.

Let $\nabla$ denote the Levi-Cività connection on $N$. This induces a
connection on $u^{\ast}TN$. The harmonic map equation is:
$d_{\nabla}^{\ast}du=0$. Since $\omega$ is balanced, Lemma 3.3 implies that
$u$ is harmonic if and only if

(4.1) $d_{\nabla}d^{c}u\wedge\omega^{n-1}=0$

Next, we follow the argument in [1, pp. 73-75]. Since $\Omega_{0}$ is closed,

$d\langle d_{\nabla}d^{c}u\wedge d^{c}u\rangle\wedge\Omega_{0}=\langle
R_{N}(d^{c}u)\wedge d^{c}u\rangle\wedge\Omega_{0}+\langle
d_{\nabla}d^{c}u\wedge d_{\nabla}d^{c}u\rangle\wedge\Omega_{0}$

and so,

$0=\int_{X}\left\\{\langle R_{N}(d^{c}u)\wedge
d^{c}u\rangle\wedge\Omega_{0}+\langle d_{\nabla}d^{c}u\wedge
d_{\nabla}d^{c}u\rangle\wedge\Omega_{0}\right\\}$

By (4.1) and the Hodge-Riemann property, the second term is nonpositive. We
claim that also

$\langle R_{N}(d^{c}u)\wedge d^{c}u\rangle\wedge\Omega_{0}\leq 0$

Given this, it follows that $d_{\nabla}d^{c}u=0$; hence, the pluriharmonicity.
The claim follows from the assumption on $\Omega_{0}$ and the nonpositivity of
the complexified sectional curvature of $N$. We work at a point $x$. By
definition, we know that

$\Omega_{0}=\sum_{i}\mu_{i}\sqrt{-1}\alpha_{1}^{i}\wedge\overline{\alpha_{1}^{i}}\wedge\cdots\wedge\sqrt{-1}\alpha_{n-2}^{i}\wedge\overline{\alpha_{n-2}^{i}}$

where $\mu_{i}\geq 0$, and $\\{\alpha_{1}^{i},\cdots,\alpha_{n-2}^{i}\\}$ are
linearly independent $(1,0)$ forms. Denote by $P_{i}$ the complex two
dimensional subspace of $TM$ where $\alpha^{i}_{j}|_{P_{i}}=0$ for
$j=1,\cdots,n-2$. Fix $X_{i},Y_{i}$ so that $\\{X_{i},Y_{i},JX_{i},JY_{i}\\}$
form an orthogonal basis for $P_{i}$. Then

$\langle R_{N}(d^{c}u)\wedge
d^{c}u\rangle\wedge\Omega_{0}=\sum_{i}\mu_{i}^{\prime}\langle
R_{N}(d^{c}u)\wedge
d^{c}u\rangle(X_{i},Y_{i},JX_{i},JY_{i})\operatorname{dVol}$

for some $\mu_{i}^{\prime}\geq 0$. Now as in [1, p. 75], we know

$\langle R_{N}(d^{c}u)\wedge
d^{c}u\rangle(X_{i},Y_{i},JX_{i},JY_{i})=R_{N}(Z_{i},W_{i},\overline{W_{i}},\overline{Z_{i}})\leq
0$

where $Z_{i}=du(X_{i}-JX_{i})$ and $W_{i}=du(W_{i}-JW_{i})$. The claim
follows. ∎

######  Remark 4.2.

If $\widetilde{X}$ is the universal cover of $X$, then Theorem 4.1 remains
valid for harmonic maps $u:\widetilde{X}\to N$ that are equivariant with
respect to a representation $\rho:\pi_{1}(X)\to\operatorname{Iso}(N)$. These
play a role in the next section. The existence of equivariant harmonic maps to
nonpositively curved targets $N$ is guaranteed if $\rho$ is _reductive_ (or
_semisimple_) (see [5, 9, 14, 13]).

Theorem 4.1, combined with [7, Prop. III.1.11] implies

######  Corollary 4.3.

Suppose $X$ is a compact complex manifold with a balanced metric $\omega$ of
the form (1.2), where $\omega_{i}$ are positive $(1,1)$-forms and

$d(\omega_{1}\wedge\cdots\wedge\omega_{n-2})=0$

If $N$ is a Riemannian manifold with nonpositive complexified sectional
curvature, then every harmonic map $u:X\to N$ is pluriharmonic.

##  5\. The nonabelian Hodge correspondence

The goal of this section is to prove the following generalization of [21, Cor.
1.3].

######  Theorem 5.1.

Suppose $X$ is a compact complex manifold and $\omega$ is a balanced metric of
Hodge-Riemann type on $X$. Then the nonabelian Hodge correspondence holds over
$(X,\omega)$. More precisely, we have a 1-1 correspondence between

  1. (1)

semisimple flat bundles on $X$, and

  2. (2)

isomorphism classes of $\omega$-polystable Higgs bundles
$(\mathcal{E},\theta)$ with
$\operatorname{ch}_{1}(\mathcal{E})\cup[\omega^{n-1}]=0$ and
$\operatorname{ch}_{2}(\mathcal{E})\cup[\Omega_{0}]=0$.

######  Remark 5.2.

  * •

When $\omega_{0}=\omega$, $\Omega_{0}=\omega^{n-2}/(n-1)!$, then by our
assumptions $(X,\omega)$ is Kähler. Then Theorem 5.1 reduces to the well known
nonabelian Hodge correspondence for compact Kähler manifolds.

  * •

There exist many examples where $\omega$ is not a Kähler metric, even when the
underlying manifold $X$ is Kähler, or even projective algebraic. For example,
take $X$ to be projective with $[\omega_{i}]$, $i=0,\ldots,n-2$, all ample
classes, and take $\omega$ as in (1.2). Then the class $\omega^{n-1}$
represents a point in the interior of the cone of _movable curves_ (see [3]).

  * •

Notice that if $\Omega_{0}=\omega_{1}^{n-2}$, then $d\Omega_{0}=0$ implies
$d\omega_{1}=0$, and the manifold is Kähler. However, closedness of
$\Omega_{0}=\omega_{1}\wedge\cdots\wedge\omega_{n-2}$, for different
$\omega_{i}$, can occur in the non-Kähler setting. One might expect that this
will provide new insights for the study the non-Kähler complex manifolds,
since the results obtained here already put restrictions on complex manifolds
admitting such structures.

###### Proof of Theorem 5.1.

The proof, of course, closely follows the lines of the classical theorem,
taking care to avoid the Kähler condition.

First, assume $(\mathcal{E},\theta)$ is an $\omega$-polystable Higgs bundle.
If $\operatorname{ch}_{1}(\mathcal{E})\cup[\omega^{n-1}]=0$, then by Theorem
3.2 there is a hermitian metric $h$ on $\mathcal{E}$ such that (1.3) is
satisfied for

(5.1) $D=\bar{\partial}_{E}+\partial_{E}+\theta+\theta^{\ast}$

(note that $\lambda=0$ by (3.2)). Hence, $F_{D}$ is primitive. Moreover,
$\sqrt{-1}F_{D}$ is of type $(1,1)$ and hermitian. Since
$\operatorname{ch}_{2}(\mathcal{E})\cup[\Omega_{0}]=0$, we have

$0=\int_{X}\operatorname{tr}(F_{D}\wedge F_{D})\wedge\Omega_{0}$

Since $\Omega_{0}$ is a Hodge-Riemann form, we conclude that $D$ is a flat
connection, which is necessarily semisimple.

Now let $D$ be a semisimple flat connection on $E$. By Corlette’s theorem (see
Remark 4.2) there exists a _harmonic metric_ , which has the following
consequence. Decomposing into type we can express $D$ as in (5.1), where
$\theta\in\Omega^{1,0}(X,\operatorname{End}E)$, and $\theta+\theta^{\ast}$ is
essentially $du$ for an equivariant harmonic map
$u:\widetilde{X}\to\mathsf{GL}(n,\mathbb{C})/\mathsf{U}(n)$. Let

(5.2) $D^{\prime\prime}=\bar{\partial}_{E}+\theta\quad,\quad
D^{\prime}=\partial_{E}+\theta^{\ast}$

We wish to prove that $G_{D}=(D^{\prime\prime})^{2}=0$, for then
$\bar{\partial}_{E}$ is integrable, $\bar{\partial}_{E}\theta=0$, and
$\theta\wedge\theta=0$; i.e. $(\bar{\partial}_{E},\theta)$ is a Higgs bundle.

Flatness of $D$ implies,

  1. (1)

$\partial_{E}\theta=0$;

  2. (2)

$(\bar{\partial}_{E}\theta)^{\ast}=-\bar{\partial}_{E}\theta$;

  3. (3)

$\partial_{E}^{2}+\tfrac{1}{2}[\theta,\theta]=0$;

By (4.1), we have
$(\bar{\partial}_{E}\theta-(\bar{\partial}_{E}\theta)^{\ast})\wedge\omega^{n-1}=0$,
and combining this with (2) we have

(5.3)
$\bar{\partial}_{E}\theta\wedge\omega^{n-1}=\bar{\partial}_{E}\theta\wedge\omega_{0}\wedge\Omega_{0}=0$

i.e. $G_{D}$ is primitive (note that we have only used the balanced condition
for this part).

To prove that $G_{D}=0$, we argue as in the proof of Theorem 4.1 (see also [5,
proof of Thm. 5.1]). We have:

$\displaystyle
d\operatorname{tr}(\bar{\partial}_{E}\theta\wedge\theta^{\ast})\wedge\Omega_{0}$
$\displaystyle=\partial\operatorname{tr}(\bar{\partial}_{E}\theta\wedge\theta^{\ast})\wedge\Omega_{0}$
$\displaystyle=\operatorname{tr}(\partial_{E}\bar{\partial}_{E}\theta\wedge\theta^{\ast})\wedge\Omega_{0}+\operatorname{tr}(\bar{\partial}_{E}\theta\wedge(\bar{\partial}_{E}\theta)^{\ast})\wedge\Omega_{0}$
$\displaystyle=-\frac{1}{2}\operatorname{tr}([[\theta,\theta^{\ast}],\theta]\wedge\theta^{\ast})\wedge\Omega_{0}+\operatorname{tr}(\bar{\partial}_{E}\theta\wedge(\bar{\partial}_{E}\theta)^{\ast})\wedge\Omega_{0}$
$\displaystyle=-\frac{1}{4}\operatorname{tr}([\theta,\theta]\wedge[\theta,\theta]^{\ast})\wedge\Omega_{0}+\operatorname{tr}(\bar{\partial}_{E}\theta\wedge(\bar{\partial}_{E}\theta)^{\ast})\wedge\Omega_{0}$

so integrating,

$0=-\frac{1}{4}\int_{X}\operatorname{tr}([\theta,\theta]\wedge[\theta,\theta]^{\ast})\wedge\Omega_{0}+\int_{X}\operatorname{tr}(\bar{\partial}_{E}\theta\wedge(\bar{\partial}_{E}\theta)^{\ast})\wedge\Omega_{0}$

By the Hodge-Riemann property of $\Omega_{0}$, both terms on the right hand
side are nonpositive, and hence vanish. We conclude that
$\bar{\partial}_{E}\theta=0$, and $[\theta,\theta]=0$. By (3) above,
$\bar{\partial}_{E}$ is integrable, and this completes the proof. ∎

##  6\. Rigidity of representations of fundamental groups

For the sake of completeness, in this last section we point out that two
important results of Corlette and Simpson generalize to our setting. Let
$G_{\mathbb{R}}$ be a simple real algebraic group acting by isometries on the
irreducible bounded symmetric domain $G_{\mathbb{R}}/K$. We assume
$(X,\omega)$ is a compact complex manifold with a balanced metric of Hodge-
Riemann type. Let $P$ be the principle $G_{\mathbb{R}}$ bundle with structure
group reduced to $K$. As in [6], one can associate a volume
$\operatorname{vol}(P)$ to $P$ by defining it as a power of the first Chern
class of $P$ up to a conformal factor. Now the following generalizes [6, Thm.
0.1].

######  Theorem 6.1.

Suppose $P$ is flat with $\operatorname{vol}(P)\neq 0$ and $G_{\mathbb{R}}/K$
is not of the form $\mathsf{U}(n,1)/\mathsf{U}(n)\times\mathsf{U}(1)$ or
$\mathsf{SO}(2n+1,2)/\mathsf{S}(\mathsf{O}(2n+1)\oplus\mathsf{O}(2))$. Then
the monodromy homomorphism of the fundamental group of $X$ into
$G_{\mathbb{R}}$ is locally rigid as a homomorphism of the fundamental group
of $X$ into the complexification of $G_{\mathbb{R}}$.

The argument follows by replacing [6, Prop. 2.4] with argument in the proof of
Theorem 5.1 to get the holomorphic property of the harmonic sections. This is
the only place where the Kähler assumption is needed in [6]. Simpson’s
argument in the Kähler case also gives (see [21])

######  Theorem 6.2.

Suppose $\rho:\pi_{1}(X)\rightarrow\mathsf{GL}(n,\mathbb{C})$ is a locally
rigid representation of the fundamental group of $X$. Then the associated flat
vector bundle is the underlying vector bundle of a complex variation of Hodge
structure.

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# Constraints on Bosonic Dark Matter with Low Threshold Germanium Detector at
Kuo-Sheng Reactor Neutrino Laboratory

Manoj Kumar Singh  Lakhwinder Singh<EMAIL_ADDRESS>Mehmet
Agartioglu  Vivek Sharma<EMAIL_ADDRESS>Venktesh Singh  Henry
Tsz-king Wong  Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
Department of Physics, Institute of Science, Banaras Hindu University,
Varanasi 221005, India  Department of Physics, National Dong Hwa University,
Shoufeng, Hualien 97401, Taiwan

###### Abstract

We report results from searches of pseudoscalar and vector bosonic super-
weakly interacting massive particles (super-WIMP) in the TEXONO experiment at
the Kuo-Sheng Nuclear Power Station, using 314.15 kg days of data from
$n$-type Point-Contact Germanium detector. The super-WIMPs are absorbed and
deposit total energy in the detector, such that the experimental signatures
are spectral peaks corresponding to the super-WIMP mass. Measured data are
compatible with the background model, and no significant excess of super-WIMP
signals are observed. We derived new upper limits on couplings of electrons
with the pseudoscalar and vector bosonic super-WIMPs in the sub-keV mass
region, assuming they are the dominant contributions to the dark matter
density of our galaxy.

###### keywords:

Germanium Detectors , Data Acquisition , Bosonic Dark Matter , super-WIMP

††journal: Journal of LaTeX Templates

##  1 Introduction

There is ample evidence for the existence of dark matter (DM) that come from
numerous astronomical and cosmological observations at different scales. Since
all the existing evidence only probe the gravitational interaction of DM,
their non-gravitational interactions with normal matter are still elusive.
Many extensions of the standard model (SM) of particle physics predict viable
DM candidates. Weakly Interacting Massive Particles (WIMP), axions or axion-
like particles (ALPs), sterile neutrinos and millicharged particles are
leading candidates of DM. Searches for these leading candidates are in full
swing, but an experimental verification via direct, indirect detection or
production from LHC is still awaited. In the absence of a credible positive
signal, direct detection experiments have ruled out a substantial portion of
the favored WIMP parameter space in the GeV to TeV mass range [1, 2]. The
light-DM (LDM) has received more attention at both theoretical and
experimental front in recent years. Bosonic super-weakly interacting particles
(denoted by $\chi$) constitute another large category of LDM candidates with
masses at the keV-scale [3]. These bosonic $\chi$ candidates are
experimentally very interesting due to their absorption via ionization or
excitation of an electron in the target-atom of detectors. The bosonic $\chi$
would deposit the energy equivalent to their rest mass in the detector. A
photo-like peak with unexplained origin in the measured energy spectrum would
be “smoking gun” signature of bosonic LDM. Point-Contact Germanium (PCGe)
detectors [4] with their excellent energy resolution, sub-keV threshold, and
low intrinsic radioactivity background are the best candidates to study LDM
and other physics beyond the SM.

The theme of this article is to report improved direct laboratory limits on
$\chi$ coupling based on the data acquired by an n-type Point-Contact
Germanium ($n$PCGe) detector at the Kuo-Sheng Reactor Neutrino Laboratory
(KSNL). The structure of this report is as follows. Section 2 describes the
interaction and expected rate of $\chi$ at Ge target. Highlights of the
experimental setup and data acquisition at KSNL are presented in Sec. 3. The
performance parameters of the $n$PCGe detector are summarized in Sec. 4. The
detail of event selection of candidates and background suppression is
explained in Sec 5. We present our results compared with the other
representative experiments in Sec. 6. Finally, we conclude in Sec. 7.

##  2 Bosonic Dark Matter

Pseudoscalar, scalar, and vector are three generic possible candidates of non-
relativistic LDM that may have a superweak coupling with SM particles. The
correct relic density of bosonic LDM in a wider mass range could be obtained
via either thermally or non-thermally misalignment mechanism [5, 6, 7]. The
bosonic pseudoscalar ($\chi_{\mathrm{ps}}$) are excellent candidates of LDM.
The effective interaction Lagrangian of pseudoscalar-LDM particles with the
electron is given as

$\mathcal{L}_{int}=\frac{g_{aee}}{2m_{e}}(\partial_{{\mu}}a)\bar{\psi}\gamma^{\mu}\gamma^{5}\psi,$
(1)

where $a$ is the bosonic pseudoscalar field, $g_{aee}$ is the pseudoscalar-
electron coupling constant, $m_{e}$ is the mass of electron, $\gamma^{\mu}$
are the Dirac matrices, and $\bar{\psi}$, $\psi$ are dual Dirac spinors. The
phenomenology behind $\chi_{\mathrm{ps}}$ is similar to nonrelativistic ALPs.
The $\chi_{\mathrm{ps}}$ have coupling to atoms through the axioelectric
effect:

$\chi+A\rightarrow A^{+}+e^{-},$ (2)

which is analogous to the photoelectric effect with the absorption of
$\chi_{\mathrm{ps}}$ instead of a photon [8]. The only difference between
photoelectric effect and axioelectric effect is the wave function of absorbed
photon and $\chi_{\mathrm{ps}}$. The wave function of absorbed photon contains
a space-dependent factor exp($i\bf{kr}$), which is replaced by
exp(${im_{\mathrm{ps}}\bf{v_{\mathrm{\chi}}r}}$) in case of massive
$\chi_{\mathrm{ps}}$ [3]. Here $m_{\mathrm{ps}}$ and $v_{\mathrm{\chi}}$ are
the mass and velocity of the incoming $\chi_{\mathrm{ps}}$ particle,
respectively. The absorption cross section $\sigma_{abs}$ (axio-electric
effect) for $\chi_{\mathrm{ps}}$ can be written as

$\displaystyle\sigma_{abs}\simeq\frac{3m_{\mathrm{ps}}^{2}}{4\pi\alpha
f^{2}_{a}\beta}{\sigma_{pe}(w=m_{\mathrm{ps}})},$ (3)

where $\sigma_{pe}$ is the photoelectric cross section with the photon energy
$\omega$ replaced by the mass of $\chi_{\mathrm{ps}}$ ($m_{\mathrm{ps}}$),
$f_{a}$ = 2$m_{e}/g_{aee}$ is the dimensionless coupling strength of
$\chi_{\mathrm{ps}}$ to SM particles and $\beta\equiv v_{{\chi}}/c$. The
absorption cross section of $\chi_{\mathrm{ps}}$ is directly proportional to
$m_{\mathrm{ps}}^{2}$.

The best motivated model for bosonic vector dark matter ($\chi_{\mathrm{v}}$)
is kinetic mixing model, in which an extra U(1)D gauge group is introduced
into SM gauge group. The kinetic mixing with the hypercharge field strength is
responsible for the interaction between the ordinary matter and
$\chi_{\mathrm{v}}$. The interaction Lagrangian density for
$\chi_{\mathrm{v}}$ with electrons after breaking of U(1)D can be expressed as

$\mathcal{L}_{int}={\kappa}{e}V_{\mu}\bar{\psi}\gamma^{\mu}\psi,$ (4)

where $V_{\mu}$ is the bosonic vector field couples to the electromagnetic
current, $e$ is the electronic charge, $\kappa$ is the vector hypercharge. The
product $e^{\prime}$ = $e\kappa$ is analogous to electromagnetic coupling such
that $\alpha^{\prime}$ = (e$\kappa$)2/4$\pi$ is similar to a vector-electric
fine-structure constant. The Compton wavelength of $\chi_{\mathrm{v}}$ is much
larger than the linear dimension of the atom due to nonrelativistic behavior.
This allow the multipole expansion in the interaction. The contribution of
E1-transition dominates over other multipoles [9], which makes the matrix
element of photon absorption proportional to the matrix element of
$\chi_{\mathrm{v}}$ absorption. The absorption cross section of
$\chi_{\mathrm{v}}$ ($\sigma_{abs}$) can be expressed in the photoelectric
effect with the replacement of photon energy $\omega$ by the mass mv of
$\chi_{\mathrm{v}}$ and the coupling constant is scaled appropriately as [3]

$\displaystyle\frac{\sigma_{abs}}{\sigma_{pe}(\omega=m_{\mathrm{v}})}\simeq\frac{\alpha^{\prime}}{\alpha}\frac{1}{\beta}.$
(5)

The absorption cross section of $\chi_{\mathrm{v}}$ is nearly independent of
the DM mass.

The key difference between $\chi_{\mathrm{ps}}$ and $\chi_{\mathrm{v}}$ LDM is
their decay modes to photons. The two-photon decay channel is strictly
forbidden for vector boson, whereas decay into three photons is allowed at
loop level [9]. Pseudoscalar bosons are axion-like with two-photon decay as a
dominant channel. The mass of a scalar particle receives quadratically
radiative contributions from its interactions with other SM particles. The
keV-scale bosonic scalar ($\chi_{\mathrm{s}}$) requires highly model dependent
parameters in order to make it viable candidates. The phenomenology of
$\chi_{\mathrm{s}}$ is also quite similar to the $\chi_{\mathrm{ps}}$ in many
ways [3]. Experimental searches thus mainly focus on $\chi_{\mathrm{ps}}$ and
$\chi_{\mathrm{v}}$.

Assuming the $\chi$ ($\chi_{\mathrm{ps}}$ or $\chi_{\mathrm{v}}$) constitutes
all of the galactic DM with their local density $\rho_{\chi}$ = 0.3 GeV/cm3,
the total average flux of $\chi$ ($\Phi_{\chi}=\rho_{\chi}v_{\chi}/m_{\chi}$)
is given by

$\displaystyle\Phi_{\chi}=\frac{9.0\times
10^{15}}{m_{\chi}}\times\beta~{}~{}~{}\mathrm{cm}^{-2}\mathrm{s}^{-1},$ (6)

where mχ is the mass of $\chi$ in keV/c2, $v_{\chi}$ is the average velocity
of $\chi$ relative to the Earth; $\beta\equiv v_{{\chi}}/c\;\sim$ 10-3 implies
that the energy deposition by bosonic $\chi$ would be E =
$\sqrt{(m_{\chi}^{2}+\gamma^{2}m_{\chi}^{2}\beta^{2})}\sim m_{\chi}$.

The theoretically expected interaction rate of $\chi$ in a direct detection
experiment can be expressed as

$\displaystyle R_{\chi}=\rho_{d}\sigma_{abs}\Phi_{\chi},$ (7)

where $\rho_{d}$ = NA/A is the atomic number density per unit target mass of
detector and NA is the Avogadro’s number. $\sigma_{abs}$ is the absorption
cross section of $\chi$.

Figure 1: The expected pseudoscalar and vector interaction rates in Ge-
detector, for fixed values of $g_{aee}$ and $\alpha^{\prime}$/$\alpha$, as a
function of $m_{\chi}$. The axioelectric interaction is illustrated
schematically in the inset.

The interaction rate of $\chi_{\mathrm{ps}}$ in the units of kg-1day-1 can be
calculated by using the Eq. 7. Accordingly, the expected rate becomes

$\displaystyle R_{\mathrm{ps}}\simeq\frac{1.2\times
10^{19}}{A}g_{aee}^{2}\frac{m_{\mathrm{ps}}}{[\mathrm{keV}]}\frac{\sigma_{pe}}{[\mathrm{barns}]}\mathrm{kg}^{-1}\mathrm{day}^{-1},$
(8)

where A is the atomic mass of target materials. Similarly, the expected count
rate of $\chi_{\mathrm{v}}$ in the direct detection experiment can be
expressed as

$\displaystyle R_{\mathrm{v}}\simeq\frac{4\times
10^{23}}{A}\frac{\alpha^{\prime}}{\alpha}\frac{[\mathrm{keV}]}{m_{\mathrm{v}}}\frac{\sigma_{pe}}{[\mathrm{barns}]}~{}\mathrm{kg}^{-1}\mathrm{day}^{-1}.$
(9)

Point-Contact Germanium detectors [10, 4] are ideal devices for the searches
of $\chi_{\mathrm{ps}}$ and $\chi_{\mathrm{v}}$, due to their low energy
threshold which expand the sensitivities to low $m_{\chi}$ and excellent
energy resolution allows spectral peaks to be resolved. The interaction rates
of $\chi_{\mathrm{ps}}$ and $\chi_{\mathrm{v}}$ with Ge-detector follow Eqs.
(8) and (9), respectively, are illustrated in Fig. 1 for specific choices of
the coupling constants. The characteristic features in the calculated spectrum
at energies 11.10 keV (K-shell) and 1.41 keV (L-shell) correspond to
absorption edges of the photoelectric effect in the germanium atom. These
features as peaks or edges at the specific binding energies significantly
enhance the sensitivity of the experiment compared to the case of a continuum
curve.

##  3 Experimental Setup

The TEXONO collaboration [11] aims to progressively improving the
sensitivities towards electromagnetic properties of neutrino [12, 13, 14] as
well as WIMP [15, 16], axions [17] and physics searches beyond the SM at KSNL.
The experimental setup is placed at the first floor of the seven-storey
reactor building at depth of $\sim$12 m below the sea level. The concrete of
the reactor building provides 30 meters of water equivalent overburden, which
eliminates most of the hadronic components of cosmic rays. A multipurpose
4$\pi$ passive shielding house with “inner target” detector volume of 100 cm
$\times$ 80 cm $\times$ 75 cm consists of, from the inside out, 5 cm of
oxygen-free high-conductivity copper, 25 cm of boron-loaded polyethylene, 5 cm
of steel, 15 cm of lead. The shielding house is further mounted with cosmic
ray (CR) veto scintillator panels. This 50-ton passive shielding house
provides significant attenuation to the ambient neutron and $\gamma$
backgrounds. The Ge-detector surrounded by 38.3 kg well-shaped NaI(Tl) anti-
Compton (AC) detector is placed inside the inner target volume. A detailed
description of shielding and the detector development at KSNL can be found in
Refs. [11, 14, 18].

The data acquisition (DAQ) system in operation at KSNL is a hybrid design of
digital and analog electronics. The schematic block-diagram of electronics and
DAQ system is illustrated in Fig. 2. The preamplifier of PCGe is custom
designed with 5 copies of a signal. Two copies of Ge-preamplifier signal are
shaped with Canberra 2026 shaping amplifier (SA) at 6 $\mu s$ and 12 $\mu s$
with same amplification factors. The distinct behavior of self-trigger noise
at different shaping time helps to suppress noise events near the threshold.
One copy of Ge-preamplifier is processed with SA at minimum gain to study the
high energy background behavior. The timing amplifier output (TA) preserves
the rise-time information. Therefore, two copies of Ge-preamplifier signal are
processed with Canberra 2111 TA at “out” differentiate along with 10 $ns$
integration control setting with different gains to cover a wide energy range
from sub-keV to few hundreds keV.

Figure 2: Schematic diagram of the data acquisition system for TEXONO
experiment at KSNL. Figure 3: Schematic diagram of data acquisition trigger.

Figure 4: A Typical pulse shape at 450 eV from SA at 6 $\mu s$ output.

Figure 5: A Typical pulse shape at 450 eV from TA output with raw (black) and
smoothed (red) pulse.

The KSNL readout system is built around National Instruments (NI) PXIe-1065
chassis with an embedded PXI-8108 controller and extended by NI PXI-5105
analog-to-digital converter (ADC), NI PXI-5124 ADC, and NI PXIe-7961R FlexRIO
Field Programmable Gate Arrays (FPGA) cards. The NI PXI-5105 ADC with 12-bit
resolution, 60 MHz bandwidth and 60 MS/s (MS: Mega sample) real-time sample
rate is being used to digitize the signal from shaping amplifiers. The NI
PXI-5124 ADC with 12-bit resolution, 200 MS/s real-time sample rate and 150
MHz bandwidth records the signal from timing amplifiers. The NI PXIe-7961R
FlexRIO FPGA module composed with high-speed 16-channel, 14-bit and 50 MS/s NI
5751 digitizer-adapter is the heart of the DAQ system. These FPGA-modules
provide the user-defined logic to specify the trigger condition for CR veto
scintillator panels and Ge-detector. It also helps to reject some microphonic
noise online, based on distorted pulse shape. The signal from the DC-coupled
preamplifier is discriminated after being shaped with SA at 6 $\mu s$ served
as a Ge-trigger. The random-trigger is generated by a function generator at
the rate of 0.1 Hz and recorded randomly due to OR-gate logic with random Ge-
trigger. These random-trigger events help to measure the pedestal fluctuation
and dead time of the DAQ system. The inhibit output from the preamplifier is
used to veto all triggers generated during the reset of the preamplifier,
including random-trigger. A schematic diagram of DAQ trigger scheme at KSNL is
shown in Fig. 3. A typical low energy signal from SA and TA is depicted in
Fig. 5 and Fig. 5, respectively. The signal of AC NaI-detector and CR veto
panel are also recorded along with Ge-trigger to understand the cosmic induced
background, and filtered out in the offline analysis. The known amplitude
pulses are generated from the precision wave function generator and injected
into the test port of the preamplifier to study the trigger efficiency and
electronic behavior of the DAQ system. The DAQ software is developed in NI
LabVIEW programming language which provides a user-friendly environment. To
monitor the data quality, the DAQ software generates cumulative diagnostic
plots for immediate inspection.

##  4 Characterization of N-type PCGe-Detector

The energy calibration from sub-keV to 14 keV is achieved by the combination
of precise-pulser measurement and X-ray peaks (10.37 keV and 1.29 keV),
resulting from electron capture of Ga daughter induced by cosmogenic 68,71Ge-
isotope. The maximal amplitude and integrated area of the pulse are both
energy related parameters. The maximal amplitude is adopted to estimate the
energy of an event for its better energy resolution [4, 19]. The energy
resolution of $n$PCGe is comparable to p-type Point-Contact ($p$PCGe) in low
energy region, but degraded in high energy region. The energy resolution of
$n$PCGe may be enhanced by increasing the detector operating temperature,
selecting the material with lower electron trap concentrations and higher
gradients impurity concentrations to generate higher drift field [20, 10].
Figure 7 illustrates the variation of full width at half maximum (FWHM) as a
function of energy for $n$PCGe and $p$PCGe detectors. The FWHM of $n$PCGe
degrades as energy deposition increases. The performance parameters of same
modular mass $n$PCGe and $p$PCGe detectors are summarized in Table 1.

Table 1: Comparison between $n$PCGe and $p$PCGe-detector on basis of various performance parameters. The pulse maximal amplitude is adopted as energy estimator in sub-keV energy region. Item |  $p$PCGe |  $n$PCGe
---|---|---
Modular Mass (g) | 500 | 500
RESET Amplitude (V) | 6.8 | 6.8
RESET Time Interval (ms) |  $\sim$160 |  $\sim$ 170
Pedestal Noise |  |
Amplitude RMS $\sigma_{Noise}$ (eV) | 41 | 49
Pulser Width |  |
FWHM (eV) | 110 | 133
RMS (eV) | 47 | 52
X-Ray Line Width | Ga-K | Ga-K
RMS (eV) | 87 | 104
Electronic Noise-Edge for Raw Spectra (eV ) | 228 | 285

Figure 6: Comparison of energy resolution of $n$PCGe and $p$PCGe at pedestal
fluctuation (0 keV), Ga K-Shell X-ray (10.37 keV) and 241Am $\gamma$-ray (59.5
keV).

Figure 7: The transition edge of TA-pulses as a function of energy with best
fit function.

Experimental sensitivity for physics beyond the SM is not only limited by
signal detection threshold of the Ge-detectors, but also due to the
degradation of background rejection capability at sub-keV recoil energy
region. The anomalous surface events in $p$PCGe arise from a few millimeters
thick transition layer with a weak electric field. The poor charge collection
in this region reduces the ionization yield of the events and enhances the
background at sub-keV energy region. The p+ outer surface conductive contact
of $n$PCGe is fabricated with the boron implantation technique. The thickness
of p+ contact on the outer surface of the crystal is the order of $\mu$m. The
anomalous surface effect in $n$PCGe is studied by the rise time of events. In
order to measure the rise time of events, the output of TA is fitted with the
hyperbolic tangent function

$\displaystyle f(t)=\frac{A_{0}}{2}{\mathrm{tanh}}[(t-t_{0})s_{0}]+P_{0},$
(10)

where A0, P0, t0 and $s_{0}$ are, respectively, the amplitude, pedestal
offset, transition edge and slope of the TA-pulse. The values of A0 and P0 are
evaluated from the TA-pulses through the difference and mid level of
asymptotic levels, respectively. The transition edge of TA-pulses as a
function of energy is pre-determined as shown in Fig. 7 and provides
constraint on t0. The bending attribute between energy and transition edge of
TA-pulse at low energies is due to the difference in shaping times of SA-
trigger and TA pulse. The rise time ($\tau$) of TA-pulse is characterized by
the slope parameter $s_{0}$ and rise time from 5% to 95% is evaluated by
$\tau~{}=~{}\mathrm{log}(19)/{s_{0}}$. The distribution of $\tau$ as a
function of energy is depicted in Fig. 9, which demonstrates the absence of
anomalous surface events.

Figure 8: The scatter plot between $\tau$ and T for the
$\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$-tag events shows absence of anomalous
surface events. Typical single- and multiple-site events are shown in inset,
multiple-site events are characterized by kinks in rise-time profiles.

Figure 9:  The $\tau$-distribution of Ga K-Shell X-ray (10.37 keV), comparing
the candidate events with those of 241Am, 137Cs between 8 to 12 keV energy
region.

The scattered events above the curve (red) in Fig. 9 are identified as
multiple sites events. Typical single- and multiple-site events at low energy
are illustrated in the inset of Fig. 9. The $\tau$-distribution of events from
different known origins can verify the uniform timing response over the entire
detector fiducial volume. Figure 9 illustrates the uniform timing response for
external $\gamma$-ray from 241Am, 137Cs and internal X-rays at 10.37 keV from
68,71Ge, which decay via electron capture.

##  5 Data Analysis

Data taken with the $n$PCGe detector have been adopted to place constraints on
milli-charged neutrino [21] and dark matter cosmic ray [22], as well as heavy
sterile neutrinos as dark matter [13]. The data presented in this article were
acquired over 628.3 live-days (whole exposure) with 500g fiducial mass of
$n$PCGe-detector. The event information from AC and CR-detectors is recorded
along with Ge-trigger. Thus, every Ge-trigger is categorized by
AC${}^{+(-)}\otimes CR^{+(-)}$, where the superscript +(-) denotes coincidence
(anti-coincidence). The $\mathrm{AC}^{+}\otimes\mathrm{CR}^{+}$-tag events are
mostly due to energy deposited by radiations in the Ge-detector. These double
tagged events serve as the reference samples to differentiate signal from
noise.

Figure 10: Scatter plot of the correlation between maximum time bin of pulse
and corresponding energy.

Figure 11:  Scatter plot between amplitude of TA-pulse and measured energy T
to illustrate the PSD selection. The inset illustrates the trigger efficiency
as a function of energy, along with total PSD efficiency.

The $\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$-tag events are uncorrelated with
AC and CR active shielding detectors and may be the candidate events of
$\chi$, neutrino, internal cosmogenically induced background and self-trigger
(electronic noise) near the noise-edge. The threshold of Ge-detector is
restricted by self-trigger and microphonic noise. In order to avoid the
contamination of noise near the detector threshold, it is necessary to define
a precise signal acceptance. Therefore, pulse shape discrimination (PSD)
technique is adopted to remove a majority of microphonic pulses and define
noise edge for analysis. The shaped waveform of Ge-detector is characterized
by several parameters. Lowest-level, pedestal (average value of pre-trigger
baseline) and maximum time bin are used to discard the most of the noise,
which leaks to high energy bins. The maximum time bin of the good pulses lies
within a time interval after the trigger. Thus, the amplitude and the maximum
time bin of the signal pulse exhibit the strong correlation as shown in Fig.
11, and all events outside the correlated band are rejected. In order to
define the noise edge between self-trigger and physics events, the energy of
event and amplitude of TA-pulse (ATA) after process with the trapezium-filter
provide one of the best possible parameter space. Figure 11 shows the
distribution of $\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$ and
$\mathrm{AC}^{+}\otimes\mathrm{CR}^{+}$ with the noise-edge cut. The fraction
of $\mathrm{AC}^{+}\otimes\mathrm{CR}^{+}$-tag events survives as an
efficiency of the PSD cuts. The trigger efficiency measured by survival
fraction of precise pulser at given trigger threshold. The trigger efficiency
and combined PSD efficiency is 100% above 300 eV and 400 eV, respectively, as
shown in the inset of Fig. 11.

Figure 12: Measured energy spectrum of data sample,
$\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$ before and after cosmic ray
correction.

Figure 13: Measured energy spectrum of data sample,
$\mathrm{AC}^{+}\otimes\mathrm{CR}^{-}$, before and after cosmic ray
correction.

Cosmic ray muon detection efficiency of CRV system on average for a week
period is typically 93%, and the rejection efficiency of cosmic ray induced
event by timing correlation between Ge-trigger and CR-trigger is 92%. The
measured $\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$ spectrum therefore could be
contaminated with cosmic ray induced events. The presence of additional cosmic
events in the residual energy spectrum of
$\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$ events is suppressed by the
statistical background model. The real cosmic (Tr) and non-cosmic (Vr) event
rate can be correlated with measured (Vm ,Tm ) rate via following couple of
equations

$\displaystyle
V_{r}=\frac{\lambda_{crv}}{\epsilon_{crv}+\lambda_{crv}-1}.V_{m}-\frac{1-\lambda_{crv}}{\epsilon_{crv}+\lambda_{crv}-1}.T_{m}~{}~{}~{}~{}~{}\mathrm{a}nd$
(11) $\displaystyle
T_{r}=\frac{\epsilon_{crv}}{\epsilon_{crv}+\lambda_{crv}-1}.T_{m}-\frac{1-\epsilon_{crv}}{\epsilon_{crv}+\lambda_{crv}-1}.V_{m},$
(12)

where $\epsilon_{crv}$ is the rejection efficiency of cosmic ray cut and
$\lambda_{crv}$ is the detection efficiency of cosmic ray system. The
verification of the correction method is provided by the intensity of known
K-shell internal X-rays and ambient $\gamma$-ray energy spectrum. The
intensities of K-shell X-rays increased by putting back leakage events into
their own sample, as illustrated in Fig. 13. High energy $\gamma$-rays from
ambient radioactivity produce flat electron-recoil background at low energy,
which is verified by simulation and measurements (241Am and 137Cs source).
Figure 13 demonstrates the flat energy spectra after correction for
$\mathrm{AC}^{+}\otimes\mathrm{CR}^{-}$ data sample.

The observed $\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$ spectra with internal
X-ray lines from cosmogenically-induced isotopes decays via electron capture
process is shown in Fig. 14.

Figure 14: Measured Raw and $\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$ spectra
with $n$PCGe, the L-shell X-ray lines predicted from the intensities of
measured K-shell X-ray lines is shown in inset. Figure 15: The efficiency-
corrected energy spectrum near to the analysis threshold with the whole
exposure. An example of excluded rate (90% C.L.) at 0.8 keV for non-
relativistic bosonic $\chi$ signal is shown with blue curve. The assumption of
flat plus exponential background is also superimposed.

The intensity of K-shell peaks decay with time and low energy L-capture lines
corresponding to 65Zn and 68Ge behaves similar to high energy K-capture lines.
The L/K-capture ratio from the same isotope has been well understood by
theoretically and independent experimental measurements. Therefore, the
intensities of L-shell X-ray lines are estimated from measured K-shell X-ray
lines and subtracted from measured $\mathrm{AC}^{-}\otimes\mathrm{CR}^{-}$
spectra. The predicted intensities of L-shell X-ray lines are depicted in the
inset of Fig. 14.

##  6 Experimental Constraints

The signal of nonrelativistic $\chi$ would be $\delta$-function centered at
its mass, smearing by the energy resolution of the detector. The expected
signal rate of $\chi$ at measurable energy is obtained by the convolution of
interaction rate from Eq. (8) or (9) with the energy resolution of the
detector:

$\frac{dR_{\chi}}{dE}(E,m_{\chi})=\Phi_{\chi}(m_{\chi})\sigma_{\chi}\times\frac{1}{\sqrt{2\pi}\sigma_{E}(m_{\chi})}e^{-\frac{(E-m_{\chi})^{2}}{2\sigma_{E}^{2}(m_{\chi})}}$
(13)

where $\chi$ = $\chi_{\mathrm{ps}}$, $\chi_{\mathrm{v}}$ represents
pseudoscalar and vector particles, respectively. A minimal-chi-square analysis
is applied with free parameters describing a flat plus exponential background
and coupling constant of $\chi$. Our data are compatible with the background
only hypothesis, and no excess of events are observed over the background as
depicted in Fig. 15. The 90% confidence level (C.L.) excluded rate in
kg-1day-1 for a given mχ is evaluated from best fit parameters. The excluded
value of coupling constant is then derived from excluded count rate using Eqs.
(8) or (9). The 90% C.L. exclusion limits for coupling of $\chi_{\mathrm{ps}}$
as a function of $m_{\chi}$ are shown in Fig. 16, results from other
representative experiments are also superimposed.

Figure 16: Constraints on $g_{aee}$ of $\chi_{\mathrm{ps}}$ as a function of
mass. The TEXONO upper limits, at 90% C.L. is shown in red curve. Results from
representative experiments CDEX [23], CoGeNT [24], EDELWEISS [25], LUX [26],
Majorana Demonstrator [27], PANDAX-II [28], XENON100 [29] along with indirect
astrophysical bound from solar neutrinos [30] are also shown. Figure 17: The
90% C.L. bounds on vector bosonic DM coupling from the TEXONO compared with
astrophysical sources [9] as well as experimental results from EDELWEISS [25],
Majorana Demonstrator [27], XENON100 [31] and XMASS-I [32].

The upper limits on the electronic coupling of $\chi_{\mathrm{v}}$ at 90% C.L.
are derived using the same data and analysis technique. Figure 17 shows the
exclusion limits at 90% C.L. of electronic coupling of $\chi_{\mathrm{v}}$.
The laboratory upper limits on vector-electric coupling established by direct
detection experiments are potentially more stringent than limits derived from
cosmological and astronomical sources. The observed fluctuations in the
exclusion curve correspond to statistical fluctuations in the background. Our
data with sub-keV sensitivity provide the improved constraints on electronic
couplings of pseudoscalar and vector bosonic DM in sub-keV/c2 mass region.

##  7 Conclusion and prospects

New constraints on dark matter candidates $\chi_{\mathrm{ps}}$ and
$\chi_{\mathrm{v}}$ with $n$PCGe are presented in this article. This detector
technique provides low threshold, excellent energy resolution and without
complications of anomalous surface background events. It is therefore an
optimal detector to study not just $\chi$ but also milli-charged neutrino [21]
and dark matter WIMPs [33], as well as dark matter heavy sterile neutrinos
[13]. Data taking continues at KSNL with improved PCGe detectors. The goals
are to enhance the sensitivities of these measurements, as well as to study
the standard model neutrino-nucleus coherent scattering [4, 34, 11].

##  8 Acknowledgments

This work is supported by the 2017-21 Academia Sinica Investigator Award AS-
IA-106-M02, as well as contracts MOST 107-2119-M-001-028-MY3 from the Ministry
of Science and Technology, and 2017-18/ECP-2 from the National Center of
Theoretical Physics, Taiwan.

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