# Volumes of components of Lelong upper level sets II

Shuang Su and Duc-Viet Vu

###### Abstract

Let $X$ be a compact Kähler manifold of dimension $n$, and let $T$ be a closed
positive $(1,1)$-current in a nef cohomology class on $X$. We establish an
optimal upper bound for the volume of components of Lelong upper level sets of
$T$ in terms of cohomology classes of non-pluripolar self-products of $T$.

Keywords: relative non-pluripolar product, density current, Lelong number.

Mathematics Subject Classification 2010: 32U15, 32Q15.

##  1 Introduction

The aim of this paper is to investigate the singularities of closed positive
currents on compact Kähler manifolds. Let $X$ be a compact Kähler manifold of
dimension $n$, and let $T$ be a closed positive $(1,1)$-current on $X$. We are
interested in understanding the set of points where $T$ has a strictly
positive Lelong numbers. By the celebrated upper semi-continuity of Lelong
numbers by Siu [12], we know that this set is a countable union of proper
analytic subsets on $X$. Our goal is to estimate the size of this upper level
set. We note that by [13], for every closed positive current $R$ on $X$, there
always exists a closed positive $(1,1)$-current whose Lelong numbers coincide
with those of $R$. Thus, to study questions about Lelong numbers in this
paper, it suffices to consider only currents of bi-degree $(1,1)$. To delve
into details, let us first introduce some necessary notions.

Let $\omega$ be a fixed smooth Kähler form on $X$. We equip $X$ with the
Riemannian metric induced by $\omega$. Let $S$ be a closed positive
$(p,p)$-current for some $0\leq p\leq n$. We define the mass $\|S\|$ of $S$ to
be equal to $\int_{X}S\wedge\omega^{n-p}$. For an analytic set $V$ of pure
dimension $l$ in $X$, we recall that

$\mathop{\mathrm{vol}}(V)=\frac{1}{l!}\int_{\text{Reg}V}\omega^{l},$

where $\text{Reg}V$ is the regular locus of $V$. Given a closed positive
$(p,p)$-current $R$, we denote by $\\{R\\}\in H^{p,p}(X,\mathbb{R})$ the
cohomology class of $R$. We say $\alpha\in H^{p,p}(X,\mathbb{R})$ is
_pseudoeffective_ if $\alpha=\\{R\\}$ for some closed positive $(p,p)$-current
$R$. Let $\alpha,\beta\in H^{p,p}(X,\mathbb{R})$. We say $\alpha\geq\beta$ if
$\alpha-\beta$ is a pseudoeffective class.

For every $x\in X$, let $\nu(T,x)$ denote the Lelong number of $T$ at $x$. For
every irreducible analytic subset $V$ in $X$, we recall that the generic
Lelong number $\nu(T,V)$ of $T$ along $V$ is defined as $\inf_{x\in
V}\\{\nu(T,x)\\}$. The Lelong number is a notion measuring the singularities
of $T$. We refer to [4] for the basics of Lelong numbers. For every constant
$c>0$, we denote $E_{c}(T)\vcentcolon=\\{x\in X|\nu(T,x)\geq c\\}$ and
$E_{+}\vcentcolon=\\{x\in X|\nu(T,x)>0\\}$. By [12], $E_{c}(T)$ is a proper
analytic subset in $X$, and $E_{+}(T)=\cup_{m\in\mathbb{N}^{*}}E_{1/m}(T)$ is
a countable union of analytic sets.

Let $W$ be an irreducible analytic subset of dimension $m$ in $X$. We denote
by $E_{+}^{W}(T)\vcentcolon=\\{x\in W|\nu(T,x)>\nu(T,W)\\}$ the Lelong upper
level set of $T$ on $W$, which is also a countable union of proper analytic
subsets in $W$. Let $V\subset E_{+}^{W}(T)$ be an irreducible analytic set. We
say that $V$ is _maximal_ if there is no irreducible analytic subset
$V^{\prime}$ of $E_{+}^{W}(T)$ such that $V$ is a proper subset of
$V^{\prime}$. We call $V$ _a component of the Lelong upper level set of $T$
along $W$_, and let $\mathscr{V}_{T,W}$ be the set of such components $V$.
Observe that $\mathscr{V}_{T,W}$ has at most countably many elements. For
$0\leq l\leq m$, we denote by $\mathscr{V}_{l,T,W}$ the set of
$V\in\mathscr{V}_{T,W}$ such that $\dim V=l$.

Write $T=dd^{c}u$ locally, where $u$ is a plurisubharmonic (psh in short)
function. We define $T|_{\text{Reg}W}$ to be $dd^{c}(u|_{\text{Reg}W})$ if
$u\not\equiv-\infty$ on $\text{Reg}W$, and $T|_{\text{Reg}W}:=0$ otherwise.
One sees that this definition is independent of the choice of $u$. Thus,
$T|_{\text{Reg}W}$ is a current on $\text{Reg}W$. Here is our main result in
the paper.

######  Theorem 1.1.

Let $\alpha$ be a nef $(1,1)$-class and let $W$ be an irreducible analytic
subset of dimension $m$ in $X$. Let $T$ be a closed positive current in
$\alpha$ such that $\nu(T,W)=0$. Let $1\leq m^{\prime}\leq m$ be an integer.
Then, we have

$\displaystyle\sum_{V\in\mathscr{V}_{m-m^{\prime},T,W}}\nu(T,V)^{m^{\prime}}\mathop{\mathrm{vol}}(V)\leq\frac{1}{(m-m^{\prime})!}\int_{\text{Reg}W}\big{(}\alpha^{m^{\prime}}-\langle(T|_{\text{Reg}W})^{m^{\prime}}\rangle\big{)}\wedge\omega^{m-m^{\prime}},$
(1.1)

where in the integral, we identify $\alpha$ with a smooth closed form in
$\alpha$.

We have some comments on (1.1). To see why the term

$I:=\int_{\text{Reg}W}\big{(}\alpha^{m^{\prime}}-\langle(T|_{\text{Reg}W})^{m^{\prime}}\rangle\big{)}\wedge\omega^{m-m^{\prime}}$

is non-negative, one can consider the case where $W$ is smooth. Then, by a
monotonicity of non-pluripolar products (see Theorem 3.3 below), the
cohomology class
$(\alpha|_{W})^{m^{\prime}}-\\{\langle(T|_{W})^{m^{\prime}}\rangle\\}$ is
pseudoeffective. Hence the integral in the right-hand side of (1.1) is non-
negative. In the general case where $W$ is singular, one can use either a
desingularisation of $W$ or interpret $I$ as the mass of some non-pluripolar
product relative to $[W]$ (the current of integration along $W$); see Lemma
3.2 below. We underline however that in order to prove Theorem 1.1, it is not
possible to use desingularisation of $W$ to reduce to the case where $W$ is
smooth. The reason is that in the process of desingularisation, one has to
blow up submanifolds of $W$ which in general could be some of components of
the Lelong upper level sets of $T$ on $W$.

In [8], a less precise upper bound of volume of components of the Lelong upper
level set was given in terms of the volume of $W$ and the mass of $T$; see
also Theorem 3.7 for a more general statement. The estimate (1.1) is indeed
optimal. To see this, consider $n=m=m^{\prime}=1$ and $T$ is a sum of Dirac
masses. In this case, (1.1) becomes an equality.

If we consider $W=X$, then the generic Lelong number of $T$ along $W$ is zero.
Thus, by Theorem 1.1, we have the following result.

######  Corollary 1.2.

Let $\alpha$ be a nef $(1,1)$-class, and $T$ be a closed positive current in
$\alpha$. For $0\leq l\leq n$, let $\mathscr{V}_{l,T}$ be the set of
$V\in\mathscr{V}_{T,X}$ such that $\dim V=l$. Let $1\leq m^{\prime}\leq n$ be
an integer. Then, we have

$\displaystyle\sum_{V\in\mathscr{V}_{n-m^{\prime},T}}\nu(T,V)^{m^{\prime}}\mathop{\mathrm{vol}}(V)\leq\frac{1}{(n-m^{\prime})!}\int_{X}\big{(}\alpha^{m^{\prime}}-\\{\langle
T^{m^{\prime}}\rangle\\}\big{)}\wedge\omega^{n-m^{\prime}}.$

Corollary 1.2 generalizes [5, Corollary 7.6] by Demailly in which it was
assumed additionally that the components of the upper Lelong level set of $T$
are only of dimension $0$ (hence the cohomology class of $T$ is necessarily
nef, see [5, Lemma 6.3]). The feature of Corollary 1.2 is that it holds for
any current in a nef class.

This paper refines and substitutes [16]. The proof of Theorem 1.1 requires
both the theory of density currents in [7] and relative non-pluripolar
products in [17] (see also [1, 2]). One of the keys is Theorem 3.6 below
following from a general comparison of Lelong numbers for density currents
(see Corollary 2.6).

The organization of this paper is as follows. In Section 2, we recall basic
properties of density currents from [7]. In Section 3, we discuss the
connection between the non-pluripolar product and density currents, and prove
Theorem 1.1.

Acknowledgments. This research is partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation)-Projektnummer
500055552 and by the ANR-DFG grant QuaSiDy, grant no ANR-21-CE40-0016.

##  2 Density currents

We first recall some basic properties of density currents introduced by Dinh-
Sibony in [7].

Let $X$ be a complex Kähler manifold of dimension $n$, and $V$ a smooth
complex submanifold of $X$ of dimension $l.$ Let $T$ be a closed positive
$(p,p)$-current on $X,$ where $0\leq p\leq n.$ Denote by $\pi:E\to V$ the
normal bundle of $V$ in $X$ and $\overline{E}:=\mathbb{P}(E\oplus\mathbb{C})$
the projective compactification of $E.$ By abuse of notation, we also use
$\pi$ to denote the natural projection from $\overline{E}$ to $V$.

Let $U$ be an open subset of $X$ with $U\cap V\not=\varnothing.$ Let $\tau$ be
a smooth diffeomorphism from $U$ to an open neighborhood of $V\cap U$ in $E$
which is identity on $V\cap U$ such that the restriction of its differential
$d\tau$ to $E|_{V\cap U}$ is identity. Such a map is called _an admissible
map_. Note that in [7], to define an admissible map, it is required
furthermore that $d\tau$ is $\mathbb{C}$-linear at every point of $V$. This
difference doesn’t affect what follows. When $U$ is a small enough tubular
neighborhood of $V,$ there always exists an admissible map $\tau$ by [7, Lemma
4.2]. In general, $\tau$ is not holomorphic. When $U$ is a small enough local
chart, we can choose a holomorphic admissible map by using suitable
holomorphic coordinates on $U$. For $\lambda\in\mathbb{C}^{*},$ let
$A_{\lambda}:E\to E$ be the multiplication by $\lambda$ on fibers of $E.$ Here
is the first fundamental result for density currents.

######  Theorem 2.1.

([7, Theorem 4.6]) Let $\tau$ be an admissible map defined on a tubular
neighborhood of $V$. Then, the family $(A_{\lambda})_{*}\tau_{*}T$ is of mass
uniformly bounded in $\lambda$ on compact subsets in $E$, and if $S$ is a
limit current of the last family as $\lambda\to\infty$, then $S$ is a current
on $E$ which can be extended trivially through $\overline{E}\backslash E$ to
be a closed positive current on $\overline{E}$ such that the cohomology class
$\\{S\\}$ of $S$ in $\overline{E}$ is independent of the choice of $S$, and
$\\{S\\}|_{V}=\\{T\\}|_{V}$, and $\|S\|\leq C\|T\|$ for some constant $C$
independent of $S$ and $T$, where $\\{S\\}|_{V}$ denotes the pull-back of
$\\{S\\}$ under the natural inclusion map from $V$ to $\overline{E}$.

We call $S$ _a tangent current to $T$ along $V$_, and its cohomology class is
called _the total tangent class of $T$ along $V$_ and is denoted by
$\kappa^{V}(T)$. Tangent currents are not unique in general. By [7, Theorem
4.6] again, if

$S=\lim_{k\to\infty}(A_{\lambda_{k}})_{*}\tau_{*}T$

for some sequence $(\lambda_{k})_{k}$ converging to $\infty$, then for every
open subset $U^{\prime}$ of $X$ and every admissible map
$\tau^{\prime}:U^{\prime}\to E$ , we also have

$S=\lim_{k\to\infty}(A_{\lambda_{k}})_{*}\tau^{\prime}_{*}T.$

This is equivalent to saying that tangent currents are independent of the
choice of the admissible map $\tau$.

######  Definition 2.2.

([7, Definition 3.1]) Let $F$ be a complex manifold and $\pi_{F}:F\to V$ a
holomorphic submersion. Let $S$ be a positive current of bi-degree $(p,p)$ on
$F$. _The h-dimension_ of $S$ with respect to $\pi_{F}$ is the biggest integer
$q$ such that $S\wedge\pi_{F}^{*}\theta^{q}\not=0$ for some Hermitian metric
$\theta$ on $V$.

By a bi-degree reason, the h-dimension of $S$ is in
$[\max\\{l-p,0\\},\min\\{\dim F-p,l\\}]$. We have the following description of
currents with minimal h-dimension.

######  Lemma 2.3.

([7, Lemma 3.4]) Let $\pi_{F}:F\to V$ be a submersion. Let $S$ be a closed
positive current of bi-degree $(p,p)$ on $F$ of h-dimension $(l-p)$ with
respect to $\pi_{F}$. Then $S=\pi^{*}S^{\prime}$ for some closed positive
current $S^{\prime}$ on $V$.

By [7, Lemma 3.8], the h-dimensions of tangent currents to $T$ along $V$ are
the same and this number is called _the tangential h-dimension of $T$ along
$V$_.

Let $m\geq 2$ be an integer. Let $T_{j}$ be a closed positive current of bi-
degree $(p_{j},p_{j})$ for $1\leq j\leq m$ on $X$ and let
$T_{1}\otimes\cdots\otimes T_{m}$ be the tensor current of
$T_{1},\ldots,T_{m}$ which is a current on $X^{m}.$ A _density current_
associated to $T_{1},\ldots,T_{m}$ is a tangent current to
$\otimes_{j=1}^{m}T_{j}$ along the diagonal $\Delta_{m}$ of $X^{m}.$ Let
$\pi_{m}:E_{m}\to\Delta$ be the normal bundle of $\Delta_{m}$ in $X^{m}$.
Denote by $[V]$ the current of integration along $V$. When $m=2$ and
$T_{2}=[V]$, the density currents of $T_{1}$ and $T_{2}$ are naturally
identified with the tangent currents to $T_{1}$ along $V$ (see [14, Lemma
2.3]).

The unique cohomology class of density currents associated to
$T_{1},\ldots,T_{m}$ is called _the total density class of
$T_{1},\ldots,T_{m}$_. We denote the last class by
$\kappa(T_{1},\ldots,T_{m})$. The tangential h-dimension of
$T_{1}\otimes\cdots\otimes T_{m}$ along $\Delta_{m}$ is called _the density
h-dimension_ of $T_{1},\ldots T_{m}$.

######  Lemma 2.4.

([7, Section 5]) Let $T_{j}$ be a closed positive current of bi-degree
$(p_{j},p_{j})$ on $X$ for $1\leq j\leq m$ such that $\sum_{j=1}^{m}p_{j}\leq
n$. Assume that the density h-dimension of $T_{1},\ldots,T_{m}$ is minimal,
_i.e_ , equals to $n-\sum_{j=1}^{m}p_{j}$. Then the total density class of
$T_{1},\ldots,T_{m}$ is equal to $\pi_{m}^{*}(\wedge_{j=1}^{m}\\{T_{j}\\})$.

Let $h_{\overline{E}}$ be the Chern class of the dual of the tautological line
bundle of $\overline{E}$. By [7, Page 535], we have

$\displaystyle\kappa^{V}(T)=\sum_{j=\max\\{0,l-p\\}}^{\min\\{l,n-p-1\\}}\pi^{*}\big{(}\kappa^{V}_{j}(T)\big{)}\wedge
h_{\overline{E}}^{p-(l-j)},$ (2.1)

where $\pi:\overline{E}\to V$ is the natural projection and
$\kappa^{V}_{j}(T)\in H^{2l-2j}(V,\mathbb{R})$. The tangential $h$-dimension
of $T$ along $V$ is exactly equal to the maximal $j$ such that
$\kappa^{V}_{j}(T)\not=0$, and it was known that the class $\kappa^{V}_{j}(T)$
is pseudoeffective ([7, Lemma 3.15]).

######  Theorem 2.5.

([7, Proposition 4.13]) Let $V^{\prime}$ be a submanifold of $V$ and let $T$
be a closed positive current on $X$. Let $T_{\infty}$ be a tangent current to
$T$ along $V$. Let $s$ be the tangential h-dimension of $T_{\infty}$ along
$V^{\prime}$. Then, the tangential h-dimension of $T$ along $V^{\prime}$ is at
most $s$, and we have

$\kappa^{V^{\prime}}_{s}(T)\leq\kappa^{V^{\prime}}_{s}(T_{\infty}).$

As a consequence, we obtain the following result generalizing the well-known
lower bound of Lelong numbers of intersection of $(1,1)$-currents due to
Demailly [6, Page 169] in the compact setting. It is probably the first result
dealing with comparison of Lelong numbers for intersection of currents of
arbitrary bi-degree.

######  Corollary 2.6.

Let $T_{j}$ be a closed positive current on $X$ for $1\leq j\leq m$. Then, for
every $x\in X$ and for every density current $S$ associated to
$T_{1},\ldots,T_{m}$, we have

$\displaystyle\nu(S,x)\geq\nu(T_{1},x)\cdots\nu(T_{m},x).$ (2.2)

###### Proof.

Let $x\in X$. Let $\Delta_{m}$ be the diagonal of $X^{m}$. Let
$\pi:E\to\Delta_{m}$ be the natural projection from the normal bundle of the
diagonal $\Delta_{m}$ of $X^{m}$ in $X^{m}$. We identify $x$ with a point in
$\Delta_{m}$ via the natural identification $\Delta_{m}$ with $X$. Put
$T:=\otimes_{j=1}^{m}T_{j}$ and $V^{\prime}:=\\{x\\}$. By [11, Lemma 2.4], we
have $\nu(T,x)\geq\nu(T_{1},x)\cdots\nu(T_{m},x)$. By [7, Proposition 5.6], we
have

$\kappa^{V^{\prime}}_{0}(S)=\nu(S,x)\delta_{x},\quad\kappa^{V^{\prime}}_{0}(T)=\nu(T,x)\delta_{x},$

where $\delta_{x}$ is the Dirac measure on $x$ (notice here $\dim
V^{\prime}=0$). This combined with Theorem 2.5 applied to $X^{m}$,
$T:=\otimes_{j=1}^{m}T_{j}$, $\Delta_{m}$ the diagonal of $X^{m}$ and
$V^{\prime}:=\\{x\\}$ implies

$\nu(S,x)\geq\nu(T,x)\geq\nu(T_{1},x)\cdots\nu(T_{m},x).$

Hence, the desired inequality follows. The proof is finished. ∎

##  3 Relative non-pluripolar products

We first recall basic facts about the relative non-pluripolar product of
currents and discuss its connection with the density current.

Let $X$ be a compact Kähler manifold of dimension $n$. Let
$T_{1},\ldots,T_{m}$ be closed positive $(1,1)$-currents on $X$, and $T$ be a
closed positive $(p,p)$-current. By [17], the $T$-relative non-pluripolar
product $\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle$ is defined in a way
similar to that of the usual non-pluripolar products in [1, 2]. For readers’
convenience, we explain briefly how to do it.

Write $T_{j}=dd^{c}u_{j}+\theta_{j}$, where $\theta_{j}$ is a smooth form and
$u_{j}$ is a $\theta_{j}$-psh function. Put

$R_{k}:=\mathbb{1}_{\cap_{j=1}^{m}\\{u_{j}>-k\\}}\wedge_{j=1}^{m}(dd^{c}\max\\{u_{j},-k\\}+\theta_{j})\wedge
T$

for $k\in\mathbb{N}$. By the strong quasi-continuity of bounded psh functions
([17, Theorems 2.4 and 2.9]), we have

$R_{k}=\mathbb{1}_{\cap_{j=1}^{m}\\{u_{j}>-k\\}}\wedge_{j=1}^{m}(dd^{c}\max\\{u_{j},-l\\}+\theta_{j})\wedge
T$

for every $l\geq k\geq 1$. One can check that $R_{k}$ is positive (see [17,
Lemma 3.2]).

As in [2], we have that $R_{k}$ is of mass bounded uniformly in $k$ and
$(R_{k})_{k}$ converges to a closed positive current as $k\to\infty$. This
limit is denoted by $\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle$. We
refer to [17, Proposition 3.5] for properties of relative non-pluripolar
products.

For every closed positive $(1,1)$-current $P$, we denote by $I_{P}$ the set of
$x\in X$ so that local potentials of $P$ are equal to $-\infty$ at $x$. Note
that $I_{P}$ is a complete pluripolar set. It is clear from the definition
that $\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle$ has no mass on
$\cup_{j=1}^{m}I_{T_{j}}$. Furthermore, for every locally complete pluripolar
set $A$ (_i.e,_ $A$ is locally equal to $\\{\psi=-\infty\\}$ for some psh
function $\psi$), if $T$ has no mass on $A$, then so does
$\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle$. The following is deduced
from [17, Proposition 3.5].

######  Proposition 3.1.

  * (i)

For $R:=\langle\wedge_{j=l+1}^{m}T_{j}\dot{\wedge}T\rangle$, we have
$\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle=\langle\wedge_{j=1}^{l}T_{j}\dot{\wedge}R\rangle$.

  * (ii)

For every complete pluripolar set $A$, we have

$\mathbb{1}_{X\backslash A}\langle T_{1}\wedge T_{2}\wedge\cdots\wedge
T_{m}\dot{\wedge}T\rangle=\big{\langle}T_{1}\wedge T_{2}\wedge\cdots\wedge
T_{m}\dot{\wedge}(\mathbb{1}_{X\backslash A}T)\big{\rangle}.$

In particular, the equality

$\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle=\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T^{\prime}\rangle$

holds, where $T^{\prime}:=\mathbb{1}_{X\backslash\cup_{j=1}^{m}I_{T_{j}}}T$.

Let $V$ be an irreducible analytic set in $X$. For the case $T=[V]$, we have
the following lemma.

######  Lemma 3.2.

([18, Lemma 2.3]) Let $T_{1},\dots,T_{m}$ be closed positive $(1,1)$-currents
on $X$. Then the following properties hold:

  * (i)

If $V$ is contained in $\cup_{j=1}^{m}I_{T_{j}}$, then $\langle
T_{1}\wedge\dotsi\wedge T_{m}\dot{\wedge}[V]\rangle=0$ and there is $1\leq
j_{0}\leq m$ so that $V\subset I_{T_{j_{0}}}$.

  * (ii)

If $V$ is not contained in $\cup_{j=1}^{m}I_{T_{j}}$, then

$\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}[V]\rangle=i_{*}\langle
T_{1,V}\wedge\dotsi\wedge T_{m,V}\rangle,$

where $i\colon\text{Reg}(V)\rightarrow X$ is the natural inclusion, and
$T_{j,V}\vcentcolon=dd^{c}(u_{j}|_{\text{Reg}(V)})$ if $dd^{c}u_{j}=T_{j}$
locally.

Let $T,T^{\prime}$ be closed positive $(1,1)$-currents in the same cohomology
class on $X$. $T^{\prime}$ is said to be less singular than $T$ if for every
local potentials $u$ of $T$ and $u^{\prime}$ of $T^{\prime}$, $u\leq
u^{\prime}+O(1)$. Here is a crucial property of relative non-pluripolar
products.

######  Theorem 3.3.

([17, Theorem 1.1]) Let $T^{\prime}_{j}$ be closed positive $(1,1)$-current in
the cohomology class of $T_{j}$ on $X$ such that $T^{\prime}_{j}$ is less
singular than $T_{j}$ for $1\leq j\leq m$. Then we have

$\\{\langle T_{1}\wedge\cdots\wedge T_{m}\dot{\wedge}T\rangle\\}\leq\\{\langle
T^{\prime}_{1}\wedge\cdots\wedge T^{\prime}_{m}\dot{\wedge}T\rangle\\}.$

Weaker versions of the above result were proved in [2, 3, 19]. Let
$\alpha_{1},\dots,\alpha_{m}$ be big $(1,1)$-classed of $X$. A current
$T_{j,\min}\in\alpha_{j}$ is said to have minimal singularities if it is less
singular than any closed positive current in $\alpha_{j}$.

By Theorem 3.3, the class $\\{\langle T_{1,\min}\wedge\dotsi\wedge
T_{m,\min}\dot{\wedge}T\rangle\\}$ is a well-defined pseudoeffective class
which is independent of the choice of $T_{j,\min}$. We denote the last class
by $\\{\langle\alpha_{1}\wedge\dotsi\wedge\alpha_{m}\dot{\wedge}T\rangle\\}$.
When $T\equiv 1$, we simply write
$\langle\alpha_{1}\wedge\dotsi\wedge\alpha_{m}\rangle$ for
$\\{\langle\alpha_{1}\wedge\dotsi\wedge\alpha_{m}\dot{\wedge}T\rangle\\}$. In
this case, the product $\langle\alpha_{1}\wedge\dotsi\wedge\alpha_{m}\rangle$
was introduced in [2].

Regarding the relation between relative non-pluripolar products and density
currents, the following fact was proved in [15, Theorem 3.5], see also [9,
10].

######  Theorem 3.4.

Let $R_{\infty}$ be a density current associated to $T_{1},\ldots,T_{m},T$.
Then we have

$\displaystyle\pi_{m+1}^{*}\langle\wedge_{j=1}^{m}T_{j}\dot{\wedge}T\rangle\leq
R_{\infty},$ (3.1)

where $\pi_{m+1}$ is the natural projection from the normal bundle of the
diagonal $\Delta$ of $X^{m+1}$ to $\Delta$, and as usual we identified
$\Delta$ with $X$.

We will need the following to estimate the density h-dimension of currents.

######  Proposition 3.5.

([15, Proposition 3.6]) Let $T_{1},\ldots,T_{m},T$ be as above. Let $A$ be a
Borel subset of $X$. Assume that for every $J\subset\\{1,\ldots,m\\}$ and for
every density current $R_{J}$ associated to $(T_{j})_{j\in J},T$ with
$m_{J}:=|J|<m$, we have that $R_{J}$ has no mass on the set

$\pi_{m_{J}+1}^{*}(\cap_{j\not\in J}\\{x\in A:\nu(T_{j},x)>0\\}).$

Then for every density current $S$ associated to $T_{1},\ldots,T_{m},T$, the
h-dimension of the current $\mathbb{1}_{A}S$ is equal to $n-p-m$, where
$\mathbb{1}_{A}$ denotes the characteristic function of $A$.

For every pseudoeffective $(p,p)$-class $\gamma$ on $X$, we put
$\|\gamma\|:=\int_{X}\Theta\wedge\omega^{n-p}$, where $\Theta$ is any closed
smooth form in $\gamma$. This definition is independent of the choice of
$\Theta$ and is nonnegative because of the pseudoeffectivity of $\gamma$.

######  Theorem 3.6.

Let $P$ and $T$ be closed positive currents of bi-degree $(1,1)$ and $(p,p)$
respectively on $X$, where $1\leq p\leq n-1$. Assume that $T$ has no mass on
$I_{P}$. Then, the cohomology class

$\gamma:=\\{P\\}\wedge\\{T\\}-\\{\langle P\dot{\wedge}T\rangle\\}$

is pseudoeffective and we have

$\displaystyle\|\gamma\|\geq\sum_{V}\nu(P,V)\nu(T,V)n_{V}!\mathop{\mathrm{vol}}(V),$
(3.2)

where the sum is taken over every irreducible subset $V$ of dimension at least
$n-p-1$ in $X$, and $n_{V}:=\dim V$.

###### Proof.

Let $\mathscr{V}$ be the set of irreducible analytic subsets $V$ of dimension
at least $n-p-1$ in $X$ such that $\nu(T,V)>0$ and $\nu(P,V)>0$. We note that
in (3.2), it is enough to consider $V\in\mathscr{V}$. We will see below that
$\mathscr{V}$ has at most countable elements.

Observe that if $\nu(P,x)>0$, then $x\in I_{P}$. Hence, by hypothesis, the
trace measure of $T$ has no mass on the set $\\{x\in X:\nu(P,x)>0\\}$. This
allows us to apply Proposition 3.5 to $P$ and $T$ to obtain that the density
h-dimension of $P$ and $T$ is minimal. Using this and Lemma 2.4 gives

$\displaystyle\kappa(P,T)=\pi^{*}(\\{P\\}\wedge\\{T\\}),$ (3.3)

where $\pi$ is the natural projection from the normal bundle of the diagonal
$\Delta$ of $X^{2}$ to $\Delta$.

Let $S$ be a density current associated to $P$ and $T$. Since the h-dimension
of $S$ is minimal, using Lemma 2.3, we get that there exists a current
$S^{\prime}$ on $X$ such that $S=\pi^{*}S^{\prime}$ (recall $\Delta$ is
identified with $X$). Since the relative non-pluripolar product is dominated
by density currents (Theorem 3.4), the current $S^{\prime}-\langle
P\dot{\wedge}T\rangle$ is closed and positive. Moreover, by (3.3), the
cohomology class of the last current is equal to $\gamma$. It follows that
$\gamma$ is pseudoeffective.

It remains to prove (3.2). Let $V\in\mathscr{V}$. By definition, the generic
Lelong number of $T$ along $V$ is positive. Since $T$ is of bi-degree $(p,p)$,
the dimension of $V$ must be at most $n-p$. Hence, we have two possibilities:
either $\dim V=n-p-1$ or $\dim V=n-p$. The latter case cannot happen. If $T$
has no mass on $V$, then $\nu(T,V)=0$, which leads to a contradiction. If $T$
has mass on $V$, which is contained in $I_{P}$ (for $\nu(P,V)>0$), then this
contradicts the hypothesis that $T$ has no mass on $I_{P}$.

Let $V\in\mathscr{V}$. Since the Lelong numbers are preserved by submersion
maps ([11, Proposition 2.3]), by applying Corollary 2.6 to $P,T$ and generic
$x\in V$, we obtain

$\nu(S^{\prime},V)=\nu(S,V)\geq\nu(P,V)\nu(T,V).$

This combined with the fact that $\dim V=n-p-1$ implies
$S^{\prime}\geq\nu(P,V)\nu(T,V)\,[V]$. We deduce that

$\displaystyle S^{\prime}$ $\displaystyle\geq\langle
P\dot{\wedge}T\rangle+\mathbb{1}_{I_{P}}S^{\prime}$ $\displaystyle\geq\langle
P\dot{\wedge}T\rangle+\sum_{V\in\mathscr{V}}\nu(P,V)\nu(T,V)\,[V].$

The second inequality comes from Siu’s decomposition theorem ([6, 2.18]), and
this also shows that $\mathscr{V}$ has at most countable elements. The desired
assertion follows and the proof is finished. ∎

We now prove Theorem 1.1.

###### Proof of Theorem 1.1.

It suffices to consider the case where $\alpha$ is Kähler by using
$\alpha+\epsilon\\{\omega\\},T+\epsilon\omega$ instead of $\alpha,T$ and
letting $\epsilon\to 0$. Hence we assume from now on that $\alpha$ is Kähler.
By abuse of notation, we also denote by $\alpha$ a smooth Kähler form in
$\alpha$. By Lemma 3.2, the right hand-hand side of (1.1) can be written as

$\displaystyle\frac{1}{(m-m^{\prime})!}\int_{\text{Reg}W}\big{(}\alpha^{m^{\prime}}-\langle(T|_{\text{Reg}W})^{m^{\prime}}\rangle\big{)}\wedge\omega^{m-m^{\prime}}=\frac{1}{(m-m^{\prime})!}\big{\|}\langle\alpha^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle
T^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}.$

Step 1. We first focus on the case that $T$ has analytic singularities. Set
$S=\langle T^{m^{\prime}-1}\dot{\wedge}[W]\rangle$. Since $\alpha$ is Kähler,
by the monotonicity of non-pluripolar product (Theorem 3.3) and Proposition
3.1 (i), we get

$\displaystyle\big{\|}\langle\alpha^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle
T^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
$\displaystyle\geq\big{\|}\langle\alpha\wedge
T^{m^{\prime}-1}\dot{\wedge}[W]\rangle-\langle
T^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$ (3.4)
$\displaystyle=\|\alpha\wedge S-\langle T\dot{\wedge}S\rangle\|$

We now show that $S$ has no mass on $I_{T}$. For $m^{\prime}>1$, this directly
follows from the definition of non-pluripolar product. For $m^{\prime}=1$, the
current $S$ is just $[W]$. Since we assume that $T$ has analytic
singularities, the polar locus $I_{T}$ is an analytic subset and it doesn’t
contain $W$. Hence, $[W]$ also has no mass on $I_{T}$. Therefore, we can apply
Lemma 3.6 to $T,S$, and get

$\displaystyle\|\alpha\wedge\\{S\\}-\\{\langle T\dot{\wedge}S\rangle\\}\|$
$\displaystyle\geq(m-m^{\prime})!\sum_{V\in\mathscr{V}_{m-m^{\prime},T,W}}\nu(T,V)\nu(S,V)\mathop{\mathrm{vol}}(V)$
(3.5)

Let $V\in\mathscr{V}_{m-m^{\prime},T,W}$ and let $\text{Sing}(I_{T}\cap W)$ be
the singular locus of the analytic set $I_{T}\cap W$. Put
$X^{\prime}:=X\backslash\text{Sing}(I_{T}\cap W)$ which is an open subset in
$X$. Now, we claim that the classical intersection $T^{m^{\prime}-1}\wedge[W]$
is well-defined on some open neighborhood $U$ of
$V\backslash\text{Sing}({I_{T}\cap W)}$ in $X^{\prime}$, in the sense given in
[4, Chapter III]. Since $T$ has analytic singularities, the Lelong number
$\nu(T,x)$ is strictly positive if and only if $x$ belongs to $I_{T}$. This
implies that $V$ is contained in $I_{T}\cap W$. Let $K_{1},\dots,K_{s}$ be the
irreducible components of $I_{T}\cap W$. Since $V$ is a maximal irreducible
proper analytic set such that $\nu(T,V)>0$, $V$ must be one of the irreducible
components. By rearranging the index, we may assume $V=K_{1}$. Let $U$ be an
open neighborhood of $V\backslash\text{Sing}(I_{T}\cap W)$ such that

$U\cap K_{j}=\emptyset,\mbox{ for }j=2,\dots,s.$

Notice that $V\backslash\text{Sing}(I_{T}\cap W)$is contained in
$\text{Reg}(V)$, and is of dimension $m-m^{\prime}$. Consequently, for $0\leq
j^{\prime}\leq m^{\prime}-1$,

$\displaystyle\mathscr{H}_{2m-2j^{\prime}+1}(L(T)|_{U}\cap W)$
$\displaystyle=\mathscr{H}_{2m-2j^{\prime}+1}(I_{T}\cap W\cap U)$
$\displaystyle=\mathscr{H}_{2m-2j^{\prime}+1}(V\backslash\text{Sing}(I_{T}\cap
W))$ $\displaystyle=0,$

where $L(T)$ is the set of $x\in X$ such that the local potential of $T$ is
unbounded on any neighborhood of $x$. This allows us to apply [4, Chapter III,
Theorem 4.5] and get the well-definedness of $T^{m^{\prime}-1}\wedge[W]$ on
$U$. By [17, Proposition 3.6], we obtain

$\displaystyle\langle
T^{m^{\prime}-1}\dot{\wedge}[W]\rangle=\mathbb{1}_{U\backslash
I_{T}}T^{m^{\prime}-1}\wedge[W].$ (3.6)

Actually, the equality also holds on $U\cap I_{T}$. To show this, we need to
check that $T^{m^{\prime}-1}\wedge[W]$ has no mass on $U\cap I_{T}$. Since
$\dim(U\cap I_{T}\cap W)=m-m^{\prime}$ and $T^{m^{\prime}-1}\wedge[W]$ is of
bi-dimension $(m-m^{\prime}+1,m-m^{\prime}+1)$, the current
$T^{m^{\prime}-1}\wedge[W]$ must have no mass on $U\cap I_{T}\cap W$. Also, by
the fact ${\rm Supp}(T^{m^{\prime}-1}\wedge[W])\subset W$, the current
$T^{m^{\prime}-1}\wedge[W]$ also has no mass on $(U\cap I_{T})\backslash W$.
Therefore, the equality (3.6) extends to $U$. This implies that the Lelong
number $\nu(S,V)$ equals
$\nu(T^{m^{\prime}-1}\wedge[W],V\backslash\text{Sing}(I_{T}\cap W))$ (remember
that we consider the current $T^{m^{\prime}-1}\wedge[W]$ on $U$, and
$V\backslash\text{Sing}(I_{T}\cap W)$ is an analytic subset in $U$), and we
then have

$\displaystyle\nu(S,V)$
$\displaystyle=\nu(T^{m^{\prime}-1}\wedge[W],V\backslash\text{Sing}(I_{T}\cap
W))$ $\displaystyle\geq\nu(T,V\backslash\text{Sing}(I_{T}\cap
W))^{m^{\prime}-1}\nu([W],V\backslash\text{Sing}(I_{T}\cap W))$
$\displaystyle\geq\nu(T,V)^{m^{\prime}-1}.$ (3.7)

By (3.4), (3.5) and (3.7), the desired inequality follows in the case where
$T$ has analytic singularities.

Step 2. Now, we remove the assumption that $T$ has analytic singularities.
First, we write $T=dd^{c}u+\theta$, where $\theta$ is a closed smooth
$(1,1)$\- form and $u\in{\rm PSH}(X,\theta)$. Demailly’s analytic
approximation theorem (see [6, Corollary 14.13]) allows us to construct a
sequence $u_{k}^{D}\in{\rm PSH}(X,\theta+\epsilon_{k}\omega)$, where
$\epsilon_{k}$ decreases to $0$, such that

  * (1)

$u_{k}^{D}\geq u$ and $u_{k}^{D}$ converges to $u$ in $L^{1}$.

  * (2)

$u_{k}^{D}$ has analytic singularities.

  * (3)

$\nu(T_{k},x)$ converges to $\nu(T,x)$ uniformly on $X$, where
$T_{k}=dd^{c}u^{D}_{k}+(\theta+\epsilon_{k}\omega)$.

By the monotonicity property of non-pluripolar product (Theorem 3.3), we have

$\displaystyle\big{\|}\langle\alpha^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle
T^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
$\displaystyle=\lim_{k\to\infty}\big{\|}\langle(\alpha+\epsilon_{k}\omega)^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle(T+\epsilon_{k}\omega)^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
(3.8)
$\displaystyle\geq\limsup_{k\to\infty}\big{\|}\langle(\alpha+\epsilon_{k}\omega)^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle
T_{k}^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$

For every constant $r>0$, set
$A_{r}:=\\{V\in\mathscr{V}_{m-m^{\prime},T,W}|\nu(T,V)\geq r\\}$. Observe that
$A_{r}$ increases to $\mathscr{V}_{m-m^{\prime},T,W}$ as $r\to 0$. Since
$\nu(T_{k},x)$ converges to $\nu(T,x)$ uniformly and $T_{k}$ is less singular
than $T$, for every fixed $r>0$ we have

$A_{r}\subset\mathscr{V}_{m-m^{\prime},T_{k},W}$

when $k$ is large enough. By Step 1, we therefore have

$\displaystyle\big{\|}\langle(\alpha+\epsilon_{k}\omega)^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle
T_{k}^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
$\displaystyle\geq(m-m^{\prime})!\sum_{V\in\mathscr{V}_{m-m^{\prime},T_{k},W}}\nu(T_{k},V)^{m^{\prime}}\mathop{\mathrm{vol}}(V)$
$\displaystyle\geq(m-m^{\prime})!\sum_{V\in
A_{r}}\nu(T_{k},V)^{m^{\prime}}\mathop{\mathrm{vol}}(V).$

Letting $k\to\infty$ and using (3.8) give

$\displaystyle\big{\|}\langle\alpha^{m^{\prime}}\dot{\wedge}[W]\rangle-\langle
T^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
$\displaystyle\geq(m-m^{\prime})!\limsup_{k\to\infty}\sum_{V\in
A_{r}}\nu(T_{k},V)^{m^{\prime}}\mathop{\mathrm{vol}}(V)$
$\displaystyle=(m-m^{\prime})!\sum_{V\in
A_{r}}\nu(T,V)^{m^{\prime}}\mathop{\mathrm{vol}}(V),$

for every constant $r>0$. Letting $r\to 0$, we obtain the desired estimate. ∎

For the general case where $\nu(T,W)>0$. We couldn’t directly compare the
volume of Lelong upper level sets of $T$ on $W$ and the mass of
$\\{\langle\alpha^{m^{\prime}}\dot{\wedge}[W]\rangle\\}-\\{\langle
T^{m^{\prime}}\dot{\wedge}[W]\rangle\\}\\}$. In this case, we have the
following modified inequality which is stronger than [8, Theorem 1.1].

######  Theorem 3.7.

Let $\alpha$ be a nef $(1,1)$-class. Let $W$ be an irreducible analytic subset
in $X$. Let $T$ be a closed positive current in $\alpha$. Let $1\leq
m^{\prime}\leq m$ be an integer. Then we have

$(m-m^{\prime})!\sum_{V\in\mathscr{V}_{m-m^{\prime},T,W}}\big{(}\nu(T,V)-\nu(T,W)\big{)}^{m^{\prime}}\mathop{\mathrm{vol}}(V)\leq\\\
\big{\|}(\alpha+c\\{\omega\\})^{m^{\prime}}\wedge\\{[W]\\}-\\{\langle(T+c\omega)^{m^{\prime}}\dot{\wedge}[W]\rangle\\}\big{\|}$
(3.9)

where $c=c_{1}\cdot\nu(T,W)$ and $c_{1}>0$ is a constant independent of
$\alpha,T,W$. In particular, there is a constant $c_{2}>0$ independent of
$\alpha,T,W$ such that

$\displaystyle\sum_{V\in\mathscr{V}_{m-m^{\prime},T,W}}\big{(}\nu(T,V)-\nu(T,W)\big{)}^{m^{\prime}}\mathop{\mathrm{vol}}(V)\leq
c_{2}\mathop{\mathrm{vol}}(W)\|T\|^{m^{\prime}}.$ (3.10)

###### Proof.

The desired inequality (3.10) follows directly from (3.9). We prove now (3.9).
For convenience, set $c_{3}:=\nu(T,W)>0$. The regularization theorem of
currents introduced in [5] allows us to cut down the Lelong upper level set
$\\{x\in X|\nu(T,V)\geq c_{3}\\}$ from $T$. More precisely, by [5, Theorem
1.1], there exists a sequence of almost positive closed $(1,1)$-currents
$T_{c_{3},k}$ in $\alpha$ such that

  * (1)

$T_{c_{3},k}\geq-(c_{1}\cdot c_{3}+\epsilon_{k})\omega$, where
$\lim_{k\rightarrow\infty}\epsilon_{k}=0$ and $c_{1}>0$ is a constant
independent of $\alpha,T$ and $W$.

  * (2)

The global potentials of $T_{c_{3},k}$ decreases to the global potential of
$T$.

  * (3)

$\nu(T_{c_{3},k},x)=\max\\{\nu(T,x)-c_{3},0\\}$.

Set $\widetilde{T}_{c_{3},k}=T_{c_{3},k}+(c_{1}\cdot
c_{3}+\epsilon_{k})\omega$, which is a closed positive $(1,1)$-current. By
Theorem 3.3, we have

$\displaystyle\big{\|}(\alpha+c_{1}\cdot
c_{3}\\{\omega\\})^{m^{\prime}}\wedge\\{[W]\\}-\langle(T+c_{1}\cdot
c_{3}\omega)^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
$\displaystyle\geq\limsup_{k\rightarrow\infty}\big{\|}\big{(}\alpha+(c_{1}\cdot
c_{3}+\epsilon_{k})\\{\omega\\}\big{)}^{m^{\prime}}\wedge\\{[W]\\}-\langle\widetilde{T}_{c_{3},k}^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}.$

Since $\nu(\widetilde{T}_{c_{3},k},W)=0$, we can apply Theorem 1.1 to the
right-hand side of the above inequality and get

$\displaystyle\big{\|}\big{(}\alpha+(c_{1}\cdot
c_{3}+\epsilon_{k})\\{\omega\\}\big{)}^{m^{\prime}}\wedge\\{[W]\\}-\langle\widetilde{T}_{c_{3},k}^{m^{\prime}}\dot{\wedge}[W]\rangle\big{\|}$
$\displaystyle\geq(m-m^{\prime})!\sum_{V\in\mathscr{V}_{m-m^{\prime},\tilde{T}_{c_{3},k},W}}\nu(\widetilde{T}_{c_{3},k},V)^{m^{\prime}}\mathop{\mathrm{vol}}(V),$
(3.11)

By the above properties of $T_{c_{3},k}$, we have

$\mathscr{V}_{m-m^{\prime},\tilde{T}_{c_{3},k},W}=\mathscr{V}_{m-m^{\prime},T,W}.$

Therefore, the right-hand side of (3.11) is equal to

$\displaystyle(m-m^{\prime})!\sum_{V\in\mathscr{V}_{m-m^{\prime},T,W}}\nu(\widetilde{T}_{c_{2},k},V)^{m^{\prime}}\mathop{\mathrm{vol}}(V)$
$\displaystyle=(m-m^{\prime})!\sum_{V\in\mathscr{V}_{m-m^{\prime},T,W}}\big{(}\nu(T,V)-\nu(T,W)\big{)}^{m^{\prime}}\mathop{\mathrm{vol}}(V).$

This completes the proof. ∎

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Duc-Viet Vu, University of Cologne, Division of Mathematics, Department of
Mathematics and Computer Science, Weyertal 86-90, 50931, Köln, Germany

E-mail address<EMAIL_ADDRESS>
Shuang Su, University of Cologne, Division of Mathematics, Department of
Mathematics and Computer Science, Weyertal 86-90, 50931, Köln, Germany

E-mail address<EMAIL_ADDRESS>


# Complex scalar dark matter in a new gauged $\mathrm{U}(1)$ symmetry with
kinetic and direct mixings

Yu-Hang Sua Chengfeng Caia,b Yu-Pan Zengc,a<EMAIL_ADDRESS>Hong-Hao
Zhanga<EMAIL_ADDRESS>aSchool of Physics, Sun Yat-sen University,
Guangzhou 510275, China  bSchool of Science, Sun Yat-Sen University, Shenzhen
518107, China  cSchool of Electronics and Information Engineering, Guangdong
Ocean University, Zhanjiang 524088, China

###### Abstract

We propose a scalar dark matter model featuring a hidden gauge symmetry,
denoted as $\mathrm{U}(1)_{\mathrm{X}}$, with two complex scalars, $\Phi$ and
$S$. In this framework, $\Phi$ spontaneously breaks the
$\mathrm{U}(1)_{\mathrm{X}}$ gauge symmetry, while $S$ serves as a viable dark
matter candidate. Particularly, the kinetic and direct mixings between the
$\mathrm{U}(1)_{\mathrm{X}}$ and $\mathrm{U}(1)_{\mathrm{Y}}$ gauge groups
provide a portal between dark matter and the Standard Model particles. These
mixings offer a plausible explanation for the W boson mass anomaly observed by
the CDF Collaboration. We study the comprehensive phenomenological constraints
of this model from colliders and dark matter detection experiments, including
$Z^{\prime}$ searches at the LHC, the $125~{}\mathrm{GeV}$ Higgs boson
measurements, the relic density of dark matter and the indirect detection of
dark matter annihilation. By randomly scanning the parameter space, we find
that the regions where $m_{Z^{\prime}}\gtrsim 4750~{}\mathrm{GeV}$ and
$m_{Z^{\prime}}\lesssim 4750~{}\mathrm{GeV}$ for $g_{x}$ close to $1$ remain
viable and can be tested by future experiments.

###### Contents

  1. I Introduction
  2. II Model
    1. II.1 Lagrangian
    2. II.2 Gauge sector
  3. III Constraints
    1. III.1 Electroweak precision measurements
    2. III.2 $Z^{\prime}$ searches at the LHC
    3. III.3 Higgs physics
    4. III.4 DM-nucleon scattering
    5. III.5 DM annihilation
  4. IV Parameter scan
  5. V Conclusions

##  I Introduction

Numerous astrophysical and cosmological observations today strongly support
the existence of dark matter (DM) [1, 2, 3], yet direct experimental
confirmation remains elusive. It is well known that the couplings between DM
particles and those of the standard model (SM) are comparable in magnitude to
the weak force within the weakly interacting massive particle (WIMP) scenario.
WIMPs are thermally produced in the early universe and subsequently decoupled
from the thermal bath during the freeze-out epoch [4].

Currently, a lot of direct detection experiments are designed to search for
such DM particles, as the current and forthcoming experimental capabilities
include the predicted DM mass range proposed by various theories. However, no
conclusive signal has been detected so far. Consequently, many theoretical
models have been ruled out due to the stringent experimental constraints on
the scattering cross section between DM and nucleons [5, 6].

Apart from the DM problem, there are also particle physics anomalies awaiting
resolution. In the SM, the mass of the W boson serves as a crucial and
fundamental parameter. Its measurement not only provides valuable insights,
but also offers a gateway to explore physics beyond the SM (BSM). In recent
years, there has been increasing emphasis on precise measurements of this
parameter. In 2022, the Collider Detector at Fermilab (CDF) Collaboration
conducted a new measurement of the W boson mass, yielding $80.4335\pm
0.0094~{}\mathrm{GeV}$ [7]. This result deviates from the SM prediction by
seven standard deviations, prompting the proposal of numerous theories to
explain such an anomaly. An elegant example is the $Z^{\prime}$ model, which
introduces a new neutral gauge boson, denoted $Z^{\prime}$ [8, 9, 10]. This
boson interacts with the $Z$ boson through a relatively weak interaction, thus
evading the stringent constraints imposed by the searches $Z^{\prime}$ at the
LHC. It is noteworthy that in 2023, the ATLAS Collaboration reported a
different result following an improved re-analysis of their measurement data.
They determined the W boson mass to be $80.360\pm 0.016~{}\mathrm{GeV}$,
showing no deviation from the SM prediction [11]. Several papers have
discussed the implications and potential explanations arising from these two
measurements [12, 13]. Our focus will remain on further measurements of the W
boson mass, including the utilization of data from different center-of-mass
energies.

When establishing new physical models, the minimal gauge extension to the SM
is achieved by introducing an extra $\mathrm{U}(1)$ gauge symmetry, denoted
$\mathrm{U}(1)_{\mathrm{X}}$. A natural proposition is that this symmetry
originates from the dark sector, and DM is also under its representation. To
evade constraints from $Z^{\prime}$ searches and other experiments at the LHC,
the most straightforward strategy is to demand that all SM fermions carry no
$\mathrm{U}(1)_{\mathrm{X}}$ charges, which means $\mathrm{U}(1)_{\mathrm{X}}$
is a hidden gauge symmetry [14, 15, 16, 17]. In this scenario, the kinetic
mixing between $\mathrm{U}(1)_{\mathrm{X}}$ and $\mathrm{U}(1)_{\mathrm{Y}}$
bridges the dark sector with the SM sector. However, several studies have
shown that such mixing is difficult to simultaneously generate dark matter
relic abundance and satisfy the direct detection constraint [18, 19, 20],
necessitating the introduction of additional parameters for a viable DM model.

In this paper, we propose a $\mathrm{U}(1)_{\mathrm{X}}$ DM model with kinetic
and direct mixings, in which two new complex scalars are under both the
$\mathrm{U}(1)_{\mathrm{X}}$ and $\mathrm{U}(1)_{\mathrm{Y}}$ representations.
Particularly, these mixings offer a plausible explanation for the W boson mass
anomaly observed by the CDF Collaboration. We perform a parameter fitting
utilizing data from both the CDF and non-CDF sources simultaneously.
Furthermore, we investigate other constraints from colliders, including
searches for $Z^{\prime}$ and the $125~{}\mathrm{GeV}$ Higgs boson
measurements. Additionally, we study comprehensively the phenomenology of the
dark sector, particularly considering the direct detection experiments, the
relic abundance of DM, and the indirect detection of DM annihilation. In
calculating the constraint from the direct detection experiment, we consider
the isospin violation in the scattering of DM and nucleons [21, 22, 23, 24,
25, 26].

This paper is organized as follows. In Sec. II, we provide a detailed
description of our $\mathrm{U}(1)_{\mathrm{X}}$ model. In Sec. III, we discuss
the phenomenology of this model regarding the electroweak precision
measurements, $Z^{\prime}$ searches at the LHC, Higgs physics, direct
detection experiments, DM relic abundance, and indirect detection experiments.
Sec. IV presents the results of a random scan in the parameter space. At last,
we offer conclusions in Sec. V.

##  II Model

The model extends the SM with a new gauge symmetry
$\mathrm{U}(1)_{\mathrm{X}}$ and two complex scalars $S$ and $\Phi$. The
$\Phi$ develops a large vacuum expectation value $v_{\Phi}$ in the early
universe, and $S$ serves as the DM candidate. $S$ and $\Phi$ are both in the
representation (1, $n_{\Phi}$, 1) of the gauge symmetry
$\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}\times\mathrm{U}(1)_{\mathrm{X}}$.
Consequently, the supercharges of $S$ and $\Phi$ provide a direct mixing
between $\mathrm{U}(1)_{\mathrm{Y}}$ and $\mathrm{U}(1)_{\mathrm{X}}$, which,
along with the kinetic mixing, connects the dark sector with the SM particles.

###  II.1 Lagrangian

The full Lagrangian of the model is given by

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{SM}+(D^{\mu}S)^{{\dagger}}(D_{\mu}S)+(D^{\mu}\Phi)^{{\dagger}}(D_{\mu}\Phi)-\frac{1}{4}X^{\mu\nu}X_{\mu\nu}-\frac{s_{\varepsilon}}{2}B^{\mu\nu}X_{\mu\nu}$
(1)
$\displaystyle-\mu_{S}^{2}|S|^{2}-\lambda_{S}|S|^{4}+\mu_{\Phi}^{2}|\Phi|^{2}-\lambda_{\Phi}|\Phi|^{4}-\lambda_{HS}|H|^{2}|S|^{2}-\lambda_{H\Phi}|H|^{2}|\Phi|^{2}-\lambda_{S\Phi}|S|^{2}|\Phi|^{2}.$

Here the quadratic coefficient of DM is required to be negative, or it would
become unstable by acquiring a non-zero vacuum expectation value. The
covariant derivatives of $H$, $\Phi$ and $S$ are

$\displaystyle D_{\mu}H$ $\displaystyle=$
$\displaystyle(\partial_{\mu}-igT^{a}W^{a}_{\mu}-i\frac{1}{2}g^{\prime}B_{\mu})H,$
$\displaystyle D_{\mu}\Phi$ $\displaystyle=$
$\displaystyle(\partial_{\mu}-ig^{\prime}n_{\Phi}B_{\mu}-ig_{x}X_{\mu})\Phi,$
$\displaystyle D_{\mu}S$ $\displaystyle=$
$\displaystyle(\partial_{\mu}-ig^{\prime}n_{\Phi}B_{\mu}-ig_{x}X_{\mu})S,$ (2)

where $X_{\mu}$ is the new gauge boson of $\mathrm{U}(1)_{\mathrm{X}}$ and
$g_{x}$ is the gauge coupling. The scalar fields $H$ and $\Phi$ can be
expanded in the unitary gauge as

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{H}+h\end{pmatrix},\quad\Phi=\frac{1}{\sqrt{2}}(v_{\Phi}+\phi).$ (3)

Then we can obtain the stationary point conditions by minimizing the scalar
potential:

$\displaystyle\mu_{H}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{H}v_{H}^{2}+\frac{1}{2}\lambda_{H\Phi}v_{\Phi}^{2},$
$\displaystyle\mu_{\Phi}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{\Phi}v_{\Phi}^{2}+\frac{1}{2}\lambda_{H\Phi}v_{H}^{2}.$
(4)

The mass-squared matrix of real scalars ($h,\phi$) is given by

$\displaystyle\mathcal{L}^{\mathrm{mass}}_{O}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\begin{pmatrix}h&\phi\end{pmatrix}\begin{pmatrix}2\lambda_{H}v_{H}^{2}&\lambda_{H\Phi}v_{H}v_{\Phi}\\\
\lambda_{H\Phi}v_{H}v_{\Phi}&2\lambda_{\Phi}v_{\Phi}^{2}\end{pmatrix}\begin{pmatrix}h\\\
\phi\end{pmatrix}$ (5) $\displaystyle=$
$\displaystyle-\frac{1}{2}\begin{pmatrix}h_{1}&h_{2}\end{pmatrix}\begin{pmatrix}c_{\theta}&-s_{\theta}\\\
s_{\theta}&c_{\theta}\end{pmatrix}\begin{pmatrix}2\lambda_{H}v_{H}^{2}&\lambda_{H\Phi}v_{H}v_{\Phi}\\\
\lambda_{H\Phi}v_{H}v_{\Phi}&2\lambda_{\Phi}v_{\Phi}^{2}\end{pmatrix}\begin{pmatrix}c_{\theta}&s_{\theta}\\\
-s_{\theta}&c_{\theta}\end{pmatrix}\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\begin{pmatrix}h_{1}&h_{2}\end{pmatrix}\begin{pmatrix}m_{h_{1}}^{2}&0\\\
0&m_{h_{2}}^{2}\end{pmatrix}\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix},$

where the rotation angle $\theta$ satisfies

$\displaystyle\tan
2\theta=\frac{\lambda_{H\Phi}v_{H}v_{\Phi}}{\lambda_{\Phi}v_{\Phi}^{2}-\lambda_{H}v_{H}^{2}}.$
(6)

Here we define $h_{1}$ is the $125~{}\mathrm{GeV}$ SM-like Higgs, and we
choose $m_{h_{2}}$ as free parameter instead of $\lambda_{\Phi}$. Then we can
express parameters $\lambda_{H}$ and $\lambda_{\Phi}$ by using physical
observables $m_{h_{1}}$ and $m_{h_{2}}$:

$\displaystyle\lambda_{H}$ $\displaystyle=$
$\displaystyle\frac{m_{h_{1}}^{2}+m_{h_{2}}^{2}-\sqrt{(m_{h_{1}}^{2}+m_{h_{2}}^{2})^{2}-4(m_{h_{1}}^{2}m_{h_{2}}^{2}+\lambda_{H\Phi}^{2}v_{H}^{2}v_{\Phi}^{2})}}{4v_{H}^{2}},$
$\displaystyle\lambda_{\Phi}$ $\displaystyle=$
$\displaystyle\frac{m_{h_{1}}^{2}+m_{h_{2}}^{2}+\sqrt{(m_{h_{1}}^{2}+m_{h_{2}}^{2})^{2}-4(m_{h_{1}}^{2}m_{h_{2}}^{2}+\lambda_{H\Phi}^{2}v_{H}^{2}v_{\Phi}^{2})}}{4v_{\Phi}^{2}}.$
(7)

The DM mass $m_{S}$ can be obtained through the quadratic term of $S$.
Similarly, we regard it as a free parameter, thereby expressing parameter
$\mu_{S}^{2}$ as

$\displaystyle\mu_{S}^{2}=m_{S}^{2}-\frac{1}{2}\lambda_{HS}v_{H}^{2}-\frac{1}{2}\lambda_{S\Phi}v_{\Phi}^{2}.$
(8)

###  II.2 Gauge sector

From the covariant derivatives of $H$, $\Phi$ and $S$, we can obtain the mass-
squared matrix of gauge bosons ($W^{3}_{\mu},B_{\mu},X_{\mu}$) as

$\displaystyle\frac{1}{2}\frac{v_{H}^{2}}{4}(-gW_{\mu}^{3}+g^{\prime}B_{\mu})^{2}+\frac{v_{\Phi}^{2}}{2}(g^{\prime}n_{\Phi}B_{\mu}+g_{x}X_{\mu})^{2}$
(9) $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}W^{3,\mu}&B^{\mu}&X^{\mu}\end{pmatrix}\begin{pmatrix}\frac{g^{2}v_{H}^{2}}{4}&-\frac{gg^{\prime}v_{H}^{2}}{4}&0\\\
-\frac{gg^{\prime}v_{H}^{2}}{4}&\frac{g^{\prime 2}v_{H}^{2}}{4}+g^{\prime
2}n_{\Phi}^{2}v_{\Phi}^{2}&g_{x}g^{\prime}n_{\Phi}v_{\Phi}^{2}\\\
0&g_{x}g^{\prime}n_{\Phi}v_{\Phi}^{2}&g_{x}^{2}v_{\Phi}^{2}\end{pmatrix}\begin{pmatrix}W^{3}_{\mu}\\\
B_{\mu}\\\ X_{\mu}\end{pmatrix}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\begin{pmatrix}W^{3,\mu}&B^{\mu}&X^{\mu}\end{pmatrix}\mathcal{M}_{g}^{2}\begin{pmatrix}W^{3}_{\mu}\\\
B_{\mu}\\\ X_{\mu}\end{pmatrix}.$

Considering the kinetic mixing term $(s_{\epsilon}/2)B^{\mu\nu}X_{\mu\nu}$, we
can eliminate it by a linear transformation [28],

$\displaystyle\begin{pmatrix}W^{3}_{\mu}\\\ B_{\mu}\\\
X_{\mu}\end{pmatrix}=\begin{pmatrix}1&0&0\\\ 0&1&-t_{\epsilon}\\\
0&0&1/c_{\epsilon}\end{pmatrix}\begin{pmatrix}W^{3}_{\mu}\\\
B^{\prime}_{\mu}\\\
X^{\prime}_{\mu}\end{pmatrix}\equiv\mathcal{K}\begin{pmatrix}W^{3}_{\mu}\\\
B^{\prime}_{\mu}\\\ X^{\prime}_{\mu}\end{pmatrix},$ (10)

where $c_{\epsilon}=\sqrt{1-s_{\epsilon}^{2}}$ and
$t_{\epsilon}=s_{\epsilon}/c_{\epsilon}$. Then, the kinetic terms of the gauge
bosons are canonically diagonalized in the
($W^{3}_{\mu},B^{\prime}_{\mu},X^{\prime}_{\mu}$) basis. Then the mass-squared
matrix $\mathcal{M}_{g}^{2}$ can be diagonalized by an orthogonal
transformation
$m_{g}^{2}=O^{T}M_{g}^{2}O=\mathrm{diag}\\{0,m_{Z}^{2},m_{Z^{\prime}}^{2}\\}$,
where

$\displaystyle O$ $\displaystyle=$
$\displaystyle\mathcal{K}\begin{pmatrix}1&0&0\\\ 0&c_{\alpha}&s_{\alpha}\\\
0&-s_{\alpha}&c_{\alpha}\end{pmatrix}\begin{pmatrix}s_{w}^{\prime}&c_{w}^{\prime}&0\\\
c_{w}^{\prime}&-s_{w}^{\prime}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&c_{\xi}&s_{\xi}\\\ 0&-s_{\xi}&c_{\xi}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}s_{w}^{\prime}&c_{w}^{\prime}c_{\xi}&c_{w}^{\prime}s_{\xi}\\\
c_{w}^{\prime}(c_{\alpha}+s_{\alpha}t_{\epsilon})&-s_{\xi}(s_{\alpha}-c_{\alpha}t_{\epsilon})-c_{\xi}s_{w}^{\prime}(c_{\alpha}+s_{\alpha}t_{\epsilon})&c_{\xi}(s_{\alpha}-c_{\alpha}t_{\epsilon})-s_{\xi}s_{w}^{\prime}(c_{\alpha}+s_{\alpha}t_{\epsilon})\\\
-\frac{c_{w}^{\prime}s_{\alpha}}{c_{\epsilon}}&\frac{c_{\xi}s_{\alpha}s_{w}^{\prime}-c_{\alpha}s_{\xi}}{c_{\epsilon}}&\frac{s_{\xi}s_{\alpha}s_{w}^{\prime}+c_{\alpha}c_{\xi}}{c_{\epsilon}}\end{pmatrix},$

here we have defined,

$\displaystyle n_{x}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{c_{\epsilon}}-\frac{g^{\prime}}{g_{x}}n_{\Phi}t_{\epsilon},$
(12) $\displaystyle t_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime}n_{\Phi}}{g_{x}n_{x}^{\prime}},$ (13)
$\displaystyle g^{\prime\prime}$ $\displaystyle=$ $\displaystyle
g^{\prime}(c_{\alpha}+t_{\epsilon}s_{\alpha}),$ (14) $\displaystyle
s_{w}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime\prime}}{\sqrt{g^{2}+g^{\prime\prime 2}}},$ (15)
$\displaystyle c_{w}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{g^{2}+g^{\prime\prime 2}}},$ (16) $\displaystyle
t_{\epsilon}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{s_{\alpha}-t_{\epsilon}c_{\alpha}}{c_{\alpha}+t_{\epsilon}s_{\alpha}},$
(17) $\displaystyle n_{x}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime}}{g_{x}}n_{\Phi}s_{\alpha}+n_{x}^{\prime}c_{\alpha},$
(18) $\displaystyle t_{2\xi}$ $\displaystyle=$
$\displaystyle\frac{2g^{\prime\prime}t_{\epsilon}^{\prime}v_{H}^{2}\sqrt{g^{2}+g^{\prime\prime
2}}}{v_{H}^{2}(g^{2}+g^{\prime\prime 2})-g^{\prime\prime
2}t_{\epsilon}^{\prime 2}v_{H}^{2}-4g_{x}^{2}n_{x}^{\prime\prime
2}v_{\Phi}^{2}}.$ (19)

Finally, we obtain the physical gauge fields
$(A_{\mu},Z_{\mu},Z^{\prime}_{\mu})$ by canonical normalizing the kinetic
mixing term and diagonalizing the mass-squared matrix at the same time. The
masses for $Z$ and $Z^{\prime}$ are given by

$\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}+g^{\prime\prime
2}}{4}v_{H}^{2}(1+s_{w}^{\prime}t_{\epsilon}^{\prime}t_{\xi}),$ (20)
$\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{g_{x}^{2}n_{x}^{\prime\prime
2}v_{\Phi}^{2}}{1+s_{w}^{\prime}t_{\epsilon}^{\prime}t_{\xi}}.$ (21)

For the convenience of discussing the phenomenologies later, we show the
neutral current interactions in terms of $(A_{\mu},Z_{\mu},Z^{\prime}_{\mu})$
here,

$\displaystyle\mathcal{L}_{NC}=eA_{\mu}J_{EM}^{\mu}+Z_{\mu}\bar{f}\gamma^{\mu}(g^{f}_{V}-g^{f}_{A}\gamma^{5})f+Z_{\mu}^{\prime}\bar{f}\gamma^{\mu}(g^{\prime
f}_{V}-g^{\prime f}_{A}\gamma^{5})f,$ (22)

where

$\displaystyle g^{f}_{V}$ $\displaystyle=$
$\displaystyle\frac{T^{3}_{f}}{2}\Big{[}gO_{12}-g^{\prime}O_{22}\Big{]}+g^{\prime}Q_{f}O_{22},$
$\displaystyle g^{f}_{A}$ $\displaystyle=$
$\displaystyle\frac{T^{3}_{f}}{2}\Big{[}gO_{12}-g^{\prime}O_{22}\Big{]},$
$\displaystyle g^{\prime f}_{V}$ $\displaystyle=$
$\displaystyle\frac{T^{3}_{f}}{2}\Big{[}gO_{13}-g^{\prime}O_{23}\Big{]}+g^{\prime}Q_{f}O_{23},$
$\displaystyle g^{\prime f}_{A}$ $\displaystyle=$
$\displaystyle\frac{T^{3}_{f}}{2}\Big{[}gO_{13}-g^{\prime}O_{23}\Big{]}.$ (23)

Here the electromagnetic current $J_{EM}^{\mu}$ remains unchanged as the SM
form, while the eletric charge reads $e=gO_{11}=g^{\prime}O_{21}$.

We choose the set of free parameters in our model as

$\displaystyle\\{m_{h_{2}},m_{S},\lambda_{S},\lambda_{HS},\lambda_{H\Phi},\lambda_{S\Phi},g_{x},n_{\Phi},s_{\epsilon},m_{Z^{\prime}}\\},$
(24)

where the last four parameters are related to the gauge sector. And we have
chosen $m_{Z^{\prime}}$ as the free parameter instead of $v_{\Phi}$.

##  III Constraints

###  III.1 Electroweak precision measurements

In this subsection, we will explore whether our model can account for the
recently observed $W$ boson mass. Within our model framework, the couplings
between $Z^{\prime}$ and SM gauge bosons will lead to the deviation of
$m_{W}$. Alternatively, this deviation can be interpreted as an additional
contribution to the electroweak oblique parameters $S$, $T$, and $U$ [29, 30].
Given the considerable difference in measurements between CDF and ATLAS, we
fit our model parameters by utilizing the global fit results of $S$, $T$, and
$U$ derived from both the CDF [31] and non-CDF sources [32]. Then we compare
their fitting results. Table 1 shows the global fit results of $S$, $T$, and
$U$ under these two scenarios.

| CDF results | Correlation | Non-CDF results | Correlation
---|---|---|---|---
$S$ | $\phantom{+}0.005\pm 0.096$ |  +$1.00$ |  |  | $\phantom{+}0.040\pm 0.110$ |  +$1.00$ |  |
$T$ | $\phantom{+}0.040\pm 0.120$ | $0.91$ | $1.00$ |  | $\phantom{+}0.090\pm 0.140$ | $0.92$ | $1.00$ |
$U$ | $\phantom{+}0.134\pm 0.087$ | $-0.65$ | $-0.88$ | $1.00$ | $-0.020\pm 0.110$ | $-0.68$ | $-0.87$ | $1.00$

Table 1: Results of the global fit of the oblique parameters by fitting the
CDF and non-CDF sources.

Using the effective Lagrangian techniques, the electroweak charged and neutral
current interaction can be written as related to the electroweak oblique
parameters [33]:

$\displaystyle\mathcal{L}_{CC,W}$ $\displaystyle=$
$\displaystyle-\frac{e}{\sqrt{2}\hat{s}_{w}}(1-\frac{\alpha
S}{4(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})}+\frac{\hat{c}_{w}^{2}\alpha
T}{2(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})}+\frac{\alpha
U}{8\hat{s}_{w}^{2}})\sum\limits_{ij}V_{ij}\bar{f}_{i}\gamma^{\mu}\gamma_{L}f_{j}W_{\mu}^{\dagger}+\mathrm{c.c.},$
$\displaystyle\mathcal{L}_{NC,Z}$ $\displaystyle=$
$\displaystyle\frac{e}{\hat{s}_{w}\hat{c}_{w}}(1+\frac{\alpha
T}{2})\sum\limits_{f}\bar{f}\gamma^{\mu}[T^{3}_{f}\frac{1-\gamma^{5}}{2}-Q_{f}(\hat{s}_{w}^{2}+\frac{\alpha
S}{4(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})}-\frac{\hat{c}_{w}^{2}\hat{s}_{w}^{2}\alpha
T}{\hat{c}_{w}^{2}-\hat{s}_{w}^{2}})]fZ_{\mu}.$ (25)

Here $Q_{f}$ denotes the electric charge, defined by $Q_{f}=Y_{f}+T^{3}_{f}$.
$Y_{f}$ is the hypercharge and $T^{3}_{f}$ is the third component of weak
isospin. It should be noted that $\hat{s}_{w}$ and $\hat{c}_{w}$ with a hat
overhead are physical observables, defined by

$\displaystyle\frac{G_{F}}{\sqrt{2}}=\frac{4\pi\alpha}{8\hat{s}_{w}^{2}\hat{c}_{w}^{2}m_{Z}^{2}}.$
(26)

The charged current interaction of our model remains the same as SM. In
contrast, the neutral current interaction is modified as Eq. (II.2). Note that
the parameters $g$ and $g^{\prime}$ with no hat overhead are unphysical and
dependent upon the input value of other free parameters. Then we can obtain
the parameter relationships between $S$, $T$, $U$ and our model:

$\displaystyle\alpha T$ $\displaystyle=$ $\displaystyle
2\hat{c}_{w}\hat{s}_{w}\frac{-g^{\prime}O_{22}+gO_{12}}{e}-2,$
$\displaystyle\alpha S$ $\displaystyle=$
$\displaystyle\frac{-4g^{\prime}O_{22}(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})}{-g^{\prime}O_{22}+gO_{12}}-4\hat{s}_{w}^{2}(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})+4\hat{c}_{w}^{2}\hat{s}_{w}^{2}\alpha
T,$ $\displaystyle\alpha U$ $\displaystyle=$ $\displaystyle
8\hat{s}_{w}^{2}(\frac{\hat{s}_{w}}{s_{w}}-1+\frac{\alpha
S}{4(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})}-\frac{\hat{c}_{w}^{2}\alpha
T}{2(\hat{c}_{w}^{2}-\hat{s}_{w}^{2})}).$ (27)

(a)

(b)

(c)

(d)

Figure 1: Fitting results of the model parameters in the
$s_{\epsilon}$-$m_{Z^{\prime}}$ plane (Top) and $n_{\Phi}$-$g_{x}$ plane
(Bottom) by fitting CDF measurement (Left) and non-CDF measurement (Right).
The blue triangle gives the best fit of electroweak oblique parameters, and
the colored region corresponds to $\Delta\chi^{2}=6.18$ with respect to the
best fit.

Based on the calculations detailed in Sec. II, the free parameters associated
with $S$, $T$, $U$ in our model include $g_{x}$, $n_{\Phi}$, $s_{\epsilon}$
and $m_{Z^{\prime}}$. In Fig. 1, we plot the fitting results of these free
parameters. The blue triangle represents the best fit point in each figure and
we utilize gradient colors to fill the region where $\Delta\chi^{2}=6.18$
relative to the best fit value. Additionally, the $\chi^{2}$ value of each
point is visualized through the colorbar on the right side of each plot. Figs.
1(a) and 1(c) on the left depict the fitting results based on CDF measurement,
while Figs. 1(b) and 1(d) on the right correspond to fitting non-CDF
measurement. Moreover, Figs. 1(a) and 1(b) at the top display the results in
the $s_{\epsilon}$-$m_{Z^{\prime}}$ plane with the other parameters fixed to
be $n_{\Phi}=10^{-3}$ and $g_{x}=0.1$, while Figs. 1(c) and 1(d) at the bottom
present the fitting results in the $n_{\Phi}$-$g_{x}$ plane with the fixed
parameters chosen to be $m_{Z^{\prime}}=10^{3}~{}\mathrm{GeV}$ and
$s_{\epsilon}=0.2$. The parameter values corresponding to each best fit point
are provided in in Table 2.

| $g_{x}$ | $n_{\Phi}$ | $s_{\epsilon}$ | $m_{Z^{\prime}}(\mathrm{GeV})$ | $\chi^{2}$
---|---|---|---|---|---
Fig. 1(a) | $0.1$ | $10^{-3}$ | $0.27$ | $8047.6$ | 0.48
Fig. 1(b) | $0.1$ | $10^{-3}$ | $0.09$ | $1037.8$ | 0.75
Fig. 1(c) | $0.070$ | $8.9\times 10^{-4}$ | $0.2$ | $1000$ | 0.68
Fig. 1(d) | $0.063$ | $2.3\times 10^{-4}$ | $0.2$ | $1000$ | 0.81

Table 2: Parameter Values of best fit points in Fig. 1.

Due to the significant discrepancies in measurements between CDF and other
experiments, the fitting results for model parameters diverge substantially in
these two scenarios. The CDF measurement exhibits a $7\sigma$ deviation from
the SM prediction, suggesting that the interaction between $Z^{\prime}$ and SM
gauge bosons must fall within a specific range. Specifically, as depicted in
Fig. 1(a), $m_{Z^{\prime}}$ is greater than around $200~{}\mathrm{GeV}$ and
$s_{\epsilon}$ lies within $0.1\sim 0.4$. Meanwhile, Fig. 1(c) shows a linear
correlation between $g_{x}$ and $n_{\Phi}$, arising from the fact that $\Phi$
possesses both $\mathrm{U}(1)_{\mathrm{Y}}$ and $\mathrm{U}(1)_{\mathrm{X}}$
charges. As $g_{x}$ increases, the couplings between $Z^{\prime}$ and SM
particles weaken, necessitating an increase in $n_{\Phi}$. By comparison, in
the case of fitting non-CDF measurement, the interaction between $Z^{\prime}$
and SM particles must be sufficiently small. Notably, both $s_{\epsilon}$ and
$n_{\Phi}$ exhibit upper bounds, which vary with $m_{Z^{\prime}}$ and $g_{x}$
respectively.

###  III.2 $Z^{\prime}$ searches at the LHC

The couplings between $Z^{\prime}$ and SM fermions can induce the production
of $Z^{\prime}$ in the s-channel at hadron colliders, leading to its detection
through the decay products $Z^{\prime}\rightarrow l^{+}l^{-}$ ($l\equiv
e,\mu,\tau$). The ATLAS and CMS collaborations routinely conduct searches for
the $Z^{\prime}$ boson at the LHC [34, 35]. Since no new particles have been
discovered so far, experiments still impose stringent upper limits upon the
cross section for $Z^{\prime}$ production.

Using FeynRules [36] and MadGraph5_aMG@NLO [37], we calculate the cross
section $\sigma(pp\to Z^{\prime}X\to e^{+}e^{-}X)$ for producing $Z^{\prime}$
at the LHC, which subsequently decays to the $e^{+}e^{-}$ final state. We then
compare this result with the $95\%$ confidence level (CL) upper limit dilepton
data set observed by ATLAS at the $13~{}\mathrm{TeV}$ LHC [34].

In Fig. 2, we select different $m_{S}$ and $s_{\epsilon}$ as benchmark points
with other relevant parameters to be
$\lambda_{S}=\lambda_{HS}=\lambda_{H\Phi}=\lambda_{S\Phi}=10^{-3}$,
$m_{h_{2}}=1~{}\mathrm{TeV}$, $n_{\Phi}=10^{-3}$ and $g_{x}=0.1$. Figs. 2(a)
and 2(b) at the top correspond to the horizontal line $s_{\epsilon}=0.1$ in
Fig. 1(b) for $m_{S}=500~{}\mathrm{GeV}$ and $1000~{}\mathrm{GeV}$
respectively. In contrast, Figs. 2(c) and 2(d) at the bottom correspond to the
case for $s_{\epsilon}=0.05$. The green area in Fig. 2 lies outside the
$\Delta\chi^{2}=6.18$ region in Fig. 1(b). It’s worth noting that using the
fitting result in Fig. 1(a) will alter the green area, as it depends on the
electroweak oblique parameters in the fitting process. Nevertheless, the
analysis remains similar. The blue area is excluded by ensuring the stability
of DM, which can be determined through Eq. (8). The black line represents the
$Z^{\prime}$ production cross section while the red dashed line indicates the
$95\%$ CL upper limit set by ATLAS.

In Fig. 2(a), the entire $m_{Z^{\prime}}$ region is excluded. However, in Fig.
2(c), as $s_{\epsilon}$ decreases to $0.05$, the $m_{Z^{\prime}}$ region
survives from $1527~{}\mathrm{GeV}$ to $2245~{}\mathrm{GeV}$ due to the
weakening interaction between $Z^{\prime}$ and SM particles. Similarly, in
Fig. 2(b) as $m_{S}$ increases to $1000~{}\mathrm{GeV}$, the surviving
$m_{Z^{\prime}}$ region extends from $2754~{}\mathrm{GeV}$ to
$4464~{}\mathrm{GeV}$ owing to the rise of the lower limit imposed by DM
stability constraints. Additionally, when $m_{Z^{\prime}}\geq 2m_{S}$, the
cross section decreases significantly faster. This is attributed to the
kinematically allowed decay channel $Z^{\prime}\to S^{\dagger}S$, leading to a
decrease in the branching fraction for $Z^{\prime}\to e^{+}e^{-}$. This effect
becomes more pronounced with decreasing $s_{\epsilon}$, as evident in Figs.
2(c) and 2(d). Table 3 presents the surviving $m_{Z^{\prime}}$ regions
depicted in each subfigure of Figure 2, which will be further investigated by
future $Z^{\prime}$ search experiments.

(a) (b) (c) (d)

Figure 2: Cross section $\sigma(pp\to Z^{\prime}X\to e^{+}e^{-}X)$ as a
function of $m_{Z}^{\prime}$ for $s_{\epsilon}=0.1$ (Top) and $0.05$ (Bottom)
with $m_{S}=500~{}\mathrm{GeV}$ (Left) and $1000~{}\mathrm{GeV}$ (Right). The
red dashed line denotes the $95\%$ CL upper limit set by ATLAS, and the orange
vertical line corresponds to $m_{Z^{\prime}}=2m_{S}$. The green area lies
outside the $\Delta\chi^{2}=6.18$ region in Fig. 1(b) while the blue area is
excluded due to the stability of DM.

| $m_{S}(\mathrm{GeV})$ | $s_{\epsilon}$ | $m_{Z^{\prime}}(\mathrm{GeV})$
---|---|---|---
Fig. 2(a) | $500$ | $0.1$ |
Fig. 2(b) | $1000$ | $0.1$ | $2754\sim 4464$
Fig. 2(c) | $500$ | $0.05$ | $1527\sim 2245$
Fig. 2(d) | $1000$ | $0.05$ | $1908\sim 4507$

Table 3: Surviving region of $m_{Z^{\prime}}$ in each subfigure of Fig. 2.

###  III.3 Higgs physics

In this subsection, we discuss the implications upon the Higgs sector arising
from the presence of new particles in our model. As a consequence of the
mixing between the Higgs bosons $h_{1}$ and $h_{2}$, as well as between the
gauge bosons $Z$ and $Z^{\prime}$, the couplings between Higgs and SM
particles deviate from their expected values. These deviations can be
described by the following expression [39]

$\displaystyle\mathcal{L}_{h_{1}}\supset\kappa_{W}\frac{2m_{W}^{2}}{v_{H}}h_{1}W^{+}_{\mu}W^{-,\mu}+\kappa_{Z}\frac{m_{Z}^{2}}{v_{H}}h_{1}Z_{\mu}Z^{\mu}-\sum_{f}\kappa_{f}\frac{m_{f}}{v_{H}}h_{1}\bar{f}f,$
(28)

where

$\displaystyle\kappa_{W}$ $\displaystyle=$
$\displaystyle\kappa_{f}=c_{\theta},$ (29) $\displaystyle\kappa_{Z}$
$\displaystyle=$ $\displaystyle
c_{\theta}(c_{W}O_{22}-s_{W}O_{12})^{2}-(n_{\Phi}^{2}g^{\prime
2}O_{22}^{2}+2n_{\Phi}g^{\prime}g_{x}O_{22}O_{32}+g_{x}^{2}O_{32}^{2})\frac{s_{\theta}v_{\Phi}v_{H}}{m_{Z}^{2}}.$
(30)

These couplings elucidate the modifications to the Higgs interactions induced
by the incorporation of new particles. In SM, all these couplings equal to
$1$. Moreover, if kinematically allowed, there may exist additional Higgs
decay channels. If $m_{h_{1}}>2m_{h_{2}}$, the undetected decay channel
$h_{1}\to 2h_{2}$ becomes accessible, with its decay width given by

$\displaystyle\Gamma(h_{1}\to 2h_{2})=\frac{g_{h_{1}h_{2}h_{2}}^{2}}{32\pi
m_{h_{1}}}\sqrt{1-\frac{4m_{2}^{2}}{m_{h_{1}}^{2}}},$ (31)

where

$\displaystyle
g_{h_{1}h_{2}h_{2}}=3\lambda_{H}v_{H}c_{\theta}s_{\theta}^{2}-3\lambda_{\Phi}v_{\Phi}s_{\theta}c_{\theta}^{2}+\frac{\lambda_{H\Phi}}{2}\Big{[}v_{H}(c_{\theta}^{3}-2c_{\theta}s_{\theta}^{2})-v_{\Phi}(s_{\theta}^{3}-2c_{\theta}^{2}s_{\theta}))\Big{]}.$
(32)

Similarly, if $m_{h_{1}}>2m_{S}$, the invisible decay channel $h_{1}\to
S^{\dagger}S$ opens, with its decay width given by

$\displaystyle\Gamma(h_{1}\to
S^{\dagger}S)=\frac{g_{h_{1}S^{\dagger}S}^{2}}{16\pi
m_{h_{1}}}\sqrt{1-\frac{4m_{S}^{2}}{m_{h_{1}}^{2}}},$ (33)

where

$\displaystyle
g_{h_{1}S^{\dagger}S}=\lambda_{HS}v_{H}c_{\theta}-\lambda_{S\Phi}v_{\Phi}s_{\theta}.$
(34)

We utilize a numerical tool Lilith 2 [40, 41] to constrain our model parameter
space in Sec. IV. This tool, implemented as a public Python library,
facilitates the examination of constraints imposed on new physics models by
Higgs signal strength measurements, including data from ATLAS and CMS Run 2
Higgs results for $36~{}\mathrm{fb}^{-1}$. To ensure compatibility with
experimental observations at the $95\%$ CL, we exclude parameter points with
p-values below $0.05$. This rigorous methodology enables us to effectively
navigate the vast parameter space and identify regions consistent with
experimental data.

###  III.4 DM-nucleon scattering

In this model, two types of interactions facilitate spin-independent (SI) DM-
nucleon scattering. One is vector interactions, mediated by gauge bosons $Z$
and $Z^{\prime}$. The other is scalar interactions, mediated by the scalars
$h_{1}$ and $h_{2}$. Fig. 3 illustrates the Feynman diagrams corresponding to
these two interaction types.

(a) Vector interaction mediated by $Z$ and $Z^{\prime}$.

(b) Scalar interaction mediated by $h_{1}$ and $h_{2}$.

Figure 3:  DM-fermions scattering mediated by vector bosons (a) and scalars
(b).

To derive the cross section, we express the scattering interaction between DM
$S$ and SM fermions $f$ using the effective field theory presented in Ref.
[42]:

$\displaystyle\mathcal{L}_{S}$ $\displaystyle=$
$\displaystyle\sum_{f}F_{S,f}S^{\dagger}S\bar{f}f,$
$\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sum_{f}F_{V,f}(S^{\dagger}i\overleftrightarrow{\partial_{\mu}}S)\bar{f}\gamma^{\mu}f.$
(35)

Here $F_{S,f}$ represents the effective coupling for the scalar interaction
while $F_{V,f}$ corresponds to the vector interaction. Note that the axial
vector interaction can be neglected due to the velocity suppression in the
low-velocity limit [42]. To compute the cross section, we define four
effective couplings $F_{S,N}$ and $F_{V,N}$ ($N=p$, $n$). $F_{S,N}$ are given
by

$\displaystyle F_{S,N}$ $\displaystyle=$
$\displaystyle\sum_{q=u,d,s}F_{S,q}f_{q}^{N}\frac{m_{N}}{m_{q}}+\sum_{q=c,b,t}F_{S,q}f_{Q}^{N}\frac{m_{N}}{m_{q}}$
(36) $\displaystyle=$
$\displaystyle\frac{F(\theta)}{9}m_{N}[2+7(f_{u}^{N}+f_{d}^{N}+f_{s}^{N})],$
(37)

with

$\displaystyle
F(\theta)=\frac{\lambda_{HS}(c_{\theta}^{2}m_{h_{2}}^{2}+s_{\theta}^{2}m_{h_{1}}^{2})}{m_{h_{1}}^{2}m_{h_{2}}^{2}}-\frac{\lambda_{S\Phi}v_{\Phi}s_{\theta}c_{\theta}(m_{h_{2}}^{2}-m_{h_{1}}^{2})}{v_{H}m_{h_{1}}^{2}m_{h_{2}}^{2}}.$
(38)

Here, $\theta$ is the mixing angle between $h_{1}$ and $h_{2}$. $f_{q}^{N}$
are the nucleon form factors for each light quark, and
$f_{Q}^{N}=\frac{2}{27}(1-f_{u}^{N}-f_{d}^{N}-f_{s}^{N})$ for heavy quarks
[43]. $F_{V,N}$ are given by

$\displaystyle F_{V,p}=2F_{V,u}+F_{V,d},\quad F_{V,n}=F_{V,u}+2F_{V,d},$ (39)

with

$\displaystyle F_{V,f}$ $\displaystyle=$ $\displaystyle
g_{V}^{f}\frac{\mathcal{KO}_{32}}{g_{x}v_{\Phi}^{2}}+g_{V}^{\prime
f}\frac{\mathcal{KO}_{33}}{g_{x}v_{\Phi}^{2}}.$ (40)

Since these two types of interactions can interfere with each other, adding up
their cross sections directly is not allowed. Considering the interference
effect [44], the $SN$ and $S^{\dagger}N$ scatter cross sections can be
obtained as

$\displaystyle\sigma_{SN}=\frac{\mu_{SN}^{2}f_{SN}^{2}}{\pi},\quad\sigma_{S^{\dagger}N}=\frac{\mu_{SN}^{2}f_{S^{\dagger}N}^{2}}{\pi},$
(41)

where

$\displaystyle f_{SN}=\frac{F_{S,N}}{2m_{S}}+F_{V,N},\quad
f_{S^{\dagger}N}=\frac{F_{S,N}}{2m_{S}}-F_{V,N},$ (42)

and $\mu_{SN}$ is the reduced mass of $S$ and $N$, defined by $\mu_{SN}\equiv
m_{S}m_{N}/(m_{S}+m_{N})$. As we can see, $\sigma_{SN}$ and
$\sigma_{S^{\dagger}N}$ are generally different from each other due to the
isospin violation. As for the symmetric DM case, the average cross section of
DM-nucleus point-like scattering can be written as

$\displaystyle\sigma_{DM-A}=\frac{\mu_{SN}^{2}}{2\pi}\Big{\\{}\big{[}Zf_{Sp}+(A-Z)f_{Sn}\big{]}^{2}+\big{[}Zf_{S^{\dagger}p}+(A-Z)f_{S^{\dagger}n}\big{]}^{2}\Big{\\}}.$
(43)

Here, we suppose the nucleus $A$ contains $Z$ protons and $(A-Z)$ neutrons.
For the direct detection experiment, the detection material is dual-phase
xenon, of which $Z$ equals 54. Considering that xenon has several isotopes
$A_{i}$, the normalized-to-nucleon scattering cross section can be obtained as
[45]

$\displaystyle\sigma_{N}^{Z}=\sigma_{Sp}\frac{\sum_{i}\eta_{i}\mu_{SA_{i}}^{2}\Big{\\{}\big{[}Z+(A_{i}-Z)f_{Sn}/f_{Sp}\big{]}^{2}+\big{[}Zf_{S^{\dagger}p}/f_{Sp}+(A_{i}-Z)f_{S^{\dagger}n}/f_{Sp}\big{]}^{2}\Big{\\}}}{2\sum_{i}\eta_{i}\mu_{SA_{i}}^{2}A_{i}^{2}},$
(44)

where $\eta_{i}$ is the fractional abundance of isotope $A_{i}$ in nature. For
xenon, $A_{i}=[128,129,130,131,\\\ 132,134,136]$,
$\eta_{i}=[1.9\%,26\%,4.1\%,21\%,27\%,10\%,8.9\%]$ for each corresponding
isotope [46].

(a) (b) (c) (d)

Figure 4: SI DM-nucleon scattering cross section $\sigma_{N}^{Z}$ as a
function of $m_{S}$ for changed $g_{x}$ (b), $\lambda_{S\Phi}$ (c) and
$s_{\epsilon}$ (d) compared to (a). The red line denotes the $90\%$ C.L.
constraint from the LZ direct detection experiment. The blue and green regions
are excluded by the DM stability and $Z^{\prime}$ search experiment from
ATLAS. The orange, cyan and black lines indicate the scattering cross section
contributed from scalar interaction, vector interaction and the combination of
them.

(a) (b)

Figure 5: $\sigma_{N}^{Z}$ as a function of $m_{S}$ for changed
$m_{Z^{\prime}}$ to be $2000~{}\mathrm{GeV}$ (a) and $5000~{}\mathrm{GeV}$
(b).

In the subsequent discussion, we select certain benchmark scenarios outlined
in Table 4 to illustrate the SI DM-nucleon scattering cross section
$\sigma_{N}^{Z}$. We maintain other parameters unchanged:
$\lambda_{S}=\lambda_{HS}=\lambda_{H\Phi}=10^{-3}$,
$m_{h_{2}}=1~{}\mathrm{TeV}$, and $n_{\Phi}=10^{-3}$. The shaded blue area is
excluded due to the stability of DM, while the green area is ruled out by
ATLAS $Z^{\prime}$ search experiment. Additionally, the direct detection limit
from the LZ experiment at $90\%$ CL is depicted in the figure [6]. The black
line denotes $\sigma_{N}^{Z}$, whereas the orange and cyan lines correspond to
the scattering cross sections for scalar and vector interactions,
respectively. It can be seen that at low/high $m_{S}$, the scalar/vector
interaction dominates the DM-nucleon scattering process [42]. This trend is
further elucidated in Figs. 4(d) and 5(b).

In Fig. 4(a) with $m_{Z^{\prime}}=1500~{}\mathrm{GeV}$, $g_{x}=0.2$,
$\lambda_{S\Phi}=10^{-3}$ and $s_{\epsilon}=0.05$, the DM mass range between
$12$ and $626~{}\mathrm{GeV}$ remains unexcluded. As $g_{x}$ increases in Fig.
4(b), the constraints from DM stability and ATLAS weaken, but the DM-nucleon
interaction strength intensifies. Fig. 4(c) demonstrates that relaxing the
constraint from DM stability is achievable by reducing $\lambda_{S\Phi}$. The
lower limit can be estimated as $5.5~{}\mathrm{GeV}$ by setting
$\lambda_{S\Phi}=0$. In Fig. 4(d) and Fig. 5 as $s_{\epsilon}$ decreases or
$m_{Z^{\prime}}$ increases, $Z^{\prime}$ tends to decouple from SM particles,
resulting in a reduction of $\sigma_{N}^{Z}$.

| $m_{Z^{\prime}}(\mathrm{GeV})$ | $g_{x}$ | $\lambda_{S\Phi}$ | $s_{\epsilon}$ | $m_{S}(\mathrm{GeV})$
---|---|---|---|---|---
Fig. 4(a) | 1500 | 0.2 | $10^{-3}$ | $0.05$ | $12\sim 626$
Fig. 4(b) | 1500 | 0.5 | $10^{-3}$ | $0.05$ | $1280\sim 1896$
Fig. 4(c) | 1500 | 0.2 | $10^{-5}$ | $0.05$ | $202\sim 631$
Fig. 4(d) | 1500 | 0.2 | $10^{-3}$ | $0.01$ | $\gtrsim 168$
Fig. 5(a) | 2000 | 0.5 | $10^{-3}$ | $0.05$ | $\gtrsim 90$
Fig. 5(b) | 5000 | 0.5 | $10^{-3}$ | $0.05$ | $\gtrsim 224$

Table 4: Surviving region of $m_{S}$ in each benchmark.

###  III.5 DM annihilation

The relic abundance of DM is determined by the annihilation cross section
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ at the freeze-out epoch.
The primary annihilation channels include $ZZ$, $Z^{\prime}Z^{\prime}$,
$W^{+}W^{-}$ and $h_{i}h_{j}$ ($i,j=1,2$) in this model if kinematically
allowed. For the reason that the couplings between SM fermions and Higgs are
compressed by their masses, the annihilation channels into $f\bar{f}$ are
strongly suppressed except for $t\bar{t}$.

We compute the relic density and thermal-averaged cross section using
micrOMEGAs [47], a code capable of calculating DM properties within any given
new physics model. The relic density must satisfy the Planck measurement
$\Omega_{\mathrm{DM}}h^{2}=0.1200\pm 0.0012$ [48]. Parameter regions yielding
a larger DM relic abundance than this limit are excluded, while those with
smaller relic abundance are permissible, as other DM candidates or non-thermal
production mechanisms may contribute to the total relic abundance. The
annihilation cross section of DM during the freeze-out epoch is typically
expected to be around $\mathcal{O}(10^{-26})~{}\mathrm{cm^{3}s^{-1}}$, with a
thermally averaged DM velocity of approximately $\mathcal{O}(10^{-1})$, to be
consistent with the Planck result.

Currently, DM in the present universe is still undergoing annihilation,
emitting gamma rays in the process. A joint analysis of MAGIC observations for
158 hours and Fermi Large Area Telescope (LAT) observations for 6 years has
placed strong constraints on the DM annihilation cross section [49]. Further
studies indicate that DM particles in dwarf galaxies typically exhibit
velocities on the order of $\mathcal{O}(10^{-5})$ [50], suggesting that DM
annihilation today can be well approximated as s-wave annihilation. In Section
IV, we utilize both the Planck measurement and the Fermi-MAGIC result to
constrain our model parameters.

##  IV Parameter scan

In this model, there are 10 free parameters, as delineated in Eq. (24). We
conduct a random scan within the following parameter ranges:

$\displaystyle 10\ \mathrm{GeV}<m_{h_{2}},\ m_{S}<10^{4}\ \mathrm{GeV},\quad
10^{2}\ \mathrm{GeV}<m_{Z^{\prime}}<10^{4}\ \mathrm{GeV},$ $\displaystyle
10^{-2}<g_{x}<1,\quad 10^{-3}<s_{\varepsilon}<0.9,\quad 10^{-3}<n_{\Phi}<1,$
$\displaystyle 10^{-4}<\lambda_{S},\ \lambda_{HS},\ \lambda_{H\Phi},\
\lambda_{S\Phi}<1.$ (45)

Moreover, we employ the following constraints to filter each parameter point:

  * •

According to the Lilith calculation, p-value must exceed $0.05$.

  * •

The SI DM-nucleon scattering cross section $\sigma_{N}^{Z}$ must not surpass
the limit established by the LZ direct detection experiment.

  * •

The predicted relic density of DM, $\Omega_{\mathrm{DM}}h^{2}$, must fall
within the Planck result.

  * •

The thermal-avearged annihilation cross section of DM in dwarf galaxies,
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$, should satisfy the Fermi-
LAT constraint.

Figure 6:  Selected parameter points projected onto the $\sigma(pp\to
Z^{\prime}X\to e^{+}e^{-}X)$-$m_{Z^{\prime}}$ plane, with a colorbar
corresponding to $g_{x}$. The red line denotes the $95\%$ CL upper limit set
by ATLAS.

In Fig. 6, we project the parameter points satisfying the above constraints
onto the $\sigma(pp\to Z^{\prime}X\to e^{+}e^{-}X)$-$m_{Z^{\prime}}$ plane,
with a colorbar corresponding to $g_{x}$ displayed on the right. Additionally,
we include the $95\%$ CL upper limit set by ATLAS on the graph. It can be seen
that for $m_{Z^{\prime}}\gtrsim 4750~{}\mathrm{GeV}$, all parameter points can
evade detection in the $Z^{\prime}$ search experiment. Conversely, for
$m_{Z^{\prime}}\lesssim 4750~{}\mathrm{GeV}$, smaller values of $g_{x}$ are
disfavored. This trend emerges because, as $g_{x}$ increases, the interaction
between $Z^{\prime}$ and SM particles weakens. For the majority of surviving
parameter points, $g_{x}$ ranges from $0.1$ to $1$. Utilizing interpolation,
we discard parameter points that exceed the upper limit, ensuring that the
remaining points satisfy all considered constraints. Finally, the last
phenomenological requirement is

  * •

The cross section $\sigma(pp\to Z^{\prime}X\to e^{+}e^{-}X)$ for producing
$Z^{\prime}$ at the LHC, which then decays to $e^{+}e^{-}$ final state, falls
within the ATLAS limit.

These criteria are essential for ensuring that the model remains consistent
with both theoretical expectations and experimental observations.

(a) (b) (c)

Figure 7: Surviving parameter points projected onto the the
$v_{\Phi}$-$m_{Z^{\prime}}$ (a), $v_{\Phi}$-$m_{h_{2}}$ (b) and
$g_{x}$-$n_{\Phi}$ (c) planes, with colorbars corresponding to $g_{x}$,
$\lambda_{\Phi}$ and $s_{\epsilon}$, respectively.

Then we project the surviving parameter points onto the
$v_{\Phi}$-$m_{Z^{\prime}}$, $v_{\Phi}$-$m_{h_{2}}$ and $g_{x}$-$n_{\Phi}$
planes in Figs. 7(a), 7(b) and 7(c), with colorbars corresponding to $g_{x}$,
$\lambda_{\Phi}$ and $s_{\epsilon}$, respectively. Figs. 7(a) and 7(b) suggest
that $m_{Z^{\prime}}$ and $m_{h_{2}}$ have an approximate linear relationship
with $v_{\Phi}$. Particularly, this relationship appears more pronounced in
Fig. 7(a), implying $m_{Z^{\prime}}\simeq g_{x}v_{\Phi}$. This is consistent
with the case when $t^{\prime}_{\epsilon}\to 0$ in Eq. (21). In contrast, when
$m_{h_{2}}\lesssim 200~{}\mathrm{GeV}$ in Fig. 7(b), such linearity tends to
break down. According to Eq. (7), when the masses of $h_{1}$ and $h_{2}$
become close, this approximate relationship no longer holds. Fig. 7(c)
illustrates the relation among the parameters in the gauge sector. It shows
that $n_{\Phi}$ is proportional to $g_{x}$ and negatively correlated to
$s_{\epsilon}$. More precisely, $g_{x}$ ranges from approximately $0.02$ to
$1$ while $n_{\Phi}$ in $10^{-3}\sim 0.2$ and $s_{\epsilon}$ in $10^{-3}\sim
0.9$. This surviving parameter space in the gauge sector can be further
examined by future $Z^{\prime}$ searches and direct detection experiments.

(a) (b)

Figure 8: Surviving parameter points projected onto the the
$m_{S}$-$m_{Z^{\prime}}$ (a) and $m_{S}$-$m_{h_{2}}$ (b) planes, with
colorbars corresponding to $g_{x}$ and $\lambda_{S\Phi}$, respectively. The
red lines in (a) and (b) correspond to the resonant annihilation of DM via
mediators $Z^{\prime}$ and $h_{2}$, respectively.

Figs. 8(a) and 8(b) illustrate the resonant annihilation of DM in the the
$m_{S}$-$m_{Z^{\prime}}$ and $m_{S}$-$m_{h_{2}}$ planes, with colors denoting
$g_{x}$ and $\lambda_{S\Phi}$. In Fig. 8(a), a cluster of parameter points
concentrates along the line where $m_{Z^{\prime}}=2m_{S}$, representing the
resonance region where two DM particles annihilate into one $Z^{\prime}$
particle. By comparison, in Fig. 8(b), only a few points concentrate along the
line where $m_{h_{2}}=2m_{S}$, indicating that the resonant annihilation
mediated by $h_{2}$ is much weaker than that mediated by $Z^{\prime}$ for most
parameter points.

(a) (b)

Figure 9: Surviving parameter points projected onto the
$m_{h_{2}}$-$\lambda_{H}$ and $g_{x}$-$\kappa_{Z}$ planes, with colorbars
corresponding to the Lilith p-value of each point.

Fig. 9 depicts the Lilith p-values of the surviving parameter points projected
onto the $m_{h_{2}}$-$\lambda_{H}$ and $g_{x}$-$\kappa_{Z}$ planes. In Fig.
9(a) when $m_{h_{2}}\gtrsim m_{h_{1}}$, $\lambda_{H}$ tends to converge toward
$\lambda_{SM}=m_{h_{1}}^{2}/(2v_{H}^{2})\simeq 0.13$, which corresponds to the
quartic Higgs coupling in the SM. Therefore, along this linear trend lies the
region with the largest p-value. Conversely, when $m_{h_{2}}\lesssim
m_{h_{1}}$, $\lambda_{H}$ deviates significantly from $\lambda_{SM}$ and
converges along the line described by Eq. (7). In Fig. 9(b), the majority of
parameter points converge around $\kappa_{Z}=1$, consistent with the SM
prediction. Furthermore, most points tend to cluster within the region where
$g_{x}$ ranges from $0.1$ to $1$. This trend arises because smaller values of
$g_{x}$ are more likely to be excluded by $Z^{\prime}$ search experiments,
aligning with the earlier discussion.

(a) (b)

Figure 10: Surviving parameter points projected onto the
$m_{S}$-$\Omega_{\mathrm{DM}}h^{2}$ and
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$-$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}/\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$
planes, with colorbars corresponding to
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ and $m_{S}$ respectively.
The red line in (a) denotes the Planck result while the one in (b) means that
the parameter points around it are s-waves dominated.

Moreover, we present the surviving parameter points in the
$m_{S}$-$\Omega_{\mathrm{DM}}h^{2}$ plane in Fig. 10(a), with a colorbar
indicating the thermal-avearged annihilation cross section
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ at the freeze-out epoch.
The red line denotes the Planck observation, with the area above it
representing the over-production of DM. In Fig.10(b), we illustrate the
relation between $\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ and
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$. The red line denotes the
case where
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}=\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$.
Most of the parameter points cluster around the vicinity of the red line,
demonstrating that they are dominated by s-waves. The range of
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ for these points is
approximately $10^{-26}\mathrm{cm^{3}/s}$ to $10^{-25}\mathrm{cm^{3}/s}$.
However, there are also a number of parameter points far away from the red
line, indicating significant differences between their
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$ and
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$. This discrepancy arises
because these points lie within the DM resonant annihilation region, resulting
in a strong dependence on their velocity. Consequently,
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ for these points is no
longer dominated by s-waves.

##  V Conclusions

In this work, we propose a scalar dark matter model with a hidden gauge
symmetry $\mathrm{U}(1)_{\mathrm{X}}$ and two complex scalars $\Phi$ and $S$.
Two types of mixings between $\mathrm{U}(1)_{\mathrm{X}}$ and
$\mathrm{U}(1)_{\mathrm{Y}}$ are introduced to simultaneously generate dark
matter relic abundance and satisfy the direct detection constraint. As the new
gauge boson $Z^{\prime}$ interacts with SM particles, our model is constrained
from colliders as well. We discuss the constraint from electroweak precision
measurements in the effective Lagrangian framework. Considering the
significant difference between CDF measurement and other experiments, we fit
our model parameters using two sets of $S$, $T$ and $U$ parameters for
comparison. We present the fitting results for several benchmarks. We also
discuss the constraint from the search for $Z^{\prime}$ at the LHC. Our
analysis suggests that large $m_{S}$ and small $s_{\epsilon}$ are favored.
Especially, We present several surviving parameter spaces to be examined in
future experiments. Additionally, the interaction within the Higgs sector may
change in this model. Therefore, we select parameter points in the random scan
conducted in Sec.IV, to satisfy the constraints from the ATLAS and CMS Run 2
Higgs results. Considering the different couplings of $Z$ and $Z^{\prime}$ to
protons and neutrons, the isospin violation effect is considered in
calculating the DM-nucleon scattering cross sections. Our findings show a wide
range of the parameter space remains unexplored by current experiments. We
present several parameter spaces by varying $s_{\epsilon}$, $m_{Z^{\prime}}$,
$g_{x}$, and $\lambda_{S\Phi}$. These parameter spaces will be further tested
by future direct detection experiments.

By considering all the above constraints, and the Planck and Fermi-LAT
observations as well, we perform a random scan to investigate the parameter
space. We observe that DM annihilation mediated by $Z^{\prime}$ is dominated
compared with other mediators. The resonance effect renders annihilation
sharply velocity-dependent in the resonance region, where
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}\neq\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$.
In the standard scenario where
$\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}\simeq\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{D}}$,
our scan results indicate that the thermal-averaged cross section at the
freeze-out period $\langle\sigma_{\mathrm{ann}}v\rangle_{\mathrm{FO}}$ lies
between $10^{-26}\mathrm{cm^{3}/s}$ and $10^{-25}\mathrm{cm^{3}/s}$. For the
majority of surviving parameter points, $g_{x}$ ranges from $0.1$ to $1$.
Additionally, we find that the regions where $m_{Z^{\prime}}\gtrsim
4750~{}\mathrm{GeV}$ and $m_{Z^{\prime}}\lesssim 4750~{}\mathrm{GeV}$ for
$g_{x}$ close to $1$ remain viable and can be tested by future experiments.

###### Acknowledgements.

CFCai is supported by the National Natural Science Foundation of China (NSFC)
under Grant No. 11905300. HHZhang is supported by NSFC under Grant No.
12275367. YPZeng is supported by program for scientific research start-up
funds of Guangdong Ocean University. This work is also supported by the
Fundamental Research Funds for the Central Universities, and the Sun Yat-sen
University Science Foundation.

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