Let $K=\\{k\in H|\quad\beta(k,x)=0\quad\forall x\in J\\}$. Then the set of
maps of the form $\beta(h,\\_)$, $h\in H$, are naturally identified with
$H/K$. Two such maps are homologous precisely if they differ by a map of the
form $\beta(x,\\_)$, $x\in J$. Such maps are naturally identified with $J/K$.
Consequently, we have $\Theta=H/(J+K)$.

Let $\lambda_{1}:J\to J$ be the map induced by $\psi_{3}$ and let
$\lambda_{2}:J\to J$ be the dual of the map induced by $\psi_{2}$. We will now
compute $\nu(x/a,y/b)$, for $x,y\in J$, $a,b\in\mathbb{Z}$.

$x/a$ represents the element of $\Theta$ corresponding to the map
$\frac{1}{a}\beta(x,\\_)$. Hence

$\psi_{*}(x/a)=\frac{1}{a}\beta(x,\lambda_{2}\\_)$. Applying this to $y$ and
dividing by $b$ gives:

$\nu(x/a,y/b)=\frac{1}{ab}\beta(x,\lambda_{2}(y))$

From the commutativity of the square

$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{1}}$$\scriptstyle{\beta}$$\textstyle{J^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{2}^{*}}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{*}}$$\textstyle{J^{*}}$

we know that $\beta(x,\lambda_{2}(y))=\beta(y,\lambda_{1}(x))$, for all
$x,y\in J$. Hence

$\beta(x,\lambda_{2}(y))+\beta(y,\lambda_{2}(x))=\beta(x,(\lambda_{1}+\lambda_{2})(y))$.

The fact that $\nu$ is antisymmetric may therefore be stated as follows:

Lemma 4.6.3 If $a|\beta(x,\\_)$ and $b|\beta(y,\\_)$ then
$ab|\beta(x,(\lambda_{1}+\lambda_{2})(y))$.

We now suppose that $M$ is self dual. We may choose $(\mathcal{B},\beta)$ such
that we may take $\phi$ and $\psi$ to be identity chain maps. In this case
$\beta$ will be symmetric and $\lambda_{1}=\lambda_{2}=1$. Hence from lemma
4.6.3 we have that if $a|\beta(x,\\_)$ and $b|\beta(y,\\_)$ then
$ab|2\beta(x,y)$.

We will denote the kernel of $\beta$ by $H_{3}$. The inclusion of $H_{3}$ in
$J$ splits over $\mathbb{Z}$. Let $V$ denote a complementary space. By
combining a basis of $H_{3}$, with a basis of $V$, we obtain a basis of $J$,
with respect to which, we may represent $\beta$ by a matrix $B$. Then $B$ will
have the form

$B=\left(\begin{array}[]{cc}0&0\\\ 0&C\end{array}\right)$

where $\rm{Det}(C)\neq 0$.

The condition $m$ divides $\beta(x,y)$ for all $y\in J$, is equivalent to
$m|Bx$. Hence we have:

Lemma 4.6.4 $m|Bx$ and $l|By$ imply that $ml|2x^{T}By\,\,$ for all integers
$m,\,l$ and $x,\,y\in J$.

Lemma 4.6.5 $m|Cx$ and $l|Cy$ imply that $ml|2x^{T}Cy$ for all integers
$m,\,l$ and $x,\,y\in V$.

Proof: If $m|Cx$ and $l|Cy$ then $m|Bx^{\prime}$ and $l|By^{\prime}$, where
$x^{\prime}=(0,x)$ and $y^{\prime}=(0,y)$. Consequently $ml|2x^{\prime
T}By^{\prime}$. But $2x^{\prime T}By^{\prime}=2x^{T}Cy$ so $ml|2x^{T}Cy$.

$\Box$

Note that $C$ is invertible over $\mathbb{Q}$ and symmetric. Note also, that
$\Theta={{\rm coker}(C)}$.

Lemma 4.6.6 If $\Theta$ has a non-trivial element of order $k$, then there
exists some vector $x\in V$, such that $k|Cx$ and k, x are coprime.

Proof: Some vector must have order $k$ modulo the columns of $C$. Hence
multiplying that vector by $k$, gives $Cx$ for some $x$. If $l$ were some non-
trivial divisor of $x$ and $k$, then $Cx/l$ would be in the image of $C$ and
our original vector would have order less than or equal to $k/l$.

$\Box$

Suppose $\Theta$ has a non-trivial element of order $k$. Let $x^{\prime}$
denote $x$ factored out by the highest common factor of the components of $x$.
As $k$ is coprime to $x$, we still have $k|Cx^{\prime}$. We may extend the
vector ${x^{\prime}}$ to a basis of $V$. Let $D$ denote the matrix
representing the bilinear form represented by $C$, with respect to the new
basis.

We know that $D$ is a symmetric matrix with non-zero determinant. Also we know
that if $m|Dx$ and $l|Dy$ then $ml|2x^{T}Dy$, for any $x,y\in J$. Finally, we
know that the first row and the first column of $D$ are divisible by $k$.

Lemma 4.6.7 $k=2$

Proof: Let $e_{1}$ denote $\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\ .\\\ .\\\
.\\\ .\\\ 0\end{array}\right)$

As $D$ has non-zero determinant, we may choose some vector $v$ such that
$Dv=me_{1}$ for some positive integer $m$.

We have $v^{T}De_{1}=m$. As $m|Dv$ and $k|De_{1}$, we know that $km$ divides
$2m$. $m$ is positive, so $k=2$.

$\Box$

Corollary 4.6.8 $\Theta={\mathbb{Z}_{2}}^{s}$, for some $s$.

Proof: $\Theta$ is a finitely generated Abelian group, whose non-trivial
elements all have order 2.

$\Box$

We have shown that if $M$ is self dual, then
$H_{2}(\tilde{M};ZG)=\mathbb{Z}^{r}\oplus\mathbb{Z}_{2}^{s}$ for some integers
$r$ and $s$. To show that elements of ${\rm TOP}^{5}$ are not always self
dual, we need only show a manifold which does not satisfy this homological
condition.

We refer again to [1].

Proposition 4.6.9 (See [1], lemma 1.1(i)) For each integer $k>1$, there exists
a simply connected, closed, manifold $M_{k}$, with
$H_{2}(M_{k};\mathbb{Z})=\mathbb{Z}_{k}\oplus\mathbb{Z}_{k}$.

Lemma 4.5.4 tells us that for any manifold, we may choose
$(\mathcal{B},\beta)$ so that we may take both $\lambda_{1}$ and $\lambda_{2}$
to augment to $1$. If the manifold is simply connected (as above), then we are
working over $\mathbb{Z}$, so $J$ is free and the augmentation condition is
vacuous. However, it is possible that some condition on $G$ could force
closed, orientable, simply connected manifolds with fundamental group $G$, to
be self dual.

### References

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#
The CDF $W$-mass, muon $g-2$, and dark matter in a $U(1)_{L_{\mu}-L_{\tau}}$
model with vector-like leptons

Quan Zhou, Xiao-Fang Han∗00footnotetext: *<EMAIL_ADDRESS>Department of
Physics, Yantai University, Yantai 264005, P. R. China

###### Abstract

We study the CDF $W$-mass, muon $g-2$, and dark matter observables in a local
$U(1)_{L_{\mu}-L_{\tau}}$ model in which the new particles include three
vector-like leptons ($E_{1},~{}E_{2},~{}N$), a new gauge boson $Z^{\prime}$, a
scalar $S$ (breaking $U(1)_{L_{\mu}-L_{\tau}}$), a scalar dark matter $X_{I}$
and its partner $X_{R}$. We find that the CDF $W$-mass disfavors
$m_{E_{1}}=m_{E_{2}}={m_{N}}$ or $s_{L}=s_{R}=0$ where $s_{L(R)}$ is mixing
parameter of left (right)-handed fields of vector-like leptons. A large mass
splitting between $E_{1}$ and $E_{2}$ is favored when the differences between
$s_{L}$ and $s_{R}$ becomes small. The muon $g-2$ anomaly can be
simultaneously explained for appropriate difference between $s_{L}$
$(m_{E_{1}})$ and $s_{R}$ $(m_{E_{2}})$, and some regions are excluded by the
diphoton signal data of the 125 GeV Higgs. Combined with the CDF $W$-mass,
muon $g-2$ anomaly and other relevant constraints, the correct dark matter
relic density is mainly obtained in two different scenarios: (i)
$X_{I}X_{I}\to Z^{\prime}Z^{\prime},~{}SS$ for
$m_{Z^{\prime}}(m_{S})<m_{X_{I}}$ and (ii) the co-annihilation processes for
$min(m_{E_{1}},m_{E_{2}},m_{N},m_{X_{R}})$ closed to $m_{X_{I}}$. Finally, we
use the direct searches for $2\ell+E_{T}^{miss}$ event at the LHC to constrain
the model, and show the allowed mass ranges of the vector-like leptons and
dark matter.

##  I introduction

Recently, the CDF collaboration reported their new measurement of the
$W$-boson mass cdfmw

$\displaystyle\Delta m_{W}=80.4335\pm 0.0094{\rm GeV},$ (1)

which approximately has $7\sigma$ deviation from the Standard Model (SM)
value, $80.357\pm 0.006$ GeVsmmw . Besides, combined with the FNAL E989
experiment measurement mug2-exp , the BNL experiment of the muon anomalous
magnetic moment (muon $g-2$) fermig2  has an approximate $4.2\sigma$
discrepancy from the SM prediction smg2-1 ; smg2-2 ; smg2-3 ,

$\displaystyle\Delta a_{\mu}=a_{\mu}^{exp}-a_{\mu}^{SM}=(25.1\pm 5.9)\times
10^{-10}.$ (2)

The two anomalies both call for new physics beyond SM. There have been many
works explaining the CDF $W$-mass 2204.03693 ; 2204.03796 ; 2204.03996 ;
2204.04183 ; 2204.04191 ; 2204.04202 ; 2204.04204 ; 2204.04286 ; 2204.04356 ;
2204.04514 ; 2204.04559 ; 2204.04770 ; 2204.04805 ; 2204.04834 ; 2204.05024 ;
2204.05031 ; 2204.05085 ; 2204.05260 ; 2204.05267 ; 2204.05269 ; 2204.05283 ;
2204.05284 ; 2204.05285 ; 2204.05296 ; 2204.05302 ; 2204.05303 ; 2204.04688 ;
2204.05728 ; 2204.05760 ; 2204.05942 ; 2204.05962 ; 2204.05965 ; 2204.05975 ;
2204.05992 ; 2204.06327 ; 2204.06485 ; 2204.06505 ; 2204.06541 ; 2204.07144 ;
2204.07511 ; 2204.10130 ; 2204.10315 ; 2204.08390 ; 2204.07022 ; 2204.07138 ;
2204.07411 ; 2204.08266 ; 2204.08067 ; 2204.08546 ; 2204.08568 ; 2204.09001 ;
2204.09024 ; 2204.09031 ; 2204.09585 ; 2204.09671 ; 2204.10338 ; 2204.10274 ;
2204.10156 ; 2204.11570 ; 2204.11755 ; 2204.11871 ; 2204.11945 ; 2204.12018 ;
2204.12453 .

In this paper, we study the CDF $W$-mass, the muon $g-2$, and the DM
observables in a local $U(1)_{L_{\mu}-L_{\tau}}$ model in which a singlet
vector-like lepton, a doublet vector-like lepton and a complex singlet $X$
field are introduced in addition to the $U(1)_{L_{\mu}-L_{\tau}}$ gauge boson
$Z^{\prime}$ lu-lt  and a complex singlet ${\cal S}$ breaking
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry. As the lightest component of $X$, $X_{I}$
is a candidate of dark matter (DM) and its heavy partner is $X_{R}$. The gauge
boson self-energy diagrams exchanging the vector-like leptons in the loop can
give additional contributions to the oblique parameters ($S,~{}T,~{}U$), and
explain the CDF $W$-mass. The interactions between the vector-like leptons and
muon mediated by the $X_{I}~{}(X_{R})$ can enhance the muon $g-2$ 1305.3522 ;
1312.5329 ; 1906.11297 ; 1605.06298 ; 1610.06587 ; 1711.11567 ; 1808.06639 ;
1807.11484 ; 2104.03202 ; 2101.05819 ; Bharadwaj:2021tgp . These new particles
can affect the DM relic density via the DM pair-annihilation and various co-
annihilations processes.

Our work is organized as follows. In Sec. II we introduce the model. In Sec.
III and Sec. IV we study the W-boson mass, muon g-2 anomaly, and the DM
observables imposing relevant theoretical and experimental constraints.
Finally, we give our conclusion in Sec. V.

##  II The model

In addition to the $U(1)_{L_{\mu}-L_{\tau}}$ gauge boson $Z^{\prime}$, we
introduce a complex singlet ${\cal S}$ breaking $U(1)_{L_{\mu}-L_{\tau}}$, a
complex singlet $X$, and the following vector-like lepton fields,

$E^{"}_{L,R}=\left(\begin{array}[]{c}N_{L,R}\\\
E^{"}_{L,R}\end{array}\right)\,,E^{{}^{\prime}}_{L,R}\,.$ (3)

Their quantum numbers under the gauge group $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}\times U(1)_{L_{\mu}-L_{\tau}}$ are displaced in Table 1.

Table 1: The quantum numbers of $U(1)_{L_{\mu}-L_{\tau}}$. The charge of the other field is zero. | SU(3)c | SU(2)L | U(1)Y | U(1)X
---|---|---|---|---
$E^{"}_{L,R}$ | 1 | 2 | $-1/2$ | $1-q_{x}$
$E^{\prime}_{L,R}$ | 1 | 1 | $-1$ | $1-q_{x}$
$X$ | 1 | 1 | $0$ | $q_{x}$
${\cal S}$ | 1 | 1 | $0$ | $-2q_{x}$
$L_{\mu}$ | 1 | 2 | $-1/2$ | $1$
$\mu_{R}$ | 1 | 1 | $-1$ | $1$
$L_{\tau}$ | 1 | 2 | $-1/2$ | $-1$
$\tau_{R}$ | 1 | 1 | $-1$ | $-1$

The new Lagrangian respecting the $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}\times U(1)_{L_{\mu}-L_{\tau}}$ symmetry is written as

$\displaystyle{\cal L}$ $\displaystyle=-{1\over
4}Z^{\prime}_{\mu\nu}Z^{\prime\mu\nu}+g_{Z^{\prime}}Z^{\prime\mu}(\bar{\mu}\gamma_{\mu}\mu+\bar{\nu}_{\mu_{L}}\gamma_{\mu}\nu_{\mu_{L}}-\bar{\tau}\gamma_{\mu}\tau-\bar{\nu}_{\tau_{L}}\gamma_{\mu}\nu_{\tau_{L}})$
$\displaystyle+\bar{E"}(i\centernot D)E"+\bar{E^{\prime}}(i\centernot
D)E^{\prime}+(D_{\mu}X^{\dagger})(D^{\mu}X)+(D_{\mu}{\cal
S}^{\dagger})(D^{\mu}{\cal S})$ $\displaystyle-V+\mathcal{L}_{\rm Y}.$ (4)

Where $D_{\mu}$ is the covariant derivative and $g_{Z^{\prime}}$ is the gauge
coupling constant of the $U(1)_{L_{\mu}-L_{\tau}}$ group. The field strength
tensor
$Z^{\prime}_{\mu\nu}=\partial_{\mu}Z^{\prime}_{\nu}-\partial_{\nu}Z^{\prime}_{\mu}$,
and the kinetic mixing term of gauge bosons of $U(1)_{L_{\mu}-L_{\tau}}$ and
$U(1)_{Y}$ is ignored. $V$ and $\mathcal{L}_{\rm Y}$ indicate the scalar
potential and Yukawa interactions.

The scalar potential $V$ is written as

$\displaystyle V$ $\displaystyle=$
$\displaystyle-\mu_{h}^{2}(H^{\dagger}H)-\mu_{S}^{2}({\cal S}^{\dagger}{\cal
S})+m_{X}^{2}(X^{\dagger}X)+\left[\mu X^{2}{\cal S}+\rm h.c.\right]$ (5)
$\displaystyle+\lambda_{H}(H^{\dagger}H)^{2}+\lambda_{S}({\cal
S}^{\dagger}{\cal S})^{2}+\lambda_{X}(X^{\dagger}X)^{2}+\lambda_{SX}({\cal
S}^{\dagger}{\cal S})(X^{\dagger}X)$
$\displaystyle+\lambda_{HS}(H^{\dagger}H)({\cal S}^{\dagger}{\cal
S})+\lambda_{HX}(H^{\dagger}H)(X^{\dagger}X),$

where the SM Higgs doublet $H$, the complex singlet fields ${\cal S}$ and $X$
are

$H=\left(\begin{array}[]{c}G^{+}\\\
\frac{1}{\sqrt{2}}\,(h_{1}+v_{h}+iG)\end{array}\right)\,,{\cal
S}={1\over\sqrt{2}}\left(h_{2}+v_{S}+i\omega\right)\,,X={1\over\sqrt{2}}\left(X_{R}+iX_{I}\right)\,.$
(6)

Here $H$ and ${\cal S}$ respectively acquire vacuum expectation values (VEVs),
$v_{h}=246$ GeV and $v_{S}$, and the VEV of $X$ field is zero. The parameters
$\mu^{2}_{h}$ and $\mu^{2}_{S}$ are determined by the minimization conditions
for Higgs potential,

$\begin{split}&\quad\mu_{h}^{2}=\lambda_{H}v_{h}^{2}+{1\over
2}\lambda_{HS}v_{S}^{2},\\\ &\quad\mu_{S}^{2}=\lambda_{S}v_{S}^{2}+{1\over
2}\lambda_{HS}v_{h}^{2}.\\\ \end{split}$ (7)

The complex scalar $X$ is split into two real scalar fields $X_{R}$ and
$X_{I}$ by the $\mu$ term after the ${\cal S}$ field acquires VEV $v_{S}$.
Their masses are

$\displaystyle m_{X_{R}}^{2}=m_{X}^{2}+{1\over 2}\lambda_{HX}v_{H}^{2}+{1\over
2}\lambda_{SX}v_{S}^{2}+\sqrt{2}\mu v_{S}$ $\displaystyle
m_{X_{I}}^{2}=m_{X}^{2}+{1\over 2}\lambda_{HX}v_{H}^{2}+{1\over
2}\lambda_{SX}v_{S}^{2}-\sqrt{2}\mu v_{S}.$ (8)

Because the $X$ field has no VEV, there is a remnant discrete $Z_{2}$ symmetry
which makes the lightest component $X_{I}$ to be stable and as a candidate of
DM.

The $\lambda_{HS}$ term leads to a mixing of $h_{1}$ and $h_{2}$, and their
mass eigenstates $h$ and $S$ are obtained from following relation,

$\displaystyle\left(\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\
S\end{array}\right)$ (15)

with $\alpha$ being the mixing angle. From the $\lambda_{HS}$ term and
$\lambda_{HX}$ term, we can obtain the 125 GeV Higgs $h$ coupling to a pair of
DM. In order to escape the strong bounds of the DM indirect detection and
direct detection experiments, we simply assume the $hX_{I}X_{I}$ coupling to
be absent, namely choosing $\lambda_{HS}=0$ and $\lambda_{HX}=0$. Thus we
obtain

$\theta=0,~{}~{}~{}\lambda_{H}=\frac{m_{h}^{2}}{2v_{h}^{2}},~{}~{}~{}\lambda_{S}=\frac{m_{S}^{2}}{2v_{S}^{2}}.$
(16)

The gauge boson $Z^{\prime}$ acquires a mass after ${\cal S}$ breaks the
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry,

$m_{Z^{\prime}}=2g_{Z^{\prime}}\mid q_{x}\mid v_{S}.$ (17)

The Yukawa interactions with the $U(1)_{L_{\mu}-L_{\tau}}$ symmetry are given
as

$\displaystyle-\mathcal{L}_{\rm Y}=$ $\displaystyle
m_{1}\overline{E^{\prime}_{L}}E^{\prime}_{R}+m_{2}\overline{E^{"}_{L}}E^{"}_{R}+\kappa_{1}\overline{\mu_{R}}XE^{\prime}_{L}+\kappa_{2}\overline{L_{\mu}}XE^{"}_{R}$
$\displaystyle+\sqrt{2}y_{1}\overline{E^{"}_{L}}HE^{\prime}_{R}+\sqrt{2}y_{2}\overline{E^{"}_{R}}HE^{\prime}_{L}+\frac{\sqrt{2}m_{\mu}}{v}\overline{L_{\mu}}H\mu_{R}+{\rm
h.c.},$ (18)

where $L_{\mu}=\left(\nu_{\mu L},\mu_{L}\right)$.

Since the $X$ field has no VeV, there is no mixing between the vector-like
leptons and the muon lepton. However, there is a mixing between the vector-
like leptons $E"$ and $E^{\prime}$ after the $H$ acquires the VeV, $v_{h}=$
246 GeV, and their mass matrix is given as

$\displaystyle M_{E}=\begin{pmatrix}m_{1}&y_{2}v_{h}\\\
y_{1}v_{h}&m_{2}\end{pmatrix}.$ (19)

We take two unitary matrices to diagnolize the mass matrix,

$\displaystyle U_{L}=\begin{pmatrix}c_{L}&s_{L}\\\
-s_{L}&c_{L}\end{pmatrix},\quad U_{R}=\begin{pmatrix}c_{R}&s_{R}\\\
-s_{R}&c_{R}\end{pmatrix},\quad
U_{L}^{\dagger}M_{E}U_{R}=\mathrm{diag}\left(m_{E_{1}},m_{E_{2}}\right),$ (20)

where $c_{L,R}^{2}+s_{L,R}^{2}=1$. The $E_{1}$ and $E_{2}$ are the mass
eigenstates of charged vector-like leptons, and the mass of neutral vector-
like lepton $N$ is

$m_{N}=m_{2}.$ (21)

From the Eq. (18), we can obtain the interactions between the charged vector-
like leptons and muon mediated by $X_{R}$ and $X_{I}$,

$-\mathcal{L}_{\rm
X}\supset\frac{1}{\sqrt{2}}(X_{R}+iX_{I})\left[\bar{\mu}_{R}(\kappa_{1}c_{L}E_{1L}-\kappa_{1}s_{L}E_{2L})+\bar{\mu}_{L}(\kappa_{2}s_{R}E_{1R}+\kappa_{2}c_{R}E_{2R})\right]+h.c.~{},$
(22)

and the 125 GeV Higgs interactions to the charged vector-like leptons $E_{1}$
and $E_{2}$,

$\displaystyle-\mathcal{L}_{\rm h}$ $\displaystyle\supset$
$\displaystyle\frac{m_{E_{1}}(c_{L}^{2}s_{R}^{2}+c_{R}^{2}s_{L}^{2})-2m_{E_{2}}s_{L}c_{L}s_{R}c_{R}}{v_{h}}~{}h\bar{E}_{1}E_{1},$
(23)
$\displaystyle+\frac{m_{E_{2}}(s_{L}^{2}c_{R}^{2}+c_{L}^{2}s_{R}^{2})-2m_{E_{1}}s_{L}c_{L}s_{R}c_{R}}{v_{h}}~{}h\bar{E}_{2}E_{2}.$

##  III The $S,~{}T,~{}U$ parameters, $W$-mass, and muon $g-2$

In addition to $m_{h}=$ 125 GeV, $v_{h}=$ 246 GeV, $\lambda_{HS}=0$,
$\lambda_{HX}=0$, there are many new parameters in the model. We take
$g_{Z^{\prime}}$, $q_{x}$, $m_{Z^{\prime}}$, $\lambda_{X}$, $\lambda_{SX}$,
$m_{S}$, $m_{X_{R}}$, $m_{X_{I}}$, $m_{E_{1}}$, $m_{E_{2}}$, $s_{L}$, $s_{R}$,
$\kappa_{1}$, and $\kappa_{2}$ as the input parameters, which can be used to
determine other parameters.

In order to maintain the perturbativity, we conservatively take

$\displaystyle\mid\lambda_{SX}\mid\leq 4\pi,~{}~{}\mid\lambda_{X}\mid\leq
4\pi,$ $\displaystyle-1\leq\kappa_{1}\leq 1,~{}~{}-1\leq\kappa_{2}\leq 1.$
(24)

The mixing parameters $s_{L}$ and $s_{R}$ are taken as

$-\frac{1}{\sqrt{2}}\leq
s_{L}\leq\frac{1}{\sqrt{2}},~{}~{}-\frac{1}{\sqrt{2}}\leq
s_{R}\leq\frac{1}{\sqrt{2}}.$ (25)

We scan over the input mass parameters in the following ranges:

$\displaystyle 60{\rm GeV}\leq m_{X_{I}}\leq 500{\rm
GeV},~{}~{}~{}m_{X_{I}}\leq m_{X_{R}}\leq 1{\rm TeV},$ $\displaystyle
m_{X_{I}}\leq m_{E_{1}}\leq 1{\rm TeV},~{}~{}~{}m_{X_{I}}\leq m_{E_{2}}\leq
1{\rm TeV},$ $\displaystyle 100{\rm GeV}\leq m_{Z^{\prime}}\leq 1{\rm
TeV},~{}~{}~{}100{\rm GeV}\leq m_{S}\leq 1{\rm TeV}.$ (26)

The mass of neutral vector-like lepton $N$ is determined by $m_{E_{1}}$,
$m_{E_{2}}$, $s_{L}$ and $s_{R}$, we require $m_{N}>m_{X_{I}}$. We choose 0
$<g_{Z^{\prime}}/m_{Z^{\prime}}\leq$ (550 GeV)-1 to satisfy the bound of the
neutrino trident process trident . We take -2 $<q_{x}\leq 2$, and require
$\mid g_{Z^{\prime}}(1-q_{x})\mid\leq 1$ and $g_{Z^{\prime}}\leq 1$ to respect
the perturbativity of the $Z^{\prime}$ couplings.

The tree-level stability of the potential in Eq. (5) impose the following
bounds,

$\displaystyle\lambda_{H}\geq 0\,,\quad\lambda_{S}\geq
0\,,\quad\lambda_{X}\geq 0\,,\quad$
$\displaystyle\lambda_{HS}\geq-2\sqrt{\lambda_{H}\,\lambda_{S}}\,,\quad\lambda_{HX}\geq-2\sqrt{\lambda_{H}\,\lambda_{X}}\,,\quad\lambda_{SX}\geq-2\sqrt{\lambda_{S}\,\lambda_{X}}\,,\quad$
$\displaystyle\sqrt{\lambda_{HS}+2\sqrt{\lambda_{H}\,\lambda_{S}}}~{}\sqrt{\lambda_{HX}+2\sqrt{\lambda_{H}\,\lambda_{X}}}~{}\sqrt{\lambda_{SX}+2\sqrt{\lambda_{S}\,\lambda_{X}}}$
$\displaystyle+2\,\sqrt{\lambda_{H}\lambda_{S}\lambda_{X}}+\lambda_{HS}\sqrt{\lambda_{X}}+\lambda_{HX}\sqrt{\lambda_{S}}+\lambda_{SX}\sqrt{\lambda_{H}}\geq
0\,.$ (27)

The $H\to\gamma\gamma$ decay can be corrected by the loops of the charged
vector-like leptons $E_{1}$ and $E_{2}$. We impose the bound of the diphoton
signal strength of the 125 GeV Higgs smmw ,

$\mu_{\gamma\gamma}=1.11^{+0.1}_{-0.09}.$ (28)

###  III.1 The $S,~{}T,~{}U$ parameters and $W$-mass

The model contains the interactions of gauge bosons and vector-like leptons,

$\displaystyle-\mathcal{L}_{\rm VG}=$
$\displaystyle-e\gamma\bar{E}_{1,2}\gamma^{\mu}E_{1,2}+Z\bar{E}_{i}\gamma^{\mu}(L_{ij}P_{L}+R_{ij}P_{R})E_{j}+\frac{g}{2c_{W}}Z\bar{N}\gamma^{\mu}(P_{L}+P_{R})N$
(29)
$\displaystyle+\frac{1}{\sqrt{2}}W^{+}\bar{N}\gamma^{\mu}[(c_{L}P_{L}+c_{R}P_{R})E_{2}+(s_{L}P_{L}+s_{R}P_{R})E_{1}]+h.c.~{},$

where $L_{ij}$ and $R_{ij}$ are

$\displaystyle
L(R)_{11}=A_{1}c_{L(R)}^{2}+A_{2}s_{L(R)}^{2},~{}~{}L(R)_{22}=A_{1}s_{L(R)}^{2}+A_{2}c_{L(R)}^{2},$
$\displaystyle L(R)_{12}=L(R)_{21}=(A_{2}-A_{1})s_{L(R)}c_{L(R)}$ (30)

with

$A_{1}=\frac{g}{c_{W}}s_{W}^{2},~{}~{}A_{2}=\frac{g}{c_{W}}(-\frac{1}{2}+s_{W}^{2}).$
(31)

Where $s_{W}\equiv\sin\theta_{W}$ and $c_{W}=\sqrt{1-s_{W}^{2}}$, and
$\theta_{W}$ is the weak mixing angle.

The gauge boson self-energy diagrams exchanging the vector-like leptons in the
loop can give additional contributions to the oblique parameters
($S,~{}T,~{}U$) stu ; w-stu , which are calculated as stu ; w-stu ; 1305.4712

$\displaystyle\alpha(M_{Z}^{2})\,S$ $\displaystyle=$
$\displaystyle\frac{4s_{W}^{2}c_{W}^{2}}{M_{Z}^{2}}\left[\Pi^{\text{NP}}_{ZZ}(M_{Z}^{2})-\Pi^{\text{NP}}_{ZZ}(0)-\Pi^{\text{NP}}_{\gamma\gamma}(M_{Z}^{2})-\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\,\Pi^{\text{NP}}_{\gamma
Z}(M_{Z}^{2})\right],$ (32) $\displaystyle\alpha(M_{Z}^{2})\,T$
$\displaystyle=$
$\displaystyle\frac{\Pi^{\text{NP}}_{WW}(0)}{M_{W}^{2}}-\frac{\Pi^{\text{NP}}_{ZZ}(0)}{M_{Z}^{2}},$
(33) $\displaystyle\alpha(M_{Z}^{2})\,U$ $\displaystyle=$ $\displaystyle
4s_{W}^{2}\left[\frac{\Pi^{\text{NP}}_{WW}(M_{W}^{2})-\Pi^{\text{NP}}_{WW}(0)}{M_{W}^{2}}-c_{W}^{2}\left(\frac{\Pi^{\text{NP}}_{ZZ}(M_{Z}^{2})-\Pi^{\text{NP}}_{ZZ}(0)}{M_{Z}^{2}}\right)\right.$
(34) $\displaystyle\left.-2s_{W}c_{W}\,\frac{\Pi^{\text{NP}}_{\gamma
Z}(M_{Z}^{2})}{M_{Z}^{2}}-s_{W}^{2}\,\frac{\Pi^{\text{NP}}_{\gamma\gamma}(M_{Z}^{2})}{M_{Z}^{2}}\right],$

where the $\Pi^{\text{NP}}$ function is given in Appendix A.

Analyzing precision electroweak data including the new CDF $W$-mass, Ref.
2204.03796  gave the values of $S,~{}T$ and $U$,

$S=0.06\pm 0.10,~{}~{}T=0.11\pm 0.12,~{}~{}U=0.14\pm 0.09$ (35)

with correlation coefficients

$\rho_{ST}=0.9,~{}~{}\rho_{SU}=-0.59,~{}~{}\rho_{TU}=-0.85.$ (36)

The $W$-boson mass is given as w-stu ,

$\Delta m_{W}^{2}=\frac{\alpha
c_{W}^{2}}{c_{W}^{2}-s_{W}^{2}}m_{Z}^{2}(-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U).$
(37)

We perform a global fit to the values of $S,~{}T,~{}U$, and require
$\chi^{2}<\chi^{2}_{\rm min}+6.18$ with $\chi^{2}_{\rm min}$ denoting the
minimum of $\chi^{2}$. These surviving samples mean to be within the $2\sigma$
range in any two-dimension plane of the model parameters fitting to the
$S,~{}T$, and $U$ parameters.

Figure 1: The surviving samples explaining the CDF $W$-mass within $2\sigma$
range while satisfying the oblique parameters and theoretical constraints.

In Fig. 1, we show the samples explaining the CDF $W$-boson mass within
$2\sigma$ range while satisfying the constraints of the oblique parameters and
theoretical constraints. Fig. 1 shows that the explanation of the CDF $W$-mass
requires appropriate mass splittings among $E_{1},~{}E_{2}$ and $N$, which do
not simultaneously equal to zero. For example, when $m_{E_{2}}=m_{N}$, the
mass splitting between $m_{E_{1}}$ and $m_{E_{2}}(m_{N})$ is required to be
larger than 100 GeV. From the right panel of Fig. 1, we see that the
measurement of CDF $W$-mass disfavors $s_{L}$ and $s_{R}$ to approach to zero
simultaneously, and favors $E_{1}$ and $E_{2}$ to have a large mass splitting
when the difference between $s_{L}$ and $s_{R}$ becomes small.

Now we understand the reason. The function $\Pi^{\text{NP}}_{WW}(0)$ is zero
for $m_{E_{2}}=m_{N}$ and $m_{E_{1}}=m_{N}$, and the $\Pi^{\text{NP}}_{ZZ}(0)$
is zero for $m_{E_{2}}=m_{E_{1}}$. Therefore, from Eq. (33) we see that the
corrections of the model to $T$ parameter are absent for
$m_{E_{2}}=m_{E_{1}}=m_{N}$, which is disfavored by the CDF measurement of $W$
mass. Because there is no mixing between $E_{2}$ and $E_{1}$ for
$s_{L}=s_{R}=0$, both the $ZE_{2}E_{1}$ and $WE_{1}N$ couplings disappear and
$m_{E_{2}}$ equals to $m_{N}$. Therefore, for $s_{L}=s_{R}=0$, both
$\Pi^{\text{NP}}_{WW}(0)$ and $\Pi^{\text{NP}}_{ZZ}(0)$ are zero, and the
corrections to $T$ parameter are also absent. The case of $s_{L}=s_{R}=0$ is
disfavored by the CDF measurement of $W$\- mass.

###  III.2 The muon $g-2$

The model can give additional corrections to the muon $g-2$ via the one-loop
diagrams containing the interactions between muon and $E_{1}~{}(E_{2})$
mediated by $X_{R}$ and $X_{I}$, and the main corrections are calculated as
1305.3522 ; 1906.11297 ; 0902.3360

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}m_{\mu}\left(\kappa_{1}c_{L}\kappa_{2}s_{R}H(m_{E_{1}},m_{X_{R}})-\kappa_{1}s_{L}\kappa_{2}c_{R}H(m_{E_{2}},m_{X_{R}})\right.$
(38)
$\displaystyle\left.+\kappa_{1}c_{L}\kappa_{2}s_{R}H(m_{E_{1}},m_{X_{I}})-\kappa_{1}s_{L}\kappa_{2}c_{R}H(m_{E_{2}},m_{X_{I}})\right),$

where the function

$H(m_{f},m_{\phi})=\frac{m_{f}}{m_{\phi}^{2}}\frac{(r^{2}-4r+2\log{r}+3)}{(r-1)^{3}}$
(39)

with $r=\frac{m_{f}^{2}}{m_{\phi}^{2}}$. Eq. (38) shows that the correction of
the model to the muon $g-2$ is absent for $m_{E_{1}}=m_{E_{2}}$ and
$s_{L}=s_{R}$.

Figure 2: All the samples explaining the muon $g-2$ anomaly while satisfying
the constraints of ”pre-muon $g-2$”. Figure 3: All the samples explaining the
muon $g-2$ anomaly while satisfying the constraints of ”pre-muon $g-2$”. The
squares and bullets are allowed and excluded by the diphoton signal data of
125 GeV Higgs, respectively.

We respectively take $s_{L}=s_{R}$ and $m_{E_{1}}=m_{E_{2}}$, and show the
samples explaining the muon $g-2$ anomaly within $2\sigma$ range while
satisfying the constraints ”pre-muon $g-2$” (denoting the theory, the oblique
parameters, and the CDF $W$-mass) in Fig. 2. From Fig. 2, we see that the
explanation of the muon $g-2$ anomaly favors $\mid s_{L}\mid$ to decrease with
increasing of $\mid m_{E_{1}}-m_{E_{2}}\mid$ for $s_{L}=s_{R}$, and
$m_{E_{1}}$ to increase with decreasing of $\mid s_{L}-s_{R}\mid$. This
characteristic can be well understood from Eq. (38).

Figure 4: All samples satisfy the constraints of ”pre-muon $g-2$”, the
diphoton signal data of the 125 GeV Higgs, and muon $g-2$. The squares and
bullets are allowed and excluded by the DM relic density. Here
$min(m_{i},m_{j},\cdots)$ denotes the minimal value of $m_{i},m_{j},\cdots$.

After imposing the constraints of the diphoton signal data of the 125 GeV
Higgs and ”pre-muon $g-2$”, we scan over the parameter space, and project the
samples explaining the muon $g-2$ anomaly in Fig. 3. We find that the diphoton
signal data of the 125 GeV Higgs exclude some samples explaining the muon
$g-2$ anomaly, and favors $s_{L}$ and $s_{R}$ to have same sign, especially
for large $\mid s_{L}\mid$ and $\mid s_{R}\mid$. When $s_{L}$ and $s_{R}$ have
same sign, the terms of $h\bar{E}_{1}E_{1}$ ($h\bar{E}_{2}E_{2}$) coupling in
Eq. (23) are canceled to some extent, which suppresses the corrections of
$E_{1}$ and $E_{2}$ to the $h\to\gamma\gamma$ decay.

##  IV The DM observables

In the model, in addition to $X_{I}X_{I}\to\mu^{+}\mu^{-}$, and the DM pair-
annihilation processes $X_{I}X_{I}\to Z^{\prime}Z^{\prime},~{}SS$ will be open
for $m_{Z^{\prime}}~{}(m_{S})<m_{X_{I}}$. When the masses of
$E_{1},~{}E_{2},~{}N$ and $X_{R}$ are closed to $m_{X_{I}}$, their various co-
annihilation processes will play important roles in the DM relic density. We
use FeynRules feyrule  to generate model file, and employ micrOMEGAs micomega
to calculate the relic density. The Planck collaboration reported the relic
density of cold DM in the universe, $\Omega_{c}h^{2}=0.1198\pm 0.0015$ planck
.

After imposing the constraints of ”pre-muon $g-2$”, the diphoton signal data
of the 125 GeV Higgs, and the muon $g-2$ anomaly, we project the samples
achieving the DM relic density within $2\sigma$ range in Fig. 4. Due to the
constraints of muon $g-2$ on the interactions between the vector-like leptons
and muon mediated by $X_{I}$, it is not easy to obtain the correct DM relic
density only via the $X_{I}X_{I}\to\mu^{+}\mu^{-}$ annihilation process, and
other processes are needed to accelerate the DM annihilation. As shown in Fig.
4, for $min(m_{Z^{\prime}},m_{S})<m_{X_{I}}$, the $X_{I}X_{I}\to
Z^{\prime}Z^{\prime}$ or $SS$ will be open and play a main role in achieving
the correct relic density. Then the masses of $X_{R}$, $E_{1}$, $E_{2}$ and
$N$ are allowed to have sizable deviation from $m_{X_{I}}$. When
$min(m_{Z^{\prime}},m_{S})$ is larger than $m_{X_{I}}$ and the $X_{I}X_{I}\to
Z^{\prime}Z^{\prime},~{}SS$ processes are kinematically forbidden,
$min(m_{E_{1}},~{}m_{E_{2}},~{}m_{N},~{}m_{X_{R}})$ is required to be closed
to $m_{X_{I}}$ so that the correct DM relic density is obtained via their co-
annihilation processes.

The $X_{I}$ has no interactions to the SM quark, and its couplings to the muon
lepton and vector-like leptons are constrained by the muon $g-2$ anomaly.
Therefore, the model can easily satisfy the bound from the direct detection of
DM. At the LHC, the vector-like leptons are mainly produced via electroweak
processes,

$\displaystyle p~{}p\to\gamma/Z^{*}\to
E_{1}\bar{E}_{1,2},~{}E_{2}\bar{E}_{1,2},~{}N\bar{N},$ $\displaystyle p~{}p\to
W^{*}\to E_{1,2}\bar{N},~{}\bar{E}_{1,2}N,$ (40)

then the decay modes include

$E_{1,2}\to\mu X_{I},~{}~{}N\to\nu_{\mu}X_{I}.$ (41)

If kinematically allowed, the following decay modes will be open,

$E_{1,2}\to\mu X_{R},~{}WN,~{}~{}E_{1,2}\to
ZE_{2,1},~{}~{}N\to\nu_{\mu}X_{R}.$ (42)

The $2\mu+E_{T}^{miss}$ event searches at the LHC can impose strong
constraints on the vector-like leptons and DM. The production processes of
$2\mu+E_{T}^{miss}$ in our model are very similar to the electroweak
production of charginos and sleptons decaying into final states with
$2\ell+E_{T}^{miss}$ analyzed by ATLAS with 139 fb-1 integrated luminosity
data 1908.08215 . Therefore, we will use this analysis to constrain our model,
which is implemented in the MadAnalysis5 ma5-1 ; ma5-2 ; ma5-3 . We perform
simulations for the samples using MG5_aMC-3.3.2 Alwall:2014hca  with PYTHIA8
Torrielli:2010aw  and Delphes-3.2.0 deFavereau:2013fsa .

If the DM relic density is achieved via the co-annihilation processes of
vector-like lepton, the mass of vector-like lepton is required to be closed to
$m_{X_{I}}$. As a result, the $\mu$ from the vector-like lepton decay is too
soft to be distinguished at detector, and the scenario can easily satisfy the
constraints of the direct searches at LHC. Here, we employ the ATLAS analysis
of $2\ell+E_{T}^{miss}$ in Ref. 1908.08215  to constrain another scenario in
which $1<m_{X_{R}}/m_{X_{I}}<1.15$ and $m_{Z}^{\prime}~{}(m_{S})>m_{X_{R}}$ is
taken, and the co-annihilation processes of $X_{R}$ can play a main role in
achieving the correct relic density. Thus, the masses of the vector-like
leptons are allowed to be much larger than $m_{X_{I}}$.

Figure 5: For the scenario of $1<m_{X_{R}}/m_{X_{I}}<1.15$ and
$m_{Z^{\prime}}~{}(m_{S})>m_{X_{R}}$, all samples satisfy the constraints of
”pre-muon $g-2$”, the diphoton signal data of the 125 GeV Higgs, the muon
$g-2$ anomaly, and the DM relic density. The squares and bullets are allowed
and excluded by the direct searches for $2\ell+E^{miss}_{T}$ at the LHC.

We impose the constraints of ”pre-muon $g-2$”, the diphoton signal data of the
125 GeV Higgs, the muon $g-2$ anomaly, the DM relic density, and the direct
searches for $2\ell+E^{miss}_{T}$ at the LHC, and project the surviving
samples in Fig. 5. From Fig. 5 we see that the mass of the lightest charged
vector-like lepton is allowed to be as low as 120 GeV if
$min(m_{E_{1}},~{}m_{E_{2}})-m_{X_{I}}<$ 60 GeV since the muon becomes soft in
the region. As $min(m_{E_{1}},~{}m_{E_{2}})-m_{X_{I}}$ increases, the energy
of muon becomes large, and the vector-like lepton needs to be large enough to
escape the constraints of direct searches for $2\ell+E^{miss}_{T}$ at the LHC.
For example, $min(m_{E_{1}},~{}m_{E_{2}})$ is favored to be larger than 500
GeV for $min(m_{E_{1}},~{}m_{E_{2}})-m_{X_{I}}>$ 300 GeV. The DM mass is
allowed to be as low as 100 GeV if $min(m_{E_{1}},~{}m_{E_{2}})-m_{X_{I}}<$ 60
GeV or $min(m_{E_{1}},~{}m_{E_{2}})-m_{X_{I}}>$ 400 GeV.

##  V Conclusion

In this paper we discussed the CDF $W$-mass, the muon $g-2$, and the DM
observables in a local $U(1)_{L_{\mu}-L_{\tau}}$ model, and obtained the
following observations: (i) The CDF $W$-mass disfavors
$m_{E_{1}}=m_{E_{2}}={m_{N}}$ or $s_{L}=s_{R}=0$, and favors a large mass
splitting between $E_{1}$ and $E_{2}$ when the differences between $s_{L}$ and
$s_{R}$ becomes small. (ii) The muon $g-2$ anomaly can be simultaneously
explained for appropriate difference between $s_{L}$ $(m_{E_{1}})$ and $s_{R}$
$(m_{E_{2}})$, and some regions are excluded by the diphoton signal data of
the 125 GeV Higgs. (iii) Combined with the CDF $W$-mass, muon $g-2$ anomaly
and other relevant constraints, the correct DM relic density is mainly
achieved in two different scenarios: (1) $X_{I}X_{I}\to
Z^{\prime}Z^{\prime},~{}SS$ for $m_{Z^{\prime}}(m_{S})<m_{X_{I}}$ and (2) the
co-annihilation processes for $min(m_{E_{1}},m_{E_{2}},m_{N},m_{X_{R}})$
closed to $m_{X_{I}}$. (iv) The direct searches for $2\ell+E_{T}^{miss}$ event
at the LHC impose strong bounds on the masses of the vector-like leptons and
DM as well as their mass splitting.

## Acknowledgment

We thank Songtao Liu, Shuyuan Guo, Shiyu Wang, Liangliang Shang, and Yang
Zhang for the helpful discussions. This work was supported by the National
Natural Science Foundation of China under grant 11975013.

##  Appendix A The $\Pi$ function

The $\Pi_{XY}(p^{2},m_{1}^{2},m_{2}^{2})$ and
$\Pi_{XY}(0,m_{1}^{2},m_{2}^{2})$ are given as

$\displaystyle\Pi_{XY}(p^{2},m_{1}^{2},m_{2}^{2})$ $\displaystyle=$
$\displaystyle-{N_{c}\over 16\pi^{2}}\biggl{\\{}{2\over
3}\biggl{(}g_{LX}^{f_{1}f_{2}}g_{LY}^{f_{1}f_{2}}+g_{RX}^{f_{1}f_{2}}g_{RY}^{f_{1}f_{2}}\biggr{)}\biggl{[}m_{1}^{2}+m_{2}^{2}-{p^{2}\over
3}-\biggl{(}A_{0}(m_{1}^{2})+A_{0}(m_{2}^{2})\biggr{)}$ (43)
$\displaystyle+{m_{1}^{2}-m_{2}^{2}\over
2p^{2}}\biggl{(}A_{0}(m_{1}^{2})-A_{0}(m_{2}^{2})\biggr{)}$
$\displaystyle+{2p^{4}-p^{2}(m_{1}^{2}+m_{2}^{2})-(m_{1}^{2}-m_{2}^{2})^{2}\over
2p^{2}}B_{0}(p^{2},m_{1}^{2},m_{2}^{2})\biggr{]}$
$\displaystyle+2m_{1}m_{2}\biggl{(}g_{LX}^{f_{1}f_{2}}g_{RY}^{f_{1}f_{2}}+g_{RX}^{f_{1}f_{2}}g_{LY}^{f_{1}f_{2}}\biggr{)}B_{0}(p^{2},m_{1}^{2},m_{2}^{2})\biggr{\\}},$
$\displaystyle\Pi_{XY}(0,m_{1}^{2},m_{2}^{2})$ $\displaystyle=$
$\displaystyle-{N_{c}\over 16\pi^{2}}\biggl{\\{}{2\over
3}\biggl{(}g_{LX}^{f_{1}f_{2}}g_{LY}^{f_{1}f_{2}}+g_{RX}^{f_{1}f_{2}}g_{RY}^{f_{1}f_{2}}\biggr{)}\biggl{[}m_{1}^{2}+m_{2}^{2}-\biggl{(}A_{0}(m_{1}^{2})+A_{0}(m_{2}^{2})\biggr{)}$
(44) $\displaystyle-{m_{1}^{2}+m_{2}^{2}\over
2}B_{0}(0,m_{1}^{2},m_{2}^{2})-{(m_{1}^{2}-m_{2}^{2})^{2}\over
2}B_{0}^{{}^{\prime}}(0,m_{1}^{2},m_{2}^{2})\biggr{]}$
$\displaystyle+2m_{1}m_{2}\biggl{(}g_{LX}^{f_{1}f_{2}}g_{RY}^{f_{1}f_{2}}+g_{RX}^{f_{1}f_{2}}g_{LY}^{f_{1}f_{2}}\biggr{)}B_{0}(0,m_{1}^{2},m_{2}^{2})\biggr{\\}}.$

Here the coupling constants $g_{LX}^{f_{1}f_{2}}$ and $g_{RX}^{f_{1}f_{2}}$
are from

${\overline{f}}_{1}\biggl{(}g_{LX}^{f_{1}f_{2}}P_{L}+g_{RX}^{f_{1}f_{2}}P_{R}\biggr{)}\gamma_{\mu}f_{2}X^{\mu}\,,$
(45)

and the $A_{0}$, $B_{0}$, and $B_{0}^{{}^{\prime}}$ functions are

$\displaystyle A_{0}(m^{2})$ $\displaystyle=$
$\displaystyle\biggl{(}{4\pi\mu^{2}\over
m^{2}}\biggr{)}^{\epsilon}\Gamma(1+\epsilon)\biggl{(}{1\over\epsilon}+1\biggr{)}m^{2},$
$\displaystyle B_{0}(p^{2},m_{1}^{2},m_{2}^{2})$ $\displaystyle=$
$\displaystyle\biggl{(}{4\pi\mu^{2}\over
m_{2}^{2}}\biggr{)}^{\epsilon}\Gamma(1+\epsilon)\left[{1\over\epsilon}-f_{1}(p^{2},m_{1}^{2},m_{2}^{2})\right],$
$\displaystyle B_{0}^{{}^{\prime}}(p^{2},m_{1}^{2},m_{2}^{2})$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
p^{2}}B_{0}(p^{2},m_{1}^{2},m_{2}^{2}),$ (46)

and

$f_{1}(p^{2},m_{1}^{2},m_{2}^{2})=\int_{0}^{1}dx\log\biggl{(}x+{m_{1}^{2}(1-x)-p^{2}x(1-x)\over
m_{2}^{2}}\biggr{)}\,.$ (47)

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