# Flipped $U(1)$ extended Standard Model and Majorana dark matter

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Basic Science, Phenikaa University, To Huu, Yen Nghia, Ha
Dong, Hanoi 100000, Vietnam

###### Abstract

We propose a general flavor-independent extension of the Standard Model (SM)
with the minimal particle content, based on the symmetry $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y^{\prime}}\times U(1)_{X}\times Z_{2}$. In this
scenario, the charge operator is identified in terms of the charges of two
$U(1)$ gauge symmetries. The light neutrino masses are generated via Type-I
seesaw mechanism only with two heavy right-handed neutrinos acquiring their
Majorana masses through the $U(1)_{Y^{\prime}}\times U(1)_{X}$ symmetry
breaking. We study various experimental constraints on the parameters of the
model and investigate the phenomenology of the right-handed neutrino dark
matter (DM) candidate assigned a $Z_{2}$-odd parity. We find that the most
important constraints are the observed DM relic abundance, the current LHC
limits, and the ambiguity of the SM neutral gauge boson mass.

##  I Introduction

Although the Standard Model (SM) is successful in explaining most of the
observed elementary particle phenomena, there are at least two evidences which
hint new physics beyond the SM, such as nonzero but tiny neutrino masses and
the dark matter (DM). A simple extension of the SM is to introduce an exotic
$U(1)$ gauge symmetry which corresponds to a new short-range neutral gauge
boson. Various models have been proposed from the top-down approach
Langacker1981 ; Rizzo1989 ; Nam2019a ; Nam2020a ; Nam2020b  and the bottom-up
one Davidson1979 ; Mo-Marshak1980 ; XHe1991 ; Baek2001 ; Appelquist2003 ;
Tait2004 ; Khalil2008 ; VPleitez2009 ; Khalil2010 ; Heeck2011 ; Montero2011 ;
WChao2011 ; Latosinski2013 ; MDas2014 ; Schmitz2014 ; Altmannshofer2014 ;
SBaek2015 ; MaPollard2015 ; LeeYun2016 ; WChao2016 ; ABiswas2016 ; Das2017 ;
DelleRose2017 ; Biswas2017 ; Asai2017 ; Singirala2018 ; Das-Raut2017 ;
TNomura2018 ; SLee2018 ; Escudero2018 ; Nomura2018 ; DelleRose2018 ;
Kamada2018 ; Arcadi2018 ; Banerjee2018 ; Camargo2019 ; Marzo2019 ; XHou2019 ;
CWChiang2019 ; Nam2020c ; Choudhury2020 ; Dong032020 ; Loi042020 ; Borah2020 .
In the $U(1)$ extensions of the SM, the right-handed neutrinos and even exotic
chiral/Dirac fermions are introduced to achieve the anomaly cancellations,
generate the masses for the observed neutrinos via the well-known seesaw
mechanism, and provide the DM candidates. The DM phenomenology has been
investigated in various $U(1)$ extensions of the SM Belanger2008 ; Okada2010 ;
Kanemura2011 ; ZhengYu2012 ; Okada2012 ; Lindner2014 ; Alves2014 ; Agrawal2014
; Alves2015 ; Schmitz2015 ; Duerr2015 ; Berlin2015 ; Okada2016 ; Kaneta2017 ;
SingiralaPatra2018 ; Chao2017 ; Borah2017 ; Bandyopadhyay2018 ; Liu2018 ;
Hutauruk2019 ; Frank2019 ; Raut2019 ; Gu2020 .

In traditional $U(1)$ extensions of the SM, the exotic $U(1)$ charge is
usually not related to the charge operator. However, it is natural to consider
that the charge operator would be identified in terms of all charges of the
present $U(1)$ gauge symmetries. As a result, an extension of the SM in this
way should be based on the following symmetry, $\text{GSM}\equiv
SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y^{\prime}}\times U(1)_{X}$, where the
symmetry $U(1)_{Y^{\prime}}$ is in general different to the weak hypercharge
symmetry $U(1)_{Y}$ of the SM, which is called flipped $U(1)$ extended SM. The
charge operator in this scenario is identified as follows

$\displaystyle Q=T_{3}+Y^{\prime}+X$ (1)

where $Y^{\prime}$ and $X$ are the $U(1)_{Y^{\prime}}$ and $U(1)_{X}$ charges,
respectively. This is the motivation for the present work which we construct a
general flavor-independent model in a minimal way which is consistent with the
experimental constraints as well as incorporates the neutrino masses and a DM
candidate.

This work is organized as follows. In Sec. II, we present the model
construction. In this section, we introduce the field content and charge
assignment, and then we study neutrino and neutral gauge sectors and determine
the couplings of the neutral gauge bosons to the fermions. In Sec. III, we
analyze the experimental constraints on the mass and gauge coupling of the new
gauge boson. In Sec. IV, we study the DM phenomenology with the third right-
handed neutrino stabilized by the $Z_{2}$ discrete symmetry and hence playing
the role of DM candidate. Finally, we conclude in Sec. V.

##  II Model construction

We consider a model based on the symmetry $\text{GSM}\times Z_{2}$ with the
field content and charge assignment under this symmetry listed on Table 1.
Under the $Z_{2}$ discrete symmetry, only the third right-handed neutrino is
odd, whereas other fields take even parity. This would guarantee the stability
of the third right-handed neutrino and hence it can play the role of the DM
candidate.

Before proceeding, we pause here to comment on the $Z_{2}$ discrete symmetry
introduced above. We can consider this $Z_{2}$ discrete symmetry as a reduced
symmetry which arises from the high-scale symmetry breaking of a certain
$U(1)$. Under that $U(1)$, the charge assignment for the first two right-
handed neutrinos is different to that for the third right-handed neutrino,
which satisfies the cancellation of all anomalies. (An example of such a
charge assignment has been considered for the case of $U(1)_{B-L}$
VPleitez2009 .) Because of this charge assignment, the reduced $Z_{2}$ charge
of the first two right-handed neutrinos is generally different to that of the
third right-handed neutrino. With a consistent charge assignment, it may thus
lead to that the first two right-handed neutrinos and the SM particles are
even under the reduced $Z_{2}$ symmetry whereas the third right-handed
neutrino is odd.

|  $SU(3)_{C}$ |  $SU(2)_{L}$ |  $U(1)_{Y^{\prime}}$ |  $U(1)_{X}$ |  $Z_{2}$
---|---|---|---|---|---
$Q_{a}$ | 3 | 2 | $\frac{1}{6}+z_{2}-z_{1}$ | $z_{1}-z_{2}$ | $+$
$u_{aR}$ | 3 | 1 | $\frac{2}{3}-z_{1}$ |  $z_{1}$ | $+$
$d_{aR}$ | 3 | 1 | $2z_{2}-z_{1}-\frac{1}{3}$ | $z_{1}-2z_{2}$ | $+$
$L_{a}$ | 1 | 2 | $3(z_{1}-z_{2})-\frac{1}{2}$ | $3(z_{2}-z_{1})$ | $+$
$\nu_{iR}$ | 1 | 1 | $3z_{1}-4z_{2}$ | $4z_{2}-3z_{1}$ | $+$
$\nu_{3R}$ | 1 | 1 | $3z_{1}-4z_{2}$ | $4z_{2}-3z_{1}$ | $-$
$e_{aR}$ | 1 | 1 | $3z_{1}-2z_{2}-1$ | $2z_{2}-3z_{1}$ | $+$
$H$ | 1 | 2 | $\frac{1}{2}-z_{2}$ | $z_{2}$ | $+$
$\Phi$ | 1 | 1 | $2(4z_{2}-3z_{1})$ | $2(3z_{1}-4z_{2})$ | $+$
Table 1: The charge assignement for the fermions and scalars under the
symmetry $\text{GSM}\times Z_{2}$ where $z_{1}$ and $z_{2}$ are the free
parameters with $a=1,2,3$ and $i=1,2$.

Up to the gauge fixing and ghost terms, the kinetic terms of the gauge fields,
and the terms related to the $SU(3)_{C}$ sector, the most total Lagrangian is
given by

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\sum_{f}\bar{f}i\gamma^{\mu}D_{\mu}f+\left(D_{\mu}H\right)^{\dagger}(D^{\mu}H)+\left(D_{\mu}\Phi\right)^{\dagger}(D^{\mu}\Phi)-V(H,\Phi)+\mathcal{L}_{\text{Y}},$
(2)

where $f$ refers to the chiral fermions of our model,
$D_{\mu}=\partial_{\mu}+ig\frac{\sigma_{i}}{2}W_{i\mu}+ig_{1}Y^{\prime}B_{\mu}+ig_{2}XX_{\mu}$
with $\\{g,g_{1},g_{2}\\}$ to be the gauge couplings corresponding to
$\\{SU(2)_{L},U(1)_{Y^{\prime}},U(1)_{X}\\}$. The scalar potential $V(H,\Phi)$
takes the following form

$\displaystyle
V(H,\Phi)=\mu^{2}_{1}H^{\dagger}H+\lambda_{1}(H^{\dagger}H)^{2}+\mu^{2}_{2}\Phi^{\dagger}\Phi+\lambda_{2}(\Phi^{\dagger}\Phi)^{2}+\lambda_{3}(H^{\dagger}H)(\Phi^{\dagger}\Phi),$
(3)

where in order to guarantee the potential bounded from below, the coupling
constants must satisfy the condition as

$\displaystyle\lambda_{1},\lambda_{2}>0,\ \ \ \
\lambda_{3}>-2\sqrt{\lambda_{1}\lambda_{2}}.$ (4)

The Yukawa interactions are given by

$\displaystyle-\mathcal{L}_{\text{Y}}$ $\displaystyle=$
$\displaystyle\sum^{3}_{a,b=1}\left(h^{e}_{ab}\bar{L}_{a}He_{bR}+h^{d}_{ab}\bar{Q}_{a}Hd_{bR}+h^{u}_{ab}\bar{Q}_{a}\widetilde{H}u_{bR}+h^{M}_{ab}\bar{\nu}^{C}_{aR}\nu_{bR}\Phi\right)$
(5)
$\displaystyle+\sum^{3}_{a=1}\sum^{2}_{i=1}h^{\nu}_{ai}\bar{L}_{a}\widetilde{H}\nu_{iR}+\textrm{H.c},$

with $\widetilde{H}=i\sigma_{2}H^{*}$ and without loss of generality the
matrix $h^{M}$ to be diagonal for simplicity. Note that, the term
$\bar{L}_{a}\widetilde{H}\nu_{3R}$ is forbidden by the $Z_{2}$ charge
assignment.

Because the weak hypercharge symmetry $U(1)_{Y}$ is identified as the
remaining symmetry of the spontaneously $U(1)_{Y^{\prime}}\times U(1)_{X}$
symmetry breaking, one can find the relation between the gauge coupling
$g^{\prime}$ of the symmetry $U(1)_{Y}$ in terms of $g_{1}$ and $g_{2}$ as

$\displaystyle g^{\prime}=\frac{g_{1}g_{2}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}.$ (6)

We discuss the assignment of the $U(1)_{Y^{\prime}}$ and $U(1)_{X}$ charges
from the conditions of the anomaly cancellation and gauge invariance of the
Yukawa couplings. The nontrivial anomalies in our model are listed as follows:
$[SU(3)_{C}]^{2}U(1)_{Y^{\prime}}$, $[SU(2)_{L}]^{2}U(1)_{Y^{\prime}}$,
$[U(1)_{Y^{\prime}}]^{2}U(1)_{X}$, $[U(1)_{Y^{\prime}}]^{3}$,
$[\text{Gravity}]^{2}U(1)_{Y^{\prime}}$, $[SU(3)_{C}]^{2}U(1)_{X}$,
$[SU(2)_{L}]^{2}U(1)_{X}$, $[U(1)_{X}]^{2}U(1)_{Y^{\prime}}$,
$[U(1)_{X}]^{3}$, and $[\text{Gravity}]^{2}U(1)_{X}$. We can check that
because of the relation $Y=Y^{\prime}+X$ the anomaly cancellation conditions
associated with the last five anomalies are automatically satisfied if the
first five anomalies vanish, and vice versa. Therefore, we can write the
anomaly cancellation conditions as

$\displaystyle[SU(3)_{C}]^{2}U(1)_{X}:\ \ 2X_{Q}-X_{u_{R}}-X_{d_{R}}=0,$
$\displaystyle[SU(2)_{L}]^{2}U(1)_{X}:\ \ 3X_{Q}+X_{L}=0,$
$\displaystyle[U(1)_{X}]^{2}U(1)_{Y^{\prime}}:\ \
2\left(3X^{2}_{Q}Y^{\prime}_{Q}+X^{2}_{L}Y^{\prime}_{L}\right)-3(X^{2}_{u_{R}}Y^{\prime}_{u_{R}}+X^{2}_{d_{R}}Y^{\prime}_{d_{R}})-X^{2}_{e_{R}}Y^{\prime}_{e_{R}}-X^{2}_{\nu_{R}}Y^{\prime}_{\nu_{R}}=0,$
$\displaystyle[U(1)_{X}]^{3}:\ \
2\left(3X^{3}_{Q}+X^{3}_{L}\right)-3(X^{3}_{u_{R}}+X^{3}_{d_{R}})-X^{3}_{e_{R}}-X^{3}_{\nu_{R}}=0,$
$\displaystyle[\text{Gravity}]^{2}U(1)_{X}:\ \
2\left(3X_{Q}+X_{L}\right)-3\left(X_{u_{R}}+X_{d_{R}}\right)-X_{e_{R}}-X_{\nu_{R}}=0.$
(7)

In addition, the gauge invariance of the Yukawa couplings leads to

$\displaystyle-X_{Q}-X_{H}+X_{u_{R}}$ $\displaystyle=$ $\displaystyle-
X_{Q}+X_{H}+X_{d_{R}}=0,$ $\displaystyle-X_{L}-X_{H}+X_{\nu_{R}}$
$\displaystyle=$ $\displaystyle-X_{L}+X_{H}+X_{e_{R}}=0,$ $\displaystyle
2X_{\nu_{R}}+X_{\Phi}$ $\displaystyle=$ $\displaystyle 0.$ (8)

Note that, the similar conditions for the $U(1)_{Y^{\prime}}$ charges are
automatically satisfied because of the relation $Y=Y^{\prime}+X$. With (8),
the first and fifth lines in (7) are automatically satisfied. Whereas, the
remaining conditions in (7) lead to unique solution as

$\displaystyle X_{\nu_{R}}=4X_{H}-3X_{u_{R}}.$ (9)

Then, we can express the $U(1)_{Y^{\prime}}$ and $U(1)_{X}$ charges of the
fields in terms of two independent parameters which are the $U(1)_{X}$ charges
of the scalar doublet $H$ and right-handed up-type quark, as seen in Table 1.

The scalar fields develop their vacuum expectation values (VEVs) given as,
$\langle H\rangle=(\begin{array}[]{cc}0&v/\sqrt{2}\end{array})$ and
$\langle\Phi\rangle=v^{\prime}/\sqrt{2}$. The VEV of $\Phi$ would break the
symmetry $U(1)_{Y^{\prime}}\times U(1)_{X}$ down to $U(1)_{Y}$, and the VEV of
$H$ would break the electroweak symmetry $SU(2)_{L}\times U(1)_{Y}$ down to
$U(1)_{\text{em}}$. After the spontaneous symmetry breaking, the mass matrices
for the observed light neutrinos $\nu^{\prime}_{L}$ and the heavy neutrinos
${\nu^{\prime}_{R}}$ are given by

$\displaystyle m_{\nu^{\prime}_{L}}\simeq-
U_{{}_{MNS}}M_{D}M^{-1}_{M}M^{T}_{D}U^{\dagger}_{{}_{MNS}},\ \ \ \ \ \
m_{\nu^{\prime}_{R}}\simeq M_{M},$ (10)

where $\left[M_{D}\right]_{ai}=\frac{v}{\sqrt{2}}h^{\nu}_{ai}$ and
$\left[M_{D}\right]_{a3}=0$ (with $a=1,2,3$ and $i=1,2$),
$\left[M_{M}\right]_{ab}=\sqrt{2}v^{\prime}h^{M}_{ab}$, and $U_{{}_{MNS}}$ is
the Maki-Nakagawa-Sakata (MNS) matrix determined by the current neutrino
oscillation data Tanabashi2018 . Because of
$\text{det}\left(m_{\nu^{\prime}_{L}}\right)=0$ the model predicts one
massless light neutrino eigenstate, which is still consistent with the
neutrino oscillation data Frampton2002 . The mass eigenstates are related to
the flavor states as

$\displaystyle\left(\begin{array}[]{c}\nu^{\prime C}_{L}\\\
\nu^{\prime}_{R}\\\ \end{array}\right)$ $\displaystyle\approx$
$\displaystyle\left(\begin{array}[]{cc}U_{{}_{MNS}}&-\delta^{\dagger}\\\
\delta&1\\\ \end{array}\right)\left(\begin{array}[]{c}\nu^{C}_{L}\\\
\nu_{R}\\\ \end{array}\right),$ (17)

where $\delta=M^{-1}_{M}M^{T}_{D}$. Because the mixing parameter $\delta$ is
extremely small due to the sub-eV neutrino mass scale, we have the
approximation, $\nu^{\prime}_{L}\approx U_{{}_{MNS}}\nu_{L}$ and
$\nu^{\prime}_{R}\approx\nu_{R}$.

The first expression of Eq. (10) indicates that the light SM neutrino masses
are generated via Type-I seesaw mechanism with two right-handed neutrinos. If
the Dirac-Yukawa coupling constants $h^{\nu}_{ai}$ are not too small, then
very large masses of the right-handed neutrinos are necessary to generate the
sub-eV mass scale for the light SM neutrinos. For instance, with the Dirac-
Yukawa coupling constants $h^{\nu}_{ai}$ to be of order unity (the order of
the Yukawa coupling constant of the top quark), the right-handed neutrino
masses are required to be of order $\mathcal{O}(10^{14})$ GeV. The very large
mass scale of the right-handed neutrinos is well motivated from the grand
unification models. However, no any experimental evidence indicates that the
right-handed neutrino masses should be of order the grand unification or
around. The low-scale Type-I seesaw scenario with the right-handed neutrino
masses to be of order TeV is very attractive due to the potential discovery of
new physics at the LHC as well as the signature modifications on the SM
phenomenology. In this seesaw scenario, the Dirac-Yukawa coupling constants
$h^{\nu}_{ai}$ have to be very small, which are of order
$\mathcal{O}(10^{-6})$.

The mass matrix of the neutral gauge bosons is given in the basis
$(W_{3\mu},B_{\mu},X_{\mu})$ as

$\displaystyle
M^{2}_{\text{NGB}}=\left(\begin{array}[]{ccc}\frac{g^{2}v^{2}}{4}&\star&\star\\\
-\frac{gg_{1}(1-2z_{2})v^{2}}{4}&\frac{g^{2}_{1}}{4}\left[(1-2z_{2})^{2}v^{2}+16(4z_{2}-3z_{1})^{2}{v^{\prime}}^{2}\right]&\star\\\
-\frac{gg_{2}z_{2}v^{2}}{2}&\frac{g_{1}g_{2}}{2}\left[(1-2z_{2})z_{2}v^{2}-8(4z_{2}-3z_{1})^{2}v^{\prime
2}\right]&g^{2}[z^{2}_{2}v^{2}+4(4z_{2}-3z_{1})^{2}v^{\prime 2}]\\\
\end{array}\right).$ (21)

The corresponding mass eigenvalues are obtained as,
$\text{Diag}(M^{2}_{Z},M^{2}_{Z^{\prime}},0)=VM^{2}_{\text{NGB}}V^{T}$, where

$\displaystyle M^{2}_{Z}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(g^{2}+g^{\prime 2})v^{2}-\frac{g^{2}+g^{\prime
2}}{64(3z_{1}-4z_{2})^{2}}\left[\frac{2(1+t^{2})z_{2}-1}{1+t^{2}}\right]^{2}\frac{v^{4}}{v^{\prime
2}}+\mathcal{O}(v^{4}/v^{\prime 4}),$ $\displaystyle M^{2}_{Z^{\prime}}$
$\displaystyle=$ $\displaystyle
4(g^{2}_{1}+g^{2}_{2})(3z_{1}-4z_{2})^{2}v^{\prime
2}+\frac{(g^{2}_{1}+g^{2}_{2})}{4}\left[\frac{2(1+t^{2})z_{2}-1}{1+t^{2}}\right]^{2}v^{2}+\mathcal{O}(v^{2}/v^{\prime
2}),$ (23)

with $t\equiv g_{2}/g_{1}$ and the diagonalizing matrix $V$ is given by

$\displaystyle V=\left(\begin{array}[]{ccc}c_{\beta}&-s_{\beta}&0\\\
s_{\beta}&c_{\beta}&0\\\ 0&0&1\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{W}&0&-s_{W}\\\ 0&1&0\\\
s_{W}&0&c_{W}\\\ \end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\frac{g_{1}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}&-\frac{g_{2}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}\\\
0&\frac{g_{2}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}&\frac{g_{1}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}\\\
\end{array}\right),$ (33)

with $\theta_{W}$ to be the Weinberg angle and the mixing angle $\beta$ given
by

$\displaystyle\tan 2\beta$ $\displaystyle=$
$\displaystyle\frac{t\left[2(1+t^{2})z_{2}-1\right]}{8s_{W}(3z_{1}-4z_{2})^{2}(1+t^{2})^{2}}\frac{v^{2}}{v^{\prime
2}}+\mathcal{O}(v^{4}/v^{\prime 4}),$ (34) $\displaystyle=$ $\displaystyle
2s_{W}\frac{\left[2(1+t^{2})z_{2}-1\right]}{t}\frac{M^{2}_{Z}}{M^{2}_{Z^{\prime}}}+\mathcal{O}(M^{4}_{Z}/M^{4}_{Z^{\prime}}).$

The physical states are related to $(W_{3\mu},B_{\mu},X_{\mu})$ as,
$(Z_{\mu},Z^{\prime}_{\mu},A_{\mu})^{T}=V(W_{3\mu},B_{\mu},X_{\mu})^{T}$.

The couplings of the neutral gauge bosons to the SM chiral fermions and the
heavy neutrinos $\nu^{\prime}_{aR}$ are given as

$\displaystyle\mathcal{L}_{NC}$ $\displaystyle=$
$\displaystyle-\sum_{f_{i}}\left(C^{Z}_{f_{i}}\bar{f}_{i}\gamma^{\mu}f_{i}Z_{\mu}+C^{Z^{\prime}}_{f_{i}}\bar{f}_{i}\gamma^{\mu}f_{i}Z^{\prime}_{\mu}\right),$
(35)

where $i=L,R$ and the coupling factors read

$\displaystyle C^{Z}_{f_{i}}$ $\displaystyle=$
$\displaystyle\frac{gc_{\beta}}{c_{W}}\left[T_{3}(f_{i})-Q_{f}s^{2}_{W}\right]-\frac{gt_{W}s_{\beta}}{t}\left(Y^{\prime}_{f_{i}}-t^{2}X_{f_{i}}\right),$
$\displaystyle C^{Z^{\prime}}_{f_{i}}$ $\displaystyle=$
$\displaystyle\frac{gs_{\beta}}{c_{W}}\left[T_{3}(f_{i})-Q_{f}s^{2}_{W}\right]+\frac{gt_{W}c_{\beta}}{t}\left(Y^{\prime}_{f_{i}}-t^{2}X_{f_{i}}\right).$
(36)

More explicitly, we show the $Z$ and $Z^{\prime}$ couplings to the fermions in
Tables 2 and 3.

$f$ | $C^{Z}_{V,f}=(C^{Z}_{f_{L}}+C^{Z}_{f_{R}})/2$ | $C^{Z}_{A,f}=(C^{Z}_{f_{R}}-C^{Z}_{f_{L}})/2$
---|---|---
$\ \ \nu^{\prime}_{1L},\nu^{\prime}_{2L},\nu^{\prime}_{3L}\ \ $ | $\ \ \ \ \frac{gc_{\beta}}{4c_{W}}-gt_{W}s_{\beta}\frac{6(z_{1}-z_{2})(1+t^{2})-1}{4t}\ \ \ \ $ | $\ \ \ \ -\frac{gc_{\beta}}{4c_{W}}+gt_{W}s_{\beta}\frac{6(z_{1}-z_{2})(1+t^{2})-1}{4t}$
$\ \ \nu^{\prime}_{1R},\nu^{\prime}_{2R},\nu^{\prime}_{3R}\ \ $ | $\ \ \ \ -gt_{W}s_{\beta}\frac{(3z_{1}-4z_{2})(1+t^{2})}{2t}\ \ \ \ $ | $\ \ \ \ -gt_{W}s_{\beta}\frac{(3z_{1}-4z_{2})(1+t^{2})}{2t}$
$\ \ e,\mu,\tau\ \ $ | $\ \ \ \ \frac{gc_{\beta}}{4c_{W}}(4s^{2}_{W}-1)-gt_{W}s_{\beta}\frac{2(6z_{1}-5z_{2})(1+t^{2})-3}{4t}\ \ \ \ $ | $\ \ \ \ \frac{gc_{\beta}}{4c_{W}}-gt_{W}s_{\beta}\frac{2z_{2}(1+t^{2})-1}{4t}$
$\ \ u,c,t\ \ $ | $\ \ \ \ \frac{gc_{\beta}}{12c_{W}}(3-8s^{2}_{W})-gt_{W}s_{\beta}\frac{5-6(2z_{1}-z_{2})(1+t^{2})}{12t}\ \ \ \ $ | $\ \ \ \ -\frac{gc_{\beta}}{4c_{W}}-gt_{W}s_{\beta}\frac{1-2z_{2}(1+t^{2})}{4t}$
$\ \ d,s,b\ \ $ | $\ \ \ \ \frac{gc_{\beta}}{12c_{W}}(4s^{2}_{W}-3)-gt_{W}s_{\beta}\frac{6(3z_{2}-2z_{1})(1+t^{2})-1}{12t}\ \ \ \ $ | $\ \ \ \ \frac{gc_{\beta}}{4c_{W}}-gt_{W}s_{\beta}\frac{2z_{2}(1+t^{2})-1}{4t}$
Table 2: Vector and axial-vector couplings of $Z$ to the fermions. $f$ | $C^{Z^{\prime}}_{V,f}=(C^{Z^{\prime}}_{f_{L}}+C^{Z^{\prime}}_{f_{R}})/2$ | $C^{Z^{\prime}}_{A,f}=(C^{Z^{\prime}}_{f_{R}}-C^{Z^{\prime}}_{f_{L}})/2$
---|---|---
$\ \ \nu^{\prime}_{1L},\nu^{\prime}_{2L},\nu^{\prime}_{3L}\ \ $ | $\ \ \ \ \frac{gs_{\beta}}{4c_{W}}+gt_{W}c_{\beta}\frac{6(z_{1}-z_{2})(1+t^{2})-1}{4t}\ \ \ \ $ | $\ \ \ \ -\frac{gs_{\beta}}{4c_{W}}-gt_{W}c_{\beta}\frac{6(z_{1}-z_{2})(1+t^{2})-1}{4t}$
$\ \ \nu^{\prime}_{1R},\nu^{\prime}_{2R},\nu^{\prime}_{3R}\ \ $ | $\ \ \ \ gt_{W}c_{\beta}\frac{(3z_{1}-4z_{2})(1+t^{2})}{2t}\ \ \ \ $ | $\ \ \ \ gt_{W}c_{\beta}\frac{(3z_{1}-4z_{2})(1+t^{2})}{2t}$
$\ \ e,\mu,\tau\ \ $ | $\ \ \ \ \frac{gs_{\beta}}{4c_{W}}(4s^{2}_{W}-1)+gt_{W}c_{\beta}\frac{2(6z_{1}-5z_{2})(1+t^{2})-3}{4t}\ \ \ \ $ | $\ \ \ \ \frac{gs_{\beta}}{4c_{W}}+gt_{W}c_{\beta}\frac{2z_{2}(1+t^{2})-1}{4t}$
$\ \ u,c,t\ \ $ | $\ \ \ \ \frac{gs_{\beta}}{12c_{W}}(3-8s^{2}_{W})+gt_{W}c_{\beta}\frac{5-6(2z_{1}-z_{2})(1+t^{2})}{12t}\ \ \ \ $ | $\ \ \ \ -\frac{gs_{\beta}}{4c_{W}}+gt_{W}c_{\beta}\frac{1-2z_{2}(1+t^{2})}{4t}$
$\ \ d,s,b\ \ $ | $\ \ \ \ \frac{gs_{\beta}}{12c_{W}}(4s^{2}_{W}-3)+gt_{W}c_{\beta}\frac{6(3z_{2}-2z_{1})(1+t^{2})-1}{12t}\ \ \ \ $ | $\ \ \ \ \frac{gs_{\beta}}{4c_{W}}+gt_{W}c_{\beta}\frac{2z_{2}(1+t^{2})-1}{4t}$
Table 3: Vector and axial-vector couplings of $Z^{\prime}$ to the fermions.

In Fig. 1, we show the behavior of the $Z^{\prime}$ coupling strength
$\sqrt{(C^{Z^{\prime}}_{V,f})^{2}+(C^{Z^{\prime}}_{A,f})^{2}}$ to the fermion
$f$ as a function of the ratio $g_{2}/g_{1}$ for various values of $z_{1,2}$.

---
Figure 1: The $Z^{\prime}$ coupling strength
$\sqrt{(C^{Z^{\prime}}_{V,f})^{2}+(C^{Z^{\prime}}_{A,f})^{2}}$ as a function
of the ratio $g_{2}/g_{1}$ for several values of $z_{1}$ and $z_{2}$. The
blue, red, green, brown, and purple curves corresponding to
$\nu^{\prime}_{L}$, $\nu^{\prime}_{R}$, $e$, $u$, and $d$, respectively. Top-
left panel: $z_{1}=1/4$ and $z_{2}=1/8$. Top-right panel: $z_{1}=1/5$ and
$z_{2}=1/7$. Bottom panel: $z_{1}=1/5$ and $z_{2}=1/10$.

From this figure, we see that the $Z^{\prime}$ coupling strength
$\sqrt{(C^{Z^{\prime}}_{V,f})^{2}+(C^{Z^{\prime}}_{A,f})^{2}}$ should first
decrease until a minimum and then increase as increasing the parameter $t$. On
the other hand, there exists a certain value of $t$ with given $z_{1,2}$ near
which the $Z^{\prime}$ coupling strength is relatively small but becomes
larger as $t$ going far from this value of $t$.

In summary, from Eqs. (23) $\&$ (34) and Table 3, the properties of the new
gauge boson $Z^{\prime}$ in our model are primarily described by four
independent parameters which are $t$, $z_{1,2}$, and VEV $v^{\prime}$ of the
scalar field $\Phi$. One can not use the redefinition of the couplings
$g_{1,2}$ or the charges to define one of the parameters $t$ and $z_{1,2}$ to
be $1$.

Note that, the electromagnetic couplings and the charged currents of the
quarks do not get modified in our model, whereas the charged currents of the
leptons is given by
$\mathcal{L}_{CC}=-\frac{g}{\sqrt{2}}\bar{l}_{L}\gamma^{\mu}U^{\dagger}_{MNS}\nu^{\prime}_{L}W^{-}_{\mu}+\textrm{H.c.}$.

##  III Constraints

In this section, we use various current experimental constraints to impose the
bounds on the mass of the new gauge boson $Z^{\prime}$ which are in general
the functions of $t$ and $z_{1,2}$.

First, we impose the perturbativity conditions on the gauge couplings as

$\displaystyle g\leq\sqrt{4\pi},$ (37)

for the $SU(2)_{L}$ and

$\displaystyle|Y^{\prime}_{Q,u_{R},d_{R},L,e_{R},\nu_{R},H,\Phi}|g_{1}\leq\sqrt{4\pi},$
$\displaystyle|X_{Q,u_{R},d_{R},L,e_{R},\nu_{R},H,\Phi}|g_{2}\leq\sqrt{4\pi},$
(38)

for the $U(1)_{Y^{\prime}}$ and $U(1)_{X}$, respectively. In addition, the
perturbative condition on the Higgs couplings is given as Huffel1981 ;
Duerr2016

$\displaystyle|\lambda_{3}|\leq 8\pi,\ \ \ \
3(\lambda_{1}+\lambda_{2})\pm\sqrt{\lambda^{2}_{3}+9(\lambda_{1}+\lambda_{2})^{2}}\leq
8\pi.$ (39)

The precise measurement of the mass of the SM gauge boson $Z$ is given by
$M_{Z}=91.1876\pm 0.0021$ GeV Tanabashi2018 , hence we require the following
constraint

$\displaystyle|\Delta
M_{Z}|\simeq\frac{s^{2}_{W}}{2}\left[\frac{2(1+t^{2})z_{2}-1}{t}\right]^{2}\frac{M^{3}_{Z}}{M^{2}_{Z^{\prime}}}\lesssim
0.0021\ \ \text{GeV}.$ (40)

This leads to a lower bound as

$\displaystyle M_{Z^{\prime}}\gtrsim 6.46\times\frac{|2(1+t^{2})z_{2}-1|}{t}\
\ \text{TeV}.$ (41)

The modification of the mass of the SM gauge boson $Z$ also leads to the
correction to $\rho$ parameter and thus would constraint the free parameters
of the model. The corresponding correction is given by

$\displaystyle\Delta\rho$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{W}}{c^{2}_{W}M^{2}_{Z}}-\rho_{\text{SM}}$ (42)
$\displaystyle\simeq$ $\displaystyle
s^{2}_{W}\left[\frac{2(1+t^{2})z_{2}-1}{t}\right]^{2}\frac{M^{2}_{Z}}{M^{2}_{Z^{\prime}}},$

where $\rho_{\text{SM}}=1$ is the SM prediction. Using the experimental value
$\rho=1.00039\pm 0.00019$ Tanabashi2018 , we require new physics satisfying
$\Delta\rho<0.00058$ from which we obtain

$\displaystyle M_{Z^{\prime}}\gtrsim 1.82\times\frac{|2(1+t^{2})z_{2}-1|}{t}\
\ \text{TeV},$ (43)

which is clearly weaker than the bound (41).

The mixing between the SM gauge boson $Z$ and new one $Z^{\prime}$ leads to
the corrections in the $Z$ couplings to the SM fermions. Thus, we use the
precision measurement of the $Z$ decay width to constrain the relevant
corrections. First, we rewrite the $Z$ couplings to the SM chiral fermions
$f_{i}$ ($i=L,R$) as

$\displaystyle\mathcal{L}_{Z}$ $\displaystyle=$
$\displaystyle-\bar{f}_{i}\gamma^{\mu}C^{\text{SM}}_{f_{i}}\left(1+\delta_{f_{i}}\right)f_{i}Z_{\mu}$
(44)

where $C^{\text{SM}}_{f_{L}}=g\left[T_{3}(f_{L})-Q_{f}s^{2}_{W}\right]/c_{W}$
and $C^{\text{SM}}_{f_{R}}=-\frac{g}{c_{W}}Q_{f}s^{2}_{W}$ are the SM values
for the $Z$ couplings to the left- and right-handed fermions, respectively,
and the corrections to the SM predictions of the $Z$ couplings are given by

$\displaystyle\delta_{f_{i}}$ $\displaystyle=$
$\displaystyle-s^{2}_{W}\frac{2(1+t^{2})z_{2}-1}{t^{2}}\frac{Y^{\prime}_{f_{i}}-t^{2}X_{f_{i}}}{T_{3}(f_{i})-Q_{f}s^{2}_{W}}\frac{M^{2}_{Z}}{M^{2}_{Z^{\prime}}}+\mathcal{O}(M^{4}_{Z}/M^{4}_{Z^{\prime}}).$
(45)

Then, the $Z$ decay width in our model is given as,
$\Gamma_{Z}=\Gamma^{\text{SM}}_{Z}+\Delta\Gamma_{{Z}}$, where
$\Gamma^{\text{SM}}_{Z}$ is the SM value and the correction
$\Delta\Gamma_{{Z}}$ for the SM prediction reads

$\displaystyle\Delta\Gamma_{{Z}}$ $\displaystyle\simeq$
$\displaystyle\frac{M_{Z}}{12\pi}\sum_{f_{i}}N_{C}(f_{i})\left(C^{\text{SM}}_{f_{i}}\right)^{2}\delta_{f_{i}}+\frac{\Delta
M_{Z}}{24\pi}\sum_{f_{i}}N_{C}(f_{i})\left(C^{\text{SM}}_{f_{i}}\right)^{2},$
(46)

where $f_{i}$ refers to all chiral fermions in the SM and $\Delta
M_{Z}=-|\Delta M_{Z}|$ with $|\Delta M_{Z}|$ to be given in (40). From the
experimental value and the SM prediction for the total $Z$ width as,
$\Gamma^{\text{exp}}_{Z}=2.4952\pm 0.0023$ GeV and
$\Gamma^{\text{SM}}_{Z}=2.4942\pm 0.0008$ GeV Tanabashi2018 , we require
$|\Delta\Gamma_{Z}|<0.0041$ GeV which leads to

$\displaystyle M_{Z^{\prime}}\gtrsim
1.96\times\left|\frac{2(1+t^{2})z_{2}-1}{t^{2}}\left[(0.26+t^{2})z_{1}+(0.41-0.32t^{2})z_{2}-0.5\right]\right|^{1/2}\
\ \text{TeV}.$ (47)

The weak nuclear charge of Cesium has been measured to a precision given by
$Q^{\text{exp}}_{W}(^{133}_{\phantom{k}55}\text{Cs})=-73.16(29)_{\text{exp}}(20)_{\text{th}}$
Porsev09-10  which is in agreement with the SM prediction (including
electroweak radiative corrections) as
$Q^{\text{th}}_{W}(^{133}_{\phantom{k}55}\text{Cs})=-73.16(3)$ Marciano90-92 .
This suggests that the contribution of new physics for the nuclear weak charge
of Cesium must satisfy $\left|\Delta
Q_{W}(^{133}_{\phantom{k}55}\text{Cs})\right|\lesssim 0.52$. In our model, the
exchange of the new gauge boson $Z^{\prime}$ leads to an additional
contribution for the nuclear weak charge of Cesium, which is given as follows
Diener2012

$\displaystyle\Delta
Q_{W}(^{133}_{\phantom{k}55}\text{Cs})=-16\left(\frac{M_{Z}}{M_{Z^{\prime}}}\right)^{2}\left(\frac{c_{W}}{g}\right)^{2}C^{Z^{\prime}}_{A,e}\left[(2Z+N)C^{Z^{\prime}}_{V,u}+(Z+2N)C^{Z^{\prime}}_{V,d}\right],$
(48)

where
$C^{Z^{\prime}}_{V,f}=(C^{Z^{\prime}}_{f_{L}}+C^{Z^{\prime}}_{f_{R}})/2$,
$C^{Z^{\prime}}_{A,f}=(C^{Z^{\prime}}_{f_{R}}-C^{Z^{\prime}}_{f_{L}})/2$,
$Z=55$, and $N=78$. Then, one can find a lower bound on the mass of the new
gauge boson $Z^{\prime}$ as

$\displaystyle M_{Z^{\prime}}\gtrsim
0.95\times\left|\frac{2(1+t^{2})z_{2}-1}{t^{2}}\left[6.76(1+t^{2})z_{2}-6.57(1+t^{2})z_{1}+1\right]\right|^{1/2}\
\ \text{TeV}.$ (49)

At the $e^{+}e^{-}$ colliders such as the LEP, the new gauge boson
$Z^{\prime}$ would not be directly produced if it is massive enough. However,
the presence of $Z^{\prime}$ would lead to the deviation from the SM
prediction in the scattering processes $e^{+}e^{-}\rightarrow\bar{f}f$. Hence,
we study the constraint on the contact interactions which are induced by the
$Z^{\prime}$ exchange and described by the following effective Lagrangian

$\displaystyle\mathcal{L}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\delta_{ef}}\frac{1}{M^{2}_{Z^{\prime}}}\sum_{i,j=L,R}\eta_{ij}\bar{e}_{i}\gamma_{\mu}e_{i}\bar{f}_{j}\gamma^{\mu}f_{j},$
(50)

where $\delta_{ef}=1(0)$ for $f=e$ ($f\neq e$) and
$\eta_{ij}=C^{Z^{\prime}}_{e_{i}}C^{Z^{\prime}}_{f_{j}}$. From the LEP data
Schael2013 , we impose the constraint on this contact interaction as

$\displaystyle\frac{2\sqrt{2\pi}M_{Z^{\prime}}}{\sqrt{\left(C^{Z^{\prime}}_{e_{L}}\right)^{2}+\left(C^{Z^{\prime}}_{e_{R}}\right)^{2}}}\gtrsim
24.6(17.8)\ \ \text{TeV},$ (51)

for $\eta_{LR},\eta_{RL}>0(<0)$ corresponding to
$\left[1+6(z_{2}-z_{1})(1+t^{2})\right]\left[1+3(z_{2}-z_{1})+t^{2}(2z_{2}-3z_{1})\right]>0(<0)$.
Then, we obtain a lower bound on the mass of the new gauge boson $Z^{\prime}$
as

$\displaystyle M_{Z^{\prime}}\gtrsim
1.75(1.27)\times\frac{I(t,z_{1},z_{2})^{1/2}}{t}\ \ \text{TeV},$ (52)

for $\eta_{LR},\eta_{RL}>0(<0)$ where

$\displaystyle
I(t,z_{1},z_{2})=\left[\frac{1}{2}+3(z_{2}-z_{1})(1+t^{2})\right]^{2}+\left[1+3(z_{2}-z_{1})+t^{2}(2z_{2}-3z_{1})\right]^{2}.$
(53)

At the LHC, the new gauge boson $Z^{\prime}$ can be produced through the
Drell-Yan process at which the most significant decay channel is
$Z^{\prime}\rightarrow l^{+}l^{-}$ ($l=e,\mu$). The cross section of the
process $pp\rightarrow Z^{\prime}\rightarrow l^{+}l^{-}$ at a fixed collider
center-of-mass energy $\sqrt{s}$ can be written as

$\displaystyle\sigma(pp\rightarrow Z^{\prime}\rightarrow l^{+}l^{-})$
$\displaystyle=$
$\displaystyle\sum_{q}\int^{s}_{4m^{2}_{l}}d\hat{s}L_{q\bar{q}}(\hat{s})\frac{\hat{s}}{144\pi}\frac{C_{S}}{\left(\hat{s}^{2}-M^{2}_{Z^{\prime}}\right)^{2}+M^{2}_{Z^{\prime}}\Gamma^{2}_{Z^{\prime}}},$
(54)

where $\sqrt{\hat{s}}$ is the invariant mass of the dilepton system, the
parton luminosities $L_{q\bar{q}}$ are defined by

$\displaystyle
L_{q\bar{q}}(\hat{s})=\int^{1}_{\frac{\hat{s}}{s}}\frac{dx}{xs}\left[f_{q}(x,\hat{s})f_{\bar{q}}\left(\frac{\hat{s}}{xs},\hat{s}\right)+f_{q}\left(\frac{\hat{s}}{xs},\hat{s}\right)f_{\bar{q}}(x,\hat{s})\right],$
(55)

with $f_{q(\bar{q})}(x,\hat{s})$ to be the parton distribution function for
the quark $q$ (antiquark $\bar{q}$) evaluated at the scale $\hat{s}$
Stirling2009 , and
$C_{S}=[(C^{Z^{\prime}}_{q_{L}})^{2}+(C^{Z^{\prime}}_{q_{R}})^{2}][(C^{Z^{\prime}}_{l_{L}})^{2}+(C^{Z^{\prime}}_{l_{R}})^{2}]$.
In Eq. (54), the decay width of the new gauge boson $Z^{\prime}$ into the two-
body states is given under the assumption (66) as

$\displaystyle\Gamma_{Z^{\prime}}=\sum_{f\in\text{SM}}\Gamma(Z^{\prime}\rightarrow\bar{f}f)+\sum^{3}_{a=1}\Gamma(Z^{\prime}\rightarrow\bar{\nu}^{\prime}_{aR}\nu^{\prime}_{aR})+\Gamma(Z^{\prime}\rightarrow
W^{+}W^{-})+\Gamma(Z^{\prime}\rightarrow Zh_{1}),$ (56)

where the partial decay widths are given by

$\displaystyle\Gamma(Z^{\prime}\rightarrow\bar{f}f)$ $\displaystyle=$
$\displaystyle\frac{N_{C}(f)M_{Z^{\prime}}}{24\pi}\sqrt{1-\frac{4m^{2}_{f}}{M^{2}_{Z^{\prime}}}}\left\\{\left[\left(C^{Z^{\prime}}_{f_{L}}\right)^{2}+\left(C^{Z^{\prime}}_{f_{R}}\right)^{2}\right]\left(1-\frac{m^{2}_{f}}{M^{2}_{Z^{\prime}}}\right)+6C^{Z^{\prime}}_{f_{L}}C^{Z^{\prime}}_{f_{R}}\frac{m^{2}_{f}}{M^{2}_{Z^{\prime}}}\right\\},$
$\displaystyle\Gamma(Z^{\prime}\rightarrow\bar{\nu}^{\prime}_{aR}\nu^{\prime}_{aR})$
$\displaystyle=$
$\displaystyle\frac{M_{Z^{\prime}}}{24\pi}\left(C^{Z^{\prime}}_{\nu^{\prime}_{R}}\right)^{2}\left(1-\frac{4m^{2}_{\nu^{\prime}_{aR}}}{M^{2}_{Z^{\prime}}}\right)^{3/2},$
$\displaystyle\Gamma(Z^{\prime}\rightarrow W^{+}W^{-})$ $\displaystyle=$
$\displaystyle(gc_{W}s_{\beta})^{2}\frac{M^{5}_{Z^{\prime}}}{192\pi
M^{4}_{W}}\left(1-\frac{4M^{2}_{W}}{M^{2}_{Z^{\prime}}}\right)^{3/2}\left(1+\frac{20M^{2}_{W}}{M^{2}_{Z^{\prime}}}+\frac{12M^{4}_{W}}{M^{4}_{Z^{\prime}}}\right),$
$\displaystyle\Gamma(Z^{\prime}\rightarrow Zh_{1})$ $\displaystyle=$
$\displaystyle\frac{\left(g_{{}_{Z^{\prime}Zh_{1}}}\right)^{2}M_{Z^{\prime}}}{192\pi
M^{2}_{Z}}\left[1-\frac{2m^{2}_{h_{1}}-10M^{2}_{Z}}{M^{2}_{Z^{\prime}}}+\frac{(m^{2}_{h_{1}}-M^{2}_{Z})^{2}}{M^{4}_{Z^{\prime}}}\right]$
(57)
$\displaystyle\times\left[1-\frac{2(m^{2}_{h_{1}}+M^{2}_{Z})}{M^{2}_{Z^{\prime}}}+\frac{(m^{2}_{h_{1}}-M^{2}_{Z})^{2}}{M^{4}_{Z^{\prime}}}\right]^{1/2},$

where $N_{C}(f)$ is the color number of the fermion $f$, $h_{1}$ refers to the
SM Higgs, and the mass dimension coupling $g_{{}_{Z^{\prime}Zh_{1}}}$ is given
by

$\displaystyle g_{{}_{Z^{\prime}Zh_{1}}}$ $\displaystyle\simeq$ $\displaystyle
gM_{W}\left[\frac{c^{2}_{W}}{2}s_{2\beta}-t^{2}_{W}(1-2z_{2})^{2}\left(\frac{s_{W}}{t}c_{2\beta}+\frac{1-t^{2}s^{2}_{W}}{2t^{2}}s_{2\beta}\right)+4z^{2}_{2}t^{2}_{W}\left(ts_{W}c_{2\beta}-\frac{t^{2}-s^{2}_{W}}{2}s_{2\beta}\right)\right.$
(58)
$\displaystyle\left.+2z_{2}t_{W}\left(tc_{W}c_{2\beta}+\frac{s_{2\theta_{W}}}{2}s_{2\beta}\right)+2z_{2}t^{2}_{W}(1-2z_{2})\left(\frac{(t^{2}-1)s_{W}}{t}c_{2\beta}+(1+s^{2}_{W})s_{2\beta}\right)\right.$
$\displaystyle\left.-(1-2z_{2})t_{W}\left(\frac{c_{W}}{t}c_{2\beta}-\frac{s_{2\theta_{W}}}{2}s_{2\beta}\right)\right],$

with $s_{2\beta}\equiv\sin 2\beta$, $c_{2\beta}\equiv\cos 2\beta$, and
$s_{2\theta_{W}}\equiv\sin 2\theta_{W}$. Note that, the partial decay widths
of $Z^{\prime}\rightarrow W^{+}W^{-}$ and $Z^{\prime}\rightarrow Zh_{1}$ are
approximately equal together as a result of the Goldstone boson equivalence in
the high energy limit. If the decay width of $Z^{\prime}$ is very narrow, Eq.
(54) can be approximated as

$\displaystyle\sigma(pp\rightarrow Z^{\prime}\rightarrow
l^{+}l^{-})\simeq\frac{\pi}{6}\sum_{q}L_{q\bar{q}}(M^{2}_{Z^{\prime}})\left[\left(C^{Z^{\prime}}_{q_{L}}\right)^{2}+\left(C^{Z^{\prime}}_{q_{R}}\right)^{2}\right]{\rm
Br}(Z^{\prime}\rightarrow l^{+}l^{-}),$ (59)

where ${\rm Br}(Z^{\prime}\rightarrow l^{+}l^{-})$ is the branching ratio of
$Z^{\prime}$ decaying into the given pair $l^{+}l^{-}$. No evidence for the
dilepton resonances has been found in the current LHC data and the $95\%$
confidence level (CL) upper bounds on $\sigma\times BR$ of new neutral gauge
boson have been produced using $36.1$ fb-1 of proton-proton collision data at
$\sqrt{s}=13$ TeV ATLAS2017 . From these upper bounds, we can impose the
constraint on the $Z^{\prime}$ gauge boson mass and the gauge coupling ratio
$t$ for $z_{1,2}$ kept fixed.

In Fig. 2, we show the allowed parameter region in the $t-M_{Z^{\prime}}$
plane, for several values of $z_{1}$ and $z_{2}$, by combining the lower
bounds corresponding to the current LHC limits, precision measurement of the
$Z$ decay width, the ambiguity of the $Z$ boson mass, the LEP data, and weak
nuclear charge of Cesium.

---
Figure 2: The allowed parameter region given by the lightblue region in the
$g_{2}/g_{1}-M_{Z^{\prime}}$ plane for several values of $z_{1}$ and $z_{2}$,
with $m_{\nu^{\prime}_{3R}}\simeq M_{Z^{\prime}}/2$ and
$m_{\nu^{\prime}_{1R}}=m_{\nu^{\prime}_{2R}}=M_{Z^{\prime}}/3$. The black,
blue, red, purple, and orange curves denote the lower bounds obtained from the
current LHC limits, precision measurement of the $Z$ decay width, the
ambiguity of the $Z$ boson mass, the LEP data, and weak nuclear charge of
Cesium, respectively. The region which is below each of these curves is
excluded. Top-left panel: $z_{1}=1/4$ and $z_{2}=1/8$. Top-right panel:
$z_{1}=1/5$ and $z_{2}=1/7$. Bottom panel: $z_{1}=1/5$ and $z_{2}=1/10$.

This figure indicates that the current LHC limits and the ambiguity of the $Z$
boson mass impose the most stringent bounds on the relation between the
$Z^{\prime}$ boson mass and the parameter $t$. Other lower bounds are much
weaker than these bounds. In the top-left and bottom panels, we observe that
the constraint from the ambiguity of the $Z$ boson mass is stronger(weaker)
than that from the current LHC limits if the parameter $t$ is smaller(larger)
than about $1.02$ and $1.2$, respectively. In top-right panel, the constraint
from the current LHC limits is weaker than that from the ambiguity of the $Z$
boson mass for the regions of the sufficiently small and large $t$, but
stronger than for the intermediate region. In addition, we see a weakening in
the current LHC limits (as well as in other constraints) appearing around a
certain value of $t$ such as $t\approx 0.94$ and $t\approx 1.46$ for the top-
left and top-right panels of Fig. 2, respectively. This is due to the behavior
of the $Z^{\prime}$ coupling strength to each fermion which is relatively
small around a minimum but becomes larger as going far from this value, as
depicted in Fig. 1. And, thus it makes the lower bound of the new gauge boson
$Z^{\prime}$ weaker. Also, we find that the $Z^{\prime}$ boson mass must not
be lower than a minimum value which is about $3$, $0.85$, and $2.8$ TeV
corresponding to the top-left, top-right, and bottom panels, respectively.

##  IV Dark matter phenomenology

In this section, we investigate the DM phenomenology where the right-handed
neutrino $\nu^{\prime}_{3R}$ is considered as the DM candidate.111Other right-
handed neutrinos $\nu^{\prime}_{1,2R}$ can not be the DM candidate. Because of
the Yukawa couplings, they can decay into the light SM neutrinos and SM Higgs.
In order for the DM candidate, their lifetime must be above the universe age
which is around $13.8$ billion years, which corresponds to their decay width
have to be below $10^{-42}$ GeV. For the right-handed neutrino masses of a few
TeV, the Dirac-Yukawa coupling constants $h^{\nu}_{ai}$ must be below the
order $\mathcal{O}(10^{-23})$, which can not generate the observed neutrino
masses. In our model, the DM $\nu^{\prime}_{3R}$ communicates with the SM
particles through two ways: the Higgs portal and gauge boson portal. Because
the DM phenomenology with the Higgs portal has been well investigated in the
literature Okada2010 ; Okada2012 . Also, the couplings of the new gauge boson
$Z^{\prime}$ to the fermions in our model differ from other models. Therefore,
in this work we focus the $\nu^{\prime}_{3R}$ DM phenomenology corresponding
to the gauge boson portal. On the other hand, we assume that the mixing
between the SM Higgs and exotic Higgs is negligibly small, e.g.
$\lambda_{3}\ll 1$. In addition, as indicated in the previous section that the
new gauge boson $Z^{\prime}$ is heavy enough and hence the mixing between $Z$
and $Z^{\prime}$ is negligibly small. Thus, with respect to the gauge boson
portal the DM $\nu^{\prime}_{3R}$ communicates with the SM particles mainly
through the new gauge boson $Z^{\prime}$. In the following analysis, we employ
$m_{\nu^{\prime}_{1R}}=m_{\nu^{\prime}_{2R}}=M_{Z^{\prime}}/3$.

###  IV.1 Direct detection

The effective Lagrangian, which describes the DM scattering on nuclei in the
low momentum transfer limit, is obtained by the $t$-channel exchange of the
new gauge boson $Z^{\prime}$ as Barger2008

$\displaystyle\mathcal{L}_{\text{eff}}=\frac{C^{Z^{\prime}}_{A,\nu^{\prime}_{R}}}{M^{2}_{Z^{\prime}}}(\bar{\nu}^{\prime}_{3R}\gamma_{\mu}\gamma^{5}\nu^{\prime}_{3R})\bar{q}\gamma^{\mu}(C^{Z^{\prime}}_{V,q}+C^{Z^{\prime}}_{A,q}\gamma^{5})q,$
(60)

where $C^{Z^{\prime}}_{A,f}$ is the axial-vector coupling of $Z^{\prime}$ to
the fermion $f$. Note that, because the DM in this work is the self-conjugate
field, there is no vector-vector coupling in the above effective Lagrangian.
The Lagrangian (60) leads to the spin-dependent (SD) scattering and in the
case of that the nucleus target is neutron the cross-section is given as
Barger2008

$\displaystyle\sigma_{\text{SD}}=(2!)^{2}\frac{3m^{2}_{\text{DM-n}}}{\pi
M^{4}_{Z^{\prime}}}\left(C^{Z^{\prime}}_{A,\nu^{\prime}_{R}}\right)^{2}\left[C^{Z^{\prime}}_{A,u}\lambda^{(\text{n})}_{u}+C^{Z^{\prime}}_{A,d}\left(\lambda^{(\text{n})}_{d}+\lambda^{(\text{n})}_{s}\right)\right]^{2}$
(61)

where

$\displaystyle
m_{\text{DM-n}}=\frac{m_{\nu^{\prime}_{3R}}m_{\text{n}}}{m_{\nu^{\prime}_{3R}}+m_{\text{n}}},$
(62)

with $m_{\text{n}}$ to be the mass of proton and $\lambda^{(\text{n})}_{q}$
are the fractional quark-spin coefficients of proton given explicitly as
Chiang2012 , $\lambda^{(\text{n})}_{u}=-0.42$,
$\lambda^{(\text{n})}_{d}=0.85$, $\lambda^{(\text{n})}_{s}=-0.08$.

In Fig. 3, we show the DM-neutron cross-section as a function of
$M_{Z^{\prime}}$ for various values of $t$ and $z_{1,2}$ along with the $90\%$
C.L. upper limits from XENON100 experiment XENON100 . From this figure, it is
obvious that the DM-neutron cross-section associated with the allowed
parameter region obtained in the previous section is below the XENON100 $90\%$
C.L. upper limits.

---
Figure 3: The DM-neutron cross-section as a function of $M_{Z^{\prime}}$ with
$m_{\nu^{\prime}_{3R}}\simeq M_{Z^{\prime}}/2$. The blue, red, purple, orange,
green, and brown curves correspond to $t=0.2$, $0.9$, $1.8$, $2.6$, $3.4$, and
$4.0$, respectively. The black curve refers to the $90\%$ C.L. upper limits
from XENON100 experiment. Left panel: $z_{1}=1/4$ and $z_{2}=1/8$. Right
panel: $z_{1}=1/5$ and $z_{2}=1/10$.

###  IV.2 Indirect detection

During the early universe with the efficiently high temperature, the DM
particles are in the thermal equilibrium with the thermal bath of the SM
particles. However, when the temperature drops below the DM mass, the rate of
the annihilation per DM particle becomes smaller than the rate of the Hubble
expansion and hence it would lead to the freeze-out of the DM species. The DM
relic abundance is observed by the Planck satellite experiments as,
$\Omega_{\text{DM}}h^{2}=0.1198\pm 0.0015$ Ade2016 .

The DM relic density is given by the following analytic approximation
Gondolo1991

$\displaystyle\Omega_{\text{DM}}h^{2}=\frac{1.07\times 10^{9}\ \
\text{GeV}^{-1}}{g^{1/2}_{*}J(x_{f})M_{\text{Pl}}},$ (63)

where $g_{*}$ is the effective total number of degrees of freedom for the
particles at the time of freeze-out (for the following analysis we employ
$g_{*}$=106.75 which is for the SM particles), $M_{\text{Pl}}=1.22\times
10^{19}$ GeV is the Planck scale. The function $J(x_{f})$ is given by

$\displaystyle J(x_{f})$ $\displaystyle=$
$\displaystyle\int^{\infty}_{x_{f}}\frac{\langle\sigma v\rangle(x)}{x^{2}}dx,$
$\displaystyle\langle\sigma v\rangle(x)$ $\displaystyle=$
$\displaystyle\frac{x}{8m^{5}_{\nu^{\prime}_{3R}}[K_{2}(x)]^{2}}\int^{\infty}_{4m^{2}_{\nu^{\prime}_{3R}}}\sigma\times(s-4m^{2}_{\nu^{\prime}_{3R}})\sqrt{s}K_{1}\left(\frac{x\sqrt{s}}{m_{\nu^{\prime}_{3R}}}\right)ds,$
(64)

where $\sigma$ is the total cross section of the DM pair annihilation into the
SM particles as well as other exotic particles and the freeze-out parameter
$x_{f}$ is computed by the following equation

$\displaystyle
x_{f}=\ln\left(\frac{0.038gM_{\text{Pl}}m_{\text{DM}}\langle\sigma
v\rangle(x_{f})}{\left(g_{*}x_{f}\right)^{1/2}}\right),$ (65)

which is about $25$ for the DM mass at a few TeV, with $g=2$ to be the number
of degrees of freedom for the DM particle in consideration. In this work we
are only interested the DM mass satisfying

$\displaystyle
m_{\nu^{\prime}_{1R}},m_{\nu^{\prime}_{1R}}<m_{\nu^{\prime}_{3R}}$
$\displaystyle<$ $\displaystyle\frac{M_{Z^{\prime}}}{2}<\frac{m_{h_{2}}}{2},$
(66)

where $h_{2}$ refers to the exotic Higgs. This means that the two-body
annihilation channels of the DM pair through the $Z^{\prime}$ portal are

$\displaystyle\bar{\nu}^{\prime}_{3R}\nu^{\prime}_{3R}\rightarrow\bar{f}f,\bar{\nu}^{\prime}_{iR}\nu^{\prime}_{iR},W^{+}W^{-},Zh_{1},$
(67)

where $f$ represents the SM fermions and $i=1,2$. Due to the mixing angle
between $Z$ and $Z^{\prime}$ to be negligibly small, the contributions from
the DM pair annihilation into $W^{+}W^{-}$ and $Zh_{1}$ are ignored. As a
result, the total cross section of the DM pair annihilation is approximately
given by

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\sigma_{\text{SM}}+\sum_{i=1,2}\sigma_{\nu^{\prime}_{iR}},$
$\displaystyle\sigma_{\text{SM}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left(\frac{gt_{W}}{12t}\right)^{4}\left[6(1+t^{2})(3z_{1}-4z_{2})\right]^{2}\frac{\sqrt{s\left(s-4m^{2}_{\nu^{\prime}_{3R}}\right)}}{\left(s-M^{2}_{Z^{\prime}}\right)^{2}+M^{2}_{Z^{\prime}}\Gamma^{2}_{Z^{\prime}}}$
$\displaystyle\times\left[F_{1}(t,z_{1},z_{2})+F_{2}(t,z_{1},z_{2})\left(1+\frac{2m^{2}_{t}}{s}\right)\sqrt{1-\frac{4m^{2}_{t}}{s}}\right],$
$\displaystyle\sigma_{\nu^{\prime}_{iR}}$ $\displaystyle=$
$\displaystyle\frac{1}{48\pi}\left[\frac{gt_{W}(1+t^{2})(3z_{1}-4z_{2})}{t}\right]^{4}\frac{1}{s\left[\left(s-M^{2}_{Z^{\prime}}\right)^{2}+M^{2}_{Z^{\prime}}\Gamma^{2}_{Z^{\prime}}\right]}\sqrt{\frac{s-4m^{2}_{\nu^{\prime}_{iR}}}{s-4m^{2}_{\nu^{\prime}_{3R}}}}$
(68)
$\displaystyle\times\left[\left(s-4m^{2}_{\nu^{\prime}_{3R}}\right)\left(s-4m^{2}_{\nu^{\prime}_{iR}}\right)+\frac{12m^{2}_{\nu^{\prime}_{3R}}m^{2}_{\nu^{\prime}_{iR}}}{M^{4}_{Z^{\prime}}}\left(s-M^{2}_{Z^{\prime}}\right)^{2}\right]$

where $\sigma_{\text{SM}}$ and $\sigma_{\nu^{\prime}_{iR}}$ describe the DM
pair annihilation into the SM fermions and the heavy neutrino pair
$\bar{\nu}^{\prime}_{iR}\nu^{\prime}_{iR}$, respectively, and

$\displaystyle F_{1}(t,z_{1},z_{2})$ $\displaystyle=$ $\displaystyle
103+12(1+t^{2})\left[(23z_{2}-43z_{1})+3(1+t^{2})(37z^{2}_{1}+39z^{2}_{2}-70z_{1}z_{2})\right],$
$\displaystyle F_{2}(t,z_{1},z_{2})$ $\displaystyle=$ $\displaystyle
17+12(1+t^{2})\left[(z_{2}-5z_{1})+3(1+t^{2})(2z^{2}_{1}+z^{2}_{2}-2z_{1}z_{2})\right].$
(69)

The relic density of the DM $\nu^{\prime}_{3R}$ depends on five parameters
which are $t$, $z_{1,2}$, $m_{\nu^{\prime}_{3R}}$, and $M_{Z^{\prime}}$.

In Fig. 4, we show the prediction of our model for the DM relic abundance as a
function of the DM mass for various values of $t$ at which the $Z^{\prime}$
boson mass and $z_{1,2}$ are kept fixed all. In the left panel, the blue, red,
and purple curves correspond to $t=1.0$, $1.2$, and $1.4$, respectively, and
we have fixed $2z_{1}=z_{2}=1/8$ and $M_{Z^{\prime}}=4$ TeV. From this panel
and the top-left panel of Fig. 2, we can see that for $M_{Z^{\prime}}=4$ TeV
most of values of $t$ constrained in the previous section can lead to the
range of the observed DM relic abundance
$0.1183\leq\Omega_{\text{DM}}h^{2}\leq 0.1213$. In the right panel, the blue,
red, purple, and orange curves correspond to $t=1.3$, $1.415$, and $1.5$, and
$1.6$, respectively, and we have fixed $2z_{1}=z_{2}=1/10$ and
$M_{Z^{\prime}}=3.5$ TeV. This panel indicates that the value of $t$ should be
above the lower bound $1.415$ to lead to the range of the observed DM relic
abundance. In addition, Fig. 4 shows that, in order to achieve the observed DM
relic abundance, the DM mass should be around $M_{Z^{\prime}}/2$ corresponding
to the $Z^{\prime}$ boson resonance.

---
Figure 4: The DM relic abundance as a function of the DM mass for various
values of $t$. The green band refers to the range of the observed DM relic
abundance $0.1183\leq\Omega_{\text{DM}}h^{2}\leq 0.1213$. Left panel:
$2z_{1}=z_{2}=1/8$ and $M_{Z^{\prime}}=4$ TeV. Right panel:
$2z_{1}=z_{2}=1/10$ and $M_{Z^{\prime}}=3.5$ TeV.

In Fig. 5, we show the prediction of our model for the DM relic abundance as a
function of $t$ for various values of the $Z^{\prime}$ gauge boson mass with
$m_{\nu^{\prime}_{3R}}\simeq M_{Z^{\prime}}/2$.

---
Figure 5: The DM relic abundance as a function of $g_{2}/g_{1}$ for various
values of the $Z^{\prime}$ boson mass. In the left(right) panel, the blue,
red, purple, orange, and brown curevs correspond to $M_{Z^{\prime}}=3.5(2.9)$,
$3.8(3.1)$, $3.998(3.175)$, $4.2(3.3)$, and $4.5(3.4)$ TeV, respectively. The
black horizontal line represents the upper bound of the DM relic abundance.
Top-left panel: $z_{1}=1/4$ and $z_{2}=1/8$. Top-right panel: $z_{1}=1/5$ and
$z_{2}=1/10$.

From this figure, we find that if the $Z^{\prime}$ gauge boson mass is below a
certain value which is about $3998(3175)$ GeV for the left(right) panel,
almost values of $t$ satisfy the upper bound of the DM relic abundance
$\left(\Omega_{\text{DM}}h^{2}\right)_{\text{ub}}=0.1213$. On the contrary,
only the sufficiently small or large $t$ values satisfy this upper bound.

In Fig. 6, we show the allowed parameter region in the $t-M_{Z^{\prime}}$
plane which is consistent with the constraints discussed in the previous
section and the observed DM relic abundance.

---
Figure 6: Allowed parameter region given by the green region in the
$g_{2}/g_{1}-M_{Z^{\prime}}$ plane for several values of $z_{1}$ and $z_{2}$,
with $m_{\nu^{\prime}_{3R}}\simeq M_{Z^{\prime}}/2$. The region above the blue
curve is excluded by the upper bound of the DM relic abundance, whereas the
regions below the black and red curves are excluded by the current LHC limits
and the ambiguity of the $Z$ boson mass, respectively. Top-left panel:
$z_{1}=1/4$ and $z_{2}=1/8$. Top-right panel: $z_{1}=1/5$ and $z_{2}=1/10$.

The blue curve represents the upper bound on $M_{Z^{\prime}}$ as a function of
$t$ obtained from the constraint which the relic density of the DM candidate
$\nu^{\prime}_{3R}$ must not be larger than the upper bound of the DM relic
abundance. Whereas, the black and red curves represent the lower bounds on
$M_{Z^{\prime}}$ as the functions of $t$, which are obtained from the current
LHC limits and the ambiguity of the $Z$ gauge boson mass, respectively. The
green region refers to the allowed parameter region after combining these
three constraints. It indicates lower bounds for the parameter $t$ and the
$Z^{\prime}$ gauge boson mass, $t\gtrsim 0.9(1.1)$ and $M_{Z^{\prime}}\gtrsim
3.0(2.8)$ TeV for $2z_{1}=z_{2}=1/8$($2z_{1}=z_{2}=1/10$).

##  V Conclusion

In this work, we have constructed a general flavor-independent model in a
minimal way with the symmetry $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y^{\prime}}\times U(1)_{X}\times Z_{2}$ and the charge operator
identified in terms of both $U(1)_{Y^{\prime}}$ and $U(1)_{X}$ charges. We
have determined the gauge charges of the fields relying on the conditions of
the anomaly cancellation and the gauge invariance of the Yukawa couplings. The
smallness of active neutrino masses is explained through Type-I seesaw
mechanism only with two heavy right-handed neutrinos whose Majorana masses are
determined by the $U(1)_{Y^{\prime}}\times U(1)_{X}$ symmetry breaking scale.
We have obtained the physical states of the neutral gauge bosons and their
couplings to the fermions. We have studied the constraints on the mass and
gauge coupling of the new gauge boson coming from various current experiments
at which the current LHC limits and the ambiguity of the Standard Model
neutral gauge boson mass impose the most stringent bounds. Finally, we have
investigated the phenomenology of the $Z_{2}$-odd right-handed neutrino dark
matter. We have identified the allowed parameter space of the model which is
consistent with the upper bound on the dark matter relic abundance and other
bounds.

## Acknowledgements

I am grateful Theoretical Physics Group at IFIRSE for the warm hospitality
during my visit. I would like to thank Dr. LE Duc Ninh, Dr. DAO Thi Nhung, and
members in their group for the useful discussions.

## References

  * (1) P. Langacker, Phys. Rept. 72, 185 (1981).
  * (2) J. L. Hewett and T. G. Rizzo, Phys. Rept. 183, 193 (1989).
  * (3) C. H. Nam, Eur. Phys. J. C 79, 384 (2019).
  * (4) C. H. Nam, Eur. Phys. J. C 80, 231 (2020).
  * (5) C. H. Nam, arXiv: 2001.02421 [hep-ph].
  * (6) A. Davidson, Phys. Rev. D 20, 776 (1979).
  * (7) R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980); 44, 1643(E) (1980).
  * (8) X. He, G. C. Joshi, H. Lew, and R. Volkas, Phys. Rev. D 43, R22 (1991).
  * (9) S. Baek, N. G. Deshpande, X.-G. He, and P. Ko, Phys. Rev. D 64, 055006 (2001).
  * (10) T. Appelquist, B. A. Dobrescu, and A. R. Hopper, Phys. Rev. D 68, 035012 (2003).
  * (11) M. Carena, A. Daleo, B. A. Dobrescu, and T. M. P. Tait, Phys. Rev. D 70, 093009 (2004).
  * (12) S. Khalil, J. Phys. G 35, 055001 (2008).
  * (13) J. C. Montero and V. Pleitez, Phys. Lett. B 675, 64 (2009).
  * (14) S. Khalil, Phys. Rev. D 82, 077702 (2010).
  * (15) J. Heeck and W. Rodejohann, Phys. Rev. D 84, 075007 (2011).
  * (16) J. C. Montero and B. L. Sánchez-Vega, Phys. Rev. D 84, 053006 (2011).
  * (17) W. Chao, Phys. Lett. B 695, 157 (2011).
  * (18) A. Latosinski, K. A. Meissner, and H. Nicolai, Eur. Phys. J. C 73, 2336 (2013).
  * (19) M. Das and S. Mohanty, Phys. Rev. D 89, 025004 (2014).
  * (20) B. L. Sánchez-Vega, J. C. Montero, and E. R. Schmitz, Phys. Rev. D 90, 055022 (2014).
  * (21) W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Phys. Rev. D 89, 095033 (2014).
  * (22) S. Baek, H. Okada, and K. Yagyu, JHEP 1504, 049 (2015).
  * (23) E. Ma, N. Pollard, R. Srivastava, and M. Zakeri, Phys. Lett. B 750, 135 (2015).
  * (24) H.-S. Lee and S. Yun, Phys. Rev. D 93, 115028 (2016).
  * (25) W. Chao, Phys. Rev. D 93, 115013 (2016).
  * (26) A. Biswas, S. Choubey, and S. Khan, JHEP 1609, 147 (2016).
  * (27) A. Das, N. Okada, and N. Papapietro, Eur. Phys. J. C 77, 122 (2017).
  * (28) E. Accomando, L. Delle Rose, S. Moretti, E. Olaiya, and C. H. Shepherd-Themistocleous, JHEP 1704, 081 (2017).
  * (29) A. Biswas, S. Choubey, and S. Khan, Eur. Phys. J. C 77, 875 (2017).
  * (30) K. Asai, K. Hamaguchi and N. Nagata, Eur. Phys. J. C 77, 763 (2017).
  * (31) S. Singirala, R. Mohanta, and S. Patra, Eur. Phys. J. Plus 133, 477 (2018).
  * (32) A. Das, and N. Okada, and D. Raut, Eur. Phys. J. C 78, 696 (2018).
  * (33) T. Nomura, Eur. Phys. J. C 78, 189 (2018).
  * (34) S. Lee, T. Nomura, and H. Okada, Nucl. Phys. B 931, 179 (2018)
  * (35) M. Escudero, and S. J. Witte, and N. Rius, JHEP 08, 190 (2018).
  * (36) T. Nomura and H. Okada, Phys. Rev. D 97, 015015 (2018).
  * (37) E. Accomando, L. Delle Rose, S. Moretti, E. Olaiya, and C. H. Shepherd-Themistocleous, JHEP 1802, 109 (2018).
  * (38) A. Kamada, K. Kaneta, K. Yanagi, and H. B. Yu, JHEP 1806, 117 (2018).
  * (39) G. Arcadi, T. Hugle, and F. S. Queiroz, Phys. Lett. B 784, 151 (2018).
  * (40) H. Banerjee, P. Byakti, and S. Roy, Phys. Rev. D 98, 075022 (2018).
  * (41) D. A. Camargo, M. D. Campos, T. B. de Melo, and F. S. Queiroz, Phys. Lett. B 795, 319 (2019).
  * (42) C. Marzo, L. Marzola, and V. Vaskonen, Eur. Phys. J. C 79, 601 (2019).
  * (43) J.-X. Hou and C.-X. Yue, Eur. Phys. J. C 79, 983 (2019).
  * (44) C.-W. Chiang, G. Cottin, A. Das, and S. Mandal, JHEP 12,070 (2019).
  * (45) C. H. Nam, arXiv: 1912.12595 [hep-ph].
  * (46) D. Choudhury, K. Deka, T. Mandal, and S. Sadhukhan, arXiv: 2002.02349 [hep-ph].
  * (47) P. V. Dong, arXiv: 2003.13276 [hep-ph].
  * (48) D. V. Loi, C. H. Nam, N. H. Tan, and P. V. Dong, arXiv: 2004.06005 [hep-ph].
  * (49) D. Borah, S. J. Das, and A. K. Saha, arXiv: 2005.11328 [hep-ph].
  * (50) G. Belanger, A. Pukhov, and G. Servant, JCAP 0801, 009 (2008).
  * (51) N. Okada and O. Seto, Phys. Rev. D 82, 023507 (2010).
  * (52) S. Kanemura, O. Seto, and T. Shimomura, Phys. Rev. D 84, 016004 (2011).
  * (53) J.-M. Zheng, Z.-H. Yu, J.-W. Shao, X.-J. Bi, Z. Li, and H.-H. Zhang, Nucl. Phys. B 854, 350 (2012).
  * (54) N. Okada and Y. Orikasa, Phys. Rev. D 85, 115006 (2012).
  * (55) M. Lindner, D. Schmidt, and A. Watanabe, Phys. Rev. D 89, 013007 (2014).
  * (56) A. Alves, S. Profumo, and F. S. Queiroz, JHEP 04, 063 (2014).
  * (57) P. Agrawal, Z. Chacko, and C. B. Verhaaren, JHEP 08, 147 (2014).
  * (58) A. Alves, A. Berlin, S. Profumo, and F. S. Queiroz, Phys. Rev. D 92, 083004 (2015).
  * (59) B. L. Sanchez-Vega and E. R. Schmitz, Phys. Rev. D 92, 053007 (2015).
  * (60) M. Duerr, P. F. Perez, and J. Smirnov, Phys. Rev. D 92, 083521 (2015).
  * (61) A. Alves, A. Berlin, S. Profumo, and F. S. Queiroz, JHEP 10, 076 (2015).
  * (62) N. Okada and S. Okada, Phys. Rev. D 93, 075003 (2016).
  * (63) K. Kaneta, Z. Kang, and H.-S. Lee, JHEP 02, 031 (2017).
  * (64) S. Singirala, R. Mohanta, S. Patra, and S. Rao, JCAP 11, 026 (2018).
  * (65) W. Chao, H.-K. Guo, and Y. Zhang, JHEP 04, 034 (2017).
  * (66) D. Nanda and D. Borah, Phys. Rev. D 96, 115014 (2017).
  * (67) P. Bandyopadhyay, E. J. Chun, and R. Mandal, Phys. Rev. D 97, 015001 (2018).
  * (68) D. Liu, J. Liu, C. E. M. Wagner, and X.-P. Wang, Phys. Rev. D 97, 055021 (2018).
  * (69) P. T. P. Hutauruk, T. Nomura, H. Okada, and Y. Orikasa, Phys. Rev. D 99, 055041 (2019).
  * (70) M. Frank, K. Huitu, and S. Mondal, Phys. Rev. D 100, 115018 (2019).
  * (71) N. Okada, S. Okada, and D. Raut, Phys. Rev. D 100, 035022 (2019).
  * (72) P-H. Gu, Phys. Rev. D 101, 015006 (2020).
  * (73) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
  * (74) P. H. Frampton, S. L. Glashow, and T. Yanagida, Phys. Lett. B 548, 119 (2002).
  * (75) H. Huffel and G. Pocsik, Z. Phys. C8, 13 (1981).
  * (76) M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz, and S. Vogl, JHEP 09, 042 (2016).
  * (77) S. G. Porsev, K. Beloy, and A. Derevianko, Phys. Rev. Lett. 102, 181601 (2009); Phys. Rev. D 82, 036008 (2010).
  * (78) W. J. Marciano and J. L. Rosner, Phys. Rev. Lett. 65, 2963 (1990); 68, 898(E) (1992).
  * (79) R. Diener, S. Godfrey, and I. Turan, Phys. Rev. D 86, 115017 (2012).
  * (80) S. Schael et al. (ALEPH and DELPHI and L3 and OPAL and LEP Electroweak Collaborations), Phys. Rep. 532, 119 (2013).
  * (81) A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt, Eur. Phys. J. C 63, 189 (2009).
  * (82) M. Aaboud _et al._ , (ATLAS collaboration), J. High Energy Phys. 1710, 182 (2017).
  * (83) V. Barger, W.-Y. Keung, and G. Shaughnessy, Phys. Rev. D 78, 056007 (2008).
  * (84) H. Y. Cheng and C. W. Chiang, J. High Energy Phys. 1207, 009 (2012).
  * (85) E. Aprile _et al._ [XENON100 Collaboration], Phys. Rev. Lett. 111, 021301 (2013).
  * (86) P. A. R. Ade _et al._ (Planck Collaboration), Astron. Astrophys. 594, A13 (2016).
  * (87) P. Gondolo and G. Gelmini, Nucl. Phys.B 360, 145 (1991).


# Faraday spectroscopy of atoms confined in a dark optical trap

Matthew L. Terraciano  Mark Bashkansky  Fredrik K. Fatemi
<EMAIL_ADDRESS>Naval Research Laboratory, 4555 Overlook Ave. S.W.,
Washington, DC 20375

###### Abstract

We demonstrate Faraday spectroscopy with high duty cycle and sampling rate
using atoms confined to a blue-detuned optical trap. Our trap consists of a
crossed pair of high-charge-number hollow laser beams, which forms a dark,
box-like potential. We have used this to measure transient magnetic fields in
a 500$\mu$m-diameter spot over a 400 ms time window with nearly unit duty
cycle at a 500 Hz sampling rate. We use these measurements to quantify and
compensate time-varying magnetic fields to $\approx$10 nT per time sample.

###### pacs:

33.55.+b, 07.55.Ge, 37.10.Gh

##  I Introduction

A spin-polarized atom sample is strongly birefringent for near-resonant light.
This magneto-optic polarization rotation can be used for sensitive alkali-
vapor magnetometry Budker et al. (2002), and has been the subject of several
recent studies in a variety of cold atom samples Isayama et al. (1999);
Labeyrie et al. (2001); Franke-Arnold et al. (2001); Smith et al. (2003);
Geremia et al. (2005). When applied to localized cold atom ensembles, the
result can be sensitive magnetometry with linear spatial resolution of a few
tens of microns Vengalattore et al. (2007). These magnetic microscopes could
be of use for imaging fields near a variety of surfaces, including integrated
circuits Chatraphorn et al. (2000); Wildermuth et al. (2006) and atom chips
designed for cold atom interferometry Wang et al. (2005). At a more
fundamental level, Faraday spectroscopy has also been considered for searches
of atomic electric dipole moments (EDM) Romalis and Fortson (1999) and for
nondestructive quantum state estimation and preparation Smith et al. (2004);
Chaudhury et al. (2007); Geremia et al. (2005). Such measurements benefit from
large atom numbers, long interrogation times, and “field-free” confinement,
_i.e._ confinement in which the trapping potential minimally perturbs the
measurement.

A simple way to achieve field-free conditions for a cold atom sample is to
release the atoms from a trap and probe them during freefall. A drawback of
this is that the maximum interrogation time is limited to a few tens of
milliseconds as the atom cloud falls away from the interaction region. Isayama
et. al. Isayama et al. (1999) reported a Faraday signal from atoms in freefall
with a 1/$e$ decay time of 11 ms. This limitation has been overcome by
confining the atoms to the antinodes of a red-detuned optical lattice in which
one of the lattice beams also serves as a probe beam Smith et al. (2003). When
the atoms were held in the intensity nodes of a blue-detuned lattice, dark-
field confinement was achieved, although the signal was reduced because the
interaction with the probe was correspondingly diminished.

By confining atoms in a blue-detuned trap, however, it is possible to achieve
the simultaneous conditions of long interrogation time, low-field confinement,
and large atom-number Ozeri et al. (1999); Friedman et al. (2002); Kaplan et
al. (2005); Kulin et al. (2001). Blue-detuned traps produce lower light shifts
and photon scattering rates than red-detuned traps, enabling deep, large
volume traps with low power requirements. Although these traps have been
proposed for use in magnetometry Budker et al. (2002); Isayama et al. (1999)
and EDM searches Romalis and Fortson (1999), to the best of our knowledge, no
experimental demonstrations have been performed.

In this paper, we report the use of dark optical traps to confine atoms in a
submillimeter, box-like volume for dynamic magnetometry using Faraday
spectroscopy. The traps are formed from crossed, high-charge-number hollow
laser beams Fatemi and Bashkansky (2007); Fatemi et al. (2007). By
repetitively spin-polarizing the confined sample, we extend the measurement
time from only a few milliseconds to $\approx$400 ms in a single loading cycle
with up to 1 kHz sampling rate. We demonstrate the technique by measuring and
compensating ambient time-varying magnetic fields, such as those arising from
eddy currents and the AC power line. We also show that nonlinear spin dynamics
due to the probe beam Smith et al. (2004) are preserved in these traps. The
increase in duty cycle demonstrated here is promising for both magnetometry
and for efficient quantum state preparation based on these nonlinear dynamics.

Figure 1: (Color online) Layout of experiment. Crossed hollow beams confine
atoms to a 0.48mm diameter spot. Relay lenses for the hollow beams are not
shown. Faraday pump and probe beams propagate along the x-axis to a balanced
polarimeter. WP: Wollaston prism. Osc: Oscilloscope. Image of beam and cross
sectional profile are shown.

Figure 1 shows the schematic layout of the experiment. The hollow beam is
relayed to intersect itself by an $8f$ imaging relay, as described in Ref.
Fatemi et al. (2007). Helmholtz coils on all three axes control the magnetic
field. The hollow beams for our trap are formed by modifying the wavefront
phase of a Gaussian beam with a reflective spatial light modulator (SLM). SLMs
have found increasing value in cold atom manipulation experiments because of
their ability to control trap parameters in a programmable manner and to
produce traps with nontrivial intensity profiles Olson et al. (2007); Fatemi
et al. (2007); McGloin et al. (2003); Pasienski and DeMarco (2008);
Chattrapiban et al. (2006). The applied phase for the hollow beams used here
has a profile $\Psi(\rho,\phi)=n\phi+f\lambda/(\pi\rho^{2})$, where $\rho$ and
$\phi$ are cylindrical coordinates and $n$ is an integer. The second term is a
lens function of focal length $f\approx 200$mm to focus the beam of wavelength
$\lambda$ onto the atom sample. For high charge number beams ($n\geq 4$), we
usually operate the trap a few centimeters away from the focal plane, where
aberrations are reduced and the peak intensity is maximum Fatemi and
Bashkansky (2007).

The light for the hollow beam is derived from a tunable extended cavity diode
laser. It is amplified to 400 mW by a tapered amplifier, 200 mW of which is
coupled into polarization-maintaining fiber. Residual resonant light from
amplified spontaneous emission is filtered out by a heated vapor cell. The
fiber output is collimated to a 1/$e^{2}$ waist of 1.71 mm, and modified by
the SLM (Boulder Nonlinear Systems), which has $\approx$ 90% diffraction
efficiency. The SLM has been calibrated at the pixel level to correct for
wavefront distortion intrinsic to the SLM. An image of the beam is shown in
the inset to Fig. 1. Our choice of $n=8$ is driven by the practical
considerations of field-free confinement and large trap size, but these trap
parameters can be adjusted with the SLM.

Our experiment begins with cold 85Rb atoms derived from a magneto-optical trap
(MOT). We confine $\approx 10^{7}$ atoms in a $\approx 500\mu$m diameter
(1/$e^{2}$) cloud. The atoms are further cooled in a 10 ms long molasses stage
to $\approx 10\mu$K, after which all MOT-related beams are extinguished. The
hollow beam trap is on throughout the MOT loading, but can be switched off by
an acoustooptic modulator. The Faraday spectroscopy is performed by similar
technique as in Ref. Isayama et al. (1999). To perform these measurements, a
pair of laser beams is used along the $x$-axis (Fig. 1). The atoms are
optically pumped into the $F=3,m_{F}=3$ stretched state by a 20 $\mu$s
$\sigma^{+}$ pulse connecting $F=3\rightarrow F^{\prime}=3$. This beam has
1/$e^{2}$ waist of 6.0 mm and has a peak intensity of $\approx 3I_{sat}$,
where $I_{sat}$ is 1.6 mW/cm2. This beam is retroreflected to prevent
unidirectional momentum kicks, and a small amount of repumper light ($\approx
0.04I_{sat}$) is added during this pulse to keep the atoms in the $F=3$
hyperfine ground state. When this light is extinguished, the atoms begin
precessing freely at the Larmor precession frequency
$\omega_{L}=g_{F}\mu_{B}B/\hbar$, where $g_{F}$ is the gyromagnetic ratio,
$\mu_{B}$ is the Bohr magneton, and $B$ is the magnetic field. For 85Rb,
$g_{F}\mu_{B}/\hbar=466.7415$ kHz/Gauss Alexandrov et al. (2004). A linearly
polarized probe beam at a detuning $\Delta_{p}=2\pi\times 2.5$ GHz with
$\simeq$20mW and 1/$e^{2}$ waist $\omega_{p}$ = 6.0mm passes through the atom
cloud to a simple polarimeter consisting of a Wollaston prism that splits the
probe beam into two orthogonal polarization states that are detected by a
balanced photodetector. For these parameters, the photon scattering time from
the probe beam is calculated to be $\tau_{p}\approx$2ms. The MOT region is
imaged onto a pinhole along the axis of the probe beam so that only the
portion of the probe that interacts with the confined atoms reaches the
detector.

The hollow beam trap prevents the atoms from falling away from the interaction
region during the probing process. The beam has 150 mW total power at the
trap. We use a detuning $\Delta_{t}\approx 25$GHz (=0.05 nm) above the
$F=3\rightarrow F^{\prime}=4$ transition. At the MOT, the hollow beam has a
diameter of 0.48 mm, measured between maxima, and the peak intensity is
$8.2\times 10^{4}$ mW/cm2 for a trap depth $U\approx 2\hbar\Gamma\approx
3000E_{r}$, where $E_{r}$ is the recoil energy for 85Rb. The gravitational
potential energy across this trap is $\hbar\Gamma/6$. For these parameters,
the peak scattering rate from the trapping beams would be
$\gamma_{t}=1/\tau_{t}\approx 2\pi\times 3$kHz, but is reduced from this value
by being trapped in the dark. Although we do not measure this value, we
establish an upper bound to be $\gamma_{t}{\leq}2\pi\times 200$Hz.

Figure 2:  (Color online) Faraday signals for a) trapped sample; and b)
untrapped sample. The untrapped atoms fall away from the probe region within
20 ms, while the trapped atoms remain with 150 ms time constant. Insets to (a)
are expanded views of the raw data.

Each optical pumping event initiates the Larmor precession. Figure 2a shows 64
averages of 200 optical pumping cycles spaced 2 ms apart in the presence of
the hollow beam trap and a bias magnetic field of $\approx 100$ mG along the
$z$-axis. A single Larmor precession signal is shown in the lower inset to
Fig. 2a. The envelope over all Larmor precession signals decays with a 1/$e$
time constant of $\approx 150$ ms. This decay is due primarily to the steady
heating that occurs during each optical pumping cycle, which gradually boils
atoms out of the trap. In contrast, Fig. 2b shows the signals without the
hollow beam trap present. In this case, the atoms fall completely out of the
probe beam detection window within 25 ms, with a 1/$e$ decay time of 13 ms,
similar to that reported in Ref. Isayama et al. (1999). The signals in Fig. 2
are recorded immediately following the molasses phase of the MOT loading
cycle. Over the first few pumping cycles, the envelope of the individual
precession signals in Fig. 2b changes dramatically due to residual eddy
currents in the vacuum chamber. Holding the atoms in an optical trap allows
measurements to be performed after eddy currents have subsided, while also
substantially increasing both the measurement window and the overall duty
cycle.

For our parameters, each independent Larmor precession signal dephases with a
submillisecond 1/$e$ decay time. This dephasing occurs from several factors,
including spatial gradients and photon scattering from the trap and probe
beams. Nonlinear Hamiltonian terms can also shorten the decay time of the
signal, as described in Ref. Smith et al. (2004). These nonlinear terms depend
on the angle between the polarization of the probe laser and the magnetic
field. When the relative angle is $\approx$ 54∘, the effects of these terms
are eliminated. For this work, we operated at this relative orientation so
that the dephasing occurs primarily through photon scattering. From Fig. 2, we
find that the untrapped signals decay with a 1/$e$ time of $\approx$0.7ms. For
the samples trapped in the hollow beam, we observe a slight reduction in the
decay time to $\approx$0.5 ms. Thus we have an upper bound for $\gamma_{t}\leq
2\pi\times 200$Hz.

Figure 3:  (Color online) a) Polarimeter output for single-shot data (top) and
averaged data (bottom). b) Fast-Fourier transform (FFT) of data in (a). Fits
to a Lorentzian profile are shown as solid lines. Single-shot (circles) and
averaged (squares) data are shown.

In a gradient-free, static magnetic field, the voltage output of the
polarimeter is an exponentially-damped sinusoid, $V(t)=$
Aexp(-$t/\tau$)sin$(2\pi\nu_{L}t+\phi)$, where A is the initial amplitude,
$\tau$ is the 1/$e$ decay time, $\omega_{L}=2\pi\times\nu_{L}$ is the Larmor
frequency, and $\phi$ is a phase. To determine $\nu_{L}$, the averaged data in
each 2 ms probing window (Fig. 3a) are Fourier transformed (Fig. 3b). We fit
these transforms to a Lorentzian, the center of which is $\nu_{L}$.

Figure 4: (Color online) a) Larmor precession frequency as a function of time
with various levels of compensation. Dashed: no compensation; dotted:
compensation of eddy currents, and solid: full compensation. Inset to (a)
shows magnified view of compensation. Fluctuations in our Larmor precession
are dominated by uncompensated harmonics of the AC line. b) Magnetic field
spectrum with no compensation (dotted), and 60Hz compensation (solid).

In Fig. 4a, we plot $\nu_{L}(t)$ over 200 pumping cycles (64 averages) spaced
1 ms apart (dashed line). The signal displays two dominant sources of time-
dependence. First, the exponential decay occurs from the metal vacuum chamber,
which develops eddy currents when the MOT coils are extinguished. Due to the
symmetry of our chamber, the eddy currents are along the axis of the MOT coils
($z$). The bias field of $\nu_{L}\approx 50$kHz for these measurements is also
on the $z$-axis so that the eddy current field adds linearly to the bias
field. The second source of time-dependent behavior is ambient AC magnetic
fields in the room arising from power supply transformers, power strips, etc.
We note that our experiment is triggered off the AC power line. We found that
this field is also primarily along the $z$-axis, because the amplitude of the
oscillation signal is independent of this bias field. An orthogonal component
would add in quadrature and cause the amplitude to vary with the bias field.
Additionally, an orthogonal, oscillating magnetic field component added in
quadrature would show up at twice its oscillation frequency. Since the Larmor
frequencies retrieved from Faraday spectroscopy determine the scalar magnetic
field, full vector information is not acquired in a single shot, but can be
acquired through multiple measurements Terraciano et al. (2007). Some
information about magnetic field orientation can be obtained directly from the
polarimeter signal (e.g. there is no spin precession if the $B$ field is
parallel to the optical pumping axis), but that effect is outside the scope of
this work.

For many applications, control over the magnetic field is required to sub-mG
levels, especially those involving Raman transitions between magnetically
sensitive states Boyer et al. (2004); Ringot et al. (2001); Terraciano et al.
(2007); Kerman et al. (2000). As a simple application of the long measurement
time capability, we demonstrate compensation of these time varying fields. We
first compensate the effects of eddy currents, which produce an exponentially
decaying magnetic field at the atom sample. This field decays with a 1/$e$
time of $\approx$20 ms (Fig. 4a). For a given MOT coil current setting, the
eddy current amplitude is constant. We produce an opposing time-varying field
flux by using a voltage-controlled current source (Kepco ATE15-15M). This
current passes through a 20-turn Helmholtz pair of diameter 20 cm, width 2.5
cm, and separation 11.4 cm oriented along the MOT coil axis. The appropriate
time variation is done by low-pass filtering of a step function whose
amplitude is adjusted for optimum compensation. The result is shown in Fig. 4a
(dotted line). Although this source of time variation is not canceled
perfectly, the field beyond 25 ms is constant to within the 60 Hz field
amplitude.

The ambient AC magnetic fields are primarily due to 60 Hz power line sources.
By triggering our experiment from the power line, this source of magnetic
field variation is reproducible and can be compensated. Without this
triggering, the variations of a few mG observed in Fig. 4a would lead to
significant shot-to-shot fluctuations of the field measurements. We produce an
opposing field by adding a 60 Hz sinusoidally varying current to the bias
coils. The current amplitude and phase are adjusted for optimum compensation.
The result with all compensations applied is shown in the inset to Fig. 4a.
The signal remains constant to within a standard deviation of 110 Hz (230
$\mu$G). Most of this residual field is due to higher AC line harmonics; in
Fig. 4b, we show frequency spectrum of the magnetic field, which clearly shows
higher harmonics at 180, 300, and 420 Hz. We suppressed the 60 Hz component by
a factor of 20. With appropriate signal processing, the field measurements in
our setup could be made in real time (with single-shot measurements as in Fig.
3) and be used as feedback control with a bandwidth determined by the
Helmholtz compensation coils.

Figure 5:  2D image of data that exposes qualitative magnetic field features
without FFT processing. (a) uncompensated, (b) 60 Hz compensation, c) full
compensation.

Another way to visualize the time dependent signals, shown in Fig. 5a, is by
converting the 1D data set of Fig. 2 to a 2D matrix. Each successive column
contains the Larmor precession signal for subsequent triggers. This exposes
time variations in an easily identifiable way with no FFT analysis. We show
these images for the magnetic fields with no compensation, 60 Hz compensation
and full compensation. A constant magnetic field shows up as a series of
horizontal lines whose spacing is inversely proportional to $\omega_{L}$ (Fig.
5c).

Figure 6: (Color online) Comparison of 64 data runs (dashed) to a simulated
signal with white noise added to have similar SNR as the experiment (black).
For clarity, the signals are offset from each other.

Because of the uncompensated field variations in our lab, our measurement
uncertainty is dominated by systematic errors. To differentiate the systematic
error from the random error, we measure the Larmor precession frequency in a 1
ms measurement window for 64 independent loading cycles at a trapping time of
T = 100 ms. These scans are recorded at a 1 Hz rate. The result is shown in
Fig. 6. There is a long term drift in our lab on the order of several seconds.
Our time-domain signals have a signal-to-noise ratio (SNR) of $\approx$15
(Fig. 3a). For comparison, we simulated the expected Larmor signals for
exponentially-damped sinusoids with the same SNR that had additive white noise
(Fig. 6). For the experimentally measured case, we found a standard deviation,
or single-shot error, in each 1 ms optical pumping cycle of
$\delta\nu_{L}\approx$45Hz, or $\delta{B}\approx 100\mu$G (=10 nT), and for
the simulated case, the error was $\delta\nu_{L}\approx$16 Hz, or
$\delta{B}\approx 30\mu$G (=3 nT). This discrepancy is likely due to other
sources, such as unwanted variations in the MOT coil current.

Our hollow beam traps are initially loaded with $N\approx 10^{6}$ atoms. The
shot-noise-limited magnetic field measurement error due to atom number is
$\delta{B}\simeq\left(\hbar/g\mu_{B}\right)(1/\sqrt{N\tau{T_{m}}})$, where
$\tau$ is the spin-coherence time and $T_{m}$ is the measurement time Budker
et al. (2002). Because we are measuring a rapidly varying field,
$T_{m}=1\rm{-}2$ms, limiting $\delta{B}\approx 2\mu$G (=200 pT) in each
optical pumping cycle. After T = 400 ms of trapping time, when there are only
$\approx 10^{5}$ atoms remaining, this increases to $\approx 6\mu$G (=600 pT).
Our measured values are above the shot noise limit due to the simple
photodetection circuit we used and to incomplete optical pumping, which
effectively reduces N.

For static magnetic fields, each measurement cycle through the total trap time
can be averaged, effectively increasing $T_{m}$ to several hundred
milliseconds and greatly increasing the shot-noise-limited sensitivity.
Likewise, $\tau$ can be increased by using larger detunings for the probe and
trapping beams. These blue-detuned traps are capable of capturing large enough
atom numbers that measurements in the low pT range or better should be
possible in a single MOT loading cycle over the entire measurement window.

Figure 7: (Color online) Simulations of atom number remaining for different
degrees of optical pumping. With no optical pumping or probe beams (longest-
lived curve) the atoms still scattered $\approx$1 photon/ms from the trap
beams. A curve fit is shown for the case of scattering 7 photons/ms. This time
constant of 160 ms agrees well with our experimental value of 150 ms (Fig. 2).

For any trap depth, there is a trade-off between SNR and the number of
possible field measurements allowed before the signal decays. SNR improves by
increasing the number of atoms that are optically pumped or by decreasing the
pump detuning Smith et al. (2003), but these approaches also boil the atoms
out of the trap more quickly. For most of our results presented here, we only
weakly pumped the atoms to reduce heating and to increase the number of
optical pumping cycles we could achieve. In general, the dominant heating will
occur from the $20\mu$s optical pumping phase of each cycle, during which
several photons are scattered. As a rough estimate, the timescale for signal
decay should be on the order of the time required for the average atom energy
to equal the trap depth. This will occur after a time
$T_{boil}=U/(\gamma_{tot}E_{r})$, where U is the trap depth, $\gamma_{tot}$ is
the total scattering rate (including probe, trap, and optical pumping beams),
and $E_{r}$ is the recoil energy. For our trap of $U~{}\approx 3000E_{r}$, and
assuming $\approx 10$ scattered photons every 2ms optical pumping cycle, this
gives $\approx 300$ pumping cycles before the atoms are boiled away.

To examine this boiling process more accurately, we perform Monte Carlo
simulations of the atom dynamics within our trap for different total
scattering rates. Within each time step, the atom’s momentum is changed with a
probability determined by the local scattering rate for the probe and trap
beams, as calculated by the Kramers-Heisenberg formula Miller et al. (1993).
We performed these simulations for different optical pumping rates. In Fig. 7
we plot the number of atoms remaining as a function of time for varying
degrees of scattering rates. For the case of 7 photons scattered every
millisecond, we find an exponential decay of $\approx$160 ms, which is close
to our observed value of $T_{boil}=150$ms. By turning off the probe and
optical pumping beams in the simulation, we find that the trap beam scattering
rate is $\gamma_{t}\approx 2\pi\times 100$Hz which agrees with our upper bound
of $\gamma_{t}\leq 2\pi\times 200$Hz from the Faraday decay time.

Within our measurement error, we observed no effect of the trapping light on
the Larmor frequency. Optically-induced Zeeman shifts that occur with
elliptically polarized light Vengalattore et al. (2007); Romalis and Fortson
(1999); Park et al. (2002) should be small, because the trap beam
polarizations are linear and because of the low field confinement.
Furthermore, any vector light shifts from the trap beams, confined to the
$x-y$ plane, would add in quadrature to our applied magnetic field along $z$,
reducing the effect on $\omega_{L}$ Romalis and Fortson (1999). We are
currently studying the effects of trap geometry on the Larmor precession
signals.

Figure 8: (Color online) Larmor precession signals with and without the trap
light at two different relative orientations of the laser polarization with
respect to the magnetic field. These were done at a) 0 degrees (revivals
maximized) and b) 54 degrees (revivals suppressed).

The dominant source of nonlinear effects due to the laser fields is the probe
light. As discussed in Ref. Smith et al. (2004), the tensor component to the
light shift adds a nonlinear term to the spin Hamiltonian, whose magnitude is
dependent on the angle between the laser polarization and the magnetic field.
This Hamiltonian plays an important role in studies of quantum chaos and is
useful for both nondestructive quantum state preparation and measurement Smith
et al. (2004). For sufficiently large magnetic fields, the nonlinearity
vanishes when the relative angle is $\theta$=arctan$(\sqrt{2})\approx
54^{\circ}$, but is maximized for $\theta=0$. We have verified that these
nonlinear spin dynamics, which manifest themselves as revivals of the Faraday
oscillation signal, can still be observed in these hollow beam traps. In Fig.
8, we show the Larmor precession signals for $\theta=0$ and
$\theta=54^{\circ}$ both with the trap on continuously and with the trap
switched off immediately prior to the optical pumping pulse. Thus the high
duty cycle of the technique presented here may be of use for rapidly testing
quantum state preparation procedures employing this nonlinearity.

We have demonstrated Faraday spectroscopy with high repetition rate, long
measurement time, and submillimeter spatial resolution in a dark hollow beam
optical trap. We used high-charge-number hollow laser beams to provide box-
like confinement with near resonant light and low laser power. These traps can
be sufficiently deep that several hundred Faraday measurements are possible
before atoms are heated over the confining potential. We demonstrated a
continuous magnetic field measurement over a period of 400 ms which enabled us
to measure and compensate for time-varying magnetic fields. This work was
funded by the Office of Naval Research and by the Defense Advanced Research
Projects Agency.

## References

  * Budker et al. (2002) D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, Rev. Mod. Phys. 74, 1153 (2002).
  * Isayama et al. (1999) T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, Phys. Rev. A 59, 4836 (1999).
  * Labeyrie et al. (2001) G. Labeyrie, C. Miniatura, and R. Kaiser, Phys. Rev. A 64, 033402 (2001).
  * Franke-Arnold et al. (2001) S. Franke-Arnold, M. Arndt, and A. Zeilinger, J. Phys. B 34, 2527 (2001).
  * Smith et al. (2003) G. A. Smith, S. Chaudhury, and P. S. Jessen, J. Opt. B 5, 323 (2003).
  * Geremia et al. (2005) J. M. Geremia, J. K. Stockton, and H. Mabuchi, Phys. Rev. Lett. 94, 203002 (2005).
  * Vengalattore et al. (2007) M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, Phys. Rev. Lett. 98, 200801 (2007).
  * Chatraphorn et al. (2000) S. Chatraphorn, E. F. Fleet, F. C. Wellstood, L. A. Knauss, and T. M. Eiles, Appl. Phys. Lett. 76, 2304 (2000).
  * Wildermuth et al. (2006) S. Wildermuth, S. Hofferberth, I. Lesanovsky, S. Groth, P. Krüger, J. Schmiedmayer, and I. Bar-Joseph, Appl. Phys. Lett. 88, 264103 (2006).
  * Wang et al. (2005) Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, Phys. Rev. Lett. 94, 090405 (2005).
  * Romalis and Fortson (1999) M. V. Romalis and E. N. Fortson, Phys. Rev. A 59, 4547 (1999).
  * Smith et al. (2004) G. A. Smith, S. Chaudhury, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. 93, 163602 (2004).
  * Chaudhury et al. (2007) S. Chaudhury, S. Merkel, T. Herr, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. 99, 163002 (2007).
  * Ozeri et al. (1999) R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
  * Friedman et al. (2002) N. Friedman, A. Kaplan, and N. Davidson, Adv. Atom. Mol. Opt. Phys. 48, 99 (2002).
  * Kaplan et al. (2005) A. Kaplan, M. F. Andersen, T. Grunzweig, and N. Davidson, J. Opt. B 7, R103 (2005).
  * Kulin et al. (2001) S. Kulin, S. Aubin, S. Christe, B. Peker, S. L. Rolston, and L. A. Orozco, J. Opt. B 3, 353 (2001).
  * Fatemi and Bashkansky (2007) F. K. Fatemi and M. Bashkansky, Appl. Opt. 46, 7573 (2007).
  * Fatemi et al. (2007) F. K. Fatemi, M. Bashkansky, and Z. Dutton, Opt. Express 15, 3589 (2007).
  * Olson et al. (2007) S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
  * McGloin et al. (2003) D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, Opt. Express 11, 158 (2003).
  * Pasienski and DeMarco (2008) M. Pasienski and B. DeMarco, Opt. Express 16, 2176 (2008).
  * Chattrapiban et al. (2006) N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, J. Opt. Soc. Am. B 23, 94 (2006).
  * Alexandrov et al. (2004) E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, Tech. Phys. 49, 779 (2004).
  * Boyer et al. (2004) V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, Phys. Rev. A 70, 043405 (2004).
  * Ringot et al. (2001) J. Ringot, P. Szriftgiser, and J. C. Garreau, Phys. Rev. A 65, 013403 (2001).
  * Terraciano et al. (2007) M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, Phys. Rev. A 76, 053421 (2007).
  * Kerman et al. (2000) A. J. Kerman, V. Vuletić, C. Chin, and S. Chu, Phys. Rev. Lett. 84, 439 (2000).
  * Miller et al. (1993) J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
  * Park et al. (2002) C. Y. Park, J. Y. Kim, J. M. Song, and D. Cho, Phys. Rev. A 65, 033410 (2002).