#  Efficient approach to low scale Peccei-Quinn symmetry breaking without
domain wall problem

Pei-Hong Gu<EMAIL_ADDRESS>School of Physics, Jiulonghu Campus, Southeast
University, Nanjing 211189, China

###### Abstract

We propose an efficient mechanism to realize an invisible axion from a low
scale Peccei-Quinn symmetry breaking. Our basic model only contains a gauge
boson, an up-type vector-like quark, two Higgs doublets and two Higgs singlets
besides the standard model fermions and gauge bosons. The physical Peccei-
Quinn global symmetry is a result of two independent global symmetries
connected by the new gauge symmetry. Anyone of these two global symmetries
only acts on either the right-handed top quark or the left-handed new quark so
that it can avoid the domain wall problem. Thanks to the electroweak and new
gauge interactions, the Higgs doublet for the top quark mass generation and
the Higgs singlet for the new quark mass generation can only contribute a tiny
fraction in the axion. The axion decay constant can be largely enhanced by a
factor composed of the vacuum expectation values of the four Higgs scalars.

##  I Introduction

Currently the global Peccei-Quinn (PQ) symmetry pq1977  $U(1)_{PQ}$ has become
the most popular scheme to solve the so-called strong CP problem. In this
scenario, a Nambu-Goldstone (NG) boson from the spontaneous PQ symmetry
breaking picks up a mass through the color anomaly adler1969 ; bj1969 ; ab1969
, and hence becomes a pseudo NG boson, named the axion weinberg1978 ;
wilczek1978 . Since the axion has not been observed in any experiments, it can
only interact with the standard model (SM) particles at an extremely weak
level kg2010 ; dgnv2020 ; pdg2020 . For this purpose, the Kim-Shifman-
Vainstein-Zakharov (KSVZ) kim1979 ; svz1980  and Dine-Fischler-Srednicki-
Zhitnitsky (DFSZ) dfs1981 ; zhitnitsky1980  models were proposed after the
original PQ model was quickly ruled out. In the KSVZ and DFSZ models kim1979 ;
svz1980 ; dfs1981 ; zhitnitsky1980 , the PQ symmetry should be spontaneously
broken far far above the weak scale to make the axion invisible kg2010 ;
dgnv2020 ; pdg2020 .

In the invisible axion models, the accompanying new particles would be too
heavy, i.e. of the order of the PQ symmetry breaking scale, to leave any
experimental signals, unless the related couplings were artificially small. In
other words, all experimental attempts to discover the axion can only depend
on the mixing of the axion to the $\pi$ and $\eta$ mesons kg2010 ; dgnv2020 ;
pdg2020 . A high scale PQ symmetry breaking also has some theoretical
disadvantages. For example, the more high the PQ symmetry breaking scale is,
the more serious the axion quality problem becomes dgnv2020 . Moreover, for a
huge hierarchy between the PQ and electroweak symmetry breaking scales, unless
the inevitable Higgs portal has an extremely small coupling, its contribution
should have a large cancellation with the rarely quadratic term of the SM
Higgs scalar cv2016 . In order to realize a low scale PQ symmetry breaking,
people have tried to consider an anomalous gauge symmetry mrs2001  or enormous
scalar fields cky2014 ; kr2016 ; ci2016 ; gu2021 .

It was pointed out sikivie1982  after the $U(1)_{PQ}$ global symmetry was
explicitly broken by the QCD gluon anomaly, its discrete subgroup $Z_{N_{DW}}$
could remain unbroken and hence there would be $N_{DW}$ vacua with exactly
same physical properties. Here the index $N_{DW}$ is nothing but the number of
the quarks transform under the $U(1)_{PQ}$ symmetry. The early universe thus
could suffer a serious problem of domain wall formation after the spontaneous
breaking of the $U(1)_{PQ}$ global symmetry and then the $Z_{N_{DW}}$ discrete
symmetry. One solution to this problem is to render a unique vacuum by
assuming that only one quark is charged under the $U(1)_{PQ}$ symmetry pwy1986
; kw1986 , like the original KSVZ model just with one vector-like quark.
However, it seems nontrivial to do such an assignment in the DSFZ-like models
with two or more Higgs doublets, where an up-type quark must be accompanied by
a down-type quark to simultaneously couple with the axion dfs1981 ;
zhitnitsky1980 .

On the other hand, the top quark is the unique SM fermion whose Yukawa
coupling to the SM Higgs boson is of the order of unity. Being heavier than a
$W^{\pm}$ boson, it is the only quark to decay before hadronization and have a
very short lifetime. Therefore, the top quark plays a special role in the SM
pdg2020 . This special role implies that the top quark may have a different
mass origin from the other SM fermions. The heavy top quark mass can be argued
by a proper PQ symmetry, under which the right-handed top quark is the unique
SM fermion carrying a nontrivial PQ charge pwy1986 ; kw1986 ; cfty2015 ;
cfty2017 ; ctty2018  so that it can exclusively couple to one Higgs doublet
with a bigger vacuum expectation value (VEV) while the other SM fermions can
only couple to another Higgs doublet with a smaller VEV.

The scalars and fermions |  |  $\xi$ |  $\sigma$ |  $\phi$ |  $\eta$ |  |  $t_{R}$ |  $T_{L}$ |  $T_{R}$ |  The other SM fermions
---|---|---|---|---|---|---|---|---|---|---
The $U(1)_{X}$ gauge symmetry |  | $+\frac{1}{2}$ | $+\frac{1}{2}$ | $+\frac{1}{2}$ | $0$ |  | $+\frac{1}{2}$ | $+\frac{1}{2}$ | $0$ | $0$
The $U(1)_{PQ}$ global symmetry |  | $-1$ | $x\neq-1$ | $-1$ | $0$ |  | $-1$ | $x\neq-1$ | $0$ | $0$
Table 1:  The scalars and fermions in our basic models. The scalars $\xi$ and
$\sigma$ are the Higgs singlets while $\phi$ and $\eta$ are the Higgs
doublets. The new quarks $T_{L,R}$ carry an electric charge $+\frac{2}{3}$ as
same as the up-type quarks. The right-handed top quark is the unique SM
fermion carrying the nonzero quantum numbers under the $U(1)_{X}$ and
$U(1)_{PQ}$ symmetries. While the $U(1)_{X}$ gauge symmetry is free of
anomaly, the $U(1)_{PQ}$ global symmetry keeps anomalous.

In the present work we shall propose an efficient mechanism to realize an
invisible axion from a spontaneous PQ symmetry breaking at low scales. In our
basic model, the right-handed top quark is the unique SM fermion charged under
a new $U(1)_{X}$ gauge symmetry. Then a new left-handed up-type quark and its
right-handed partner are introduced to cancel the gauge anomalies. The new
left-handed and right-handed quarks construct their Yukawa coupling with a
Higgs singlet for spontaneously breaking the $U(1)_{X}$ gauge symmetry.
Another Higgs singlet dominates the spontaneous breaking of the $U(1)_{PQ}$
global symmetry. There is also a Higgs doublet coupling to the SM fermions
except for the top quark. A second Higgs doublet exclusively couples to the
top quark and dominantly leads to the electroweak symmetry breaking. Because
of the gauge interactions, the Higgs doublet for the top quark mass generation
as well as the Higgs singlet for the new quark mass generation can only
contribute a negligible fraction to the axion. The axion decay constant can be
largely enhanced by a factor composed of the VEVs of the four Higgs scalars.
In our mechanism, the $U(1)_{PQ}$ symmetry indeed is a result of two
independent $U(1)$ global symmetries connected by the $U(1)_{X}$ gauge
symmetry. Every $U(1)$ global symmetry only acts on either the right-handed
top quark or the left-handed new quark. So, the spontaneous breaking of the
two $U(1)$ global symmetries and hence the physical $U(1)_{PQ}$ global
symmetry can avoid the domain wall problem.

##  II Basic model

We now begin to demonstrate our mechanism in details. The scalars and fermions
in our basic model are summarised in Table 1. The scalars $\xi$ and $\sigma$
are the Higgs singlets while $\phi$ and $\eta$ are the Higgs doublets. The new
quarks $T_{L,R}$ carry the electric charge $+\frac{2}{3}$ as same as the up-
type quarks. The right-handed top quark is the unique SM fermion carrying the
nonzero quantum numbers under the $U(1)_{X}$ and $U(1)_{PQ}$ symmetries. While
the $U(1)_{X}$ gauge symmetry is free of anomaly, the $U(1)_{PQ}$ global
symmetry keeps anomalous. The full scalar potential is

$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu_{\sigma}^{2}\sigma^{\dagger}\sigma+\mu_{\xi}^{2}\xi^{\dagger}\xi+\mu_{\phi}^{2}\phi^{\dagger}\phi+\mu_{\eta}^{2}\eta^{\dagger}\eta+\lambda_{\sigma}(\sigma^{\dagger}\sigma)^{2}$
(1)
$\displaystyle+\lambda_{\xi}(\xi^{\dagger}\xi)^{2}+\lambda_{\phi}(\phi^{\dagger}\phi)^{2}+\lambda_{\eta}(\eta^{\dagger}\eta)^{2}+\lambda_{\sigma\xi}\sigma^{\dagger}\sigma\xi^{\dagger}\xi$
$\displaystyle+\lambda_{\sigma\phi}\sigma^{\dagger}\sigma\phi^{\dagger}\phi+\lambda_{\sigma\eta}\sigma^{\dagger}\sigma\eta^{\dagger}\eta+\lambda_{\xi\phi}\xi^{\dagger}\xi\phi^{\dagger}\phi$
$\displaystyle+\lambda_{\xi\eta}\xi^{\dagger}\xi\eta^{\dagger}\eta+\lambda_{\phi\eta}\phi^{\dagger}\phi\eta^{\dagger}\eta+\lambda^{\prime}_{\phi\eta}\eta^{\dagger}\phi\phi^{\dagger}\eta$
$\displaystyle+\rho(\xi\phi^{\dagger}\eta+\textrm{H.c.})\,,$

where the gauge invariant term
$\lambda^{\prime}_{\sigma\xi}\left[\left(\sigma^{\dagger}\xi\right)^{2}+\textrm{H.c.}\right]$
has been forbidden by the $U(1)_{PQ}$ global symmetry. All of the allowed
Yukawa interactions are

$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$ $\displaystyle-
y_{ie}\bar{l}_{Li}\eta
e_{R}-y_{i\mu}\bar{l}_{Li}\eta\mu_{R}-y_{i\tau}\bar{l}_{Li}\eta\tau_{R}$ (2)
$\displaystyle-y_{id}\bar{q}_{Li}\eta d_{R}-y_{is}\bar{q}_{Li}\eta
s_{R}-y_{ib}\bar{q}_{Li}\eta b_{R}$ $\displaystyle-
y_{iu}\bar{q}_{Li}\tilde{\eta}u_{R}-y_{ic}\bar{q}_{Li}\tilde{\eta}c_{R}-y_{iT}\bar{q}_{Li}\tilde{\eta}T_{R}$
$\displaystyle-
y_{it}\bar{q}_{Li}\tilde{\phi}t_{R}-y_{Tu}\sigma\bar{T}_{L}u_{R}-y_{Tc}\sigma\bar{T}_{L}c_{R}$
$\displaystyle-y_{TT}\sigma\bar{T}_{L}T_{R}+\textrm{H.c.}\,.$

Without loss of generality, two of the three Yukawa couplings $y_{Tu}$,
$y_{Tc}$ and $y_{TT}$ can be removed by a proper phase rotation. For
convenience we can set $y_{Tu}=y_{Tc}=0$ and $y_{TT}=y_{T}$.

We emphasize that the above $U(1)_{PQ}$ global symmetry indeed should be
treated as two independent $U(1)$ global symmetries: a $U(1)_{t}$ only acts on
the right-handed top quark $t_{R}$, the Higgs doublet $\phi$ and the Higgs
singlet $\xi$, while a $U(1)_{T}$ only acts on the left-handed new quark
$T_{L}$ and the Higgs singlet $\sigma$. Therefore, the $U(1)_{t}$ and
$U(1)_{T}$ global symmetries will not suffer the domain wall problem. As shown
later, only one of the two NG bosons from the $U(1)_{t}$ and $U(1)_{T}$ global
symmetry breaking can survive from the gauge interactions to serve as the
axion. This means the physical $U(1)_{PQ}$ global symmetry should be a result
of the $U(1)_{t}$ and $U(1)_{T}$ global symmetries.

##  III Physical scalars

After the gauge and global symmetries are all spontaneously broken, i.e.
$SU(2)_{L}\times U(1)_{Y}\times U(1)_{X}\times U(1)_{PQ}\Longrightarrow
U(1)_{em}$, we can express the Higgs singlets and doublets to be

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(v_{\sigma}+h_{\sigma}+iP_{\sigma}\right),~{}~{}\xi=\frac{1}{\sqrt{2}}\left(v_{\xi}+h_{\xi}+iP_{\xi}\right),$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\phi^{+}\\\\[5.69054pt]
\frac{1}{\sqrt{2}}\left(v_{\phi}+h_{\phi}+iP_{\phi}\right)\end{array}\right],$
(5) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\eta^{+}\\\\[5.69054pt]
\frac{1}{\sqrt{2}}\left(v_{\eta}+h_{\eta}+iP_{\eta}\right)\end{array}\right].$
(8)

In addition to the four massive Higgs bosons $h_{\xi}$, $h_{\sigma}$,
$h_{\phi}$ and $h_{\eta}$, i.e.

$\displaystyle V$ $\displaystyle\supset$
$\displaystyle\lambda_{\sigma}v_{\sigma}^{2}h_{\sigma}^{2}+\frac{1}{2}\left(2\lambda_{\xi}v_{\xi}^{2}-\frac{\rho
v_{\phi}v_{\eta}}{\sqrt{2}v_{\xi}}\right)h_{\xi}^{2}+\frac{1}{2}\left(2\lambda_{\phi}v_{\phi}^{2}-\frac{\rho
v_{\xi}v_{\eta}}{\sqrt{2}v_{\phi}}\right)h_{\phi}^{2}+\frac{1}{2}\left(2\lambda_{\eta}v_{\eta}^{2}-\frac{\rho
v_{\xi}v_{\phi}}{\sqrt{2}v_{\eta}}\right)h_{\eta}^{2}+\lambda_{\sigma\xi}v_{\sigma}v_{\xi}h_{\sigma}h_{\xi}$
(9)
$\displaystyle+\lambda_{\sigma\phi}v_{\sigma}v_{\phi}h_{\sigma}h_{\phi}+\lambda_{\sigma\eta}v_{\sigma}v_{\eta}h_{\sigma}h_{\eta}+\left(\lambda_{\xi\phi}v_{\xi}v_{\phi}+\frac{1}{\sqrt{2}}\rho
v_{\eta}\right)h_{\xi}h_{\phi}+\left(\lambda_{\xi\eta}v_{\xi}v_{\eta}+\frac{1}{\sqrt{2}}\rho
v_{\phi}\right)h_{\xi}h_{\eta}$
$\displaystyle+\left[\left(\lambda_{\phi\eta}+\lambda^{\prime}_{\phi\eta}\right)v_{\phi}v_{\eta}+\frac{1}{\sqrt{2}}\rho
v_{\xi}\right]h_{\phi}h_{\eta}\,,$

there should be five massless NG bosons,

$\displaystyle G^{\pm}_{W}$ $\displaystyle=$
$\displaystyle\frac{v_{\phi}\phi^{\pm}+v_{\eta}\eta^{\pm}}{\sqrt{v_{\phi}^{2}+v_{\eta}^{2}}}\,,~{}~{}G^{0}_{Z}=\frac{v_{\phi}P_{\phi}+v_{\eta}P_{\eta}}{\sqrt{v_{\phi}^{2}+v_{\eta}^{2}}}\,,~{}~{}G^{0}_{X}=\frac{v_{\phi}v_{\eta}^{2}P_{\phi}-v_{\phi}^{2}v_{\eta}P_{\eta}+v_{\xi}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)P_{\xi}+v_{\sigma}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)P_{\sigma}}{\sqrt{\left(v_{\phi}^{2}+v_{\eta}^{2}\right)\left[v_{\sigma}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)+v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)+v_{\phi}^{2}v_{\eta}^{2}\right]}}\,;$
(10) $\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{-v_{\sigma}\left[v_{\phi}v_{\eta}^{2}P_{\phi}-v_{\phi}^{2}v_{\eta}P_{\eta}+v_{\xi}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)P_{\xi}\right]+\left[v_{\phi}^{2}v_{\eta}^{2}+v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)\right]P_{\sigma}}{\sqrt{\left[v_{\phi}^{2}v_{\eta}^{2}+v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)\right]\left[v_{\sigma}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)+v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)+v_{\phi}^{2}v_{\eta}^{2}\right]}}\,,$
(11)

one massive pseudo scalar,

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{-v_{\xi}v_{\eta}P_{\phi}+v_{\xi}v_{\phi}P_{\eta}+v_{\phi}v_{\eta}P_{\xi}}{\sqrt{v_{\phi}^{2}v_{\eta}^{2}+v_{\xi}^{2}v_{\phi}^{2}+v_{\xi}^{2}v_{\eta}^{2}}}~{}~{}\textrm{with}$
(12) $\displaystyle m_{P}^{2}=-\frac{1}{\sqrt{2}}\rho
v_{\xi}v_{\phi}v_{\eta}\left(\frac{1}{v_{\xi}^{2}}+\frac{1}{v_{\phi}^{2}}+\frac{1}{v_{\eta}^{2}}\right),$

and one massive charged scalar,

$\displaystyle H^{\pm}$ $\displaystyle=$
$\displaystyle\frac{-v_{\eta}\phi^{\pm}+v_{\phi}\eta^{\pm}}{\sqrt{v_{\phi}^{2}+v_{\eta}^{2}}}~{}~{}\textrm{with}$
(13) $\displaystyle
m_{H^{\pm}}^{2}=-\left(\frac{1}{2}\lambda^{\prime}_{\phi\eta}+\frac{\rho
v_{\xi}}{\sqrt{2}v_{\phi}v_{\eta}}\right)\left(v_{\phi}^{2}+v_{\eta}^{2}\right)\,.$

While the four NG bosons $G^{\pm}_{W}$, $G^{0}_{Z}$ and $G^{0}_{X}$ are eaten
by the longitudinal components of the SM gauge bosons $W^{\pm}$ and $Z^{0}$ as
well as the $U(1)_{X}$ gauge boson $X$, the fifth NG boson $a$ keeps available
to be an axion. Actually, the two NG bosons $G^{0}_{X}$ and $a$ are the linear
combinations of the two NG bosons from the $U(1)_{t}$ and $U(1)_{T}$ global
symmetry breaking, i.e. the $P_{\phi}$, $P_{\eta}$ and $P_{\xi}$ part in the
$G^{0}_{X}$ and $a$ formula is according to the $U(1)_{t}$ NG boson while the
$P_{\sigma}$ part is according to the $U(1)_{T}$ NG boson.

We can deduce the coupling of the axion $a$ to the gluons $G^{a}_{\mu\nu}$ by

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{P_{\sigma}}{2v_{\sigma}}\partial_{\mu}\left(\bar{T}\gamma^{\mu}\gamma_{5}T\right)-\frac{P_{\phi}}{2v_{\phi}}\partial_{\mu}\left(\bar{t}\gamma^{\mu}\gamma_{5}t\right)$
(14) $\displaystyle=$
$\displaystyle\frac{a}{f_{a}}\frac{\alpha_{s}}{8\pi}G^{a}_{\mu\nu}\tilde{G}^{a\mu\nu}\,,$

where the axion decay constant is given by

$\displaystyle\\!\\!\\!\\!f_{a}$ $\displaystyle=$ $\displaystyle
2v_{\sigma}\sqrt{v_{\sigma}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)+v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)+v_{\phi}^{2}v_{\eta}^{2}}$
(15)
$\displaystyle\times\frac{\sqrt{v_{\phi}^{2}v_{\eta}^{2}+v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)}}{\left[v_{\sigma}^{2}v_{\eta}^{2}-v_{\xi}^{2}\left(v_{\phi}^{2}+v_{\eta}^{2}\right)-v_{\phi}^{2}v_{\eta}^{2}\right]}\,.$

Remarkably, only the top quark $t$ coupling to the Higgs doublet $\phi$ and
the new up-type quark $T$ coupling to the Higgs singlet $\sigma$ contribute to
the axion-gluon coupling while the contributions from the other quarks
coupling to the Higgs doublet $\eta$ have been exactly cancelled by each
other. This result is consistent with our starting point: the right-handed top
quark $t_{R}$ and the left-handed new quark $T_{L}$ are the unique fermions
carrying the nontrivial PQ charges. Moreover, the top quark and the new quark
are much heavier than the QCD scale so that the axion in the present model
should obtain its mass like the KSVZ axion rather than the DFSZ axion.

##  IV Symmetry breaking scales

Now the bottom quark mass $m_{b}=y_{b}v_{\eta}/\sqrt{2}$ and the perturbation
requirement $y_{b}<\sqrt{4\pi}$ puts a low limit on the VEV $v_{\eta}$, i.e.
pdg2020

$\displaystyle
v_{\eta}=\frac{m_{b}}{y_{b}/\sqrt{2}}>\frac{4.18\,\textrm{GeV}}{\sqrt{4\pi}/\sqrt{2}}=1.18\,\textrm{GeV}\,,$
(16)

and hence offers a large factor pdg2020 ,

$\displaystyle\frac{v_{\phi}^{2}}{v_{\eta}^{2}}<\frac{\left(246\,\textrm{GeV}\right)^{2}-\left(1.18\,\textrm{GeV}\right)^{2}}{\left(1.18\,\textrm{GeV}\right)^{2}}=4.35\times
10^{4}\,.$ (17)

We then assume

$\displaystyle v_{\eta}^{2}\ll v_{\phi,\xi}^{2}\ll v_{\sigma}^{2}\,,$ (18)

to simplify Eqs. (10) and (11), i.e.

$\displaystyle G_{W}^{\pm}$ $\displaystyle\simeq$
$\displaystyle\phi^{\pm}+\frac{v_{\eta}}{v_{\phi}}\eta^{\pm}\,,~{}~{}G_{Z}^{0}\simeq
P_{\phi}+\frac{v_{\eta}}{v_{\phi}}P_{\phi}\,,$ $\displaystyle G_{X}^{0}$
$\displaystyle\simeq$
$\displaystyle\frac{v_{\eta}^{2}}{v_{\sigma}v_{\phi}}P_{\phi}-\frac{v_{\eta}}{v_{\sigma}}P_{\eta}+\frac{v_{\xi}}{v_{\sigma}}P_{\xi}+P_{\sigma}\,;$
(19) $\displaystyle a$ $\displaystyle\simeq$
$\displaystyle-\frac{v_{\eta}^{2}}{v_{\xi}v_{\phi}}P_{\phi}+\frac{v_{\eta}}{v_{\xi}}P_{\eta}-P_{\xi}+\frac{v_{\xi}}{v_{\sigma}}P_{\sigma}\,,$
(20)

as well as Eqs. (12) and (13), i.e.

$\displaystyle P$ $\displaystyle\simeq$
$\displaystyle\frac{v_{\eta}}{v_{\xi}}P_{\xi}-\frac{v_{\eta}}{v_{\phi}}P_{\phi}+P_{\eta}~{}~{}\textrm{with}$
(21) $\displaystyle m_{P}^{2}\simeq-\frac{\sqrt{2}\rho
v_{\xi}v_{\phi}}{v_{\eta}}\simeq m_{h_{\eta}}^{2}\,;$ $\displaystyle H^{\pm}$
$\displaystyle\simeq$
$\displaystyle-\frac{v_{\eta}}{v_{\phi}}\phi^{\pm}+\eta^{\pm}~{}~{}\textrm{with}$
(22) $\displaystyle m_{H^{\pm}}^{2}\simeq
m_{h_{\eta}}^{2}-\frac{1}{2}\lambda^{\prime}_{\phi\eta}v_{\phi}^{2}\,.$

As for the axion decay constant $f_{a}$, it can be also given by a simple
formula,

$\displaystyle f_{a}\simeq
2v_{\xi}\left(\frac{v_{\phi}^{2}}{v_{\eta}^{2}}\right)\left/\left(1-\frac{v_{\phi}^{2}/v_{\eta}^{2}}{v_{\sigma}^{2}/v_{\xi}^{2}}\right)\right..$
(23)

In the above simplification, the Higgs singlet $\sigma$ dominates the
$U(1)_{X}$ gauge symmetry breaking while the Higgs singlet $\xi$ dominates the
$U(1)_{PQ}$ global symmetry breaking. In consequence, the PQ symmetry breaking
scale should be determined by the VEV $v_{\xi}$. To mach the experimental
constraint on the axion decay constant, i.e. $f_{a}\gtrsim
10^{9}\,\textrm{GeV}$ pdg2020 , the PQ symmetry breaking scale $v_{\xi}$ thus
should have the low bound as follows,

$\displaystyle v_{\xi}$ $\displaystyle=$ $\displaystyle 1.15\times
10^{4}\,\textrm{GeV}\left(\frac{f_{a}}{10^{9}\,\textrm{GeV}}\right)$ (24)
$\displaystyle\textrm{for}~{}~{}\frac{v_{\phi}^{2}}{v_{\eta}^{2}}=4.35\times
10^{4}\ll\frac{v_{\sigma}^{2}}{v_{\xi}^{2}}\,.$

A more appealing possibility is to consider the technically natural parameter
choice as follows,

$\displaystyle
c=1-\frac{v_{\phi}^{2}/v_{\eta}^{2}}{v_{\sigma}^{2}/v_{\xi}^{2}}\simeq\mathcal{O}\left(0.001-0.1\right),$
(25)

and hence break the PQ symmetry near the weak scale,

$\displaystyle v_{\xi}$ $\displaystyle\simeq$ $\displaystyle
f_{a}\left(\frac{c}{v_{\phi}^{2}/v_{\eta}^{2}}\right)$ (26) $\displaystyle=$
$\displaystyle
115\,\textrm{GeV}\left(\frac{f_{a}}{10^{9}\,\textrm{GeV}}\right)\left(\frac{4.35\times
10^{4}}{v_{\phi}^{2}/v_{\eta}^{2}}\right)\left(\frac{c}{0.01}\right).~{}~{}~{}~{}$

Accordingly, the $U(1)_{X}$ gauge symmetry breaking scale is determined to be

$\displaystyle v_{\sigma}$ $\displaystyle\simeq$ $\displaystyle
v_{\xi}\sqrt{\frac{v_{\phi}^{2}}{v_{\eta}^{2}}}$ $\displaystyle=$
$\displaystyle 2.40\times
10^{4}\,\textrm{GeV}\left(\frac{f_{a}}{10^{9}\,\textrm{GeV}}\right)\sqrt{\frac{4.35\times
10^{4}}{v_{\phi}^{2}/v_{\eta}^{2}}}\left(\frac{c}{0.01}\right).$

This means the $U(1)_{X}$ gauge boson which has the mass and kinetic mixing
with the SM gauge bosons can be also near the weak scale for a proper
$U(1)_{X}$ gauge coupling.

##  V Summary and discussion

In this work we have demonstrated a simple, economical and practical mechanism
to efficiently lower the breaking scale of PQ symmetry. Our basic model
contains one gauge boson, one up-type vector-like quark, two Higgs doublets
and two Higgs singlets besides the SM fermions and gauge bosons. In
particular, the right-handed top quark is the unique SM fermion carrying a
nontrivial PQ charge. Such special role can be well motivated by the fact that
the top quark is much heavier than the other SM fermions. Thanks to the SM and
new gauge interactions, the Higgs doublet for the top quark mass generation as
well as the Higgs singlet for the new quark mass generation can only
contribute a tiny fraction to the axion. The axion decay constant can be
enhanced by a huge ratio between the two Higgs doublet VEV squares. This ratio
can reduce the PQ symmetry breaking scale by four orders of magnitude, saying
from the order $10^{9}\,\textrm{GeV}$ to the order $10^{4-5}\,\textrm{GeV}$.
If the PQ symmetry breaking scale is further expected at the weak scale, we
can take the ratio between the two Higgs singlet VEV squares to be almost the
ratio between the two Higgs doublet VEV squares. In our mechanism, two
independent global symmetries connected by the new gauge interaction are the
origin of the physical PQ global symmetry, meanwhile, everyone of these two
global symmetries only acts on either the right-handed top quark or the left-
handed new quark so that the PQ symmetry can avoid the domain wall problem.

In the usual invisible axion models, the related new particles should be of
the order of the PQ symmetry breaking scale and hence they could not leave any
signals to be verified experimentally. However, our mechanism can allow the PQ
symmetry, which is now accompanied by the $U(1)_{X}$ gauge symmetry, to be
broken at low scales even near the weak scale. The new particles in our
mechanism could be near the weak scale and hence they may be verified at
colliders. Furthermore, our basic model can easily accommodate the seesaw
mechanism for small neutrino mass. Specifically, the PQ symmetry can enforce
the Higgs doublet $\eta$ rather than the Higgs doublet $\phi$ to take part in
the seesaw mechanism. Since the Higgs doublet $\eta$ acquires a VEV of the GeV
order, the low scale seesaw such as the TeV scale seesaw can become more
motivated.

Acknowledgement: This work was supported in part by the Fundamental Research
Funds for the Central Universities.

## References

  * (1) R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
  * (2) S.L. Adler, Phys. Rev. 177, 2426 (1969).
  * (3) J.S. Bell and R. Jackiw, Nuovo Cim. 60A, 47 (1969).
  * (4) S.L. Adler and W.A. Bardeen, Phys. Rev. 182, 1517 (1969).
  * (5) S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
  * (6) F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
  * (7) J.E. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010).
  * (8) L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, Phys. Rep. 870, 1 (2020).
  * (9) P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) and 2021 update.
  * (10) J.E. Kim, Phys. Rev. Lett. 43, 103 (1979).
  * (11) M.A. Shifman, A. Vainshtein, and V.I. Zakharov, Nucl. Phys. B 166, 493 (1980).
  * (12) M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981).
  * (13) A. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980).
  * (14) J.D. Clarke and R.R. Volkas, Phys. Rev. D 93, 035001 (2016).
  * (15) E. Ma, M. Raidal, and U. Sarkar, Phys. Lett. B 504, 296 (2001).
  * (16) K. Choi, H. Kim, and S. Yun, Phys. Rev. D 90, 023545 (2014).
  * (17) D.E. Kaplan and R. Rattazzi, Phys. Rev. D 93, 085007 (2016).
  * (18) K. Choi and S.H. Im, JHEP 1601, 149 (2016).
  * (19) P.H. Gu, arXiv:2106.13010 [hep-ph].
  * (20) P. Sikivie, Phys. Rev. Lett. 48, 1156 (1982).
  * (21) R.D. Peccei, T.T. Wu, and T. Yanagida, Phys. Lett. B 172, 435 (1986).
  * (22) L.M. Krauss and F. Wilczek, Phys. Lett. B 173, 189 (1986).
  * (23) C.W. Chiang, H. Fukuda, M. Takeuchi, and T.T. Yanagida, JHEP 1511, 057 (2015)
  * (24) C.W. Chiang, H. Fukuda, M. Takeuchi, and T.T. Yanagida, Phys. Rev. D 97, 035015 (2018).
  * (25) C.W. Chiang, M. Takeuchi, P.Y. Tseng, and T.T. Yanagida, Phys. Rev. D 98, 095020 (2018).