# Hidden $\mathrm{SU}(2)_{\mathrm{D}}$ vector dark matter with a scalar
septuplet

Zhao Zhang School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
Chengfeng Cai<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  Hong-Hao Zhang<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

We propose a vector dark matter model from a hidden
$\mathrm{SU}(2)_{\mathrm{D}}$ gauge symmetry at
$\mathrm{T}\mathrm{e}\mathrm{V}$ scale. A scalar septuplet is introduced to
break the $\mathrm{SU}(2)_{\mathrm{D}}$ symmetry spontaneously. The septuplet
also play the role of a portal between the standard model and the dark sector.
We find that there are two different vacuum configurations corresponding to
the sign of the quartic coupling $\lambda_{3}$, which yields different mass
spectrum for the gauge bosons. For a $\lambda_{3}<0$, the masses of gauge
bosons are splitting, while for a $\lambda_{3}\geq 0$, the masses are
degenerate. We also study the RG evolutions of the couplings, and find that
the perturbativity and vacuum stability can set a stringent bound on the
parameter space. For the phenomenological aspect, we consider the experimental
constraints including dark matter direct detection, indirect detection, relic
density, and Higgs couplings measurements. We find that there are parameter
space survive from all the constraints, and they can be tested in future dark
matter direct and indirect detection experiments.

###### Contents

  1. I Introduction
  2. II The setup of the model
    1. II.1 The 7-dimensional representation of $\mathrm{SU}(2)_{\mathrm{D}}$ and the Lagrangian
    2. II.2 Vacuum configurations and mass spectrum
      1. II.2.1 The $\lambda_{3}<0$ case (splitting spectrum)
      2. II.2.2 The $\lambda_{3}\geq 0$ case (degenerate spectrum)
  3. III Vacuum stability and perturbativity
  4. IV Constraining the model by phenomenologies, perturbativity, and vacuum stability
    1. IV.1 Splitting case $(\lambda_{3}<0)$
    2. IV.2 Degenerate case $(\lambda_{3}\geq 0)$
  5. V Conclusion and discussion
  6. A Analytic solution of Landau pole

##  I Introduction

Cosmology and astrophysics have provided many hints for the existence of dark
matter (DM). According to the observation of cosmic microwave background, the
energy density of cold dark matter is about
$\Omega_{\text{CDM}}h^{2}=0.1200\pm 0.0012$ Planck:2018vyg , which cannot be
explained by the standard model (SM) of particle physics. In order to unravel
the DM mystery, people have tried many scenarios beyond SM. One of the most
famous paradigms is the weakly interacting massive particles (WIMPs), which
can realize the observed DM relic density through the thermal freeze-out
mechanism. Based on the interaction properties of WIMPs, people have designed
many underground experiments aiming to directly detect the recoil signal from
the collision of WIMP and nucleus in detectors LUX:2016ggv ; XENON:2018voc ;
PandaX-4T:2021bab ; LUX-ZEPLIN:2022qhg  . However, no convincing signal has
been detected so far, resulting in a stringent bound on the DM-nucleon cross
section.

From the view of particle physics, the WIMP DM candidate can be an elementary
particle with various spins. Tons of scalar and fermionic DM models have been
studied in literatures, while the vector DM models are less explored. The most
natural way to introduce an elementary vector DM field is basing on a new
gauge symmetry, which can be abelian or non-abelian. For a gauged U(1) vector
DM model, a discrete symmetry should be imposed manually in order to stabilize
the DM particle Lebedev:2011iq ; Abe:2012hb ; Farzan:2012hh ; Baek:2012se ;
Domingo:2013tna ; Yu:2014pra ; Chen:2014cbt ; Duch:2015jta ; DiFranzo:2015nli
; Azevedo:2018oxv ; Mohamadnejad:2019vzg ; Glaus:2019itb ; Arcadi:2020jqf ;
Delaunay:2020vdb ; Salehian:2020asa . On the other hand, in the non-abelian
group cases, the discrete symmetry can be induced as a remnant of spontaneous
symmetry breaking (SSB) naturally, and thus the vector bosons can be stable.
The simplest setup of non-abelian model is to consider a gauged
$\mathrm{SU}(2)_{\mathrm{D}}$ with some multiplet fields, such as a scalar
doublet Hambye:2008bq ; Boehm:2014bia ; Gross:2015cwa ; Karam:2015jta ;
Khoze:2016zfi ; Baouche:2021wwa ; Borah:2021ftr , a scalar triplet
Baek:2013dwa ; Khoze:2014woa ; Ghosh:2020ipy ; Nomura:2020zlm , two scalar
triplets Ghosh:2020ipy ; Hu:2021pln , scalar doublet and triplet Cai:2021wmu ,
scalar and Dirac fermion doublet Belyaev:2022shr ; Elahi:2022hgj , scalar
triplet and quintet Nomura:2020zlm , etc Barman:2018esi ; Barman:2019lvm ;
Chowdhury:2021tnm . The multiplets can be a portal to the SM through a
renormalizable or non-renormalizable operators.

A real inert scalar septuplet under the SM $\mathrm{SU}(2)_{\mathrm{L}}$ in
the framework of minimal dark matter model Cirelli:2005uq  had been studied in
Ref. Cai:2015kpa . In that case, the septuplet does not obtain any vacuum
expectation value (VEV), and the neutral component of the septuplet is
regarded as the DM candidate. It has been proved that the quartic couplings of
the septuplet in this model have Landau pole below the scale due to the high
dimensional representation Hamada:2015bra ; Cai:2015kpa . In this work, we
consider a dark sector to be a new $\mathrm{SU}(2)_{\mathrm{D}}$ gauge theory
with a real scalar septuplet. The septuplet can obtain a nonzero VEV for
breaking the $\mathrm{SU}(2)_{\mathrm{D}}$ spontaneously and making the gauge
bosons massive. We will prove that there are two different vacuum
configurations, corresponding to different sign of the quartic coupling
$\lambda_{3}$. In addition, after the SSB of $\mathrm{SU}(2)_{\mathrm{D}}$,
there are remnant discrete symmetries, $Z_{6}\times Z_{2}$ and $Z_{4}\times
Z_{2}$ in these two cases respectively, which keep the vector DM stable. The
vector DMs interact with SM particles through Higgs-portal and thus the vector
DMs can be the cold dark matter candidate which produced via the freeze-out
mechanism. The direct detection LUX-ZEPLIN:2022qhg ; LZ:2015kxe  and indirect
detection MAGIC:2016xys ; CTAConsortium:2012fwj  bounds for WIMPs DM can be
used to constrain the model as well. In addition, with the help of the
threshold effect Elias-Miro:2012eoi , the portal interaction can help to
alleviate the problem of unstable vacuum generated by the SM Higgs quartic
coupling. Since the gauge coupling of $\mathrm{SU}(2)_{\mathrm{D}}$ can be
smaller than the one of $\mathrm{SU}(2)_{\mathrm{L}}$, the Landau pole problem
can be avoided in our model. On the other hand, the requirement of coupling
perturbativity and vacuum stability is capable to put stringent theoretical
constraints on the the parameter space.

The paper is organized as follows. In Sec. II, we will introduce the setup of
our model, including the vacuum configurations and the mass spectra of
particles. The RG running of the couplings and the vacuum stability conditions
are introduced in Sec. III. The constraints from DM phenomenologies, Higgs
couplings measurements, perturbativity and vacuum stability will be presented
in Sec. IV. Finally, we will conclude and discuss our result in Sec. V and
give the analytic solution of Landau pole in Appendix A.

##  II The setup of the model

In Ref. Cai:2021wmu , a vector dark matter has been studied under the
framework of gauged $\mathrm{SU}(2)_{\mathrm{D}}$ completely broken by two
real triplets. In that case, a splitting mass spectrum of the vector fields
can be obtained, however there are many couplings in the potential and make
the model less predictive. In this work, we consider the gauged
$\mathrm{SU}(2)_{\mathrm{D}}$ symmetry to be completely broken by a scalar
septuplet. This model has a much simpler potential term, but the vacuum
structures are non-trivial. We will show that there are two different of
vacuum configurations depending on the sign of a quartic coupling. The gauge
bosons, which are assumed to play the role of cold dark matter candidates,
have different mass spectrum in these different cases.

###  II.1 The 7-dimensional representation of $\mathrm{SU}(2)_{\mathrm{D}}$
and the Lagrangian

The 7-dimensional representation of $\mathrm{SU}(2)$ generators are defined by

$\displaystyle\tau^{1}=\frac{\tau^{+}+\tau^{-}}{2},\quad\tau^{2}=\frac{\tau^{+}-\tau^{-}}{2i},\quad\tau^{3}=\text{diag}\\{3,2,1,0,-1,-2,-3\\},$
(1)

where

$\displaystyle\tau^{+}=\begin{pmatrix}0&-\sqrt{6}&&&&&\\\ &0&-\sqrt{10}&&&&\\\
&&0&-\sqrt{12}&&&\\\ &&&0&\sqrt{12}&&\\\ &&&&0&\sqrt{10}&\\\
&&&&&0&\sqrt{6}\\\ &&&&&&0\\\
\end{pmatrix},\quad\tau^{-}=(\tau^{+})^{\text{T}},$ (2)

It is well known that there exists a unitary symmetric (anti-symmetric) matrix
$S$ transforming the $\mathrm{SU}(2)$ generators into their complex
conjugation as follows

$\displaystyle
S_{(n)}\tau^{a}S^{-1}_{(n)}=-(\tau^{a})^{*},\quad\mathrm{where}\quad
S_{(n)ij}=\left\\{\begin{array}[]{rl}\delta_{i,n+1-j}&\qquad\text{if}\quad
i\leqslant\frac{n}{2},\\\ (-1)^{n+1}\delta_{i,n+1-j}&\qquad\text{if}\quad
i>\frac{n}{2}.\end{array}\right.$ (5)

The $\Phi$ field transforms as $\Phi\to S\Phi$ correspondingly, behaves like a
field in the complex conjugate representation. If $\Phi^{\ast}=S\Phi$, then
$\Phi$ is said to be a real scalar. A real septuplet $\Phi$ can be written as

$\Phi=\frac{1}{\sqrt{2}}(\Delta_{3},\Delta_{2},\Delta_{1},\Delta_{0},\Delta_{-1},\Delta_{-2},\Delta_{-3})^{\text{T}},$
(6)

where $\Delta_{i}^{*}=\Delta_{-i}$.

The potential terms for the scalar fields are given by

$\displaystyle V(H,\Phi)$ $\displaystyle=$
$\displaystyle-\mu^{2}H^{{\dagger}}H+\lambda(H^{{\dagger}}H)^{2}-m^{2}\Phi^{{\dagger}}\Phi+\lambda_{2}(\Phi^{{\dagger}}\Phi)^{2}+\frac{\lambda_{3}}{48}(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2}+\lambda_{4}(H^{{\dagger}}H)(\Phi^{{\dagger}}\Phi)$
$\displaystyle=$
$\displaystyle-\mu^{2}H^{{\dagger}}H+\lambda(H^{{\dagger}}H)^{2}-m^{2}\Phi^{{\dagger}}\Phi+\lambda^{\prime}_{2}(\Phi^{{\dagger}}\Phi)^{2}+\frac{\lambda_{3}}{48}(\Phi^{{\dagger}}T^{ab}\Phi)^{2}+\lambda_{4}(H^{{\dagger}}H)(\Phi^{{\dagger}}\Phi).$

where $H$ is the SM $\mathrm{SU}(2)_{\mathrm{L}}$ Higgs doublet. In the second
line, we have separated out the traceless part of $\tau^{a}\tau^{b}$ and
define $T^{ab}\equiv\\{\tau^{a},\tau^{b}\\}/2-4\delta^{ab}$,
$\lambda_{2}^{\prime}\equiv\lambda_{2}+\lambda_{3}$. The explicit expression
of the $\lambda_{3}$ term in Eq.(II.1) is given as follows,

$\displaystyle\quad\frac{1}{48}(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ (8)
$\displaystyle=$
$\displaystyle\frac{1}{8}\Delta_{0}^{4}+\frac{21}{32}(\Delta_{-1}\Delta_{1})^{2}+\frac{25}{32}(\Delta_{-3}\Delta_{3})^{2}+\frac{1}{2}\Delta_{0}^{2}\Delta_{-1}\Delta_{1}+\frac{5}{4}\Delta_{0}^{2}\Delta_{-2}\Delta_{2}-\frac{5}{8}\Delta_{0}^{2}\Delta_{-3}\Delta_{3}$
$\displaystyle+\frac{15}{16}\Delta_{-1}\Delta_{1}\Delta_{-2}\Delta_{2}-\frac{5}{16}\Delta_{-1}\Delta_{1}\Delta_{-3}\Delta_{3}+\frac{25}{16}\Delta_{-2}\Delta_{2}\Delta_{-3}\Delta_{3}+$
$\displaystyle\left(-\frac{\sqrt{15}}{8}\Delta_{1}^{3}\Delta_{-3}-\frac{\sqrt{30}}{16}\Delta_{0}\Delta_{1}^{2}\Delta_{-2}+\frac{5\sqrt{15}}{16}\Delta_{1}\Delta_{-2}^{2}\Delta_{3}+\frac{15\sqrt{2}}{16}\Delta_{0}\Delta_{1}\Delta_{2}\Delta_{-3}+\text{h.c.}\right).$

The $\lambda_{4}$ term in Eq.(II.1) is a portal connecting the SM and the dark
sectors, and thus all the DM phenomenologies relate to it. It is also
responsible for prevent the vacuum becomes unstable in the high energy by
raising the self-coupling of the SM Higgs via the threshold effect.

The kinetic terms of our dark sector are given by

$\displaystyle\mathcal{L}_{\text{kin}}$
$\displaystyle=-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}+(D^{\mu}\Phi)^{{\dagger}}D_{\mu}\Phi$
$\displaystyle=\frac{1}{2}X^{a\mu}(g_{\mu\nu}\partial^{2}-\partial_{\mu}\partial_{\nu})X^{a\nu}-g_{4}\varepsilon^{abc}(\partial_{\mu}X_{\nu}^{a})X^{b\mu}X^{c\nu}-\frac{1}{4}g_{4}^{2}\varepsilon^{abc}\varepsilon^{ade}X^{b}_{\mu}X^{c}_{\nu}X^{d\mu}X^{e\nu}$
$\displaystyle\quad+(\partial^{\mu}\Phi^{{\dagger}})\partial_{\mu}\Phi+ig_{4}X^{a}_{\mu}[\Phi^{{\dagger}}\tau^{a}\partial_{\mu}\Phi-(\partial_{\mu}\Phi^{{\dagger}})\tau^{a}\Phi]+g_{4}^{2}X^{a}_{\mu}X^{b\mu}\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi,$
(9)

where $F^{a}_{\mu\nu}$ is the field strength tensor of gauge bosons $X^{a}$,
$D_{\mu}=\partial_{\mu}-ig_{4}X^{a}_{\mu}\tau^{a}$ is the covariant derivative
and $g_{4}$ is the gauge coupling coefficient.

In this work, we consider the lightest gauge bosons $X^{a}_{\mu}$ as WIMP DM
candidate. They become massive after the septuplet breaking
$\mathrm{SU}(2)_{\mathrm{D}}$ symmetry completely, which are presented in next
subsection. The VEV of septuplet $v^{\prime}$ is set around a few
$\mathrm{T}\mathrm{e}\mathrm{V}$ to $10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$.
Some discrete symmetries is retained and make the DM stable.

###  II.2 Vacuum configurations and mass spectrum

The vacuum configurations of the septuplet can be obtained by minimizing the
the potential term in Eq.(II.1). We only consider the tree level potential for
simplicity, and assume that it could drive the spontaneous symmetry breaking
of SU(2)${}_{L}\times$U(1)Y and $\mathrm{SU}(2)_{\mathrm{D}}$. Before
proceeding, we want to make some comments on the $\lambda_{3}$ term in the
potential, Eq.(II.1). If this term is absent, the tree-level potential
respects a global SO(7) symmetry. A negative mass-squared term for $\Phi$
induce the spontaneous breaking from SO(7) to SO(6), and then there is a
6-dimensional degenerate vacuum (the coset SO(7)/SO(6) is an $S^{6}$)
corresponding to six Goldstones. Three (or two) of the Goldstones become the
longitudinal polarizations of the gauge fields, while there still remains
three (or four) flat directions for the potential. Radiative corrections from
gauge boson loops can lift the potential since gauge interactions explicitly
break the SO(7), and then the vacuum is aligned in the $\Delta_{0}$ direction.
It implies that the gauged $\mathrm{SU}(2)_{\mathrm{D}}$ can not be completely
broken in this case, and a U(1) subgroup is restored. If the $\lambda_{3}$
term in Eq.(II.1) is turned on, the SO(7) is explicitly broken into an SO(3) (
SU(2) ) subgroup. It leads to a vacuum configuration which completely breaks
$\mathrm{SU}(2)_{\mathrm{D}}$.

In a generic case, every components of the septuplet can acquire nonzero VEV

$\displaystyle\Phi=\frac{1}{\sqrt{2}}\left(v_{3},v_{2},v_{1},v_{0},v_{1}^{*},v_{2}^{*},v_{3}^{*}\right)^{\text{T}}.$
(10)

If we ignore the $\lambda_{3}$ terms in the potential, the potential is a
function of radial component $v^{\prime}$ defined by
$v^{\prime}\equiv\sqrt{v_{0}^{2}+2|v_{1}|^{2}+2|v_{2}|^{2}+2|v_{3}|^{2}}$. We
can perform a $\text{SO}(7)$ rotation to keep only one of 7 components
nonzero. The potential is flat along a $S^{6}$ perpendicular to the radial
direction, which means it only depends on $v^{\prime}$. Since the
$\lambda_{3}$ term explicitly breaks the SO(7) symmetry to SU(2), the
potential becomes a function of the $S^{6}$ coordinates. Therefore, for a
fixed $v^{\prime}$, the $\lambda_{3}$ term forces the minima to locate in some
specific directions of to the $S^{6}$.

We can use the $\mathrm{SU}(2)_{\mathrm{D}}$ transformation to eliminate some
degrees of freedom in Eq. (10). Applying a transformation
$U=e^{i\frac{\pi}{2}\tau_{1}}e^{i\varphi\tau^{3}}$ to $\Phi$ as
$\Phi^{\prime}=U\Phi$, the $v_{0}$ component becomes

$\displaystyle
v_{0}^{\prime}=-\frac{\sqrt{5}}{2}|v_{3}|\left[\sin(3\varphi+\varphi_{3})+\sqrt{\frac{3}{5}}\frac{|v_{1}|}{|v_{3}|}\sin(\varphi+\varphi_{1})\right].$
(11)

where the $\varphi_{1}$ and $\varphi_{3}$ are the complex phase angles of
$v_{1}$ and $v_{3}$, respectively. We can tune $\varphi$ to set the expression
in square bracket vanishing and eliminate the $v_{0}$ component. We can
further perform another $\mathrm{SU}(2)_{\mathrm{D}}$ transformation
$e^{i\varphi^{\prime}\tau_{3}}$ after $U$ to eliminate the complex phase of
the $v_{1}$ component, and then the vacuum can be rewritten as

$\displaystyle\Phi=\frac{1}{\sqrt{2}}(v_{3},v_{2},v_{1},0,v_{1},v_{2}^{*},v_{3}^{*})^{\text{T}}.$
(12)

Substituting Eq.(12) into $(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$, one obtains

$\displaystyle(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[3v_{1}^{2}+5|v_{3}|^{2}-\sqrt{15}v_{1}(v_{3}+v_{3}^{*})\right]^{2}$
(13)
$\displaystyle+\frac{1}{4}\left[9v_{1}^{2}-5|v_{3}|^{2}-\sqrt{15}v_{1}(v_{3}+v_{3}^{*})\right]^{2}$
$\displaystyle+(3v_{1}^{2}-5|v_{3}|^{2})^{2}-\frac{15}{2}v_{1}^{2}(v_{3}-v_{3}^{*})^{2}$
$\displaystyle+\frac{1}{8}\left[3\sqrt{10}v_{1}(v_{2}+v_{2}^{*})+5\sqrt{6}(v_{2}^{*}v_{3}+v_{2}v_{3}^{*})\right]^{2}$
$\displaystyle-\frac{1}{8}\left[3\sqrt{10}v_{1}(v_{2}-v_{2}^{*})+5\sqrt{6}(v_{2}^{*}v_{3}-v_{2}v_{3}^{*})\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{63}{2}v_{1}^{4}+\frac{75}{2}|v_{3}|^{4}-15v_{1}^{2}|v_{3}|^{2}-6\sqrt{15}v_{1}^{3}(v_{3}+v_{3}^{*})$
$\displaystyle+45v_{1}^{2}|v_{2}|^{2}+75|v_{2}|^{2}|v_{3}|^{2}+15\sqrt{15}v_{1}(v_{2}^{2}v_{3}^{*}+v_{2}^{*2}v_{3}).$

Note that $(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ is a sum of squares and thus is
non-negative. The first three lines after the first equal sign depend on the
$v_{1}$ and $v_{3}$, while the last two lines after the first equal sign
depend on $v_{1}$, $v_{2}$ and $v_{3}$. We find that there are two
inequivalent vacuum configurations corresponding to different signs of
$\lambda_{3}$:

  * •

For $\lambda_{3}\geq 0$, it is easy to see that solutions
$(|v_{2}|=v^{\prime}/\sqrt{2},v_{1}=v_{3}=0)$ and
$(v_{2}=0,v_{3}=\sqrt{15}v_{1}/5)$ minimize the potential since
$(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ becomes zero. These two solutions are
equivalent and related by a specific $\mathrm{SU}(2)_{\mathrm{D}}$
transformation $e^{-i(\pi/2)\tau_{2}}e^{-i(\text{Arg}(v_{2})/2)\tau_{3}}$.
Actually, there are infinite equivalent solutions generated by applying a
continuous $\mathrm{SU}(2)_{\mathrm{D}}$ transformation on
$(|v_{2}|=v^{\prime}/\sqrt{2},v_{1}=v_{3}=0)$.

  * •

For $\lambda_{3}<0$, to find the minimum of the potential, we should maximize
$(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ with a fixing
$v^{\prime}=\sqrt{2v_{1}^{2}+2|v_{2}|^{2}+2|v_{3}|^{2}}$. Let us rewrite
$(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ with $v_{2}=|v_{2}|e^{i\varphi_{2}}$ and
$v_{3}=|v_{3}|e^{i\varphi_{3}}$,

$\displaystyle(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ $\displaystyle=$
$\displaystyle\frac{63}{2}v_{1}^{4}+\frac{75}{2}|v_{3}|^{4}-15v_{1}^{2}|v_{3}|^{2}-12\sqrt{15}v_{1}^{3}|v_{3}|\cos(\varphi_{3})$
(14)
$\displaystyle+45v_{1}^{2}|v_{2}|^{2}+75|v_{2}|^{2}|v_{3}|^{2}+30\sqrt{15}v_{1}|v_{2}|^{2}|v_{3}|\cos(2\varphi_{2}-\varphi_{3}).$

Obviously, this term is maximized when $\varphi_{3}=\pi$ and
$\varphi_{2}=\pi/2$, so we can pin down these two angles. Next, we can rewrite
$|v_{2}|=v^{\prime}\cos\theta/\sqrt{2}$,
$v_{1}=v^{\prime}\sin\theta\sin\phi/\sqrt{2}$,
$|v_{3}|=v^{\prime}\sin\theta\cos\phi/\sqrt{2}$, yielding

$\displaystyle(\Phi^{{\dagger}}T^{ab}\Phi)^{2}$ $\displaystyle=$
$\displaystyle{v^{\prime}}^{4}\left(\frac{63}{8}\sin^{4}\theta\sin^{4}\phi+\frac{75}{8}\sin^{4}\theta\cos^{4}\phi-\frac{15}{4}\sin^{4}\theta\sin^{2}\phi\cos^{2}\phi\right.$
(15)
$\displaystyle\quad+3\sqrt{15}\sin^{4}\theta\sin^{3}\phi\cos\phi+\frac{45}{4}\cos^{2}\theta\sin^{2}\theta\sin^{2}\phi+\frac{75}{4}\cos^{2}\theta\sin^{2}\theta\cos^{2}\phi$
$\displaystyle\quad+\left.\frac{15\sqrt{15}}{2}\cos^{2}\theta\sin^{2}\theta\sin\phi\cos\phi\right).$

This function reaches its maximum $75/8$ at a specific configuration
$(v_{1}=v_{2}=0,v_{3}=v^{\prime}/\sqrt{2})$. The other equivalent vacuum, e.g.
$\left(\theta=\pi/2,\tan\phi=\sqrt{15}\right)$ can be found by applying an
$\mathrm{SU}(2)_{\mathrm{D}}$ transformation on the specific vacuum.

After spontaneous symmetry breaking, the gauge bosons become massive. We can
now study the mass spectrum for these two inequivalent vacua.

####  II.2.1 The $\lambda_{3}<0$ case (splitting spectrum)

For the $\lambda_{3}<0$ case, we consider the vacuum configuration to be
$\Phi=\left(v^{\prime}/2,0,0,0,0,0,v^{\prime}/2\right)^{\text{T}}$. The mass-
squared of gauge bosons are

$\displaystyle m_{X^{\pm}}^{2}=\frac{3g_{4}^{2}}{2}{v^{\prime}}^{2},\quad
m_{Z^{\prime}}^{2}=9g_{4}^{2}{v^{\prime}}^{2},$ (16)

where $X^{\pm}\equiv(X^{1}\mp iX^{2})/\sqrt{2}$, $Z^{\prime}\equiv X^{3}$. In
this case, there are mass splitting between $X^{\pm}$ and $Z^{\prime}$. Since
$Z^{\prime}$ is heavier than twice of $X^{\pm}$’s mass, it can decay into a
pair of $X^{\pm}$.

The Goldstone bosons eaten by gauge bosons are $\Delta_{2}$ and
$-\textrm{Im}(\Delta_{3})$. The physical scalar fields are $\Delta_{0}$,
$\Delta_{1}$ and a mixture of the neutral $CP$-even component, $h$ of SM Higgs
doublets, and the CP-even mode around the VEV of $\Delta_{3}$. We can find
their mass-squared as follows

$\displaystyle
m_{\Delta_{0}}^{2}=-\frac{45\lambda_{3}}{32}{v^{\prime}}^{2},\quad
m_{\Delta_{1}}^{2}=-\frac{15\lambda_{3}}{16}{v^{\prime}}^{2},$ (17)

and

$\displaystyle\mathcal{L}_{M_{\Delta_{3}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\begin{array}[]{cc}h&\rho_{3}\end{array}\right)\left(\begin{array}[]{cc}2\lambda
v^{2}&\lambda_{4}vv^{\prime}\\\
\lambda_{4}vv^{\prime}&\left(2\lambda^{\prime}_{2}+\frac{25\lambda_{3}}{16}\right){v^{\prime}}^{2}\end{array}\right)\left(\begin{array}[]{c}h\\\
\rho_{3}\end{array}\right)$ (24) $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\lambda
v^{2}+\left(\lambda_{2}+\frac{25\lambda_{3}}{32}\right){v^{\prime}}^{2}-\sqrt{\left[\lambda
v^{2}-\left(\lambda^{\prime}_{2}+\frac{25\lambda_{3}}{32}\right){v^{\prime}}^{2}\right]^{2}+\lambda_{4}^{2}v^{2}v^{\prime
2}}\right]h_{1}^{2}$ $\displaystyle-\frac{1}{2}\left[\lambda
v^{2}+\left(\lambda_{2}+\frac{25\lambda_{3}}{32}\right){v^{\prime}}^{2}+\sqrt{\left[\lambda
v^{2}-\left(\lambda^{\prime}_{2}+\frac{25\lambda_{3}}{32}\right){v^{\prime}}^{2}\right]^{2}+\lambda_{4}^{2}v^{2}v^{\prime
2}}\right]h_{2}^{2},$

where $v\approx 246~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ is the electroweak VEV
of SM Higgs doublet and the mass eigenstates $(h_{1},~{}h_{2})$ relates to the
definition states by a rotation

$\displaystyle\left(\begin{array}[]{c}h\\\
\rho_{3}\end{array}\right)=\left(\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\right),$ (31)

where

$\displaystyle\tan\alpha=\frac{\sqrt{\left[\left(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\right){v^{\prime}}^{2}-\lambda
v^{2}\right]^{2}+(\lambda_{4}vv^{\prime})^{2}}-\left[\left(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\right){v^{\prime}}^{2}-\lambda
v^{2}\right]}{\lambda_{4}vv^{\prime}}.$ (32)

If we identify $h_{1}$ as the $125~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ SM-like
Higgs field, then $\lambda$ and the mass square of $h_{2}$ can be expressed as

$\displaystyle\lambda$ $\displaystyle\approx$
$\displaystyle\frac{m_{h_{1}}^{2}}{2v^{2}}+\frac{\lambda_{4}^{2}{v^{\prime}}^{2}}{2\left[2\left(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\right){v^{\prime}}^{2}-m_{h_{1}}^{2}\right]},$
(33) $\displaystyle m_{h_{2}}^{2}$ $\displaystyle\approx$ $\displaystyle
2\left(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\right){v^{\prime}}^{2}+\frac{\lambda_{4}^{2}v^{2}{v^{\prime}}^{2}}{2\left(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\right){v^{\prime}}^{2}-m_{h_{1}}^{2}}.$
(34)

In this work, we only focus on the situation that the scalar $\Delta_{0}$ and
$\Delta_{1}$ are massive enough for decaying into the gauge bosons. It
requires the parameters to satisfy the condition that

$\displaystyle\lambda_{3}\leq-48g_{4}^{2}/5~{},$ (35)

and thus complex vector boson $X^{\pm}$ is the lightest stable particle in the
dark sector. Since we will regard the $X^{\pm}$ (and $X^{\ast}$) as the dark
matter candidate in this case, we should ensure that they stable. We find that
there is a remnant $Z_{6}\times Z_{2}$ symmetry in the dark sector after the
$\mathrm{SU}(2)_{\mathrm{D}}$ symmetry breaking:

$\displaystyle Z_{6}$ $\displaystyle:$
$\displaystyle\Delta_{0}\rightarrow-\Delta_{0},\quad\Delta_{1}\rightarrow
e^{-i\frac{2\pi}{3}}\Delta_{1},\quad\Delta_{2}\rightarrow
e^{i\frac{5\pi}{3}}\Delta_{2},\quad\Delta_{3}\rightarrow\Delta_{3},$ (36)
$\displaystyle X^{+}\rightarrow e^{-i\frac{5\pi}{3}}X^{+},\quad
Z^{\prime}\rightarrow Z^{\prime}.$ $\displaystyle Z_{2}$ $\displaystyle:$
$\displaystyle\Delta_{0}\rightarrow-\Delta_{0}\quad\Delta_{1}\rightarrow\Delta_{-1},\quad\Delta_{2}\rightarrow-\Delta_{-2},\quad\Delta_{3}\rightarrow\Delta_{-3},$
(37) $\displaystyle X^{+}\rightarrow X^{-},\quad
Z^{\prime}\rightarrow-Z^{\prime}$

which can prevent the dark matter candidate $X^{\pm}$ from decaying to SM
particles.

####  II.2.2 The $\lambda_{3}\geq 0$ case (degenerate spectrum)

For the $\lambda_{3}\geq 0$ case, we define
$\Phi=\left(0,v^{\prime}/2,0,0,0,v^{\prime}/2,0\right)^{\text{T}}$. The mass-
squared of gauge bosons are given by

$\displaystyle m_{X^{\pm}}^{2}=m_{Z^{\prime}}^{2}=4g_{4}^{2}{v^{\prime}}^{2}.$
(38)

We can see that the masses of $X^{\pm}$ and $Z^{\prime}$ are degenerate in
this case.

One of the Goldstone bosons is $-\textrm{Im}(\Delta_{2})$, the other one is a
mixture of $\Delta_{1}$ and $\Delta_{3}$

$\displaystyle\mathcal{L}_{M_{\Delta_{1,3}}}$ $\displaystyle=$
$\displaystyle-\frac{5\lambda_{3}{v^{\prime}}^{2}}{32}\begin{pmatrix}\Delta_{-1}&\Delta_{3}\end{pmatrix}\begin{pmatrix}3&\sqrt{15}\\\
\sqrt{15}&5\end{pmatrix}\begin{pmatrix}\Delta_{1}\\\ \Delta_{-3}\end{pmatrix}$
(39) $\displaystyle=$
$\displaystyle-\begin{pmatrix}G^{+}&\Delta^{\ast}_{13}\end{pmatrix}\begin{pmatrix}0&\\\
&\frac{5\lambda_{3}{v^{\prime}}^{2}}{4}\end{pmatrix}\begin{pmatrix}G^{-}\\\
\Delta_{13}\end{pmatrix},$

with

$\displaystyle\begin{pmatrix}\Delta_{1}\\\
\Delta_{-3}\end{pmatrix}=\begin{pmatrix}\cos\beta&\sin\beta\\\
-\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}G^{-}\\\
\Delta_{13}\end{pmatrix},\quad\tan\beta=\frac{\sqrt{15}}{5}.$ (40)

The masses of $\Delta_{0}$ and $\Delta_{13}$ are given by

$\displaystyle
m_{\Delta_{0}}^{2}=m_{\Delta_{13}}^{2}=\frac{5\lambda_{3}}{4}{v^{\prime}}^{2}~{},$
(41)

which are also degenerate.

$\displaystyle\mathcal{L}_{M_{h,\Delta_{2}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\begin{pmatrix}h&\rho_{2}\end{pmatrix}\begin{pmatrix}2\lambda
v^{2}&\lambda_{4}vv^{\prime}\\\
\lambda_{4}vv^{\prime}&2\lambda^{\prime}_{2}{v^{\prime}}^{2}\end{pmatrix}\begin{pmatrix}h\\\
\rho_{2}\end{pmatrix}$ (42) $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\lambda
v^{2}+\lambda^{\prime}_{2}{v^{\prime}}^{2}-\sqrt{\left(\lambda
v^{2}-\lambda^{\prime}_{2}{v^{\prime}}^{2}\right)^{2}+\lambda_{4}^{2}v^{2}v^{\prime
2}}\right)h_{1}^{2}$ $\displaystyle-\frac{1}{2}\left(\lambda
v^{2}+\lambda^{\prime}_{2}{v^{\prime}}^{2}+\sqrt{\left(\lambda
v^{2}-\lambda^{\prime}_{2}{v^{\prime}}^{2}\right)^{2}+\lambda_{4}^{2}v^{2}v^{\prime
2}}\right)h_{2}^{2},$

with

$\displaystyle\left(\begin{array}[]{c}h\\\
\rho_{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\right),$ (49)

where

$\displaystyle\tan\alpha=\frac{\sqrt{\left(\lambda^{\prime}_{2}{v^{\prime}}^{2}-\lambda
v^{2}\right)^{2}+(\lambda_{4}vv^{\prime})^{2}}-\left(\lambda^{\prime}_{2}{v^{\prime}}^{2}-\lambda
v^{2}\right)}{\lambda_{4}vv^{\prime}}.$ (50)

If we identify $h_{1}$ as the $125$\mathrm{G}\mathrm{e}\mathrm{V}$$ SM-like
Higgs field, then its quartic coupling and the mass of the heavier Higgs field
$h_{2}$ are

$\displaystyle\lambda\approx\frac{m_{h_{1}}^{2}}{2v^{2}}+\frac{\lambda_{4}^{2}{v^{\prime}}^{2}}{2\left(2\lambda^{\prime}_{2}{v^{\prime}}^{2}-m_{h_{1}}^{2}\right)},\quad
m_{h_{2}}^{2}\approx
2\lambda^{\prime}_{2}{v^{\prime}}^{2}+\frac{\lambda_{4}^{2}v^{2}{v^{\prime}}^{2}}{2\lambda^{\prime}_{2}{v^{\prime}}^{2}-m_{h_{1}}^{2}}.$
(51)

In this case, both $X^{\pm}$ and $Z^{\prime}$ are dark matter candidates.
There is a $Z_{4}\times Z_{2}$ symmetry:

$\displaystyle Z_{4}$ $\displaystyle:$ $\displaystyle\Delta_{0}\rightarrow
e^{i\pi}\Delta_{0},\quad\Delta_{1}\rightarrow
e^{i\frac{\pi}{2}}\Delta_{1},\quad\Delta_{2}\rightarrow\Delta_{2},\quad\Delta_{3}\rightarrow
e^{-i\frac{\pi}{2}}\Delta_{3},$ (52) $\displaystyle X^{+}\rightarrow
e^{-i\frac{\pi}{2}}X^{+},\quad Z^{\prime}\rightarrow Z^{\prime}.$
$\displaystyle Z_{2}$ $\displaystyle:$
$\displaystyle\Delta_{0}\rightarrow\Delta_{0}\quad\Delta_{1}\rightarrow-\Delta_{-1},\quad\Delta_{2}\rightarrow\Delta_{-2},\quad\Delta_{3}\rightarrow-\Delta_{-3},$
(53) $\displaystyle X^{+}\rightarrow X^{-},\quad
Z^{\prime}\rightarrow-Z^{\prime}~{},$

which can prevent $X^{\pm}$ and $Z^{\prime}$ from decaying into SM particles.
Since we only focus on the situation that vector bosons are the only DM
components, a condition:

$\displaystyle\lambda_{3}\geq 64g_{4}^{2}/5~{},$ (54)

is required such that the new scalars $\Delta_{0}$ and $\Delta_{13}$ are
massive enough to decay.

##  III Vacuum stability and perturbativity

As the energy scale raises, all the couplings of interaction evolve according
to renormalization group equations. Therefore, it is possible that some of the
couplings change their sign at a critical scale. It means even the vacuum is
stable in the low energy effective theory, it might become unstable in higher
energy scale since the potential has no lower bound in the infinity of the
vacuum configuration. In the SM, the quartic coupling of the Higgs field is an
example that running to a negative value at the scale around $10^{9}$ GeV. It
tells us that if the SM is valid up to Planck scale, then the vacuum
configuration corresponding to the Higgs field is unstable or metastable.
However, if there are some new physics appearing below the scale of switching
sign, they can modify the running behaviors of the couplings, and thus the
vacuum is capable to keep stable in high energy.

The co-positivity criteria Kannike:2012pe  are used to ensure that the tree-
level potential is bounded from below. There are two cases which coincide with
the categories of two inequivalent vacua, giving respectively Cai:2015kpa

$\displaystyle\lambda_{3}<0:\begin{cases}\lambda\geq 0,\\\
\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\geq 0,\\\
\lambda_{4}+2\sqrt{\lambda(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3})}\geq
0,\end{cases}\qquad\lambda_{3}\geq 0:\begin{cases}\lambda\geq 0,\\\
\lambda^{\prime}_{2}\geq 0,\\\
\lambda_{4}+2\sqrt{\lambda\lambda^{\prime}_{2}}\geq 0.\end{cases}$ (55)

These conditions are required to be satisfied in the whole RG evolution
processes till the cutoff scale. Another theoretical concern of the running
couplings is their perturbativity in high energy. We expect them to keep
finite below the Planck scale, otherwise they should be embedded in a more
fundamental theory below that scale. The blowing up scale for the couplings is
so called Landau pole. The Landau pole problem of the
$\mathrm{SU}(2)_{\mathrm{L}}$ scalar multiplets has been studied in Ref.
Hamada:2015bra . They showed that if the dimension of representation is larger
than 4, the quartic couplings of the scalars blow up soon below the Planck
scale even they are set to vanishing at the EW scale, and thus the same
problem can be raised in our model. However, since the gauge coupling of
$\mathrm{SU}(2)_{\mathrm{D}}$ is not necessary to be the same as of
$\mathrm{SU}(2)_{\mathrm{L}}$, the Landau pole scale can be totally different
from the EW multiplet cases. In order to determine the RG running of the
couplings, we calculate their beta functions. To the one-loop level, the beta
functions of couplings are shown as follows,

$\displaystyle\beta_{g_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(-\frac{8}{3}\right)g_{4}^{3},$ (56)
$\displaystyle\beta_{\lambda^{\prime}_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left({30\lambda^{\prime}_{2}}^{2}+\frac{75}{8}\lambda^{2}_{3}+2\lambda_{4}^{2}+15\lambda^{\prime}_{2}\lambda_{3}-144\lambda^{\prime}_{2}g_{4}^{2}+288g_{4}^{4}\right),$
(57) $\displaystyle\beta_{\lambda_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\frac{63}{8}\lambda_{3}^{2}+24\lambda^{\prime}_{2}\lambda_{3}-144\lambda_{3}g_{4}^{2}+288g_{4}^{4}\right),$
(58) $\displaystyle\beta_{\lambda_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(4\lambda_{4}^{2}+12\lambda\lambda_{4}+18\lambda^{\prime}_{2}\lambda_{4}+\frac{15}{2}\lambda_{3}\lambda_{4}-\frac{9}{10}\lambda_{4}g_{1}^{2}-\frac{9}{2}\lambda_{4}g_{2}^{2}-72\lambda_{4}g_{4}^{2}+6\lambda_{4}y_{t}^{2}\right),$
(59) $\displaystyle\beta_{\lambda}$ $\displaystyle=$
$\displaystyle\beta_{\lambda}^{\text{SM}}+\frac{1}{16\pi^{2}}\frac{7}{2}\lambda_{4}^{2}.$
(60)

where $g_{1}$ and $g_{2}$ are the gauge coupling coefficients of
$\mathrm{U}(1)_{\mathrm{Y}}$ and $\mathrm{SU}(2)_{\mathrm{L}}$, $y_{t}$ is the
top quark Yukawa coupling. Since their beta functions receive no modification
from the dark sector, we refer their explicit expressions in the
Ref.Cai:2015kpa .

Before proceeding the numerical computation, it is useful to make some
qualitative senses on the running behavior of each coupling. The gauge
coupling $g_{4}$ is assumed to be smaller than the gauge coupling of
$\mathrm{SU}(2)_{\mathrm{L}}$, and thus its beta function $\beta_{g_{4}}$ is
negligible implying $g_{4}$ approximately keep constant. Landau pole is
inevitable to appear in $\beta_{\lambda^{\prime}_{2}}$ and
$\beta_{\lambda_{3}}$ if $g_{4}$ is not small enough. Note that there are
$g_{4}^{4}$ terms in these two beta functions, which means even we turn off
$\lambda^{\prime}_{2}$ and $\lambda_{3}$ in low energy, they still increase
fast owing to a nonzero $g_{4}$. On the other hand, since all the terms in
$\beta_{\lambda_{4}}$ has a factor $\lambda_{4}$, $\beta_{\lambda_{4}}$ has a
fixed point at $\lambda_{4}=0$. It means the running behavior of $\lambda_{4}$
is under control if it is small at the scale of the dark sector. For achieving
the DM freeze-out and vacuum stability, we need a small but nonzero
$\lambda_{4}$. Note that the vacuum instability is usually brought out by
$\lambda$ in the high energy due to a negative contribution of $y_{t}^{4}$
term in its beta function. The portal coupling $\lambda_{4}$ is helpful for
easing the problem in two aspect. One is that the mixing between the SM Higgs
doublet and the scalar septuplet leads to a ”jump” of $\lambda$ around the
mass scale of the septuplet due to the threshold effect Elias-Miro:2012eoi .
The other one is the positive contribution of $\lambda_{4}^{2}$ in
$\beta_{\lambda}$ can help to retard the decreasing of $\lambda$. To be
precise, below the septuplet mass scale $\Lambda_{\text{HS}}$ which is assumed
to about $\Lambda_{\text{HS}}=0.2v^{\prime}$, septuplet decouples from the
theory and thus the evolution of $\lambda$ is determined by the SM beta
function of $\lambda$ (dropping the $\lambda_{4}^{2}$ term in Eq.(60)). At the
scale $\Lambda_{\text{HS}}$, the threshold effect leads to the matching
conditions as

$\displaystyle\lambda(\Lambda_{\text{HS}})=\lambda_{\text{SM}}(\Lambda_{\text{HS}})+\frac{\lambda_{4}^{2}}{4\left(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\right)},\quad$
$\displaystyle\text{for}~{}\lambda_{3}<0,$ (61)
$\displaystyle\lambda(\Lambda_{\text{HS}})=\lambda_{\text{SM}}(\Lambda_{\text{HS}})+\frac{\lambda_{4}^{2}}{4\lambda^{\prime}_{2}},\quad$
$\displaystyle\text{for}~{}\lambda_{3}\geq 0,$ (62)

From the scale $\Lambda_{\text{HS}}$ toward higher energy, the
$\lambda_{4}^{2}$ term is turned on in Eq. (60).

In our numerical analysis, the input values of the SM parameters at $Z$ pole
applying the $\overline{\text{MS}}$ scheme are given in
ParticleDataGroup:2020ssz ; Hambye:1996wb . As for the couplings coefficients
$g_{4},\lambda^{\prime}_{2},\lambda_{3}$ and $\lambda_{4}$ are newly
introduced, their input values at $\Lambda_{\text{HS}}$ are free parameters in
our model. Since the phenomenologies are mostly related to $\lambda_{4}$,
while the Landau pole problem is related to $\lambda^{\prime}_{2}$, we will
mainly perform the experimental and theoretical constraints on the
$\lambda^{\prime}_{2}\text{-}\lambda_{4}$ plane. For the other couplings like
$g_{4}$, $\lambda_{3}$, and $v^{\prime}\sim$\mathrm{T}\mathrm{e}\mathrm{V}$$,
we select some typical values and then runs the RGE to determine the
evolutions of all couplings toward high energy scale. If the vacuum stability
conditions in Eq. (55) are violated at any energy scale below the Planck
scale, then the corresponding parameter set are marked down as unstable vacuum
region. On the other hand, if any coupling excesses $4\pi$ below Planck scale,
it becomes non-perturbative and meets Landau pole soon.

The numerical results show that the energy scale of Landau pole decreases as
$|\lambda_{3}|$ growing up, and have a maximum at
$g_{4}(\Lambda_{\text{HS}})\sim$ $0.1$–$0.3$ when other parameters are fixed.
However, as we are focusing on the case that the vector bosons are the only
stable DM candidates, we will further assume some relations between
$g_{4}^{2}$ and $\lambda_{3}$ (see Eq.(35) and (54)). In those cases, we find
that the perturbative region of parameters is enlarged as $g_{4}$ decreases.
The upper limit for the absolute value of the input portal coupling is found
to be $|\lambda_{4}(\Lambda_{\text{HS}})|\lesssim 0.24$, while the upper limit
of the input gauge coupling is found to be
$|g_{4}(\Lambda_{\text{HS}})|\lesssim 0.15$ for
$\lambda_{3}(\Lambda_{\text{HS}})\geq 0$, or
$g_{4}(\Lambda_{\text{HS}})\lesssim 0.19$ for
$\lambda_{3}(\Lambda_{\text{HS}})<0$.

Naively thinking, the Landau pole may decrease monotonically as $g_{4}$
growing. However, an analytical treatment of solving the RGEs in Eq. (84)
shows that if $\lambda^{\prime}_{2}+\eta\lambda_{3}>0$ (see more details in
Appendix A), the Landau pole scale is not a monotonic function of
$g_{4}(\Lambda_{\text{HS}})$. For a given set of other parameters, the Landau
pole scale $\Lambda_{\text{LP}}$ reaches its maximal value at a non-zero value
of $g_{4}(\Lambda_{\text{HS}})$ (see Fig. 1). This behavior is also confirmed
by the full numerical calculations. We can utilize the approximate analytical
solution to estimate the upper limit of $g_{4}(\Lambda_{\text{HS}})$ under the
scalar decay conditions. For $\lambda_{3}\geq 0$, the analytic estimation
gives $g_{4}(\Lambda_{\text{HS}})\lesssim 0.15$ which agrees with the
numerical calculation very well. For $\lambda_{3}<0$, it gives
$g_{4}(\Lambda_{\text{HS}})\lesssim 0.37$ which deviates from the numerical
calculation by a factor of 2.

Figure 1: The analytic solutions of Landau pole as a function of
$g_{4}(\Lambda_{\text{HS}})$ with different values of
$\lambda^{\prime}_{2}+\eta\lambda_{3}$. The blue, green and red lines are the
solutions at $\lambda^{\prime}_{2}+\eta\lambda_{3}=0.1,0.2$ and $0.3$
respectively, with $\eta\approx 0.9$.

##  IV Constraining the model by phenomenologies, perturbativity, and vacuum
stability

In this section, we study the phenomenological constraints for the vector
bosons DM, associated with the theoretical constraints from perturbativity and
vacuum stability. Similar to most of the the other cold dark matter models,
the vector DMs in our model are produced via the freeze-out mechanism. We
utilize the FeynRules 2 Alloul:2013bka  and micrOMEGAs Belanger:2014vza
packages to determine the relic density and annihilation cross section of DMs.
We compare the numerical result of DM relic density with the experimental
result, $\Omega_{\text{CDM}}h^{2}=0.12$, from the Planck experiment
Planck:2018vyg . We also consider the indirect detection bounds on DM
annihilation cross section provided by Fermi-LAT MAGIC:2016xys , which
combines the 6 years data of gamma-ray observations from Segue 1 with 158
hours and 15 dwarf satellite galaxies. In our numerical calculation of the
annihilation cross section, DMs in the dwarf satellite galaxies are assumed to
be at rest for simplicity. The future prospect on the sensitivity of DM
indirect detection from the Cherenkov Telescope Array (CTA)
CTAConsortium:2012fwj  is included for comparison. In addition, we consider
the most recent direct detection constraints provided by first result of LUX-
ZEPLIN (LZ) experiment LUX-ZEPLIN:2022qhg , and its prospective sensitivity of
1000 days data LZ:2015kxe . Apart from the DM phenomenologies, there are LHC
bounds on the the mixing angle between the Higgs field and the scalar
$\rho_{2,3}$. In our model, the trilinear coupling between Higgs-like field
and all other SM fields are suppressed by a factor $\cos\alpha$ due to the
mixing. We fit the measurements of couplings
$\kappa_{i},~{}i=\gamma,W,Z,g,t,b,\tau$ from ATLAS ATLAS:2019nkf  and CMS
CMS:2018uag  with a common $\kappa$, and find $\kappa>0.9875$ at 95% CL. We
will set this bound for $\kappa=\cos\alpha$ in our model.

In the theoretical aspect, we assume that there are no other new sectors
directly couple to the SM and dark sector below the Planck scale, and thus the
perturbativity and vacuum stability conditions should be satisfied below the
Planck scale. The RGEs are solved numerically with certain input of coupling
parameters at the mass scale of the septuplet (around TeV).

###  IV.1 Splitting case $(\lambda_{3}<0)$

Since $X^{\pm}$ couples to the SM particles through Higgs-portal, their
annihilation cross section at freeze-out epoch closely relate to the mixing
angle $\alpha$ between $h$ and $\rho_{3}$. The dominant DM annihilation
channels are $X^{+}+X^{-}\to W^{+}+W^{-},~{}Z+Z,~{}h_{1}+h_{1}$, as displayed
in Fig. 2. The thermal averaged annihilation cross section corresponding to
these processes are given by

$\displaystyle\sum_{ij=WW,ZZ,h_{1}h_{1}}\langle\sigma_{X^{+}X^{-}\to
ij}v\rangle\approx\frac{\sin^{2}(2\alpha)m_{X}^{2}}{48\pi v^{2}v^{\prime
2}}\left(1-\frac{4m_{X}^{2}}{m_{h_{2}}^{2}}\right)^{-2}$ (63)

in the $m_{h_{2}},m_{X}\gg m_{W},m_{Z},m_{h_{1}}$ limit. This approximately
yields a relic density of DM:

$\displaystyle\Omega_{\text{DM}}h^{2}\approx
2.59\times\left(\frac{0.1}{g_{4}}\right)^{2}\left(\frac{0.1}{\sin^{2}(2\alpha)}\right)\left(1-\frac{4m_{X}^{2}}{m_{h_{2}}^{2}}\right)^{2}.$
(64)

Since the Higgs couplings measurements imply a $\sin^{2}(2\alpha)\lesssim
0.1$, we need a $g_{4}(\Lambda_{\text{HS}})\approx 0.4$ with $4m_{X}^{2}\ll
m_{h_{2}}^{2}$ or a $g_{4}(\Lambda_{\text{HS}})<0.4$ with $4m_{X}^{2}\approx
m_{h_{2}}^{2}$ (closed to the resonance) in order to obtain an
$\Omega_{\text{DM}}h^{2}\approx 0.12$. As we have estimated in the previous
section, there is an upper bound for $g_{4}(\Lambda_{\text{HS}})$ from the
perturbativity as $g_{4}(\Lambda_{\text{HS}})\lesssim 0.19$, and thus the
correct relic density can only be achieved near the resonance.

(a) DM annihilate to $W$ and $Z$.  (b) DM annihilate to SM-like Higgs $h_{1}$.

Figure 2: The dominant annihilation channels of DM.

(a)
$g_{4}=0.05,~{}\lambda_{3}=-0.2,~{}v^{\prime}=10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$.
(b)
$g_{4}=0.1,~{}\lambda_{3}=-0.2,~{}v^{\prime}=10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$.

Figure 3: Splitting case: constraints on $\lambda^{\prime}_{2}$-$\lambda_{4}$
plane from perturbativity, vacuum stability and experiments including DM relic
density, direct and indirect detection, and LHC bounds on Higgs couplings. The
VEV of the septuplet $v^{\prime}$ is fixed as
$10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$, while $\\{g_{4},\lambda_{3}\\}$ are
set to be $\\{0.05,-0.2\\}$ (left panel) and $\\{0.1,-0.2\\}$ (right panel),
respectively. The gray regions correspond to parameter points that violating
the vacuum stability conditions given in Eq. (55), while the orange regions
correspond to the running couplings becoming non-perturbative below the Planck
scale. The regions shaded in blue are excluded by the upper limit on DM-
nucleon scattering cross section from LZ $90\%$ C.L. LUX-ZEPLIN:2022qhg ,
while the regions shaded in green are excluded by the upper limit on
$\left<\sigma_{\mathrm{ann}}v\right>$ in the $W^{\pm}$ channel from Fermi-LAT
at $95\%$ C.L. MAGIC:2016xys . The blue dashed lines are the sensitivity
curves from the prospect of LZ with 1000-days data LZ:2015kxe , while the
green dashed lines are the sensitivity of CTA CTAConsortium:2012fwj . The red
lines indicate a DM relic density value $\Omega_{\text{DM}}h^{2}=0.12$
Planck:2018vyg . The region shaded in cyan are excluded by the Higgs couplings
measurements in LHC ATLAS:2019nkf ; CMS:2018uag

In Fig. 3, we show the perturbativity, vacuum stability and DM
phenomenological constraints in the $\lambda^{\prime}_{2}$-$\lambda_{4}$
plane, with other parameters fixed. The DM mass in Fig. 3(a) and Fig. 3(b) are
$612.4~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$1.225~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$, respectively. The regions shaded
in gray are excluded since they lead to an unstable vacuum. Note that, these
regions separate in two pieces. One is in the area that
$\lambda^{\prime}_{2}+25\lambda_{3}/32<0$, which violates the second condition
given in the left part of Eq.(55). Another area of exclusion (bullet shape) is
mainly caused by $\lambda(\mu)<0$ at some higher energy scale (the first
condition in the left part of Eq.(55)). We can see that a small but non-
vanishing $\lambda_{4}$ is needed for preventing $\lambda(\mu)$ from running
into negative value. The boundary of the bullet shape area can be estimated by
$\lambda_{4}^{2}/(\lambda^{\prime}_{2}+25\lambda_{3}/32)\approx 0.1$, which is
given by the increment of $\lambda$ due to the threshold effect. As we have
mentioned previously, high dimensional representation of
$\mathrm{SU}(2)_{\mathrm{D}}$ leads to a large positive contribution in the
beta functions, and thus leads to stringent constraints of perturbativity for
the parameters. The regions shaded in orange are excluded due to some coupling
becoming non-perturbative ($>4\pi$) in high energy scale. The white region in
the plots satisfy both the perturbativity and vacuum stability conditions.
Numerical solutions of RGEs show that the allowed regions (white) shrink when
$|\lambda_{3}(\Lambda_{\text{HS}})|$ increases. Note that although the pattern
in Fig. 3 looks almost symmetric about the $\lambda_{4}=0$ axis, there is a
slight asymmetry due to the $\lambda_{4}^{2}$ term in $\beta_{\lambda_{4}}$
(see Eq. (59)). We can also see that the allowed region in Fig. 3(a) with
$g_{4}=0.05$ is smaller than the allowed region in Fig. 3(b) with $g_{4}=0.1$,
which qualitatively agrees with our previous analytical analysis. The Landau
pole scale is not monotonically depending on the $g_{4}$, and thus a smaller
$g_{4}$ does not always mean a better situation for keeping perturbative in
high energy.

In the DM phenomenological aspect, the DM observables mainly rely on
$\\{v^{\prime},g_{4},\lambda^{\prime}_{2}+25\lambda_{3}/32,\lambda_{4}\\}$ at
tree level. The red contours in Fig. 3 corresponds to DM relic density
$\Omega_{\text{DM}}h^{2}\approx 0.12$, while the region shaded in green and
blue are excluded by current DM indirect and direct detection experiments,
respectively. The green and blue dashed lines show the future prospect from
the CTA and LZ sensitivities. The region shaded in cyan are excluded by the
LHC measurement on the Higgs couplings. Note that the red contour concentrates
in a narrow region of $\lambda^{\prime}_{2}$ since the correct relic density
is obtained around the resonance point of the internal propagators. It can be
used to estimate the value of $\lambda^{\prime}_{2}$ if all other parameters
are input. To be precise, the resonance peak appears at the parameter point
such that $m_{h_{2}}\approx 2m_{X^{\pm}}$, which gives

$\displaystyle\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}\approx\frac{3}{2}g_{4}^{2}+\sqrt{\left(\frac{3}{2}g_{4}^{2}\right)^{2}-\frac{\lambda_{4}^{2}v^{2}}{4v^{\prime
2}}},$ (65)

where we have dropped the $m_{h_{1}}$ since it is negligible comparing to the
dark sector masses.

For the direct detection, the spin-independent cross section at tree level
between DM and nucleon $N=p,n$ is given by Hambye:2008bq

$\displaystyle\sigma_{\text{SI}}=\frac{f^{2}m_{N}^{4}}{4\pi(m_{X}+m_{N})^{2}}\frac{\sin^{2}(2\alpha)m_{X}^{4}}{v^{2}{v^{\prime}}^{2}}\frac{(m_{h_{2}}^{2}-m_{h_{1}}^{2})^{2}}{m_{h_{1}}^{4}m_{h_{2}}^{4}},$
(66)

where $f\equiv\langle N|\sum_{q}m_{q}\bar{q}q|N\rangle$. Since $m_{h_{2}}\gg
m_{h_{1}}$, we can eliminate $m_{h_{2}}$ in the numerator and denominator in a
rough estimation. On the other hand, from Eq. (32), the mixing angle $\alpha$
approximately depends on $(\lambda^{\prime}_{2}+25\lambda_{3}/32)$ and
$\lambda_{4}$ as follows

$\displaystyle|\sin\alpha|\approx|\tan\alpha|\approx\sqrt{\left(\frac{\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}}{|\lambda_{4}|}\right)^{2}\frac{v^{\prime
2}}{v^{2}}+1}-\frac{\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3}}{|\lambda_{4}|}\frac{v^{\prime}}{v}\approx\frac{|\lambda_{4}|}{2(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3})}\frac{v}{v^{\prime}},$
(67)

where $\lambda v^{2}$ has been neglected when compare with the large
$(\lambda^{\prime}_{2}+25\lambda_{3}/32)v^{\prime 2}$. Therefore, we can
estimate

$\displaystyle\sigma_{\text{SI}}\approx\left(\frac{f^{2}}{4\pi}\right)\left(\frac{m_{N}^{4}}{m_{h_{1}}^{4}}\right)\left(\frac{m_{X}^{2}}{v^{\prime
4}}\right)\frac{|\lambda_{4}|^{2}}{(\lambda^{\prime}_{2}+\frac{25}{32}\lambda_{3})^{2}}.$
(68)

If $m_{X},~{}g_{4},~{}v^{\prime}$ are fixed, then the combination
$|\lambda_{4}|/(\lambda^{\prime}_{2}+25\lambda_{3}/32)$ completely determines
the cross section $\sigma_{\text{SI}}$. Consequently, the boundaries of blue
region are nearly straight lines. The bounds from the Higgs couplings
measurements have similar behaviors since they only depends on the mixing
angle $\alpha$. In both panels of Fig. 3, there are regions consistent with
all the current experimental and theoretical constraints. We find that the
$\lambda^{\prime}_{2}$ is almost pinned down in a narrow region by the DM
indirect detection and relic density, when $g_{4}$, $\lambda_{3}$ and
$v^{\prime}$ are fixed. On the other hand, perturbativity and vacuum stability
conditions are available to constrain the $|\lambda_{4}|$. Current direct
detection is less effective as the perturbativity bound, however the
sensitivity of LZ with 1000-days data might be powerful enough to probe the
survival area. The Higgs couplings bounds are usually stronger than current
direct detection bounds, but still not effective enough to probe the region
allowed by perturbativity. From the prospect of the CTA experiment, we find it
has potential to confirm or exclude a part of the contour implied by the relic
density.

###  IV.2 Degenerate case $(\lambda_{3}\geq 0)$

In this case, both the complex vector field $X^{\pm}$ and the real vector
$Z^{\prime}$ are DM candidates.The dominant annihilation processes for the
$X^{\pm}$ are the same as in Fig. 2. For the $Z^{\prime}Z^{\prime}$
annihilation, we have similar diagrams and amplitudes (a factor $1/2$ appears
since it only has a half degree of freedom of $X^{\pm}$). The semi-
annihilations like $X^{+}X^{-}\to Z^{\prime}h_{1}$ and $X^{\pm}Z^{\prime}\to
X^{\pm}h_{1}$ channels and other processes are sub-dominant. The annihilation
cross section of DM are similar to the previous case, and then the relic
density of DM is

$\displaystyle\Omega_{\text{DM}}h^{2}\approx
1.46\times\left(\frac{0.1}{g_{4}}\right)^{2}\left(\frac{0.1}{\sin^{2}(2\alpha)}\right)\left(1-\frac{4m_{X}^{2}}{m_{h_{2}}^{2}}\right)^{2}.$
(69)

Note that there is an upper bound for the gauge coupling,
$g_{4}(\Lambda_{\text{HS}})\lesssim 0.15$, required by the perturbativity, and
an upper bound for $\sin^{2}(2\alpha)$, $\sin^{2}(2\alpha)\lesssim 0.1$,
required by the Higgs couplings measurements. Therefore, in order to achieve
observed relic density of DM, $\Omega_{\text{DM}}h^{2}\approx 0.12$, the
masses of $X^{\pm},Z^{\prime}$ and $h_{2}$ should be near the resonance
$2m_{X,Z^{\prime}}\approx m_{h_{2}}$:

$\displaystyle\lambda^{\prime}_{2}\approx
4g_{4}^{2}+\sqrt{\left(4g_{4}^{2}\right)^{2}-\frac{\lambda_{4}^{2}v^{2}}{4v^{\prime
2}}}.$ (70)

The experimental and theoretical constraints on the
$\lambda^{\prime}_{2}$-$\lambda_{4}$ plane for this case are presented in Fig.
4. The DM masses in Fig. 4(a), 4(b) and 4(c) are set to
$2~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$, $1~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$
and $1~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$ respectively. Note that the DM
observables in this case only rely on
$(v^{\prime},g_{4},\lambda^{\prime}_{2},\lambda_{4})$, without $\lambda_{3}$
dependency. The spin independent direct detection cross section and Higgs
couplings depend on the mixing angle given by

$\displaystyle|\tan\alpha|\approx\sqrt{\left(\frac{\lambda^{\prime}_{2}}{|\lambda_{4}|}\right)^{2}\frac{v^{\prime
2}}{v^{2}}+1}-\frac{\lambda^{\prime}_{2}}{|\lambda_{4}|}\frac{v^{\prime}}{v}\approx\frac{|\lambda_{4}|v}{2\lambda^{\prime}_{2}v^{\prime}},$
(71)

and thus their bounds are straight lines on the
$\lambda^{\prime}_{2}$-$\lambda_{4}$ plane. Note that in Fig. 4(a), the Higgs
coupling bound nearly coincides with the direct detection bound, and thus the
blue area is nearly covered by the cyan area.

Varying $v^{\prime}$ from $\mathrm{T}\mathrm{e}\mathrm{V}$ to
$10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$ does not change the perturbativity and
vacuum stability bounds significantly. However, both the direct and indirect
detection experiment bounds are sensitive to $v^{\prime}$. By comparing Fig.
4(a) and Fig. 4(b), we can see that the excluded area are enlarged when
$v^{\prime}$ decreases. Finally, we find that both prospective sensitivities
of LZ and CTA experiments can probe all the available region shown in Fig.
4(b).

(a)
$g_{4}=0.1,~{}\lambda_{3}=0.13,~{}v^{\prime}=10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$.
(b)
$g_{4}=0.1,~{}\lambda_{3}=0.13,~{}v^{\prime}=5~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$.
(c)
$g_{4}=0.05,~{}\lambda_{3}=0.2,~{}v^{\prime}=10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$.

Figure 4: Degenerate case: constraints in $\lambda^{\prime}_{2}$-$\lambda_{4}$
parameter plane by vacuum stablity, perturbativity and DM experiments
including relic density, direct and indirect detection. The parameters
$\\{g_{4},\lambda_{3},v^{\prime}\\}$ are set to be
$\\{0.1,0.13,10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$\\}$ (top left panel),
$\\{0.1,0.13,5~{}$\mathrm{T}\mathrm{e}\mathrm{V}$\\}$ (top right panel) and
$\\{0.05,0.2,10~{}$\mathrm{T}\mathrm{e}\mathrm{V}$\\}$ (bottom panel),
respectively. The meanings of different colors are the same as Fig. 3.

##  V Conclusion and discussion

The vector dark matter originates from a hidden gauged
$\mathrm{SU}(2)_{\mathrm{D}}$ is investigated in this work. The
$\mathrm{SU}(2)_{\mathrm{D}}$ is spontaneously broken by the vacuum
configuration of a real scalar septuplet. The hidden dark sector communicate
with the SM via the Higgs portal, and thus all the phenomenologies are related
to the mixing angle between the SM Higgs and the radial component of the
septuplet. We find that there are two different vacuum configurations
corresponding to different sign of the quartic coupling $\lambda_{3}$. For the
case of $\lambda_{3}<0$, the mass spectrum of gauge fields and scalar fields
are splitting, and the lightest particle is assumed to be the complex vactor
field $X^{\pm}$. We find a remnant discrete symmetry $Z_{6}\times Z_{2}$ after
the SSB, which can prevent the DM candidate from decaying into the SM
particles. For the case of $\lambda_{3}\geq 0$, the mass spectrum of gauge
fields are degenerate. Both the complex vector $X^{\pm}$ and the real vector
$Z^{\prime}$ are dark matter candidates. A remnant discrete symmetry
$Z_{4}\times Z_{2}$ is found in this case, and thus the DM candidates are
stable.

We also calculate the one-loop beta functions of couplings and let the
couplings evolve to higher energy scale. We find that $\lambda^{\prime}_{2}$
and $\lambda_{3}$ will soon blow up in higher energy if $g_{4}$,
$\lambda^{\prime}_{2}$ and $\lambda_{3}$ are not small enough. If we assume
that all the couplings should keep perturbative below Planck energy scale,
only a very restricted parameter space can survive. Generally, the scale of
Landau pole decreases when $|\lambda_{3}|$ grows. On the other hand, the scale
of Landau pole is not monotonically dependent on $g_{4}$. We find that the
scale of Landau pole reaches its maximum around $g_{4}\sim 0.1$–$0.3$, which
is confirmed by both analytical and numerical calculations. However, if we
assume that all the dark sector scalar can completely decay into the vector
dark matter, which corresponds to $\lambda_{3}\leq-48g_{4}^{2}/5$ for
$\lambda_{3}<0$ and $\lambda_{3}\geq 64g_{4}^{2}/5$ for $\lambda_{3}\geq 0$,
then the perturbative parameter regions always shrink when $g_{4}$ increases
in the region of our interests. As a result, there are upper limits for the
absolute value of the portal coupling, $|\lambda_{4}|\lesssim 0.24$, and the
gauge coupling $g_{4}\lesssim 0.15~{}(0.19)$ for the $\lambda_{3}\geq
0~{}(<0)$ case.

The range of the DM masses is from several hundreds of
$\mathrm{G}\mathrm{e}\mathrm{V}$ to a few $\mathrm{T}\mathrm{e}\mathrm{V}$. To
achieve the observed relic density, the masses of DM and the heavy Higgs
should be closed to the resonance region, $2m_{X}\approx m_{h_{2}}$ in both
cases. We also find that current direct and indirect detection of DM and the
LHC measurements of Higgs couplings can barely exclude the region recommended
by the DM relic density and perturbativity, unless the DM mass is around
hundreds of $\mathrm{G}\mathrm{e}\mathrm{V}$. However, these regions can be
tested by the future searches in the LZ and CTA experiments.

In this work, we only consider the case that vector fields constitute all the
components of DM. However, some scalar particles in the dark sector can be
parts of the DM as well if they are not massive enough for decaying into
vector particles. In this case, the co-annihilation between the vectors and
scalars can play a considerable role during the DM freeze-out. We will leave
this more complicated situation as a future research.

###### Acknowledgements.

We thank Zhao-Huan Yu for helpful discussions. This work is supported by the
National Natural Science Foundation of China (NSFC) under Grant Nos. 11875327
and 11905300, the Fundamental Research Funds for the Central Universities, the
Natural Science Foundation of Guangdong Province, and the Sun Yat-Sen
University Science Foundation.

##  Appendix A Analytic solution of Landau pole

The one-loop beta functions of gauge couplings
$\beta_{g_{i}}=\frac{b_{i}}{16\pi^{2}}g_{i}^{3}$ can be solved as,

$\displaystyle
g_{i}(\mu)=\left(\frac{1}{g_{i}^{2}(\Lambda_{0})}-\frac{b_{i}}{8\pi^{2}}\ln\frac{\mu}{\Lambda_{0}}\right)^{-\frac{1}{2}}.$
(72)

At the one-loop level, gauge and Yukawa couplings are the same as the SM and
they do not meet Landau pole below the Planck scale.

The couplings $\lambda^{\prime}_{2}$ or $\lambda_{3}$ are easy to blow up in
high energy even they are set to vanish in low energy scale. Their beta
functions are given by

$\displaystyle\frac{d\lambda^{\prime}_{2}}{dt^{\prime}}$ $\displaystyle=$
$\displaystyle{30\lambda^{\prime}_{2}}^{2}+\frac{75}{8}\lambda^{2}_{3}+15\lambda^{\prime}_{2}\lambda_{3}-144\lambda^{\prime}_{2}g_{4}^{2}+288g_{4}^{4},$
(73) $\displaystyle\frac{d\lambda_{3}}{dt^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{63}{8}\lambda_{3}^{2}+24\lambda^{\prime}_{2}\lambda_{3}-144\lambda_{3}g_{4}^{2}+288g_{4}^{4}.$
(74)

where $t^{\prime}\equiv\frac{\ln\mu}{16\pi^{2}}$. We have assumed that
$\lambda_{4}$ is small and the $\lambda_{4}^{2}$ term are neglected for
simplicity. To solve these differential equations, we define functions
$f_{1}(t^{\prime})$ and $f_{2}(t^{\prime})$ as follows,

$\displaystyle\lambda^{\prime}_{2}(t^{\prime})=f_{1}(t^{\prime})g_{4}^{2}(t^{\prime}),\qquad\lambda_{3}(t^{\prime})=f_{2}(t^{\prime})g_{4}^{2}(t^{\prime}),$
(75)

and then replace them to above two beta functions:

$\displaystyle\frac{df_{1}}{dG}$ $\displaystyle=$ $\displaystyle
30f_{1}^{2}+\frac{75}{8}f_{2}^{2}+15f_{1}f_{2}-(144+2b_{4})f_{1}+288,$ (76)
$\displaystyle\frac{df_{2}}{dG}$ $\displaystyle=$
$\displaystyle\frac{63}{8}f_{2}^{2}+24f_{1}f_{2}-(144+2b_{4})f_{2}+288,$ (77)

where $G(t^{\prime})$ is given by

$\displaystyle dG\equiv
g_{4}^{2}dt^{\prime}=\frac{1}{b_{4}g_{4}}\frac{dg_{4}}{dt^{\prime}}dt^{\prime}\quad$
$\displaystyle\Rightarrow$ $\displaystyle\quad
G(t^{\prime})=\frac{1}{b_{4}}\ln\frac{g_{4}(t^{\prime})}{g_{4}(t^{\prime}_{0})}.$
(78)

The next step is to introduce another function, $F=f_{1}+\eta f_{2}$, and
adjust $\eta$ to eliminate the $Ff_{1}$ term in

$\displaystyle\frac{dF}{dG}$ $\displaystyle=$
$\displaystyle\left(\frac{75}{8\eta^{2}}+\frac{63}{8\eta}\right)F^{2}-(144+2b_{4})F+288(1+\eta)+\left(\frac{75}{8\eta^{2}}-\frac{57}{8\eta}+6\right)f_{1}^{2}$
(79)
$\displaystyle-\left(\frac{75}{4\eta^{2}}+\frac{3}{4\eta}-24\right)Ff_{1}.$

There are two roots for the equation, $75/(4\eta^{2})+3/(4\eta)-24=0$, and
$\eta=\left(1+\sqrt{3201}\right)/64\approx 0.899647$ is chosen since it gives
a smaller coefficient of $f_{1}^{2}$ term. As a result, the evolution equation
of $F$ can be written as

$\displaystyle\frac{dF}{dG}=c_{2}F^{2}-c_{1}F+c_{0}+c^{\prime}f_{1}^{2},$ (80)

where

$c_{2}=\frac{24\left(1621+21\sqrt{3201}\right)}{\left(\sqrt{3201}+1\right)^{2}}\approx
20.3366,\quad c_{1}=144+2b_{4}\approx 138.667,$ $\displaystyle\quad
c_{0}=\frac{9\left(65+\sqrt{3201}\right)}{2}=547.098,\quad
c^{\prime}=\frac{6\left(9526-74\sqrt{3201}\right)}{\left(\sqrt{3201}+1\right)^{2}}=9.66339.$
(81)

If we ignore the $f_{1}^{2}$ term and $F$ can be solved analytically

$\displaystyle
F(G)=\frac{c_{1}}{2c_{2}}+\hat{d}\tan\left\\{c_{2}\hat{d}G(t^{\prime})+\tan^{-1}\left[\frac{1}{\hat{d}}\left(F(t^{\prime}_{0})-\frac{c_{1}}{2c_{2}}\right)\right]\right\\},$
(82)

with
$\hat{d}\equiv\sqrt{\frac{c_{0}}{c_{2}}-\frac{c_{1}^{2}}{4c_{2}^{2}}}\approx$5.1543.
When $\lambda^{\prime}_{2}$ or $\lambda_{3}$ (equivalently $f_{1}$ or
$f_{2}$), meets Landau pole at some scale $\Lambda_{\text{LP}}$, $F$ also
blows up. Therefore we can extract the Landau pole $\Lambda_{\text{LP}}$ in
this model by analyzing the divergence of $F$. Since $G(t^{\prime})$ increases
monotonically with $t^{\prime}$ and $G(t^{\prime}_{0})=0$, we can find
$F(t^{\prime})$ diverging at

$\displaystyle
c_{2}\hat{d}G(t^{\prime})+\tan^{-1}\left[\frac{1}{\hat{d}}\left(F(t^{\prime}_{0})-\frac{c_{1}}{2c_{2}}\right)\right]=\frac{\pi}{2}.$
(83)

In other words, the Landau pole is at

$\displaystyle\Lambda_{\text{LP}}$ $\displaystyle=$
$\displaystyle\Lambda_{0}\exp\left\\{\frac{8\pi^{2}}{b_{4}g_{4}^{2}(t^{\prime}_{0})}\left[1-\exp\left(\frac{-\pi
b_{4}}{c_{2}\hat{d}}\left[1-\frac{2}{\pi}\tan^{-1}\left(\frac{F(t^{\prime}_{0})}{\hat{d}}-\frac{c_{1}}{2c_{2}\hat{d}}\right)\right]\right)\right]\right\\}$
(84) $\displaystyle=$
$\displaystyle\Lambda_{0}\exp\left\\{\frac{8\pi^{2}}{b_{4}g_{4}^{2}(t^{\prime}_{0})}\left[1-\exp\left(\frac{-\pi
b_{4}}{c_{2}\hat{d}}\left[1-\frac{2}{\pi}\tan^{-1}\left(\frac{\lambda^{\prime}_{2}(t^{\prime}_{0})+\eta\lambda_{3}(t^{\prime}_{0})}{\hat{d}g_{4}^{2}(t^{\prime}_{0})}-\frac{c_{1}}{2c_{2}\hat{d}}\right)\right]\right)\right]\right\\}$
$\displaystyle=$
$\displaystyle\Lambda_{\text{LP}}^{(g_{4})}\exp\left[\frac{-8\pi^{2}}{b_{4}g_{4}^{2}(t^{\prime}_{0})}\exp\left(\frac{-\pi
b_{4}}{c_{2}\hat{d}}\left[1-\frac{2}{\pi}\tan^{-1}\left(\frac{\lambda^{\prime}_{2}(t^{\prime}_{0})+\eta\lambda_{3}(t^{\prime}_{0})}{\hat{d}g_{4}^{2}(t^{\prime}_{0})}-\frac{c_{1}}{2c_{2}\hat{d}}\right)\right]\right)\right],$

with
$\Lambda_{\text{LP}}^{(g_{4})}=\exp\left(\frac{-8\pi^{2}}{b_{4}g_{4}^{2}(t^{\prime}_{0})}\right)$.

In this approximate solution, we neglect the $\lambda_{4}$ term and thus the
Landau pole is independent with $\lambda_{4}$. However, when $\lambda_{4}$ too
large to be neglected, it should be taken into account. Another simplification
in our analytical method is to neglect the $c^{\prime}f_{1}^{2}$ term in Eq.
(80). Since $f_{1}^{2}$ may become large so that can not be neglected in
higher energy scale, the evolutions of $F$ and $f_{1}$ couple with each other
leading a complicated associated differential equations. If we want to get a
reliable quantitative result, we should solve the equations numerically.

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