IPMU21-0047

# Gamma-ray line from electroweakly interacting
non-abelian spin-1 dark matter

Tomohiro Abe<EMAIL_ADDRESS>Department of Physics, Faculty of
Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda,
Chiba 278-8510, Japan  Motoko Fujiwara<EMAIL_ADDRESS>Department of Physics, Nagoya University, Furo-cho Chikusa-ku, Nagoya,
464-8602 Japan  Junji Hisano<EMAIL_ADDRESS>Kobayashi-
Maskawa Institute for the Origin of Particles and the Universe, Nagoya
University, Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan  Department of
Physics, Nagoya University, Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan  Kavli
IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, 277-8584, Japan  Kohei
Matsushita<EMAIL_ADDRESS>Department of Physics, Nagoya
University, Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan

###### Abstract

We study gamma-ray line signatures from electroweakly interacting non-abelian
spin-1 dark matter (DM). In this model, $Z_{2}$-odd spin-1 particles including
a DM candidate have the SU(2)L triplet-like features, and the Sommerfeld
enhancement is relevant in the annihilation processes. We derive the
annihilation cross sections contributing to the photon emission and compare
with the SU(2)L triplet fermions, such as Wino DM in the supersymmetric
Standard Model. The Sommerfeld enhancement factor is approximately the same in
both systems, while our spin-1 DM predicts the larger annihilation cross
sections into $\gamma\gamma/Z\gamma$ modes than those of the Wino by
$\frac{38}{9}$. This is because a spin-1 DM pair forms not only $J=0$ but also
$J=2$ partial wave states where $J$ denotes the total spin angular momentum.
Our spin-1 DM also has a new annihilation mode into $Z_{2}$-even extra heavy
vector and photon, $Z^{\prime}\gamma$. For this mode, the photon energy
depends on the masses of DM and the heavy vector, and thus we have a chance to
probe the mass spectrum. The latest gamma-ray line search in the Galactic
Center region gives a strong constraint on our spin-$1$ DM. We can probe the
DM mass for $\lesssim 25.3\leavevmode\nobreak\ $TeV by the Cherenkov Telescope
Array experiment even if we assume a conservative DM density profile.

###### Contents

  1. I Introduction
  2. II Model
    1. II.1 Physical spectrum
    2. II.2 Couplings of $V$-particles
  3. III Annihilation cross section for $V$-particles
    1. III.1 Non-relativistic field operators for $V$-particles
    2. III.2 Two-body effective action
    3. III.3 Annihilation cross section
  4. IV Gamma-ray line signatures
    1. IV.1 Line cross section
    2. IV.2 Constraint from gamma-ray line signatures
  5. V Conclusions
  6. A Derivation of effective action
    1. A.1 Real part of potential
    2. A.2 Imaginary part of potential
      1. A.2.1 Matching procedure
      2. A.2.2 Derivation of $\leavevmode\nobreak\ \hat{\Gamma}_{\gamma\gamma}^{J}$

## I Introduction

We have overwhelming evidence that suggests dark matter (DM) in our universe.
Although the nature of DM remains unrevealed, we have one quantitative piece
of information about DM, the DM energy density in the current universe. This
energy density is determined to be $\Omega h^{2}=0.120\pm 0.001$ by the Planck
collaboration assuming the $\Lambda$CDM cosmological model 1807.06209 .

One of the most promising DM candidates is the Weakly Interacting Massive
Particles (WIMPs). If we assume the WIMPs in our expanding universe, we can
explain the current DM energy density as the thermal relic abundance through
the Freeze-out mechanism Lee:1977ua . In this scenario, the velocity-weighted
annihilation cross section for DM, $\Braket{\sigma v_{\rm rel}}$,
characterizes the prediction of DM abundance. To reach the observed DM energy
density, we need the value of the canonical cross section, $\Braket{\sigma
v_{\rm rel}}\simeq 3\times 10^{-26}$ cm3/s. This value is obtained once we
assume DM mass and couplings to be the typical values of those in the
electroweak theory in the Standard Model (SM). This fact is our motivation to
consider DM candidates with electroweak interactions.

Spin-$0$ and spin-$1/2$ DM with electroweak interactions are systematically
studied in the contexts of the Minimal DM hep-ph/0512090 ; 0706.4071 ;
0903.3381 .111The references for the other models are given in Ref. 2105.11574
. The Minimal Supersymmetric SM predicts concrete DM candidates stabilized by
the symmetry called R-parity, which is studied in many contexts hep-ph/9906527
; hep-ph/0601041 ; 1410.4549 ; 1601.04718 . The model of spin-$1$ DM with
electroweak interactions is also studied in the extra-dimensional models
0811.1598 ; 1803.01274 ; 1702.02949 ; 1808.10464 .

To reveal the general features of spin-$1$ DM, we construct the renormalizable
four-dimensional model of electroweakly interacting non-abelian spin-$1$ DM
2004.00884 . This model is the minimal setup that realizes the fundamental
features of the five-dimensional model of the spin-$1$ DM such as the
$Z_{2}$-parity and the degenerated mass spectrum. We impose the exchange
symmetry between the gauge groups to realize the above features, which is
inspired by the method of deconstructing dimensions hep-th/0104005 ; hep-
th/0105239 . Our spin-$1$ DM couples to the electroweak gauge bosons with the
electroweak gauge couplings, and the DM mass should be $\gtrsim{\cal O}(1)$
TeV to explain the correct DM energy density. In this region, the electroweak
bosons form the approximately long-range force potential. Consequently, the
Sommerfeld enhancement is relevant in the DM annihilation processes as
discussed in the other electroweakly interacting DM models hep-ph/0212022 ;
hep-ph/0307216 ; hep-ph/0412403 ; 0810.0713 ; 1603.01383 ; 2105.07650 . Due to
this enhancement, monochromatic spectral gamma-ray lines from the DM
annihilation are the striking signals for our spin-1 DM.

In this paper, we study gamma-ray line signatures from electroweakly
interacting non-abelian spin-1 DM. We make a comparison with the pure Wino DM,
which is the spin-$\frac{1}{2}$ DM candidate of the SU(2)L triplet in
supersymmetric models, and clarify the differences between these systems. We
derive the constraint from the latest gamma-ray observation in the Galactic
Center region. We also reveal the region where we can probe in the future
gamma-ray observation.

The rest of the paper is organized as follows. In Sec. II, we briefly review
the electroweakly interacting non-abelian spin-1 DM. In Sec. III, we show the
two-body effective action for the spin-$1$ DM system. We also show the
annihilation cross section contributing to the monochromatic gamma-ray
signals. In Sec. IV, we present our numerical results. We compare the
predicted annihilation cross section with that of the Wino DM and figure out
distinctive features of our spin-$1$ DM. We show the constraints from the
latest search for a monochromatic spectral line and the prospect sensitivity
in the future gamma-ray observation. Our conclusions are given in Sec. V. We
show the derivation of the two-body effective action in Appendix A.

## II Model

We briefly introduce a model of electroweakly interacting non-abelian vector
DM 2004.00884 . We extend the electroweak gauge symmetry in the SM into
SU(2)${}_{0}\times$SU(2)${}_{1}\times$SU(2)${}_{2}\times$U(1)Y. The gauge
bosons for SU(2)0, SU(2)1, SU(2)2, and U(1)Y are denoted as $W^{a}_{0\mu}$,
$W^{a}_{1\mu}$, $W^{a}_{2\mu}$, and $B_{\mu}$, respectively ($a=1,2,3$). The
gauge couplings for each symmetry are denoted as $g_{0}$, $g_{1}$, $g_{2}$,
and $g^{\prime}$, respectively. We summarize the matter fields and Higgs
fields in Table 1. Each fermion field corresponds to the SM fermion with the
same SU(3)c and U(1)Y charge. We introduce bi-fundamental scalar fields,
$\Phi_{j}$ $(j=1,2)$, expressed as the two-by-two matrices. We impose the real
conditions for $\Phi_{j}$ to reduce the degrees of freedom.

$\displaystyle\Phi_{j}=-\epsilon\Phi_{j}^{*}\epsilon,\quad\text{where}\quad\epsilon=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},$ (II.1)

and each of $\Phi_{j}$ contains four real degrees of freedom. The gauge
transformations of $\Phi_{j}$ and $H$ are shown below.

$\displaystyle\Phi_{1}$ $\displaystyle\mapsto U_{0}\Phi_{1}U_{1}^{\dagger},$
$\displaystyle\Phi_{2}$ $\displaystyle\mapsto U_{2}\Phi_{2}U_{1}^{\dagger},$
$\displaystyle H$ $\displaystyle\mapsto e^{i\theta_{Y}}U_{1}H,$ (II.2)

where $U_{n}$ denote the two-by-two gauge transformation matrices of SU(2)n
$(n=0,1,2)$ and $\theta_{Y}$ is the phase of the U(1)Y. We also impose a
discrete symmetry under the following transformation.

$\displaystyle\Phi_{1}\mapsto\Phi_{2},$
$\displaystyle\Phi_{2}\mapsto\Phi_{1},$ $\displaystyle W_{0\mu}^{a}\mapsto
W_{2\mu}^{a},$ $\displaystyle W_{2\mu}^{a}\mapsto W_{0\mu}^{a},$ (II.3)

where all the other fields remain unchanged. This transformation is equivalent
to the exchange of SU(2)0 and SU(2)2, and thus it also requires $g_{0}=g_{2}$.

Table 1: The matter and Higgs fields and their gauge charges. The generation indices for the matter fields are implicit. field | spin | SU(3)c | SU(2)0 | SU(2)1 | SU(2)2 | U(1)Y
---|---|---|---|---|---|---
$q_{L}$ | $\frac{1}{2}$ | 3 | 1 | 2 | 1 | $\frac{1}{6}$
$u_{R}$ | $\frac{1}{2}$ | 3 | 1 | 1 | 1 | $\frac{2}{3}$
$d_{R}$ | $\frac{1}{2}$ | 3 | 1 | 1 | 1 | $-\frac{1}{3}$
$\ell_{L}$ | $\frac{1}{2}$ | 1 | 1 | 2 | 1 | $-\frac{1}{2}$
$e_{R}$ | $\frac{1}{2}$ | 1 | 1 | 1 | 1 | $-1$
$H$ | 0 | 1 | 1 | 2 | 1 | $\frac{1}{2}$
$\Phi_{1}$ | 0 | 1 | 2 | 2 | 1 | 0
$\Phi_{2}$ | 0 | 1 | 1 | 2 | 2 | 0

The Lagrangian for the extended bosonic sector is

$\displaystyle{\cal L}\supset$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\sum_{j=0}^{2}\sum_{a=1}^{3}\frac{1}{4}W_{j\mu\nu}^{a}W_{j}^{a\mu\nu}$
$\displaystyle+D_{\mu}H^{\dagger}D^{\mu}H+\frac{1}{2}\mbox{tr}{D_{\mu}\Phi_{1}^{\dagger}D_{\mu}\Phi_{1}}+\frac{1}{2}\mbox{tr}{D_{\mu}\Phi_{2}^{\dagger}D_{\mu}\Phi_{2}}$
$\displaystyle-V_{\text{scalar}},$ (II.4)

where $V_{\text{scalar}}$ is the scalar potential as shown below.

$\displaystyle V_{\text{scalar}}=$ $\displaystyle
m^{2}H^{\dagger}H+m_{\Phi}^{2}\left[\mbox{tr}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)+\mbox{tr}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\right]$
$\displaystyle+\lambda(H^{\dagger}H)^{2}+\lambda_{\Phi}\left[\left(\mbox{tr}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\right)^{2}+\left(\mbox{tr}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\right)^{2}\right]$
$\displaystyle+\lambda_{h\Phi}H^{\dagger}H\left[\mbox{tr}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)+\mbox{tr}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\right]+\lambda_{12}\mbox{tr}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\mbox{tr}\left(\Phi_{2}^{\dagger}\Phi_{2}\right).$
(II.5)

We assume the following vacuum expectation values (VEVs) to realize the U(1)em
symmetry in the SM.

$\displaystyle\langle\Phi_{1}\rangle$
$\displaystyle=\langle\Phi_{2}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}v_{\Phi}&0\\\
0&v_{\Phi}\end{pmatrix},$ $\displaystyle\langle H\rangle=$
$\displaystyle\begin{pmatrix}0\\\ \frac{v}{\sqrt{2}}\end{pmatrix},$ (II.6)

where we take $v_{\Phi}\gg v$. We define component fields around the VEVs.

$\displaystyle\Phi_{j}$
$\displaystyle=\begin{pmatrix}\frac{v_{\Phi}+\sigma_{j}+i\pi_{j}^{0}}{\sqrt{2}}&i\pi_{j}^{+}\\\
i\pi_{j}^{-}&\frac{v_{\Phi}+\sigma_{j}-i\pi_{j}^{0}}{\sqrt{2}}\end{pmatrix},$
$\displaystyle H=$ $\displaystyle\begin{pmatrix}i\pi_{3}^{+}\\\
\frac{v+\sigma_{3}-i\pi_{3}^{0}}{\sqrt{2}}\end{pmatrix}.$ (II.7)

After $\Phi_{1}$ and $\Phi_{2}$ develop the nonzero VEVs, we still have the
residual SU(2)$\times$U(1)Y gauge symmetry whose gauge transformation is shown
below.

$\displaystyle\Phi_{1}$ $\displaystyle\mapsto U\Phi_{1}U^{\dagger},$
$\displaystyle\Phi_{2}$ $\displaystyle\mapsto U\Phi_{2}U^{\dagger},$
$\displaystyle H$ $\displaystyle\mapsto e^{i\theta_{Y}}UH,$ (II.8)

where $U$ is the SU(2) two-by-two gauge transformation matrix. This SU(2)
transformation corresponds to the SU(2)${}_{0}\times$SU(2)${}_{1}\times$SU(2)2
transformation with $U_{0}=U_{1}=U_{2}\equiv U$. This SU(2) is regarded as the
SU(2)L in the SM and simply called the SU(2)L in the following discussion. The
SU(2)L$\times$U(1)Y symmetry is broken by $\langle H\rangle$ to the U(1)em
symmetry.

After all the Higgs fields develop nonzero VEVs, we still have the exchange
symmetry whose symmetry transformations are shown below.

$\displaystyle\sigma_{1}\mapsto\sigma_{2},$
$\displaystyle\sigma_{2}\mapsto\sigma_{1},$ $\displaystyle W_{0\mu}^{a}\mapsto
W_{2\mu}^{a},$ $\displaystyle W_{2\mu}^{a}\mapsto W_{0\mu}^{a}.$ (II.9)

To find out a $Z_{2}$ parity from this residual discrete symmetry, we anti-
symmetrize the exchanged fields and define the following states.

$\displaystyle h_{D}$
$\displaystyle\equiv\frac{\sigma_{1}-\sigma_{2}}{\sqrt{2}},$ (II.10)
$\displaystyle V^{0}_{\mu}$
$\displaystyle\equiv\frac{W^{0}_{0\mu}-W^{0}_{2\mu}}{\sqrt{2}},$ (II.11)
$\displaystyle V^{\pm}_{\mu}$
$\displaystyle\equiv\frac{W^{\pm}_{0\mu}-W^{\pm}_{2\mu}}{\sqrt{2}},$ (II.12)

where

$\displaystyle W^{\pm}_{n\mu}=\frac{W^{1}_{n\mu}\mp
iW^{2}_{n\mu}}{\sqrt{2}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (n=0,2).$ (II.13)

These fields are eigenstates of the U(1)em charge.222The U(1)em symmetry
generator is expressed as $Q=T^{3}_{0}+T^{3}_{1}+T^{3}_{2}+Y$ where
$T^{3}_{n}$ denote the third generators of SU(2)n ($n=0,1,2$). These states
acquire $(-1)$ factors under the transformation in Eq. (II.9) while all the
other states remain unchanged. This is nothing but a $Z_{2}$ parity
assignment. Note that the $Z_{2}$-odd states are automatically mass
eigenstates since the mass mixing terms with other states are forbidden by
this $Z_{2}$ parity.

We refer to $Z_{2}$-odd spin-1 particles, $V^{0}$ and $V^{\pm}$, as
“$V$-particles.” The $V$-particles are approximately regarded as the spin-$1$
triplet of SU(2)L and have the degenerated masses at the tree-level.

$\displaystyle
m_{V^{0}}^{2}=m_{V_{\pm}}^{2}=\frac{g_{0}^{2}v_{\Phi}^{2}}{4}\equiv
m_{V}^{2}.$ (II.14)

The electroweak radiative corrections break the degeneracy, which makes the
charged component slightly heavier than the neutral one. We find the following
value for mass splitting at the one-loop level.

$\displaystyle\delta m_{V}\equiv m_{V_{\pm}}-m_{V_{0}}\simeq
168\leavevmode\nobreak\ \text{MeV}.$ (II.15)

We assume $h_{D}$ is heavier than $V^{0}$ to focus on the spin-$1$ DM
scenario. Therefore, $V^{0}$ is the lightest $Z_{2}$-odd particle and our
stable spin-1 DM candidate.

### II.1 Physical spectrum

We summarize the physical spectra in our model. Diagonalizing the mass
matrices, we obtain the following mass eigenstates in the Higgs sector.

$\displaystyle\begin{pmatrix}h_{D}\\\ h\\\
h^{\prime}\end{pmatrix}=\begin{pmatrix}1&0&0\\\ 0&\sin\phi_{h}&\cos\phi_{h}\\\
0&\cos\phi_{h}&-\sin\phi_{h}\end{pmatrix}\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\ 0&0&1\\\
\end{pmatrix}\begin{pmatrix}\sigma_{1}\\\ \sigma_{2}\\\
\sigma_{3}\end{pmatrix}.$ (II.16)

We take the scalar masses, $\\{m_{h},m_{h^{\prime}},m_{h_{D}}\\}$, and the
mixing angle, $\phi_{h}$, as input parameters. We can express the
dimensionless couplings of $V_{\rm scalar}$ in these input parameters as shown
below.

$\displaystyle\lambda=$
$\displaystyle\frac{m_{h}^{2}\cos^{2}\phi_{h}+m_{h^{\prime}}^{2}\sin^{2}\phi_{h}}{2v^{2}},$
(II.17) $\displaystyle\lambda_{h\Phi}=$
$\displaystyle-\frac{\sin\phi_{h}\cos\phi_{h}}{2\sqrt{2}vv_{\Phi}}(m_{h^{\prime}}^{2}-m_{h}^{2}),$
(II.18) $\displaystyle\lambda_{\Phi}=$
$\displaystyle\frac{m_{h}^{2}\sin^{2}\phi_{h}+m_{h^{\prime}}^{2}\cos^{2}\phi_{h}+m_{h_{D}}^{2}}{16v_{\Phi}^{2}},$
(II.19) $\displaystyle\lambda_{12}=$
$\displaystyle\frac{m_{h}^{2}\sin^{2}\phi_{h}+m_{h^{\prime}}^{2}\cos^{2}\phi_{h}-m_{h_{D}}^{2}}{8v_{\Phi}^{2}}.$
(II.20)

In the following discussion, we focus on $\phi_{h}\lesssim 0.1$ to evade the
direct detection constraints 2004.00884 .

For the charged gauge bosons, we obtain

$\displaystyle\begin{pmatrix}V^{\pm}_{\mu}\\\ W^{\pm}_{\mu}\\\
W^{\prime\pm}_{\mu}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}1&0&0\\\
0&\cos\phi_{\pm}&\sin\phi_{\pm}\\\
0&-\sin\phi_{\pm}&\cos\phi_{\pm}\end{pmatrix}\begin{pmatrix}\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\\
0&1&0\\\
\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}W_{0\mu}^{\pm}\\\
W_{1\mu}^{\pm}\\\ W_{2\mu}^{\mu}\end{pmatrix},$ (II.21)

where

$\displaystyle\cos\phi_{\pm}=\sqrt{\frac{m_{V^{\pm}}^{2}-m_{W}^{2}}{m_{W^{\prime}}^{2}-m_{W}^{2}}},\quad\sin\phi_{\pm}=\sqrt{\frac{m_{W^{\prime}}^{2}-m_{V^{\pm}}^{2}}{m_{W^{\prime}}^{2}-m_{W}^{2}}}.$
(II.22)

The masses of charged $Z_{2}$-even vectors are obtained as

$\displaystyle m_{W}^{2}=$
$\displaystyle\frac{1}{8}\left\\{g_{1}^{2}v^{2}+(g_{0}^{2}+2g_{1}^{2})v_{\Phi}^{2}-\sqrt{-4g_{0}^{2}g_{1}^{2}v^{2}v_{\Phi}^{2}+\left[g_{1}^{2}v^{2}+(g_{0}^{2}+2g_{1}^{2})v_{\Phi}^{2}\right]^{2}}\right\\},$
(II.23) $\displaystyle m_{W^{\prime}}^{2}=$
$\displaystyle\frac{1}{8}\left\\{g_{1}^{2}v^{2}+(g_{0}^{2}+2g_{1}^{2})v_{\Phi}^{2}+\sqrt{-4g_{0}^{2}g_{1}^{2}v^{2}v_{\Phi}^{2}+\left[g_{1}^{2}v^{2}+(g_{0}^{2}+2g_{1}^{2})v_{\Phi}^{2}\right]^{2}}\right\\}.$
(II.24)

For the neutral gauge bosons, we obtain

$\displaystyle\begin{pmatrix}V^{0}_{\mu}\\\ A_{\mu}\\\ Z_{\mu}\\\
Z^{\prime}_{\mu}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}&0\\\
\frac{e}{g_{0}}&\frac{e}{g_{1}}&\frac{e}{g_{0}}&\frac{e}{g^{\prime}}\\\
\omega^{0}_{Z}&\omega^{1}_{Z}&\omega^{0}_{Z}&\omega^{B}_{Z}\\\
\omega^{0}_{Z^{\prime}}&\omega^{1}_{Z^{\prime}}&\omega^{0}_{Z^{\prime}}&\omega^{B}_{Z^{\prime}}\\\
\end{pmatrix}\begin{pmatrix}W_{0\mu}^{3}\\\ W_{1\mu}^{3}\\\ W_{2\mu}^{3}\\\
B_{\mu}\end{pmatrix},$ (II.25)

where

$\displaystyle e=$
$\displaystyle\left(\frac{2}{g_{0}^{2}}+\frac{1}{g_{1}^{2}}+\frac{1}{g^{\prime
2}}\right)^{-1/2},$ (II.26) $\displaystyle\omega^{0}_{Z}=$
$\displaystyle\frac{eg_{1}}{\sqrt{g_{0}^{2}+2g_{1}^{2}}g^{\prime}}\cos\phi_{0}+\frac{g_{0}}{\sqrt{2(g_{0}^{2}+2g_{1}^{2})}}\sin\phi_{0},$
(II.27) $\displaystyle\omega^{1}_{Z}=$
$\displaystyle\frac{eg_{0}}{\sqrt{g_{0}^{2}+2g_{1}^{2}}g^{\prime}}\cos\phi_{0}-\frac{\sqrt{2}g_{1}}{\sqrt{g_{0}^{2}+2g_{1}^{2}}}\sin\phi_{0},$
(II.28) $\displaystyle\omega^{B}_{Z}=$
$\displaystyle-\frac{e\sqrt{g_{0}^{2}+2g_{1}^{2}}}{g_{0}g_{1}}\cos\phi_{0},$
(II.29) $\displaystyle\omega^{0}_{Z^{\prime}}=$
$\displaystyle\frac{g_{0}}{\sqrt{2(g_{0}^{2}+2g_{1}^{2})}}\cos\phi_{0}-\frac{eg_{1}}{\sqrt{g_{0}^{2}+2g_{1}^{2}}g^{\prime}}\sin\phi_{0},$
(II.30) $\displaystyle\omega^{1}_{Z^{\prime}}=$
$\displaystyle-\frac{\sqrt{2}g_{1}}{\sqrt{g_{0}^{2}+2g_{1}^{2}}}\cos\phi_{0}-\frac{eg_{0}}{\sqrt{g_{0}^{2}+2g_{1}^{2}}g^{\prime}}\sin\phi_{0},$
(II.31) $\displaystyle\omega^{B}_{Z^{\prime}}=$
$\displaystyle\frac{e\sqrt{g_{0}^{2}+2g_{1}^{2}}}{g_{0}g_{1}}\sin\phi_{0}.$
(II.32)

We define $\phi_{0}$ that satisfies the following relation.

$\displaystyle\frac{1}{4}\begin{pmatrix}\frac{g_{0}^{2}g_{1}^{2}g^{\prime
2}}{e^{2}(g_{0}^{2}+2g_{1}^{2})}v^{2}&-\frac{\sqrt{2}g_{0}g_{1}^{3}g^{\prime}}{e(g_{0}^{2}+2g_{1}^{2})}v^{2}\\\
-\frac{\sqrt{2}g_{0}g_{1}^{3}g^{\prime}}{e(g_{0}^{2}+2g_{1}^{2})}v^{2}&\frac{(g_{0}^{2}+2g_{1}^{2})^{2}v_{\Phi}^{2}+2g_{1}^{4}v^{2}}{(g_{0}^{2}+2g_{1}^{2})}\end{pmatrix}\begin{pmatrix}\cos\phi_{0}&-\sin\phi_{0}\\\
\sin\phi_{0}&\cos\phi_{0}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}\cos\phi_{0}&-\sin\phi_{0}\\\
\sin\phi_{0}&\cos\phi_{0}\end{pmatrix}\begin{pmatrix}m_{Z}^{2}&0\\\
0&m_{Z^{\prime}}^{2}\end{pmatrix}.$ (II.33)

The parameters in the Lagrangian in Eq. (II.4) are summarized below.

$\displaystyle\left\\{g_{0},\ g_{1},\ g^{\prime},\ m^{2},\ m_{\Phi}^{2},\
\lambda,\ \lambda_{\Phi},\ \lambda_{h\Phi},\lambda_{12}\right\\}.$ (II.34)

We choose the following input parameters for our phenomenological study.

$\displaystyle\left\\{e,\ m_{Z},\ v,\ m_{h},\ m_{Z^{\prime}},\ m_{V},\
m_{h^{\prime}},\ m_{h_{D}},\phi_{h}\right\\}.$ (II.35)

The first four parameters are already fixed by the experiments, while the last
five parameters are free. As we mentioned before, we consider the small
$\phi_{h}$ regime. Therefore, the SM Higgs coupling with the $V$-particles are
suppressed by the small mixing factor, $\sin\phi_{h}$. Although we do not
specify the values of $m_{h^{\prime}}$ and $m_{h_{D}}$ in the following
discussion, we assume these scalar masses are in the TeV scale to focus on the
phenomenological aspects resulting from the electroweak interactions. The
constraints on these parameters in the scalar sector are studied in Ref.
2004.00884 . For later convenience, we introduce $g_{W}$ as defined below.

$\displaystyle
g_{W}\equiv\left(\frac{2}{g_{0}^{2}}+\frac{1}{g_{1}^{2}}\right)^{-1/2}.$
(II.36)

This coupling corresponds to the SU(2)L gauge coupling approximately, and we
obtain $g_{W}\simeq 0.65$, which is the same value as the SU(2)L gauge
coupling in the SM, in the limit of $v_{\Phi}\gg v$.

We assume $v_{\Phi}\gg v$ throughout our study, and it is useful to derive the
approximated forms for the physical values in this limit. For $m_{W}$ and
$m_{W^{\prime}}$, we obtain

$\displaystyle m_{W}$ $\displaystyle\simeq\frac{g_{W}v}{2},$ (II.37)
$\displaystyle m_{W^{\prime}}$ $\displaystyle\simeq m_{Z^{\prime}},$ (II.38)

and thus we easily obtain the correct value of $m_{W}$. The gauge couplings,
$g_{0}$ and $g_{1}$, are expressed as

$\displaystyle g_{0}\simeq$
$\displaystyle\sqrt{2}g_{W}\frac{m_{Z^{\prime}}}{m_{V}}\frac{1}{\sqrt{\frac{m_{Z^{\prime}}^{2}}{m_{V}^{2}}-1}},$
(II.39) $\displaystyle g_{1}\simeq$ $\displaystyle
g_{W}\frac{m_{Z^{\prime}}}{m_{V}}.$ (II.40)

These couplings are constrained by the perturbative unitarity bounds. We
obtain the following constraints in the high energy regime 1202.5073 .

$\displaystyle g_{j}<\sqrt{\frac{16\pi}{\sqrt{6}}}\simeq
4.53\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (j=0,1).$ (II.41)

From these bounds, we can narrow down the viable range of the mass ratio,
$\frac{m_{Z^{\prime}}}{m_{V}}$. If we take $g_{W}\simeq 0.65$, we obtain

$\displaystyle 1.02\lesssim\frac{m_{Z^{\prime}}}{m_{V}}\lesssim 6.97.$ (II.42)

More detailed explanations of our model are given in Ref. 2004.00884 .

### II.2 Couplings of $V$-particles

We show the couplings of the $V$-particles in the limit of $v_{\Phi}\gg v$.
The vector triple couplings are shown below.

$\displaystyle{\cal L}$ $\displaystyle\supset
iC_{XYZ}\biggl{[}\left(X_{\nu}^{+}\overleftrightarrow{\partial}^{\mu}Y^{-\nu}\right)Z_{\mu}^{0}+\left(Y_{\nu}^{-}\overleftrightarrow{\partial}^{\mu}Z^{0\nu}\right)X_{\mu}^{+}+\left(Z_{\nu}^{0}\overleftrightarrow{\partial}^{\mu}X^{+\nu}\right)Y_{\mu}^{-}\biggr{]},$
(II.43)

where

$\displaystyle C_{W^{+}V^{-}V^{0}}=C_{V^{+}W^{-}V^{0}}$ $\displaystyle\simeq
g_{W},$ (II.44) $\displaystyle C_{V^{+}V^{-}A}$ $\displaystyle=e,$ (II.45)
$\displaystyle C_{V+V-Z}$ $\displaystyle\simeq g_{W}c_{W},$ (II.46)
$\displaystyle C_{V^{+}V^{-}Z^{\prime}}$
$\displaystyle\simeq\frac{g_{W}}{\sqrt{\frac{m_{Z^{\prime}}^{2}}{m_{V}^{2}}-1}}\equiv
g_{Z^{\prime}}.$ (II.47)

We define $s_{W}\equiv\sin\theta_{W}$ and $c_{W}\equiv\cos\theta_{W}$ where
$\theta_{W}$ is the Weinberg angle. We also define $g_{Z^{\prime}}$ in Eq.
(II.47) to characterize the couplings between $Z^{\prime}$ and the
$V$-particles. The vector quartic couplings are shown below.

$\displaystyle{\cal L}$ $\displaystyle\supset C_{XYZW}\leavevmode\nobreak\
X^{+}_{\rho}Y^{-}_{\sigma}Z^{0}_{\mu}W^{0}_{\nu}\left(g^{\rho\mu}g^{\sigma\nu}+g^{\rho\nu}g^{\sigma\mu}-2g^{\rho\sigma}g^{\mu\nu}\right),$
(II.48)

where

$\displaystyle C_{V^{+}V^{-}AA}$ $\displaystyle=\frac{e^{2}}{2},$ (II.49)
$\displaystyle C_{V^{+}V^{-}AZ}$ $\displaystyle\simeq eg_{W}c_{W},$ (II.50)
$\displaystyle C_{V^{+}V^{-}AZ^{\prime}}$
$\displaystyle=eC_{V^{+}V^{-}Z^{\prime}}.$ (II.51)

## III Annihilation cross section for $V$-particles

As studied in the previous work 2004.00884 , we need $m_{V}\gtrsim{\cal O}(1)$
TeV to explain the correct DM relic abundance in our model. In this region,
the $W$ and $Z$ bosons effectively form the long-range force potential for the
$V$-particles. Consequently, we have the sizable Sommerfeld enhancement in the
DM annihilation processes. For this class of models, the monochromatic gamma-
ray line from the DM annihilation is an excellent probe.

In this section, we summarize the formulas to discuss the gamma-ray line
signature from our spin-$1$ DM. We apply the same formalism as those for the
spin-$0$ or spin-$1/2$ DM with electroweak interactions hep-ph/0412403 ;
0706.4071 . First, we define the non-relativistic (NR) field operators for the
$V$-particles to fix our notations. Second, we show the NR two-body effective
action for the $V$-particles. We focus on the electrically neutral two-body
states to study the DM annihilation signals. We derive the effective action in
Appendix A.

### III.1 Non-relativistic field operators for $V$-particles

The asymptotic field operators for the $V$-particles are defined below.

$\displaystyle V^{0}_{\mu}(x)$
$\displaystyle=\sum_{A=1}^{3}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{\rm{p}}}}\left[a^{A}(\bm{p})e^{-ip\cdot
x}\epsilon^{A}_{\mu}(p)+a^{A\dagger}(\bm{p})e^{ip\cdot
x}\epsilon^{A*}_{\mu}(p)\right],$ (III.52) $\displaystyle V^{-}_{\mu}(x)$
$\displaystyle=\sum_{A=1}^{3}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{\rm{p}}}}\left[b^{A}(\bm{p})e^{-ip\cdot
x}\epsilon^{A}_{\mu}(p)+d^{A\dagger}(\bm{p})e^{ip\cdot
x}\epsilon^{A*}_{\mu}(p)\right],$ (III.53) $\displaystyle V^{+}_{\mu}(x)$
$\displaystyle=\sum_{A=1}^{3}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{\rm{p}}}}\left[d^{A}(\bm{p})e^{-ip\cdot
x}\epsilon^{A}_{\mu}(p)+b^{A\dagger}(\bm{p})e^{ip\cdot
x}\epsilon^{A*}_{\mu}(p)\right],$ (III.54)

where $\epsilon_{\mu}^{A}(p)$ denote the physical polarization vectors. These
polarization vectors satisfy the transverse conditions,

$\displaystyle p^{\mu}\epsilon_{\mu}^{A}(p)=0,$ (III.55)

and the orthogonal relations.

$\displaystyle\epsilon_{\mu}^{A*}(p)\epsilon^{B\mu}(p)=-\delta^{AB}.$ (III.56)

We impose the canonical commutation relations between the annihilation and
creation operators.

$\displaystyle[a^{A}(\bm{p}),a^{B\dagger}(\bm{p}^{\prime})]$
$\displaystyle=(2\pi)^{3}\delta^{3}(\bm{p}-\bm{p}^{\prime})\delta^{AB},$
(III.57) $\displaystyle[b^{A}(\bm{p}),b^{B\dagger}(\bm{p}^{\prime})]$
$\displaystyle=(2\pi)^{3}\delta^{3}(\bm{p}-\bm{p}^{\prime})\delta^{AB},$
(III.58) $\displaystyle[d^{A}(\bm{p}),d^{B\dagger}(\bm{p}^{\prime})]$
$\displaystyle=(2\pi)^{3}\delta^{3}(\bm{p}-\bm{p}^{\prime})\delta^{AB}.$
(III.59)

To derive the NR two-body effective action, we perform the NR expansion for
these operators and integrate out the large momentum modes. Since the zeroth
component of the polarization tensor is sub-leading, ${\cal
O}(|\bm{p}|/m_{V})$, we focus on the spacial components ($i=1,2,3$) for the
$V$-particle operators.

$\displaystyle V^{0}_{i}(x)$
$\displaystyle\simeq\frac{1}{\sqrt{2m_{V}}}\left[e^{-im_{V}t}{\cal
A}_{i}(x)+e^{im_{V}t}{\cal A}_{i}^{\dagger}(x)\right],$ (III.60)
$\displaystyle V^{-}_{i}(x)$
$\displaystyle\simeq\frac{1}{\sqrt{2m_{V}}}\left[e^{-im_{V}t}{\cal
B}_{i}(x)+e^{im_{V}t}{\cal D}_{i}^{\dagger}(x)\right],$ (III.61)
$\displaystyle V^{+}_{i}(x)$
$\displaystyle\simeq\frac{1}{\sqrt{2m_{V}}}\left[e^{-im_{V}t}{\cal
D}_{i}(x)+e^{im_{V}t}{\cal B}_{i}^{\dagger}(x)\right],$ (III.62)

where we define

$\displaystyle{\cal A}_{i}(x)$
$\displaystyle=\sum_{A=1}^{3}\int\frac{d^{3}p}{(2\pi)^{3}}a^{A}(\bm{p})e^{-ip\cdot
x}\epsilon^{A}_{i}(p),$ (III.63) $\displaystyle{\cal B}_{i}(x)$
$\displaystyle=\sum_{A=1}^{3}\int\frac{d^{3}p}{(2\pi)^{3}}b^{A}(\bm{p})e^{-ip\cdot
x}\epsilon^{A}_{i}(p),$ (III.64) $\displaystyle{\cal D}_{i}(x)$
$\displaystyle=\sum_{A=1}^{3}\int\frac{d^{3}p}{(2\pi)^{3}}d^{A}(\bm{p})e^{-ip\cdot
x}\epsilon^{A}_{i}(p).$ (III.65)

We also define the NR one particle states for the $V$-particles with the
momentum $\bm{p}$ and the polarization $A$.

$\displaystyle\Ket{V^{0};\bm{p},A}$
$\displaystyle\simeq\sqrt{2m_{V}}a^{A\dagger}(\bm{p})\Ket{0},$ (III.66)
$\displaystyle\Ket{V^{-};\bm{p},A}$
$\displaystyle\simeq\sqrt{2m_{V}}b^{A\dagger}(\bm{p})\Ket{0},$ (III.67)
$\displaystyle\Ket{V^{+};\bm{p},A}$
$\displaystyle\simeq\sqrt{2m_{V}}d^{A\dagger}(\bm{p})\Ket{0}.$ (III.68)

### III.2 Two-body effective action

We show the two-body effective action for the NR $V$-particles. The
electrically neutral two-body states are composed of
$V^{-}(t,\bm{x})V^{+}(t,\bm{y})$ or $V^{0}(t,\bm{x})V^{0}(t,\bm{y})$. We
change the space-time coordinates into the center of mass coordinate,
$R=(R^{0},\bm{R})$, and the relative coordinate, $\bm{r}$.

$\displaystyle R^{0}$ $\displaystyle=t,$ $\displaystyle\bm{R}$
$\displaystyle\equiv\frac{\bm{x}+\bm{y}}{2},$ $\displaystyle\bm{r}$
$\displaystyle\equiv\bm{x}-\bm{y}.$ (III.69)

As shown later, the NR leading-order potential has a spherically symmetric
form. Therefore, the total spin angular momentum, $J$, and $z$-component of
the spin angular momentum, $J_{z}$, are good quantum numbers. We obtain the
following effective action in the decomposed form into each partial wave
state.

$\displaystyle S_{\rm eff}$ $\displaystyle=\sum_{J,J_{z}}\int
d^{4}Rd^{3}r\leavevmode\nobreak\
\Phi^{J,J_{z}\dagger}(R,\bm{r})\cdot\left[\left(i\partial_{R^{0}}+\frac{\nabla_{R}^{2}}{4m_{V}}+\frac{\nabla_{r}^{2}}{m_{V}}\right)-\hat{V}(r)+i\frac{9}{2}\hat{\Gamma}^{J}\delta^{3}(\bm{r})\right]\cdot\Phi^{J,J_{z}}(R,\bm{r}),$
(III.70)

where $r\equiv|\bm{r}|$ and $J=0,1,2$ with $|J_{z}|\leq J$. The $\hat{V}$ and
$\hat{\Gamma}^{J}$ denote the real part and imaginary part of the potential,
respectively. We define the two-body fields for each $(J,J_{z})$ as shown
below.

$\displaystyle\Phi^{J,J_{z}}(R,\bm{r})$
$\displaystyle=\begin{pmatrix}\phi_{C}^{J,J_{z}}(R,\bm{r})\\\
\phi_{N}^{J,J_{z}}(R,\bm{r})\end{pmatrix}.$ (III.71)

The upper and lower components are formed by $V^{-}V^{+}$ and $V^{0}V^{0}$,
respectively.

$\displaystyle\phi_{C}^{J,J_{z}}(R,\bm{r})$ $\displaystyle={\cal
B}_{i}(R^{0},\bm{R}+\bm{r}/2)\leavevmode\nobreak\
S_{ij}^{J,J_{z}}\leavevmode\nobreak\ {\cal D}_{j}(R^{0},\bm{R}-\bm{r}/2)$
$\displaystyle(J=0,1,2),$ (III.72) $\displaystyle\phi_{N}^{J,J_{z}}(R,\bm{r})$
$\displaystyle=\frac{1}{\sqrt{2}}{\cal
A}_{i}(R^{0},\bm{R}+\bm{r}/2)\leavevmode\nobreak\
S_{ij}^{J,J_{z}}\leavevmode\nobreak\ {\cal A}_{j}(R^{0},\bm{R}-\bm{r}/2)$
$\displaystyle(J=0,2).$ (III.73)

In the above definition, we introduce the basis of three-by-three matrices.

$\displaystyle\hat{S}^{J,J_{z}}\equiv S^{J,J_{z}}_{ij}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (i,j=1,2,3),$
(III.74)

where

$\displaystyle\hat{S}^{0,0}$
$\displaystyle=\frac{-1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},$ (III.75) $\displaystyle\hat{S}^{1,1}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&-1\\\ 0&0&-i\\\
1&i&0\end{pmatrix},$ $\displaystyle\hat{S}^{1,0}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&i&0\\\ -i&0&0\\\
0&0&0\end{pmatrix},$ $\displaystyle\hat{S}^{1,-1}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&-1\\\ 0&0&i\\\
1&-i&0\end{pmatrix},$ (III.76) $\displaystyle\hat{S}^{2,2}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1&i&0\\\ i&-1&0\\\
0&0&0\end{pmatrix},$ $\displaystyle\hat{S}^{2,1}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&-1\\\ 0&0&-i\\\
-1&-i&0\end{pmatrix},$ $\displaystyle\hat{S}^{2,0}$
$\displaystyle=\frac{1}{\sqrt{6}}\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\
0&0&2\end{pmatrix},$ $\displaystyle\hat{S}^{2,-1}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&1\\\ 0&0&-i\\\
1&-i&0\end{pmatrix},$ $\displaystyle\hat{S}^{2,-2}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1&-i&0\\\ -i&-1&0\\\
0&0&0\end{pmatrix}.$ (III.77)

These matrices satisfy the following orthogonal relation.

$\displaystyle\mbox{tr}{\left[\hat{S}^{J,J_{z}}\hat{S}^{J^{\prime},J^{\prime}_{z}*}\right]}=(-1)^{J}\delta^{JJ^{\prime}}\delta^{J_{z}J^{\prime}_{z}}.$
(III.78)

Note that $\hat{S}^{J,J_{z}}$ is symmetric matrices for $J=0,2$ and anti-
symmetric for $J=1$. We have no $\phi^{1,J_{z}}_{N}$ because $\phi_{N}$ is
composed of identical particles. As mentioned before, $J$ is conserved as long
as we focus on the NR leading order contributions. Therefore, $J=1$ partial
wave modes are irrelevant to discuss the DM annihilation signals. We only
consider $J=0,2$ states in the following discussion. The normalization factors
for the two-body states are fixed to realize the canonical weights for the
two-body propagators. The two-body propagator of $\phi_{C}^{J,J_{z}}$ is
defined as the time-ordered product.

$\displaystyle\Braket{0}{T\phi_{C}^{J,J_{z}}(R,\bm{r})\phi_{C}^{J,J_{z}\dagger}(0,\bm{0})}{0}$
$\displaystyle=\int\frac{d^{3}P}{(2\pi)^{3}}\frac{d^{3}k}{(2\pi)^{3}}\leavevmode\nobreak\
e^{-i\Bigl{(}\frac{|\bm{P}|^{2}}{4m}+\frac{\bm{k}^{2}}{m}\Bigr{)}R^{0}+i\bm{P}\cdot\bm{R}}\leavevmode\nobreak\
e^{+i\bm{k}\cdot\bm{r}}\leavevmode\nobreak\ \theta(R^{0}),$ (III.79)

where $\theta(R^{0})$ is the Heaviside step function. This expression is
obtained by substituting the explicit forms of $\phi_{C}^{J,J_{z}}$ and
$\phi_{C}^{J,J_{z}\dagger}$ and using the canonical commutation relation
between them.333Note that
$(\hat{S}^{J,J_{z}})^{\dagger}=(-1)^{J}(\hat{S}^{J,J_{z}})^{*}$ because it is
symmetric for $J=0,2$ but anti-symmetric for $J=1$. The definition of
$\phi^{J,J_{z}}_{N}$ has another normalization factor of $\frac{1}{\sqrt{2}}$
because $\phi^{J,J_{z}}_{N}$ is composed of identical particles.

The leading-order expression for $\hat{V}$ is obtained as follows.

$\displaystyle\hat{V}(r)$ $\displaystyle=\begin{pmatrix}\displaystyle 2\delta
m_{V}-\frac{\alpha_{2}s_{W}^{2}}{r}-\frac{\alpha_{2}c_{W}^{2}e^{-m_{Z}r}}{r}&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\displaystyle-\frac{\sqrt{2}\alpha_{2}e^{-m_{W}r}}{r}\\\ \\\
\displaystyle-\frac{\sqrt{2}\alpha_{2}e^{-m_{W}r}}{r}&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \displaystyle
0\end{pmatrix},$ (III.80)

where $\alpha_{2}\equiv\frac{g_{W}^{2}}{4\pi}$. This potential is induced by
the NR processes between $V$-particles. The off-diagonal elements are induced
by the $W$ boson exchange processes, which cause the mixing between $\phi_{C}$
and $\phi_{N}$. Compared with the leading-order contributions, the Higgs
exchange contributions are suppressed by small $\phi_{h}$. The $W^{\prime}$
and $Z^{\prime}$ exchange contributions are exponentially suppressed by
$m_{W^{\prime}}$ and $m_{Z^{\prime}}$. The contributions from the vector
quadratic couplings are suppressed by $\frac{1}{m_{V}^{2}}$. The leading-order
expression for $\hat{V}$ is the same as that of the SU(2)L triplet DM with
spin-$0$/spin-$\frac{1}{2}$. This is because the spin-dependent features
decouple from the NR processes. We show the derivation of $\hat{V}$ in
Appendix A.1.

The imaginary part of the potential, $\hat{\Gamma}^{J}$, is derived by
operator matching between the two-body field operators and the calculations of
the loop amplitudes of the $V$-particles.444We choose the factor of
“$\frac{9}{2}$” in Eq. (III.70) so that the (1,1)-component of
$\hat{\Gamma}^{J}$ is equal to the tree-level spin-averaged velocity-weighted
annihilation cross sections of $V^{-}V^{+}$ with $J$. We focus on the one-loop
diagrams with the intermediate states,
$XX^{\prime}=\gamma\gamma,Z\gamma,Z^{\prime}\gamma$, involving the gamma-ray
line signatures. The annihilation modes into $h\gamma$ and $h^{\prime}\gamma$
only induce $J=1$ partial wave contributions, which is irrelevant to the DM
annihilation signals. We decompose $\hat{\Gamma}^{J}$ into each contribution
of the intermediate state, $XX^{\prime}$, as denoted by
$\Gamma^{J}_{XX^{\prime}}$. The leading-order expressions are shown below.

$\displaystyle\hat{\Gamma}^{J=0}_{\gamma\gamma}$
$\displaystyle=\frac{2}{3}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}\begin{pmatrix}s_{W}^{4}&0\\\
0&0\end{pmatrix},$ (III.81) $\displaystyle\hat{\Gamma}^{J=2}_{\gamma\gamma}$
$\displaystyle=\frac{32}{45}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}\begin{pmatrix}s_{W}^{4}&0\\\
0&0\end{pmatrix},$ (III.82) $\displaystyle\hat{\Gamma}^{J=0}_{Z\gamma}$
$\displaystyle=\frac{2}{3}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}\begin{pmatrix}2c_{W}^{2}s_{W}^{2}&0\\\
0&0\end{pmatrix},$ (III.83) $\displaystyle\hat{\Gamma}^{J=2}_{Z\gamma}$
$\displaystyle=\frac{32}{45}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}\begin{pmatrix}2c_{W}^{2}s_{W}^{2}&0\\\
0&0\end{pmatrix},$ (III.84)
$\displaystyle\hat{\Gamma}^{J=0}_{Z^{\prime}\gamma}$
$\displaystyle=\frac{1}{27}\frac{\alpha_{2}g_{Z^{\prime}}^{2}}{m_{V}^{2}}(1-r_{Z^{\prime}})(3-2r_{Z^{\prime}})^{2}\begin{pmatrix}s_{W}^{2}&0\\\
0&0\end{pmatrix},$ (III.85)
$\displaystyle\hat{\Gamma}^{J=2}_{Z^{\prime}\gamma}$
$\displaystyle=\frac{8}{135}\frac{\alpha_{2}g_{Z^{\prime}}^{2}}{m_{V}^{2}}(1-r_{Z^{\prime}})(6+3r_{Z^{\prime}}+r_{Z^{\prime}}^{2})\begin{pmatrix}s_{W}^{2}&0\\\
0&0\end{pmatrix},$ (III.86)

where $r_{Z^{\prime}}\equiv\frac{m_{Z^{\prime}}^{2}}{4m_{V}^{2}}$, and
$g_{Z^{\prime}}$ is defined in Eq. (II.47). We show the derivation of
$\hat{\Gamma}^{J}$ in Appendix A.2.

### III.3 Annihilation cross section

We derive the $s$-wave spin-averaged velocity-weighted cross section for
$V^{0}V^{0}\to XX^{\prime}$ annihilation process through the optical theorem.

$\displaystyle\braket{\sigma v_{\rm rel}}_{XX^{\prime}}$
$\displaystyle=C\sum_{\alpha,\beta}\sum_{J,J_{z}}\left(\Gamma^{J}_{XX^{\prime}}\right)_{\alpha\beta}d_{2\alpha}(E)\leavevmode\nobreak\
d_{2\beta}^{*}(E),$ (III.87)

where $C=2$ for the initial states composed of the identical particles. We
introduce $E\simeq\frac{m_{V}v^{2}_{\rm rel}}{4}$ as the NR kinetic energy of
the $V$-particles, and $d_{\alpha\beta}(E)\leavevmode\nobreak\
(\alpha,\beta=1,2)$ as the Sommerfeld enhancement factor. We numerically
obtain $d_{\alpha\beta}(E)$ by solving the Schrödinger equation hep-ph/0412403
.

Before we show the numerical results, we remark the distinctive features of
our spin-$1$ DM compared with the Wino DM, which is SU(2)L triplet
spin-$\frac{1}{2}$ DM. We focus on the leading-order results in both DM
systems to show the comparison.555For the Wino DM, the Sudakov log corrections
1307.4082 ; 1409.8294 ; 1612.04814 ; 1805.07367 ; 1903.08702  are precisely
evaluated. In our spin-$1$ DM system, those potentially large corrections have
not been evaluated yet, and thus our predictions have the uncertainty of
${\cal O}(1)$ factors. The one-loop correction for the real part of the
potential in the Wino DM system is studied in Ref. 1909.04584 ; 2009.00640 .
The Sommerfeld enhancement factor is approximately the same in both systems
since we have the same $\hat{V}$ and the mass splitting at the leading-order.
This is because the leading-order interactions with $W$ and $Z$ bosons in the
NR limit are independent of the DM spin. We have contributions other than $W$
and $Z$ interactions but they are sub-leading as mentioned above. The SM Higgs
contributions for $\hat{V}$, which is zero for pure Wino DM at the leading-
order, are suppressed by the small mixing angle $\phi_{h}$. The contributions
from $W^{\prime}$, $Z^{\prime}$, and $h^{\prime}$ are suppressed by their
masses.666Even if we consider a relatively light $h^{\prime}$ and take
$m_{h^{\prime}}\sim 1$ TeV, the contribution to the potential is suppressed by
$\exp(-m_{h^{\prime}}/m_{Z})\sim\exp(-10)$. If we take a much lighter value of
$m_{h^{\prime}}$, we have to study the constraint on the $h^{\prime}$ from the
low energy experiments, which is beyond the scope of this paper. Consequently,
we have the same resonance structure as that in the Wino system. On the other
hand, the annihilation cross section into $\gamma\gamma$ and $Z\gamma$ are
larger than those for the Wino by $\frac{38}{9}\simeq 4.22\cdots$. This is
because spin-$1$ DM forms both $J=0$ and $J=2$ partial wave states while
spin-$\frac{1}{2}$ DM forms only $J=0$ state. Our spin-$1$ DM also has a new
annihilation mode into $Z^{\prime}\gamma$. If this new channel is
kinematically opened, a photon is emitted with the energy depending on both
$m_{V}$ and $m_{Z^{\prime}}$. This channel also contributes to the
monochromatic gamma-ray line signals, which we study further in Sec. IV.

## IV Gamma-ray line signatures

We study the gamma-ray line signatures from our spin-$1$ DM. We derive the
constraints from the search for monochromatic gamma-ray lines in the Galactic
Center region. The derived constraints highly depend on the DM density
profiles, and thus we show our result by taking some types of benchmark
profiles to show the uncertainty.

### IV.1 Line cross section

The latest analysis for the gamma-ray line signals is performed by the
H.E.S.S. collaboration using the ten years data of the gamma-ray observation
in the Galactic Center region 1805.05741 . We constrain the gamma-ray line
signatures from the DM annihilation by using this result. In our model, we
have three annihilation modes involving the gamma-ray line signals,
$\\{\gamma\gamma,Z\gamma,Z^{\prime}\gamma\\}$. For the $\gamma\gamma/Z\gamma$
modes, the photon energy is approximately equal to the DM mass,
$E_{\gamma}\simeq m_{V}$, where we take the NR limit for the initial DM pair
and neglect $m_{Z}$. For the $Z^{\prime}\gamma$ mode, we can not neglect
$m_{Z^{\prime}}$ because $m_{Z^{\prime}}$ must be heavier than $m_{V}$, see
Eq. (II.42). The photon energy in the $Z^{\prime}\gamma$ annihilation mode
depends on both $m_{V}$ and $m_{Z^{\prime}}$.

$\displaystyle E_{\gamma}$ $\displaystyle\simeq
m_{V}\left(1-\frac{m_{Z^{\prime}}^{2}}{4m_{V}^{2}}\right)\equiv m_{V}-\Delta
E_{\gamma},$ (IV.88)

where $\Delta E_{\gamma}\equiv\frac{m_{Z^{\prime}}^{2}}{4m_{V}}$. If the
$Z^{\prime}\gamma$ mode is kinematically allowed, namely for
$m_{Z^{\prime}}\lesssim 2m_{V}$, we have a chance to observe the double-peak
gamma-ray spectrum at $E_{\gamma}\simeq m_{V}-\Delta E_{\gamma}$ and
$E_{\gamma}\simeq m_{V}$. To distinguish between these two peaks,
$\frac{\Delta E_{\gamma}}{m_{V}}$ should be larger than the instrumental
energy resolution. In the H.E.S.S. experiment, the energy resolution is about
$10$ $\%$ for $m_{\rm DM}\gtrsim 300\leavevmode\nobreak\ \text{GeV}$. Our
interesting region is

$\displaystyle 1.02\lesssim\frac{m_{Z^{\prime}}}{m_{V}}<2,$ (IV.89)

to discuss the signal discrimination. The lower and upper values come from the
$g_{0}$ perturbative unitarity and the kinematical suppression of the
$Z^{\prime}\gamma$ annihilation mode, respectively. If we focus on this
region, the condition $\frac{\Delta E_{\gamma}}{m_{V}}\gtrsim 0.1$ is always
satisfied. Therefore, we can discriminate the gamma-ray peak originated from
the $\gamma\gamma/Z\gamma$ modes and the peak from the $Z^{\prime}\gamma$
mode. This double-peak gamma-ray spectrum is an outstanding feature of our DM
model, and we can read out the values of $m_{V}$ and $m_{Z^{\prime}}$ from
this double-peak spectrum.777Similar signals are predicted in the context of
the extra-dimensional models. See Ref. 0904.1442  for the discussion in the
model with six-dimensions.

We define the “line cross section”, which contributes to the gamma-ray line
signal. We introduce two types of line cross sections that predict the
different final photon energy.

$\displaystyle\Braket{\sigma v_{\rm rel}}^{\rm line}_{\gamma\gamma,Z\gamma}$
$\displaystyle=\Braket{\sigma v_{\rm
rel}}_{\gamma\gamma}+\frac{1}{2}\Braket{\sigma v_{\rm rel}}_{Z\gamma}$
$\displaystyle\text{Energy peak: $E_{\gamma}=m_{V}$},$ (IV.90)
$\displaystyle\Braket{\sigma v_{\rm rel}}^{\rm line}_{Z^{\prime}\gamma}$
$\displaystyle=\frac{1}{2}\Braket{\sigma v_{\rm rel}}_{Z^{\prime}\gamma}$
$\displaystyle\text{Energy peak:
$E_{\gamma}=m_{V}\left(1-\frac{m_{Z^{\prime}}^{2}}{4m_{V}^{2}}\right)$}.$
(IV.91)

We derive the excluded region by using the experimental bound shown in Fig. 6
of Ref. 1805.05741 . In their analysis, all the final state particles are
assumed to be massless, and thus we can directly compare these constraints
with $\Braket{\sigma v_{\rm rel}}^{\rm line}_{\gamma\gamma,Z\gamma}$. We can
also derive the constraint on $\Braket{\sigma v_{\rm rel}}^{\rm
line}_{Z^{\prime}\gamma}$ by noting that the horizontal axis of Fig. 6 in Ref.
1805.05741  corresponds to $E_{\gamma}$. These constraints are derived by
assuming the three cuspy DM density profiles as defined below.

  * •

Einasto profile 0904.1442 /Einasto2 profile 1012.4515

$\displaystyle\rho_{\rm
Einasto}(r)\equiv\rho_{s}\exp\left[-\frac{2}{\alpha_{s}}\left(\left(\frac{r}{r_{s}}\right)^{\alpha_{s}}-1\right)\right].$
(IV.92)

  * •

Navarro-Frenk-White (NFW) profile Navarro:1996gj

$\displaystyle\rho_{\rm
NFW}(r)\equiv\rho_{s}\left(\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}\right)^{-1}.$
(IV.93)

In Table. 2, we summarize the parameters for the cuspy DM density profiles
used in Ref. 1805.05741 . Another choice is the cored DM density profile. The
core radius, $r_{c}$, depends on the model of baryonic physics, and cores
extending to $r_{c}\sim 5$ kpc can potentially be obtained 1405.4318 . The
cored Einasto profile is defined as shown below.

$\displaystyle\rho(r)=\left\\{\begin{array}[]{cc}\rho_{\rm
Einasto}(r)&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ r>r_{c},\\\ \rho_{\rm
Einasto}(r_{c})&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
r<r_{c},\end{array}\right.$ (IV.96)

where $\rho_{\rm Einasto}(r)$ is defined in Eq. (IV.92). For the cored
profile, $\rho_{s}$ is chosen to realize the value of the local DM density. We
show the current excluded region with the cusped DM density profiles assumed
in the analysis of the H.E.S.S. collaboration 1805.05741 . The sensitivity of
the H.E.S.S. experiment is studied for the cored profiles in Ref. 1808.04388 ,
which is focusing on the pure Wino DM search.888See the left panel of Fig. 7
in Ref. 1808.04388 . From this study, the upper bound on the line cross
section will be weakened by a factor of ${\cal O}(10-100)$ if we use the cored
DM density profile.

Table 2: The cuspy DM density profiles used in Ref. 1805.05741 . Profiles | Einasto 0904.1442  | NFW Navarro:1996gj  | Einasto2 1012.4515
---|---|---|---
$\rho_{\rm s}$ [GeV cm-3] | $0.079$ | $0.307$ | $0.033$
$r_{\rm s}$ [kpc] | $20.0$ | $21.0$ | $28.4$
$\alpha_{\rm s}$ | $0.17$ | – | $0.17$

### IV.2 Constraint from gamma-ray line signatures

Figure 1: The comparison of the line cross sections between our spin-$1$ DM
and the Wino DM. The solid (dashed) black curve shows $\Braket{\sigma v_{\rm
rel}}^{\rm line}_{\gamma\gamma,Z\gamma}$ for our model (pure Wino). The green
shaded region shows the constraint from the H.E.S.S. observation in the
Galactic Center region 1805.05741  for the Einasto profile 0904.1442 . The
green dotted curve and dashed curve show the upper limit for the NFW profile
Navarro:1996gj  and Einasto2 profile 1012.4515 , respectively. The upper bound
is expected to be weakened by a factor of ${\cal O}(10-100)$ for the cored DM
density profile.

We compare the predicted line cross section between our spin-1 DM and the Wino
system. In Fig. 1, we show the predicted value of $\Braket{\sigma v_{\rm
rel}}^{\rm line}_{\gamma\gamma,Z\gamma}$ and experimental upper bounds. The
solid (dashed) black curve shows the line cross section including Sommerfeld
enhancement for our spin-$1$ DM (the pure Wino DM). The green region with
solid boundary is the excluded region by the H.E.S.S. observation 1805.05741
for the Einasto profile 0904.1442 . We also show green dashed and green dotted
curves as the upper limits on the cross section for the cusped NFW profile
Navarro:1996gj  and the Einasto2 profile 1012.4515 , respectively. Since our
spin-$1$ DM and the Wino DM have the SU(2)L triplet features, the Sommerfeld
resonance structures are almost the same. The line cross section for the
spin-$1$ DM is larger than that for the Wino DM by $\frac{38}{9}$, and thus we
obtain more severe constraints on the spin-$1$ DM. We find the following
excluded regions for our spin-$1$ DM depending on the DM density
profiles.999If we take $m_{h^{\prime}}\simeq m_{V}$, the parameter region in
Fig. 2 may be constrained by the perturbative unitarity bounds on the scalar
couplings, which is studied in our previous collaboration 2004.00884 . We can
evade these unitarity bounds by taking sufficiently small values of $\phi_{h}$
and $m_{h^{\prime}}$.

  * •

Einasto profile

$\displaystyle 300\leavevmode\nobreak\ \text{GeV}\lesssim m_{V}\lesssim
14.4\leavevmode\nobreak\ \text{TeV}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{(Excluded region 1)},$ (IV.97) $\displaystyle 16.5\leavevmode\nobreak\
\text{TeV}\lesssim m_{V}\lesssim 22.9\leavevmode\nobreak\
\text{TeV}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(Excluded region
2)},$ (IV.98) $\displaystyle 33.8\leavevmode\nobreak\ \text{TeV}\lesssim
m_{V}\lesssim 37.5\leavevmode\nobreak\ \text{TeV}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{(Excluded region 3)}.$ (IV.99)

  * •

Cusped NFW profile

$\displaystyle 300\leavevmode\nobreak\ \text{GeV}\lesssim m_{V}\lesssim
12.5\leavevmode\nobreak\ \text{TeV}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{(Excluded region 1)},$ (IV.100) $\displaystyle 17.9\leavevmode\nobreak\
\text{TeV}\lesssim m_{V}\lesssim 22.2\leavevmode\nobreak\
\text{TeV}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(Excluded region
2)},$ (IV.101) $\displaystyle 34.8\leavevmode\nobreak\ \text{TeV}\lesssim
m_{V}\lesssim 36.7\leavevmode\nobreak\ \text{TeV}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{(Excluded region 3)}.$ (IV.102)

  * •

Einasto2 profile

$\displaystyle 300\leavevmode\nobreak\ \text{GeV}\lesssim m_{V}\lesssim
11.7\leavevmode\nobreak\ \text{TeV}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{(Excluded region 1)},$ (IV.103) $\displaystyle 18.6\leavevmode\nobreak\
\text{TeV}\lesssim m_{V}\lesssim 21.6\leavevmode\nobreak\
\text{TeV}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(Excluded region
2)},$ (IV.104) $\displaystyle 35.0\leavevmode\nobreak\ \text{TeV}\lesssim
m_{V}\lesssim 36.1\leavevmode\nobreak\ \text{TeV}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{(Excluded region 3)}.$ (IV.105)

The lower value of $300$ GeV comes from the limitation for the energy
resolution in the H.E.S.S experiment.

Figure 2: The current bound on the line cross section by the H.E.S.S.
observation for the Einasto profile 0904.1442 . The green regions are excluded
by $\Braket{\sigma v_{\rm rel}}^{\rm line}_{\gamma\gamma,Z\gamma}$. The blue
regions are excluded by $\Braket{\sigma v_{\rm rel}}^{\rm
line}_{Z^{\prime}\gamma}$. The black solid, dashed, dot-dashed, and dotted
contours show the predicted value of $\Braket{\sigma v_{\rm rel}}^{\rm
line}_{Z^{\prime}\gamma}$ for $10^{-24}$, $10^{-25}$, $10^{-26}$, and
$10^{-27}$ in the unit of cm3/s, respectively. The red solid lines show the
$g_{0}$ contours.

Figure 3: The constraints on the line cross section for the different DM
density profiles. The upper and lower panels show the constraints for the NFW
profile Navarro:1996gj  and the Einasto2 profile 1012.4515 , respectively. The
explanations for each plot are given in the caption of Fig. 2.

In Fig. 2, we show the current bound on the line cross section by the H.E.S.S.
observation for the Einasto DM density profiles 0904.1442  focusing on the
region in Eq. (IV.89). The green regions are excluded by $\Braket{\sigma
v_{\rm rel}}^{\rm line}_{\gamma\gamma,Z\gamma}$ shown in Eqs. (IV.97)-(IV.99).
These green regions are extended for $\frac{m_{Z^{\prime}}}{m_{V}}\geq 2$
where the annihilation into $Z^{\prime}\gamma$ is forbidden kinematically. The
blue region is excluded by $\Braket{\sigma v_{\rm rel}}^{\rm
line}_{Z^{\prime}\gamma}$. The black solid, dashed, dot-dashed, and dotted
contours show the predicted value of $\Braket{\sigma v_{\rm rel}}^{\rm
line}_{Z^{\prime}\gamma}$ for $10^{-24}$, $10^{-25}$, $10^{-26}$, and
$10^{-27}$ in the unit of cm3/s, respectively. The red solid lines show the
$g_{0}$ contours. In Fig. 3, we show the constraints for the NFW
Navarro:1996gj , and Einasto2 DM density profiles 1012.4515 . The derived
constraints depend on the DM density profiles. The blue excluded regions from
$\Braket{\sigma v_{\rm rel}}^{\rm line}_{Z^{\prime}\gamma}$ give stronger
constraints for $m_{Z^{\prime}}\simeq m_{V}$. This is because we have the
enhancement factor in the coupling of $Z^{\prime}$, $g_{Z^{\prime}}$, defined
in Eq. (II.47). Note that $g_{0}$ gets larger in the same region, and thus we
expect relatively large higher-order correction for our perturbative
calculations.

We have a further chance to explore the parameter region in the upcoming
Cherenkov Telescope Array (CTA) 1208.5356 ; 2007.16129 . The prospect
sensitivity for the line gamma-ray signals is studied by Ref. 2008.00692  for
the Wino and Higgsino DM. In Fig. 4, we show the current bound and the future
sensitivity expected in the CTA. The green region is excluded by the H.E.S.S.
observation for the cusped Einasto2 profile shown in the lower panel of Fig.
3. We use the prospect bound derived in Ref. 2008.00692  to show the CTA
sensitivity. The orange region with the dashed boundary shows the most
conservative sensitivity assuming the core radius $r_{c}=5$ kpc, and we will
obtain $m_{V}\gtrsim 25.3$ TeV as the prospect bound. The whole mass range in
Fig. 4 will be covered if we take $r_{c}\lesssim 2$ kpc.

Figure 4: The current bound from the H.E.S.S. observation and the prospect in
the CTA experiment. The green region is excluded by the H.E.S.S. collaboration
for the Einasto2 profile 1012.4515  shown in the lower panel of Fig. 3. The
orange region with the dashed boundary is the prospect sensitivity in the CTA
experiment for the cored Einasto profile with $r_{c}=5$ kpc, which is the most
conservative one 2008.00692 . If we take $r_{c}\lesssim 2$ kpc, the prospect
sensitivity will cover the whole region of this figure.

## V Conclusions

In this paper, we study the gamma-ray line signatures from the electroweakly
interacting non-abelian vector DM model. We derive the two-body effective
action for the spin-$1$ DM in the NR limit. Since our DM has the SU(2)L
triplet-like features, the Sommerfeld enhancement factor is almost the same as
that for the pure Wino DM. The predicted annihilation cross sections into
$\gamma\gamma/Z\gamma$ are larger by $\frac{38}{9}$ than those of the Wino.
This is because our spin-$1$ DM pair has the additional partial wave
contributions with the total spin angular momentum $J=2$. Therefore, the
gamma-ray line signals provide stronger constraints on our spin-$1$ DM
compared with the Wino DM.

We show the constraints on the line cross section for the $\gamma\gamma$,
$Z\gamma$, and $Z^{\prime}\gamma$ modes by the H.E.S.S. observation in the
Galactic Center region. The photon from the $\gamma\gamma/Z\gamma$ modes and
the $Z^{\prime}\gamma$ mode are separable through the final photon energy.
Therefore, we may observe a double-peak in the gamma-ray energy spectrum,
which provides a unique signature of our spin-$1$ DM. The constraints strongly
depend on the DM density profiles, and we obtain $m_{V}\gtrsim 14.4$ TeV
($11.7$ TeV) for the Einasto (Einasto2) profile. The annihilation into
$Z^{\prime}\gamma$ also gives a strong constraint for $m_{V}\simeq
m_{Z^{\prime}}$ where couplings between DM and $Z^{\prime}$ are enhanced. We
also show the future sensitivity in the CTA experiment. We can probe
$m_{V}\gtrsim 25.3$ TeV even if we take the conservative cored DM density
profile with the core radius of $5$ kpc.

We have possible extensions of our study. We can derive a more robust
constraint on our model by studying the continuous gamma-ray search in
observations of dwarf spheroidal galaxies by the Fermi-LAT experiment
1310.0828 ; 1812.06986 . We also expect a viable constraint from the
antiproton observations as studied for the pure Wino DM 1504.05554 . We need
to follow the decay chain including the decay of the $W^{\prime}/Z^{\prime}$
bosons in our model. Although we expect the strongest bound comes from the
gamma-ray line signatures studied in this paper, we suffer a large uncertainty
in the DM density profiles. Therefore, it is worth evaluating the more robust
constraints from the other channels. We also expect the Sommerfeld enhancement
affects the calculations of the thermal relic abundance, which we need to
derive the effective action for the electrically charged two-body states hep-
ph/0610249 ; 2009.00640 . These studies are completed in our future
collaborations.

## Acknowledgments

JH thanks Seong Chan Park for useful discussions. This work was supported by
JSPS KAKENHI Grant Numbers 19H04615 and 21K03549 (TA), and JSPS Grant-in-Aid
for Scientific Research KAKENHI Grant Numbers JP20J12392 (MF), JP20H01895
(JH), and JP21K03572 (JH). The work is also supported by JSPS Core-to-Core
Program (grant number:JPJSCCA20200002). The work of JH was supported by Grant-
in-Aid for Scientific research from the Ministry of Education, Science,
Sports, and Culture (MEXT), Japan (Grant Numbers 16H06492). The work of JH was
also supported by World Premier International Research Center Initiative (WPI
Initiative), MEXT, Japan.

##  Appendix A Derivation of effective action

We give the derivation of the effective action shown in Sec. III.

### A.1 Real part of potential

Figure 5: The relevant tree-level diagrams to derive the real part of the
potential. The diagrams for $V^{-}V^{+}\to V^{0}V^{0}$ mode are implicit.

The real part of the potential is derived to realize the NR amplitudes of the
$V$-particles. The leading-order contributions are induced from the diagrams
shown in Fig. 5. Compared with the leading-order contributions, the Higgs
exchange contributions are suppressed by small $\phi_{h}$. The $W^{\prime}$
and $Z^{\prime}$ exchange contributions are exponentially suppressed by
$m_{W^{\prime}}$ and $m_{Z^{\prime}}$. The contributions from the vector
quadratic couplings are suppressed by $\frac{1}{m_{V}^{2}}$. In the NR limit,
we obtain the following amplitude at the leading-order.

$\displaystyle i{\cal M}_{Z,\gamma}$ $\displaystyle\simeq
i4m_{V}^{2}\left(\frac{e^{2}}{|\bm{p}-\bm{p}^{\prime}|^{2}}+\frac{g_{W}^{2}c_{W}^{2}}{|\bm{p}-\bm{p}^{\prime}|^{2}+m_{Z}^{2}}\right)\epsilon_{i}(p)\epsilon_{i}^{*}(k)\epsilon_{j}(p^{\prime})\epsilon_{j}^{*}(k^{\prime}),$
(A.106) $\displaystyle i{\cal M}_{W}$ $\displaystyle\simeq
i4m_{V}^{2}\Biggl{(}\frac{g_{W}^{2}}{|\bm{p}-\bm{p}^{\prime}|^{2}+m_{W}^{2}}\epsilon_{i}(p)\epsilon_{i}^{*}(k)\epsilon_{j}(p^{\prime})\epsilon_{j}^{*}(k^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{g_{W}^{2}}{|\bm{p}-\bm{k}^{\prime}|^{2}+m_{W}^{2}}\epsilon_{i}(p)\epsilon_{i}^{*}(k^{\prime})\epsilon_{j}(p^{\prime})\epsilon_{j}^{*}(k)\Biggr{)},$
(A.107)

where ${\cal M}_{Z,\gamma}$ and ${\cal M}_{W}$ corresponds to the neutral and
charged boson exchange processes, respectively. The labels of the polarization
are implicit. We obtain the effective action composed of the NR $V$-particle
operators, which are defined in Eqs. (III.63)-(III.65), to realize the above
amplitudes.

$\displaystyle S_{\rm eff}$ $\displaystyle=\int
d^{4}Rd^{3}r\frac{\alpha_{2}s_{W}^{2}+\alpha_{2}c_{W}^{2}e^{-m_{Z}r}}{r}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times\left[{\cal B}_{i}^{\dagger}(R^{0},\bm{R}+\bm{r}/2)\leavevmode\nobreak\
{\cal B}_{i}(R^{0},\bm{R}+\bm{r}/2)\leavevmode\nobreak\ \right]\left[{\cal
D}_{j}^{\dagger}(R^{0},\bm{R}-\bm{r}/2)\leavevmode\nobreak\ {\cal
D}_{j}(R^{0},\bm{R}-\bm{r}/2)\right]$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\Biggl{\\{}\int d^{4}Rd^{3}r\frac{\alpha_{2}e^{-m_{W}r}}{r}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times\left[{\cal A}_{i}^{\dagger}(R^{0},\bm{R}+\bm{r}/2)\leavevmode\nobreak\
{\cal B}_{i}(R^{0},\bm{R}+\bm{r}/2)\right]\left[{\cal
A}_{j}^{\dagger}(R^{0},\bm{R}-\bm{r}/2)\leavevmode\nobreak\ {\cal
D}_{j}(R^{0},\bm{R}-\bm{r}/2)\right]$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +h.c.\Biggr{\\}}.$ (A.108)

To express the action in the decomposed form into each partial wave mode, we
reform the vector indices ($i$, $j$) by using the Fierz identity for
Grassmann-even operators.

$\displaystyle\delta_{ij}\delta_{k\ell}=\sum_{J,J_{z}}(-1)^{J}S_{i\ell}^{J,J_{z}}S_{kj}^{J,J_{z}*}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (i,j,k,\ell=1,2,3),$ (A.109)

where $S_{ij}^{J,J_{z}}$ is defined in Eqs. (III.75)-(III.77). After this
decomposition, we can express the action in terms of the two-body states
defined in Eqs. (III.72)-(III.73) and read out $\hat{V}$ as given in Eq.
(III.80).

### A.2 Imaginary part of potential

#### A.2.1 Matching procedure

To derive the imaginary part of the potential, $\hat{\Gamma}^{J}$, we perform
the operator matching between the two-body field operators and the
calculations of the one-loop amplitudes of the $V$-particles. Since we focus
on the annihilation modes into the neutral vectors, the $(1,1)$-component of
$\hat{\Gamma}^{J}$ only has the nonzero value. Therefore, we can use the
optical theorem to calculate the imaginary part of the one-loop amplitude.

$\displaystyle\left.{\rm Im}{\cal M}\right|_{V^{-}V^{+}\to XX^{\prime}\to
V^{-}V^{+}}^{J}$ $\displaystyle=2m_{V}^{2}\Braket{\sigma v_{\rm
rel}}^{J}_{V^{-}V^{+}\to XX^{\prime}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
(XX^{\prime}=\gamma\gamma,Z\gamma,Z^{\prime}\gamma),$ (A.110)

where the left-hand side denotes the imaginary part of the forward scattering
amplitude for $V^{-}V^{+}\to XX^{\prime}\to V^{-}V^{+}$ with $J$ which is
expressed by $(\hat{\Gamma}^{J}_{XX^{\prime}})_{11}$. In the right-hand side,
$\Braket{\sigma v_{\rm rel}}^{J}_{V^{-}V^{+}\to XX^{\prime}}$ denotes the
partial wave annihilation cross section of $V^{-}V^{+}\to XX^{\prime}$ for the
initial state with $J$. In the evaluation of the annihilation cross section,
we only leave the leading-order terms in the NR limit and take the massless
limit for all the SM particles in the final states while we leave the masses
of $Z^{\prime}$. We derive $\hat{\Gamma}^{J}$ through the above procedure as
summarized in Eqs. (III.81)-(III.86). In the succeeding section, we show how
to determine $\hat{\Gamma}^{J}_{\gamma\gamma}$ as a demonstration.

#### A.2.2 Derivation of $\leavevmode\nobreak\
\hat{\Gamma}_{\gamma\gamma}^{J}$

Figure 6: The tree-level diagrams which contribute to
$V^{-}V^{+}\to\gamma\gamma$ annihilation.

We show the derivation of $\hat{\Gamma}_{\gamma\gamma}^{J}$ as a concrete
example. We can focus on the nonzero component,
$(\hat{\Gamma}_{\gamma\gamma}^{J})_{11}$. First, we calculate the velocity-
weighted annihilation cross section for $V^{-}V^{+}\to\gamma\gamma$. The tree-
level diagrams in the unitarity gauge are shown in Fig. 6. The amplitudes that
correspond to each diagram are obtained as follows.

$\displaystyle i\mathcal{M}_{4}=$ $\displaystyle
ie^{2}(g^{\mu\rho}g^{\nu\sigma}+g^{\nu\rho}g^{\mu\sigma}-2g^{\mu\nu}g^{\rho\sigma})\epsilon_{\sigma}(p_{-})\epsilon_{\rho}(p_{+})\epsilon^{*}_{\mu}(k_{1})\epsilon^{*}_{\nu}(k_{2}),$
(A.111) $\displaystyle i\mathcal{M}_{t}=$ $\displaystyle
ie[(p_{-}+q)^{\mu}g^{\alpha\sigma}+(-p_{-}-k_{1})^{\alpha}g^{\mu\sigma}+(k_{1}-q)^{\sigma}g^{\mu\alpha}]\frac{-i}{q^{2}-m_{V}^{2}}\left(g_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{m_{V}^{2}}\right)$
(A.112) $\displaystyle\times
ie[(q-p_{+})^{\nu}g^{\beta\rho}+(-q-k_{2})^{\rho}g^{\beta\nu}+(k_{2}+p_{+})^{\beta}g^{\rho\nu}]\epsilon_{\sigma}(p_{-})\epsilon_{\rho}(p_{+})\epsilon^{*}_{\mu}(k_{1})\epsilon^{*}_{\nu}(k_{2}),$
(A.113) $\displaystyle i\mathcal{M}_{u}=$ $\displaystyle
ie[(p_{-}+q^{\prime})^{\nu}g^{\alpha\sigma}+(-p_{-}-k_{2})^{\alpha}g^{\nu\sigma}+(k_{2}-q^{\prime})^{\sigma}g^{\nu\alpha}]\frac{-i}{q^{\prime
2}-m_{V}^{2}}\left(g_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{m_{V}^{2}}\right)$
(A.114) $\displaystyle\times
ie[(q^{\prime}-p_{+})^{\mu}g^{\beta\rho}+(-q^{\prime}-k_{1})^{\rho}g^{\beta\mu}+(k_{1}+p_{+})^{\beta}g^{\rho\mu}]\epsilon_{\sigma}(p_{-})\epsilon_{\rho}(p_{+})\epsilon^{*}_{\mu}(k_{1})\epsilon^{*}_{\nu}(k_{2}),$
(A.115)

where $q\equiv p_{-}-k_{1}$ and $q^{\prime}\equiv p_{-}-k_{2}$.

In the NR limit for the initial $V$-particles, the zeroth components of the
polarization vectors are sub-leading, and we focus on the spatial component,
$\epsilon_{i}(p_{-})\epsilon_{j}(p_{+})$. We decompose the amplitude into each
partial wave contribution by replacing the initial state polarization vectors
with $S^{J,J_{z}}_{ij}$ defined in Eqs. (III.75)-(III.77).

$\displaystyle\epsilon_{i}(p_{-})\epsilon_{j}(p_{+})\to S^{J,J_{z}}_{ij}.$
(A.116)

Note that $S^{J,J_{z}}_{ij}$ is symmetric matrices for $J=0,2$ and anti-
symmetric for $J=1$. Thanks to these properties, the amplitudes in the center-
of-mass frame are obtained in the decomposed form, ${\cal M}^{J,J_{z}}$.

$\displaystyle\mathcal{M}^{0,0}=$ $\displaystyle
2e^{2}\left[4\epsilon_{i}^{*}(k_{1})S^{0,0}_{ij}\epsilon_{j}^{*}(k_{2})-\epsilon_{i}^{*}(k_{1})\epsilon_{i}^{*}(k_{2})\left(S^{0,0}_{ii}-\frac{2}{m_{V}^{2}}k_{j}S^{0,0}_{jk}k_{k}\right)\right],$
(A.117) $\displaystyle\mathcal{M}^{1,J_{z}}=$ $\displaystyle 0,$ (A.118)
$\displaystyle\mathcal{M}^{2,J_{z}}=$ $\displaystyle
2e^{2}\left[4\epsilon_{i}^{*}(k_{1})S^{2,J_{z}}_{ij}\epsilon_{j}^{*}(k_{2})+\epsilon_{i}^{*}(k_{1})\epsilon_{i}^{*}(k_{2})\frac{2}{m_{V}^{2}}k_{j}S^{2,J_{z}}_{jk}k_{k}\right],$
(A.119)

where $|J_{z}|\leq J$ and $k_{i}$ denotes the spatial components of
${k_{1}}_{\mu}$. We use $S^{2,J_{z}}_{ii}=0$ to obtain the above result. The
polarization vectors for the photons are given by

$\displaystyle\epsilon^{\pm}_{i}(k_{1})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\mp\cos\theta\\\ -i\\\
\pm\sin\theta\end{pmatrix},$ $\displaystyle\epsilon^{\pm}_{i}(k_{2})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm\cos\theta\\\ -i\\\
\mp\sin\theta\end{pmatrix},$ (A.120)

where $\theta$ is the scattering angle for the photon with its momentum
$k_{1}$.

After taking the sum over the final state spins, the squared amplitudes can be
expressed as follow.

$\displaystyle\sum_{\rm final}|\mathcal{M}^{0,0}|^{2}$
$\displaystyle=24e^{4},$ (A.121) $\displaystyle\sum_{\rm
final}|\mathcal{M}^{1,J_{z}}|^{2}$ $\displaystyle=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (J_{z}=0,\pm 1),$ (A.122) $\displaystyle\sum_{\rm
final}|\mathcal{M}^{2,J_{z}}|^{2}$
$\displaystyle=\left\\{\begin{array}[]{ll}48e^{4}\sin^{4}\theta&\hskip
20.0pt(J_{z}=0),\\\ 16e^{4}\sin^{2}\theta(3+\cos 2\theta)&\hskip
20.0pt(J_{z}=\pm 1),\\\ 8e^{4}(1+6\cos^{2}\theta+\cos^{4}\theta)&\hskip
20.0pt(J_{z}=\pm 2).\end{array}\right.$ (A.126)

We obtain the cross section for each partial wave mode.

$\displaystyle\Braket{\sigma v_{\rm rel}}^{J=0}_{V^{-}V^{+}\to\gamma\gamma}$
$\displaystyle=6\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}s_{W}^{4},$ (A.127)
$\displaystyle\Braket{\sigma v_{\rm rel}}^{J=1}_{V^{-}V^{+}\to\gamma\gamma}$
$\displaystyle=0,$ (A.128) $\displaystyle\Braket{\sigma v_{\rm
rel}}^{J=2}_{V^{-}V^{+}\to\gamma\gamma}$
$\displaystyle=\frac{32}{5}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}s_{W}^{4}.$
(A.129)

The spin-averaged total cross section is obtained by adding up all the partial
wave cross sections.

$\displaystyle\overline{\Braket{\sigma v_{\rm rel}}}^{\rm
tot}_{V^{-}V^{+}\to\gamma\gamma}$
$\displaystyle=\frac{38}{9}\frac{\pi\alpha^{2}_{2}}{m_{V}^{2}}s_{W}^{4}.$
(A.130)

This cross section is larger than that of the Wino DM by a factor of
$\frac{38}{9}$. For the $Z\gamma$ mode, we take the massless limit of the $Z$
boson, and the calculation procedure is the same. For the $Z^{\prime}\gamma$
mode, we do not neglect $m_{Z^{\prime}}$, and the longitudinal mode also
contributes to the final result.

Next, we calculate the imaginary part of the forward scattering amplitude
using the two-body effective action shown in Eq. (III.70). Namely, we use the
following coupling to calculate the $\gamma\gamma$ contribution to ${\rm
Im}{\cal M}$.

$\displaystyle S_{\rm eff}$
$\displaystyle\supset\sum_{J,J_{z}}i\frac{9}{2}(\hat{\Gamma}_{\gamma\gamma}^{J})_{11}\int
d^{4}R\leavevmode\nobreak\
\phi_{C}^{J,J_{z}\dagger}(R,\bm{0})\leavevmode\nobreak\
\phi_{C}^{J,J_{z}}(R,\bm{0}).$ (A.131)

Using this coupling, we obtain

$\displaystyle\left.{\rm Im}{\cal M}\right|_{V^{-}V^{+}\to\gamma\gamma\to
V^{-}V^{+}}^{J}$ $\displaystyle=2m_{V}^{2}\cdot
9(\hat{\Gamma}^{J}_{\gamma\gamma})_{11}.$ (A.132)

Comparing with (A.110), we can determine
$(\hat{\Gamma}^{J}_{\gamma\gamma})_{11}$ as shown below.

$\displaystyle(\hat{\Gamma}^{J}_{\gamma\gamma})_{11}$
$\displaystyle=\frac{1}{9}\Braket{\sigma v_{\rm
rel}}_{V^{-}V^{+}\to\gamma\gamma}^{J}=\left\\{\begin{array}[]{ll}\frac{2}{3}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}s_{W}^{4},\\\
0,\\\ \frac{32}{45}\frac{\pi\alpha_{2}^{2}}{m_{V}^{2}}s_{W}^{4}.\\\
\end{array}\right.$ (A.136)

This result is summarized in Eqs. (III.81)-(III.82). We can also determine
$\hat{\Gamma}^{J}_{Z\gamma}$ and $\hat{\Gamma}^{J}_{Z^{\prime}\gamma}$ in the
same procedure.

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