In this study, we have focused on a specific mass spectrum of the dark pion,
$m_{A^{\prime}}<m_{\pi^{\prime}}<2m_{A^{\prime}}$, which is relevant for
visible decay searches. For the dark pions with
$2m_{A^{\prime}}<m_{\pi^{\prime}}$, produced dark pions promptly decay into
dark photons through the anomaly-induced interaction, $\pi^{\prime 3}\to
A^{\prime}A^{\prime}$, and then we lose the future sensitivity and the
existing constraints from fixed-target experiments. When the dark pions are
the lightest particle in the dark sector, $m_{\pi^{\prime}}<m_{A^{\prime}}$,
dark pions can decay into the visible particle with lifetime longer than
$1\,[\mathrm{s}]$. Instead, dark photon would dominantly decay into dark pions
providing invisible signals of dark photons at the fixed-target experiments
such as LDMX [22, 23]. Furthermore, since the dark sector consists of the
confining gauge dynamics, dark vector mesons would exist with the mass of dark
dynamical scale. If the dark vector mesons have a broad resonance similar to
the SM $\rho$ mesons, the production of darkly charged pions would change due
to the broad resonance and kinetic mixing between dark photons and dark $\rho$
mesons. We leave the long-lived particle searches with different mass spectra
for future work.

### Acknowledgement

The work of A. K. and T. K. is partly supported by IBS under the project code,
IBS-R018-D1. A. K. acknowledges partial support from Grant-in-Aid for
Scientific Research from the Ministry of Education, Culture, Sports, Science,
and Technology (MEXT), Japan, 18K13535 and 19H04609; from World Premier
International Research Center Initiative (WPI), MEXT, Japan; from Norwegian
Financial Mechanism for years 2014-2021, grant nr 2019/34/H/ST2/00707; and
from National Science Centre, Poland, grant DEC-2018/31/B/ST2/02283. T. K. is
grateful to Nagoya University for their hospitality during the COVID-19
pandemic.

##  Appendix A Linear Sigma Model

We analyze chiral symmetry breaking and scalar dark mesons by use of the
linear sigma model (LSM) that is a low-energy effective theory of two-flavor
dark QCD. The Lagrangian density of the LSM is constructed to be invariant
under the flavor transformation, $SU(2)_{L}\times SU(2)_{R}\times U(1)_{V}$,
which reflects the symmetries of the dark QCD. $U(1)_{A}$ is explicitly broken
by an axial anomaly term. Let $\Phi$ be an LSM field that is an $2\times 2$
matrix scalar field. $\Phi$ is identified with the dark-quark bilinear as
$\Phi_{ij}\sim\bar{q}_{j}q_{i}$ with the flavor subscripts $i,j$. According to
this identification, $\Phi$ transforms under the flavor transformation,
$SU(2)_{L}\times SU(2)_{R}\times U(1)_{A}$, as follows.

$\begin{split}\Phi\to e^{i\alpha}L\Phi R^{\dagger}\,,\end{split}$ (61)

where $L$ and $R$ are the unitary matrices of $SU(2)_{L}\times SU(2)_{R}$.
$e^{i\alpha}$ is an elements of $U(1)_{A}$. $\Phi$ does not transform under
$U(1)_{V}$. Transformation properties of the dark-quark bilinear determine $C$
and $P$ transformation of $\Phi$ as follows.

$\begin{split}C\,:\,&\hat{\Phi}\to\mathcal{C}\hat{\Phi}\mathcal{C}^{-1}=\hat{\Phi}^{T}\,,\\\
P\,:\,&\hat{\Phi}\to\mathcal{P}\hat{\Phi}\mathcal{P}^{-1}=\hat{\Phi}^{\dagger}\,.\end{split}$
(62)

Here, we define the discrete transformation for the field in the charge basis,
which is denoted by the hatted fields.

We decompose the matrix field $\Phi$ into scalar fields $\sigma^{a}$ and
pseudo-scalar fields $\pi^{a}$ as

$\begin{split}\Phi=(\sigma^{a}+i\pi^{a})T^{a}\,,\end{split}$ (63)

where $T^{a}~{}(a=0,1,\cdots,3)$ are a unit matrix and $SU(2)$ generators that
satisfy

$\begin{split}[T^{a}\,,T^{b}]=if_{abc}T^{c}\,,~{}~{}~{}~{}~{}\\{T^{a}\,,T^{b}\\}=d_{abc}T^{c}\,,~{}~{}~{}~{}~{}\mathrm{tr}(T^{a}T^{b})=\frac{1}{2}\delta^{ab}\,.\end{split}$
(64)

$f_{abc}$ is totally antisymmetric in all indices, while $d_{abc}$ is totally
symmetric. They are just structure constants of $SU(2)$ when $a,b,c\neq 0$,
while we have $f_{ab0}=0\,,d_{ab0}=\delta_{ab}$.

The Lagrangian density is

$\begin{split}\mathcal{L}=\mathrm{tr}(\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi)-V(\Phi)+\mathrm{tr}(H\Phi+H^{\dagger}\Phi^{\dagger})\,.\end{split}$
(65)

In this sub section, we do not include the gauge interactions of the LSM
field. Here, the second term is the potential for $\Phi$ that is given by

$\begin{split}V(\Phi)&=-\mu^{2}\mathrm{tr}(\Phi^{\dagger}\Phi)+\lambda_{1}[\mathrm{tr}(\Phi^{\dagger}\Phi)]^{2}+\lambda_{2}\mathrm{tr}(\Phi^{\dagger}\Phi)^{2}-\left[c\det\Phi+c^{\ast}\det\Phi^{\dagger}\right]\,.\end{split}$
(66)

We assume parameters $\lambda_{1}\,,\lambda_{2}$ , and $\mu^{2}$ to be
positive. The last term breaks $U(1)_{A}$, and the coefficient $c$ is a
complex number with mass dimension two in general. Throughout this paper, we
fix the coefficient $c$ to be positive by $U(1)_{A}$ rotation of $\Phi$.

The third term of the Lagrangian reflects the explicit breaking of the chiral
symmetry by the dark quark masses. We refer to $H\equiv j^{a}T^{a}$ as a
source matrix that is proportional to the mass matrix of the current dark
quarks. It would be sufficient to consider the linear term in $H$ as far as
the dark quark masses are sufficiently small and its perturbation works well.

The Lagrangian is rewritten in terms of the components scalars, $\sigma^{a}$
and $\pi^{a}$, as follows.

$\begin{split}\mathcal{L}=&\frac{1}{2}\left[\partial_{\mu}\sigma^{a}\partial^{\mu}\sigma^{a}+\partial_{\mu}\pi^{a}\partial^{\mu}\pi^{a}\right]-U(\sigma,\pi)\,.\end{split}$
(67)

The scalar potential $U(\sigma,\pi)$ is

$\begin{split}U(\sigma,\pi)\equiv&\frac{1}{2}\sigma^{a}\left[(-\mu^{2}+c)\delta_{ab}-2c\delta_{a,0}\delta_{b,0}\right]\sigma^{b}+\frac{1}{2}\pi^{a}\left[-(\mu^{2}+c)\delta_{ab}+2c\delta_{a,0}\delta_{b,0}\right]\pi^{b}\\\
&+2\mathcal{H}_{abcd}\sigma^{a}\sigma^{b}\pi^{c}\pi^{d}+\frac{1}{3}\mathcal{F}_{abcd}\left[\sigma^{a}\sigma^{b}\sigma^{c}\sigma^{d}+\pi^{a}\pi^{b}\pi^{c}\pi^{d}\right]\\\
&-\frac{1}{2}\left[(\sigma^{a}+i\pi^{a})j^{a}+(\sigma^{a}-i\pi^{a})j^{a\ast}\right]\,,\\\
\end{split}$ (68)

with the coefficients

$\begin{split}\mathcal{H}_{abcd}&=\frac{\lambda_{1}}{4}\delta_{ab}\delta_{cd}+\frac{\lambda_{2}}{8}(d_{abn}d_{cdn}+f_{acn}f_{bdn}+f_{adn}f_{bcn})\,,\\\
\mathcal{F}_{abcd}&=\frac{\lambda_{1}}{4}(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})+\frac{\lambda_{2}}{8}(d_{abn}d_{cdn}+d_{acn}d_{bdn}+d_{adn}d_{bcn})\,.\end{split}$
(69)

$\sigma^{0}$ and $\pi^{a}~{}(a=1,2,3)$ may obtain the dominant vacuum
expectation value (VEV) since they apparently have negative mass squared. We
simply set $\pi^{a}$ that obtains its VEV to be $\pi^{3}$ by use of the
vectorial flavor rotation $SU(2)_{V}$ of $\Phi$. We parametrize masses of
scalars after the symmetry breaking as follows.

$\begin{split}U(\sigma+\bar{\sigma},\pi+\bar{\pi})=\frac{1}{2}(m_{\pi}^{2})_{ab}\pi^{a}\pi^{b}+\frac{1}{2}(m_{\sigma}^{2})_{ab}\sigma^{a}\sigma^{b}+(m_{\pi\sigma}^{2})_{ab}\pi^{a}\sigma^{b}+\cdots\,.\end{split}$
(70)

Here, letters with bars denote their VEVs, and the higher terms are
abbreviated.

Let us compute the mass spectra under some assumptions. We can generally
diagonalize the source term by use of $SU(2)_{L}\times SU(2)_{R}$
transformation of $\Phi$.

$\begin{split}H=e^{i\alpha_{H}}(j^{0}T^{0}+j^{3}T^{3})\,.\end{split}$ (71)

We can utilize $U(1)_{A}$ transformation to remove the overall phase
$\alpha_{H}$, but the phase of the anomaly term $c$ appears, again. To avoid
the phase of $c$, we keep $\alpha_{H}$ unless we assume the source term to be
real. The kinetic term and the potential $V(\Phi)$ is invariant under
$\Phi\to-\Phi$ as the number of flavors $N_{f}$ is even. Therefore, we can set
$j^{0}>0$ without loss of generality for $N_{f}=2$.

###  A.1 Decay Constant

Let us discuss the decay constant in the LSM. In particular, we clarify the
normalization of the decay constant by use of the commutation relations of the
charges and partially-conserved axial current (PCAC) relation. The
infinitesimal axial-vector ($L=R^{\dagger}=e^{i\alpha^{a}T^{a}}$) and vector
($L=R=e^{i\beta^{a}T^{a}}$) transformations define their Noether currents,
$J^{\mu}_{Aa}$ and $J^{\mu}_{Va}$.

$\displaystyle J^{\mu}_{Aa}$
$\displaystyle=-\frac{1}{2}d_{abc}(i\phi^{b\ast}\partial^{\mu}\phi^{c}-i\phi^{b}\partial^{\mu}\phi^{c\ast})\,,$
$\displaystyle J^{\mu}_{Va}$
$\displaystyle=\frac{1}{2}f_{abc}(\phi^{b\ast}\partial^{\mu}\phi^{c}+\phi^{b}\partial^{\mu}\phi^{c\ast})\,.$
(72)

Here, we define $\Phi\equiv\phi^{a}T^{a}$. The PCAC relation gives

$\begin{split}\langle
0|J^{\mu}_{Aa}(x)|\pi^{b}(p)\rangle=ip^{\mu}f^{ab}e^{-ip\cdot
x}\,,\end{split}$ (73)

where $f^{ab}$ is the decay constant. In terms of the component fields,
$\sigma^{a}$ and $\pi^{a}$, the axial current is
$J^{\mu}_{Aa}=d_{abc}(\sigma^{b}\partial^{\mu}\pi^{c}-\pi^{b}\partial^{\mu}\sigma^{c})$.
When the sigma fields get their VEVs,
$\langle\sigma^{a}\rangle=\bar{\sigma}^{a}$, the PCAC relation gives the decay
constant as follows.

$\begin{split}f^{ab}=\sum_{c}d_{abc}\bar{\sigma}^{c}\,.\end{split}$ (74)

In particular, when the VEV is universal, $\bar{\sigma}^{a}=\bar{\sigma}$, the
decay constant is diagonalized: $f^{ab}=\bar{\sigma}^{0}\delta^{ab}$.

We can also define the currents in terms of quarks denoted by $Q$:

$\displaystyle j^{\mu a}_{V}(x)$
$\displaystyle=\overline{Q}\gamma^{\mu}T^{a}Q\,,$ $\displaystyle j^{\mu
a}_{A}(x)$ $\displaystyle=\overline{Q}\gamma^{\mu}\gamma_{5}T^{a}Q\,.$ (75)

This quark axial current annihilates the one-pion state, and it also defines
the pion decay constant.

$\begin{split}\langle 0|j^{\mu
a}_{A}(x)|\pi^{b}(p)\rangle=ip^{\mu}f_{\pi}\delta^{ab}e^{-ip\cdot
x}\,,\end{split}$ (76)

In the SM, the observation of the charged pion decay, $\pi^{+}\to\mu^{+}\nu$,
determines the decay constant. In this normalization, we have
$f_{\pi}=92\,\mathrm{MeV}$ from the SM pion decay. This normalization provides
a commutation relation of the charges as follows.

$\displaystyle[q_{A}^{a},q_{A}^{b}]$ $\displaystyle=if_{abc}q^{c}_{V}\,,$
$\displaystyle q^{a}_{V}$ $\displaystyle\equiv\int d^{3}xj^{0a}_{V}(x)\,,$
$\displaystyle q^{a}_{A}$ $\displaystyle\equiv\int d^{3}xj^{0a}_{A}(x)\,.$
(77)

In the LSM, the axial charge and the vector charge are defined by

$\begin{split}Q_{A}^{a}\equiv\int
d^{3}x\,J^{0}_{Aa}(x)\,,~{}~{}~{}Q_{V}^{a}\equiv\int
d^{3}x\,J^{0}_{Va}(x)\,.\end{split}$ (78)

These charges obey the following commutation relation.

$\displaystyle[Q_{A}^{a},Q_{A}^{b}]$ $\displaystyle=if_{abc}Q^{c}_{V}\,,$ (79)

The charges in the LSM satisfy the same algebra as the charges defined in
terms of quark currents. Thus, we conclude that $f^{ab}$ defined in Eq. 73
follows the same normalization as the decay constant defined by quark
currents.

###  A.2 Universal Source

In this subsection, we compute the spectrum with the source term
$H=e^{i\alpha}j^{0}T^{0}~{}(j^{0}>0)$. First, we consider the case with the
real source ($\alpha=0$). We set
$\langle\sigma^{a}\rangle=\bar{\sigma}^{0}\delta_{a,0}$ and
$\langle\pi^{a}\rangle=0$.

The tree-level effective potential is given by

$\begin{split}U(\bar{\sigma}^{0})=-\frac{1}{2}(\mu^{2}+c)(\bar{\sigma}^{0})^{2}+\frac{1}{4}\lambda(\bar{\sigma}^{0})^{4}-j^{0}\bar{\sigma}^{0}\,,~{}~{}~{}~{}~{}\lambda\equiv\lambda_{1}+\frac{\lambda_{2}}{2}\,.\end{split}$
(80)

Since the potential is tilted in the presence of $j^{0}>0$, the stationary
point condition for this potential gives the VEV up to $\mathcal{O}(j^{0})$ as

$\begin{split}\bar{\sigma}^{0}=\sqrt{\frac{c+\mu^{2}}{\lambda}}+\frac{j^{0}}{2(c+\mu^{2})}\,.\end{split}$
(81)

We define the decay constant $f\equiv\bar{\sigma}^{0}$. We get the mass
spectrum up to $\mathcal{O}(j^{0})$ as follows.

$\displaystyle(m_{\sigma}^{2})_{00}$ $\displaystyle=2\lambda
f^{2}+\frac{j^{0}}{f}\,,$ $\displaystyle(m_{\sigma}^{2})_{ii}$
$\displaystyle=2c+\lambda_{2}f^{2}+\frac{j^{0}}{f}\,,$ (82)
$\displaystyle(m_{\pi}^{2})_{00}$ $\displaystyle=2c+\frac{j^{0}}{f}\,,$
$\displaystyle(m_{\pi}^{2})_{ii}$ $\displaystyle=\frac{j^{0}}{f}\,.$ (83)

Here, the subscript $i$ runs $1\,,2\,,$ and $3$. There is no off diagonal
components of mass matrices, and thus the mass spectrum is already
diagonalized. Since the source $j^{0}$ is proportional to the current quark
mass, $\pi^{i}~{}(i=1,2,3)$ are identified to be the pseudo Nambu-Goldstone
bosons. We can identify $\sigma^{0}$ to be the so-called $\sigma$ meson, and
$\pi^{0}$ to be the so-called $\eta^{\prime}$ meson. The $U(1)_{A}$ anomaly
provides the sizable contribution to the mass of $\pi^{0}$, and therefore
$\pi^{0}$ is heavier than the other pseudo scalars.

When the phase $\alpha_{H}$ is turned on, the potential is also tilted to the
$\pi^{0}$ direction. Therefore, $\pi^{0}$ acquires its VEV that proportional
to the source, and we set
$\langle\sigma^{a}\rangle=\bar{\sigma}^{0}\delta_{a,0}$ and
$\langle\pi^{a}\rangle=\bar{\pi}^{0}\delta_{a,0}$. The effective potential is
given by

$\begin{split}U(\bar{\sigma}^{0},\bar{\pi}^{0})&=-\frac{1}{2}(\mu^{2}+c)(\bar{\sigma}^{0})^{2}+\frac{1}{2}(-\mu^{2}+c)(\bar{\pi}^{0})^{2}+\frac{\lambda}{4}[(\bar{\sigma}^{0})^{2}+(\bar{\pi}^{0})^{2}]^{2}\\\
&\quad-(j^{0}\cos\alpha_{H}\bar{\sigma}^{0}-j^{0}\sin\alpha_{H}\bar{\pi}^{0})\,.\end{split}$
(84)

As we mentioned above, we can restrict the parameter space to
$j^{0}\cos\alpha_{H}>0$ by flipping the sign of $\Phi$ without loss of
generality. The stationary point condition for the potential gives their VEVs
as follows.

$\displaystyle\bar{\sigma}^{0}$
$\displaystyle=\sqrt{\frac{c+\mu^{2}}{\lambda}}+\frac{j^{0}}{2(c+\mu^{2})}\cos\alpha_{H}\,,$
$\displaystyle\bar{\pi}^{0}$
$\displaystyle=-\frac{j^{0}}{2c}\sin\alpha_{H}\,.$ (85)

Since both $\sigma^{0}$ and $\pi^{0}$ get their VEVs, there is mass mixing
among $\sigma^{a}$ and $\pi^{a}$ at the order of $j^{0}$. This leads
corrections of $\mathcal{O}[(j^{0})^{2}]$ to the mass eigenvalues. At the
leading order of $j^{0}$, we have the similar mass spectrum of scalars to the
one without the phase $\alpha_{H}$.

$\displaystyle(m_{\sigma}^{2})_{00}$ $\displaystyle=2\lambda
f^{2}+\frac{j^{0}}{f}\cos\alpha_{H}\,,$ $\displaystyle(m_{\sigma}^{2})_{ii}$
$\displaystyle=2c+\lambda_{2}f^{2}+\frac{j^{0}}{f}\cos\alpha_{H}\,,$ (86)
$\displaystyle(m_{\pi}^{2})_{00}$
$\displaystyle=2c+\frac{j^{0}}{f}\cos\alpha_{H}\,,$
$\displaystyle(m_{\pi}^{2})_{ii}$
$\displaystyle=\frac{j^{0}}{f}\cos\alpha_{H}\,.$ (87)

We define the decay constant $f\equiv\bar{\sigma}^{0}$, again.

###  A.3 Isospin-violating Source

In this subsection, we compute the spectrum with the source term
$H=e^{i\alpha}j^{3}T^{3}~{}(j^{3}>0)$. As with the previous subsection, we
first consider the case with the real source ($\alpha=0$). Since the potential
is tilted in the $\sigma^{3}$ direction, $\sigma^{3}$ must have its VEV that
is proportional to the source. However, the dominant VEV is determined
irrespective to the source term since the mass squared of $\sigma^{3}$ is
positive. As discussed in the introduction of this appendix, there are two
possible dominant VEVs: one is $\bar{\sigma}^{0}$ and another is
$\bar{\pi}^{3}$.

First, we discuss the former case;
$\langle\sigma^{a}\rangle=\bar{\sigma}^{0}\delta_{a,0}+\bar{\sigma}^{3}\delta_{a,3}$
and $\langle\pi^{a}\rangle=0$. The effective potential for this vacuum choice
is

$\begin{split}U(\bar{\sigma}^{0},\bar{\sigma}^{3})&=-\frac{1}{2}(\mu^{2}+c)(\bar{\sigma}^{0})^{2}+\frac{1}{2}(-\mu^{2}+c)(\bar{\sigma}^{3})^{2}+\frac{1}{4}\left(\lambda+\frac{\lambda_{2}}{2}\right)[(\bar{\sigma}^{0})^{4}+(\bar{\sigma}^{3})^{4}]\\\
&\quad+\frac{1}{2}\left(\lambda+\frac{3\lambda_{2}}{2}\right)(\bar{\sigma}^{0})^{2}(\bar{\sigma}^{3})^{2}-j^{3}\bar{\sigma}^{3}\,.\end{split}$
(88)

From the stationary point condition of this potential, we determine the VEVs
at the order of $\mathcal{O}[(j^{3})^{2}]$.

$\displaystyle\bar{\sigma}^{0}$
$\displaystyle=\sqrt{\frac{c+\mu^{2}}{\lambda}}-\sqrt{\frac{\lambda}{c+\mu^{2}}}\left(1+\frac{\lambda_{2}}{\lambda}\right)\frac{(\bar{\sigma}^{3})^{2}}{2}\,,$
$\displaystyle\bar{\sigma}^{3}=\frac{\lambda}{2c\lambda+\lambda_{2}(c+\mu^{2})}j^{3}\,.$
(89)

We note that $\bar{\sigma}^{3}$ is positive for $j^{3}>0$ and is proportional
to $j^{3}$. There is no linear correction of $j^{3}$ to the scalar masses
since the source $j^{3}$ appears in $\bar{\sigma}^{0}$ from the second orders
of $j^{3}$. The pseudo Nambu-Goldstone bosons are massless up to
$\mathcal{O}(j^{3})$. The scalar mass spectrum up to $\mathcal{O}(j^{3})$ is

$\displaystyle(m_{\sigma}^{2})_{00}$ $\displaystyle=2\lambda f^{2}\,,$
$\displaystyle(m_{\sigma}^{2})_{ii}$ $\displaystyle=2c+\lambda_{2}f^{2}\,,$
(90) $\displaystyle(m_{\pi}^{2})_{00}$ $\displaystyle=2c\,,$
$\displaystyle(m_{\pi}^{2})_{ii}$ $\displaystyle=0\,.$ (91)

Since there is the mass mixing of $\mathcal{O}(j^{3})$ between $\pi^{0}$ and
$\pi^{3}$, we have to take care of corrections of $\mathcal{O}[(j^{3})^{2}]$.
In particular, since the dark pions are massless even at the order of
$\mathcal{O}(j^{3})$, it is not obvious if the potential for pseudoscalars is
stabilized. The lightest pseudoscalar mass is indeed tachyonic:

$\begin{split}m_{\pi}^{2}=-\frac{\lambda_{2}j^{3}}{2c}\bar{\sigma}^{3}\,,\end{split}$
(92)

and hence, the lightest pseudoscalar developes its VEV. We conclude that the
choice of VEVs,
$\langle\sigma^{a}\rangle=\bar{\sigma}^{0}\delta_{a,0}+\bar{\sigma}^{3}\delta_{a,3}$
and $\langle\pi^{a}\rangle=0$, does not provide a proper vacuum.

Instead, we discuss the latter case;
$\langle\sigma^{a}\rangle=\bar{\sigma}^{3}\delta_{a,3}$ and
$\langle\pi^{a}\rangle=\bar{\pi}^{3}\delta_{a,3}$. The effective potential for
this vacuum choice is given by

$\begin{split}U(\bar{\sigma}^{3},\bar{\pi}^{3})=\frac{1}{2}(-\mu^{2}+c)(\bar{\sigma}^{3})^{2}-\frac{1}{2}(\mu^{2}+c)(\bar{\pi}^{3})^{2}+\frac{1}{4}\lambda[(\bar{\sigma}^{3})^{2}+(\bar{\pi}^{3})^{2}]^{2}-j^{3}\bar{\sigma}^{3}\,.\end{split}$
(93)

From the stationary point condition of this potential, we determine the VEVs
at the order of $\mathcal{O}[(j^{3})^{2}]$.

$\displaystyle\bar{\sigma}^{3}$ $\displaystyle=\frac{j^{3}}{2c}\,,$
$\displaystyle\bar{\pi}^{3}=\sqrt{\frac{c+\mu^{2}}{\lambda}}-\sqrt{\frac{\lambda}{c+\mu^{2}}}\frac{(\bar{\sigma}^{3})^{2}}{2}\,.$
(94)

There is no linear correction of $j^{3}$ to the scalar masses since the source
$j^{3}$ appears in $\bar{\pi}^{3}$ from the second orders of $j^{3}$. The
pseudo Nambu-Goldstone bosons are massless up to $\mathcal{O}(j^{3})$. The
scalar mass spectrum up to $\mathcal{O}(j^{3})$ is given by

$\displaystyle(m_{\pi}^{2})_{33}$ $\displaystyle=2\lambda f^{2}\,,$
$\displaystyle(m_{\pi}^{2})_{00}$
$\displaystyle=(m_{\sigma}^{2})_{11}=(m_{\sigma}^{2})_{22}=2c+\lambda_{2}f^{2}\,,$
(95) $\displaystyle(m_{\sigma}^{2})_{33}$ $\displaystyle=2c\,,$
$\displaystyle(m_{\sigma}^{2})_{00}$
$\displaystyle=(m_{\pi}^{2})_{11}=(m_{\pi}^{2})_{22}=0\,.$ (96)

Here, we take $f=\bar{\pi}^{3}$ as with the previous case:
$f^{2}\simeq\lambda^{-1}(c+\mu^{2})$ of the order of
$\mathcal{O}[(j^{3})^{0}]$. This spectrum coincides with that with the
previous vacuum choice up to $\mathcal{O}(j^{3})$. As with the previous case,
we have to care about correction up to $\mathcal{O}[(j^{3})^{2}]$ since we
have massless scalars. There is mass mixing between $\sigma$ and $\pi$ since
both $\sigma$ and $\pi$ get their VEVs in this vacuum choice. Every massless
scalars obtains the mass of $\mathcal{O}[(j^{3})^{2}]$:

$\displaystyle m^{2}$
$\displaystyle=\frac{j^{3}\bar{\sigma}^{3}\lambda_{2}}{M^{2}}\,,$
$\displaystyle M^{2}$ $\displaystyle\equiv 2c+\lambda_{2}f^{2}\,.$ (97)

Here, $j^{3}$ and $\bar{\sigma}^{3}$ have the same sign, and therefore all
massless scalars get positive mass of $\mathcal{O}[(j^{3})^{2}]$. We conclude
that the vacuum choice,
$\langle\sigma^{a}\rangle=\bar{\sigma}^{3}\delta_{a,3}$ and
$\langle\pi^{a}\rangle=\bar{\pi}^{3}\delta_{a,3}$, provides a proper vacuum up
to $\mathcal{O}[(j^{3})^{2}]$ for the isospin-breaking source
$H=j^{3}T^{3}~{}(j^{3}>0)$.

We note that mass spectrum Eqs. 95 and 96 is similar to the spectrum Eqs. 86
and 87 with $\alpha_{H}=\pi/2$. In particular, the spectrums coincide with
each other by replacing $\pi^{3}\leftrightarrow\sigma^{0}$ and
$\pi^{0}\leftrightarrow\sigma^{3}$. The universal mass with $\alpha_{H}=\pi/2$
implies the source term is pure imaginary. Since we can rotate the LSM field
by a specific flavor rotation:

$\displaystyle\widetilde{\Phi}$ $\displaystyle=U_{\Phi}\Phi\,,$ $\displaystyle
U_{\Phi}$ $\displaystyle=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}\,,$ (98)

mass spectrums have correspondence between real and pure imaginary sources.
This rotation implies that the mass spectrum of pure imaginary source $j^{3}$
coincides with Eqs. 82 and 83. Let us confirm this correspondence explicitly
by compute the spectrum with the complex isospin breaking source
$H=e^{i\alpha_{H}}j^{3}T^{3}~{}(j^{3}>0)$. We again take the same vacuum
$\langle\sigma^{a}\rangle=\bar{\sigma}^{3}\delta_{a,3}$ and
$\langle\pi^{a}\rangle=\bar{\pi}^{3}\delta_{a,3}$ as before.

$\begin{split}U(\bar{\sigma}^{3},\bar{\pi}^{3})&=\frac{1}{2}(-\mu^{2}+c)(\bar{\sigma}^{3})^{2}-\frac{1}{2}(\mu^{2}+c)(\bar{\pi}^{3})^{2}+\frac{1}{4}\lambda[(\bar{\sigma}^{3})^{2}+(\bar{\pi}^{3})^{2}]^{2}\\\
&-j^{3}\cos\alpha_{H}\bar{\sigma}^{3}-j^{3}\sin\alpha_{H}\bar{\pi}^{3}\,,\end{split}$
(99)

This determines the VEVs from the stationary point condition of the potential
as follows:

$\displaystyle\bar{\sigma}^{3}$
$\displaystyle=\frac{j^{3}}{2c}\cos\alpha_{H}\,,$
$\displaystyle\bar{\pi}^{3}=\sqrt{\frac{c+\mu^{2}}{\lambda}}+\frac{j^{3}}{2(c+\mu^{2})}\sin\alpha_{H}\,.$
(100)

These VEVs give the mass spectrum of scalars up to $\mathcal{O}(j^{3})$ as
follows:

$\displaystyle(m_{\pi}^{2})_{33}$ $\displaystyle=2\lambda
f^{2}+\frac{3j^{3}}{f}\sin\alpha_{H}\,,$ $\displaystyle(m_{\pi}^{2})_{00}$
$\displaystyle=(m_{\sigma}^{2})_{11}=(m_{\sigma}^{2})_{22}=2c+\lambda_{2}f^{2}+\frac{j^{3}}{f}\left(1+\frac{\lambda_{2}}{\lambda}\right)\sin\alpha_{H}\,,$
(101) $\displaystyle(m_{\sigma}^{2})_{33}$
$\displaystyle=2c+\frac{j^{3}}{f}\sin\alpha_{H}\,,$
$\displaystyle(m_{\sigma}^{2})_{00}$
$\displaystyle=(m_{\pi}^{2})_{11}=(m_{\pi}^{2})_{22}=\frac{j^{3}}{f}\sin\alpha_{H}\,.$
(102)

When we take $\alpha_{H}=\pi/2$ (pure imaginary $j^{3}$ source), the mass
spectrum coincides with the mass spectrum with the real $j^{0}$ source (see
Eqs. 82 and 83).

###  A.4 Mixture

In the last of this section, we consider more generic source term. In general,
quark masses violate the isospin symmetry, and hence the source term in the
effective theory should reflect the violation. In particular, the isospin
violation in the mass spectrum is important for the pion mass difference in
our study. The diagonalized quark mass matrix corresponds to the following
source term in the LSM:

$\begin{split}H&=j^{0}T^{0}+j^{3}T^{3}\,,~{}~{}~{}(j^{0},j^{3}>0)\,.\end{split}$
(103)

In the presence of two sources $j^{0}$ and $j^{3}$, the vectorial flavor
symmetry $SU(2)_{V}$ is also broken. The vacuum choice should reflect the
breaking, and hence we assume the vacuum as follows:

$\begin{split}\langle\sigma^{a}\rangle=\bar{\sigma}^{0}\delta_{a,0}+\bar{\sigma}^{3}\delta_{a,3}\,,~{}~{}~{}\langle\pi^{a}\rangle=\bar{\pi}^{0}\delta_{a,0}+\bar{\pi}^{3}\delta_{a,3}\,.\end{split}$
(104)

In the following, we find analytic formulae of the mass spectra. We compute
the spectrum by treating either $j^{0}$ or $j^{3}$ as a dominant source and by
taking into account another as perturbation, and then we check the absence of
tachyonic modes.

First, we consider the case with $j^{0}\gg j^{3}$. From the dominant source
$j^{0}$, the VEV $\bar{\sigma}^{0}$ is given by Eq. 81, and mass spectrum is
given in Eqs. 82 and 83. The potential for $\sigma^{3}$ (mass:
$2c+\lambda_{2}f^{2}$ with $f^{2}=\lambda^{-1}(c+\mu^{2})$) is tilted by the
source term $j^{3}$, and therefore $\sigma^{3}$ gets its VEV as
$\bar{\sigma}^{3}\simeq j^{3}(2c+\lambda_{2}f^{2})^{-1}$ as far as the
potential is mainly lifted up by the mass term. Thus, the VEVs in the case
with $j^{0}\gg j^{3}$ are given by

$\displaystyle\bar{\sigma}^{0}$ $\displaystyle=f+\frac{j^{0}}{2\lambda
f^{2}}\,,$ $\displaystyle\bar{\sigma}^{3}$
$\displaystyle\simeq\frac{j^{3}}{2c+\lambda_{2}f^{2}}\,,$ (105)
$\displaystyle\bar{\pi}^{0}$ $\displaystyle=0\,,$ $\displaystyle\bar{\pi}^{3}$
$\displaystyle=0\,.$ (106)

The presence of $\bar{\sigma}^{3}$ leads to the modification of the mass
spectrum of the order $\mathcal{O}[(j^{3})^{2}]$, in particular the mass
difference between $\pi^{3}$ and $\pi^{1,2}$. $\pi^{0}$–$\pi^{3}$ mixing
arises from $\bar{\sigma}^{0}$ and $\bar{\sigma}^{3}$, and then the mass
eigenvalues are

$\displaystyle m_{\pi^{0}}^{2}$
$\displaystyle=2c+\frac{j^{0}}{f}-\frac{\lambda_{2}(j^{3})^{2}(2c-\lambda_{2}f^{2})}{2c(2c+\lambda_{2}f^{2})^{2}}\,,$
$\displaystyle m_{\pi^{1}}^{2}$ $\displaystyle=\frac{j^{0}}{f}\,,$
$\displaystyle m_{\pi^{3}}^{2}$
$\displaystyle=\frac{j^{0}}{f}-\frac{\lambda_{2}(j^{3})^{2}}{2c(2c+\lambda_{2}f^{2})}\,.$
(107)

Here, $\tilde{\pi}^{0}$ and $\tilde{\pi}^{3}$ are mass eigenvalues of the
$\pi^{0}$–$\pi^{3}$ system. The presence of $j^{3}$ leads to the mass
difference among $\pi^{a}\,(a=1,2,3)$ though they form the isospin triplet. We
note that the isospin violation in the pion mass arises of order
$\mathcal{O}[(j^{3})^{2}]$, not $\mathcal{O}[j^{3}]$. The condition that the
lightest pion has a positive mass squared puts the upper bound on $j^{3}$:
$j^{3}_{\mathrm{max}}\geq j^{3}$ where

$\begin{split}j^{3}_{\mathrm{max}}=\sqrt{\frac{j^{0}}{\lambda_{2}}\frac{2c}{f}\left(2c+\lambda_{2}f^{2}\right)}\,.\end{split}$
(108)

Next, we consider the case with $j^{3}\gg j^{0}$, i.e., we treat $j^{3}$ as
the dominant source. Eq. 94 gives the VEVs when the source term contains only
$j^{3}$. Both $\sigma^{3}$ and $\pi^{3}$ get their VEVs in this case since
there is the mass mixing among $\sigma$ and $\pi$. Therefore, both
$\sigma^{0}$ and $\pi^{0}$ get their VEVs via $j^{0}$ though $j^{0}$ makes the
potential only for $\sigma^{0}$ tilted. The mass matrix for $\sigma^{0}$ and
$\pi^{0}$ has components up to $\mathcal{O}[(j^{3})^{2}]$ as

$\displaystyle m_{\sigma}^{2}$
$\displaystyle=\frac{\lambda_{2}}{4c^{2}}(j^{3})^{2}\,,$ $\displaystyle
m_{\pi}^{2}$
$\displaystyle=2c+\lambda_{2}f^{2}-\frac{\lambda_{2}}{4c^{2}}(j^{3})^{2}\,,$
$\displaystyle m_{\pi\sigma}^{2}$
$\displaystyle=\frac{\lambda_{2}}{2c}fj^{3}\,,$ (109)

where we define $f^{2}=\lambda^{-1}(c+\mu^{2})$, again. The mass basis and the
mass eigenvalues are defined by

$\displaystyle\begin{pmatrix}s_{-}\\\ s_{+}\end{pmatrix}$
$\displaystyle=R^{T}\begin{pmatrix}\sigma^{0}\\\ \pi^{0}\end{pmatrix}\,,$
$\displaystyle\mathrm{diag}(m_{-}^{2},m_{+}^{2})$
$\displaystyle=R^{T}\begin{pmatrix}m_{\sigma}^{2}&m_{\pi\sigma}^{2}\\\
m_{\pi\sigma}^{2}&m_{\pi}^{2}\end{pmatrix}R^{T}\,,$ (110)

where the mixing matrix $R$ and the mass eigenvalues $m^{2}_{\pm}$ are

$\displaystyle R(\theta)$
$\displaystyle\equiv\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\,,$ $\displaystyle\tan 2\theta$
$\displaystyle=\frac{2m^{2}_{\pi\sigma}}{m_{\sigma}^{2}-m_{\pi}^{2}}\,,$ (111)
$\displaystyle m^{2}_{\pm}$
$\displaystyle=\frac{1}{2}\left[m_{\sigma}^{2}+m_{\pi}^{2}\pm\sqrt{(m_{\sigma}^{2}-m_{\pi}^{2})^{2}+4m_{\pi\sigma}^{2}}\right]\,.$
(112)

Both $s_{-}$ and $s_{+}$ get their VEVs, and hence we obtain the VEVs in the
original basis as follows.

$\begin{split}\bar{\sigma}^{0}=j^{0}\left(\frac{1}{m_{-}^{2}}\cos^{2}\theta-\frac{1}{m_{+}^{2}}\sin^{2}\theta\right)\,,\qquad\bar{\pi}^{0}=j^{0}\cos\theta\sin\theta\left(\frac{1}{m_{-}^{2}}+\frac{1}{m_{+}^{2}}\right)\,.\end{split}$
(113)

In the limit of small sources, we obtain the VEVs:

$\displaystyle\bar{\sigma}^{0}$ $\displaystyle\simeq
j^{0}\left[\frac{\lambda_{2}f^{2}}{2c(2c+\lambda_{2}f^{2})}+\frac{2c}{(j^{3})^{2}\lambda_{2}}\left(2c+\lambda_{2}f^{2}\right)\right]\,,$
$\displaystyle\bar{\sigma}^{3}$ $\displaystyle=\frac{j^{3}}{2c}\,,$ (114)
$\displaystyle\bar{\pi}^{0}$ $\displaystyle\simeq-\frac{j^{0}}{j^{3}}f\,,$
$\displaystyle\bar{\pi}^{3}$ $\displaystyle=f-\frac{(j^{3})^{2}}{8c^{2}f}\,.$
(115)

###  A.5 Higgs-Scalar Meson Mixing

In this subsection, we consider back-reaction from the VEV of the dark Higgs
$\phi_{D}$. The potential including the LSM field and $\phi_{D}$ is given by

$\begin{split}V(\Phi,\phi_{D})=V_{\phi_{D}}(\phi_{D})+V_{\mathrm{LSM}}(\Phi)+V_{\mathrm{mix}}(\Phi,\phi_{D})\,,\end{split}$
(116)

where each part of the potential is

$\displaystyle V_{\phi_{D}}(\phi_{D})$
$\displaystyle=-\mu^{2}_{\phi}|\phi_{D}|^{2}+\frac{\lambda_{\phi}}{4}|\phi_{D}|^{4}\,,$
(117) $\displaystyle V_{\mathrm{LSM}}(\Phi)$
$\displaystyle=-\mu^{2}\mathrm{tr}(\Phi^{\dagger}\Phi)+\lambda_{1}[\mathrm{tr}(\Phi^{\dagger}\Phi)]^{2}+\lambda_{2}\mathrm{tr}(\Phi^{\dagger}\Phi)^{2}$
$\displaystyle\qquad-c\left[\det\Phi+\det\Phi^{\dagger}\right]-\mathrm{tr}(j^{a}T^{a}\Phi+j^{a\ast}T^{a}\Phi^{\dagger})\,,$
(118)

where $j^{a}T^{a}$ is a diagonalized source term that originates from the dark
quark mass matrix. $V_{\mathrm{mix}}$ denotes the mixing term of the LSM field
and $\phi_{D}$

$\begin{split}V_{\mathrm{mix}}(\Phi,\phi_{D})&=\frac{1}{v}\mathrm{tr}(\Phi_{D}j_{Y}\Phi)+\mathrm{h.c.}\,.\end{split}$
(119)

Here, $v$ is the VEV of the dark Higgs that is determined only by the Higgs
potential, $v=\sqrt{2\mu_{\phi}^{2}/\lambda_{\phi}}$, and $j_{Y}$ and
$\Phi_{D}$ are the matrix-form coupling and the dark Higgs field defined by

$\begin{split}j_{Y}=\begin{pmatrix}0&j_{1}\\\
j_{2}^{\ast}&0\end{pmatrix}\,,\qquad\Phi_{D}=\begin{pmatrix}\phi_{D}&0\\\
0&\phi_{D}^{\ast}\end{pmatrix}\,.\end{split}$ (120)

$j_{Y}$ originates from the Yukawa coupling between the dark quarks and the
dark Higgs. $j_{1}$ and $j_{2}$ are, respectively, proportional to Yukawa
couplings $y_{1}$ and $y_{2}$ in Eq. 5, and $j_{a}\,(a=1,2)$ have mass
dimension of three. A combination $\Phi_{D}j_{Y}$ has only off-diagonal
entries, and gives a source term of the off-diagonal component of $\Phi$ after
$\phi_{D}$ gets its VEV. We define the Higgs-dependent source term $H$ as
follows:

$\begin{split}\mathcal{L}\supset\mathrm{tr}(j^{a}T^{a}\Phi+j^{a\ast}T^{a}\Phi^{\dagger})-\frac{1}{v}\mathrm{tr}(\Phi_{D}j_{Y}\Phi+j_{Y}^{\dagger}\Phi_{D}^{\dagger}\Phi^{\dagger})\equiv\mathrm{tr}(H\Phi+H^{\dagger}\Phi^{\dagger})\,,\end{split}$
(121)

where

$\begin{split}H(\phi_{D})\equiv
j^{a}T^{a}-\frac{1}{v}\Phi_{D}j_{Y}\,.\end{split}$ (122)

When the source term is hermitian, $H=H^{\dagger}$ (i.e., $j_{1}=j_{2}$), the
source matrix $H$ is factored out $\mathrm{tr}H(\Phi+\Phi^{\dagger})$. In this
case, the dark Higgs potential is tilted if the charged scalar meson obtains
its VEV in the charged basis. The Lagrangian possesses the parity invariance,
$\Phi\to\Phi^{\dagger}$. As shown in previous subsection, the dominant VEVs of
the LSM field are $\bar{\sigma}^{0}$ and $\bar{\pi}^{3}$ in mass basis when
parity is conserved. When the system possesses the $SU(2)_{V}$ flavor
symmetry, no $U(1)_{D}$-charged sigma meson gets its VEV in charge basis from
$\bar{\sigma}^{0}$. Due to the parity invariance, again, no $U(1)_{D}$-charged
sigma meson obtains its VEV in charge basis from $\bar{\pi}^{3}$ in mass
basis. Therefore, we conclude that we have no back-reaction from the LSM field
to the dark Higgs VEV when we have $P$-invariance and $SU(2)_{V}$ global
flavor symmetry.

We therefore consider the following cases that the source term is not
hermitian.

  * •

$H=\frac{J}{2}U_{A}^{2}$ with an $SU(2)$ matrix $U_{A}$

  * •

$H=JU_{R}^{\dagger}T^{3}U_{L}$ with $SU(2)$ matrices $U_{L},U_{R}$

  * •

$H=U_{R}^{\dagger}(J_{0}T^{0}+J_{3}T^{3})U_{L}$ with $SU(2)$ matrices
$U_{L},U_{R}$

Since the quark mass matrix is diagonalized before the $U(1)_{D}$ breaking,
the original source term $j^{a}T^{a}$ is diagonalized and we parametrize the
source term as follows.

$\begin{split}H(\phi_{D})=\frac{1}{2}\begin{pmatrix}j_{0}+j_{3}&-\frac{2}{v}\phi_{D}j_{1}\\\
-\frac{2}{v}\phi_{D}^{\ast}j_{2}^{\ast}&j_{0}-j_{3}\end{pmatrix}\,.\end{split}$
(123)

#### $H=\frac{J}{2}U_{A}^{2}$

We consider the first case that $H=\frac{J}{2}U_{A}^{2}$ with an $SU(2)$
matrix $U_{A}$ and a real-positive parameter $J>0$. The matrix $U_{A}$ is
parametrize phase $\alpha_{A}$ and angle $\theta_{A}$ as follows:

$\begin{split}U_{A}=\begin{pmatrix}\cos\theta_{A}&-e^{-i\alpha_{A}}\sin\theta_{A}\\\
e^{i\alpha_{A}}\sin\theta_{A}&\cos\theta_{A}\end{pmatrix}\,.\end{split}$ (124)

By comparing with Eq. 123, we determine parameters as follows.

$\begin{split}j_{0}&=J\cos 2\theta_{A}\,,\qquad j_{3}=0\,,\\\
\frac{\phi_{D}}{v}j_{1}&=\frac{J}{2}e^{-i\alpha_{A}}\sin
2\theta_{A}\,,\qquad\frac{\phi_{D}^{\ast}}{v}j_{2}^{\ast}=-\frac{J}{2}e^{i\alpha_{A}}\sin
2\theta_{A}\,.\end{split}$ (125)

The parameters depend on the Higgs field as follows.

$\begin{split}J=\sqrt{j_{0}^{2}-\frac{4|\phi_{D}|^{2}j_{1}j_{2}^{\ast}}{v^{2}}}\,,\quad\tan
2\theta_{A}=\frac{2}{j_{0}}\sqrt{-\frac{|\phi_{D}|^{2}j_{1}j_{2}^{\ast}}{v^{2}}}\,,\quad\phi_{D}j_{1}e^{i\alpha_{A}}=-\phi_{D}^{\ast}j_{2}e^{-i\alpha_{A}}\,.\end{split}$
(126)

By use of the $SU(2)_{L}\times SU(2)_{R}$ rotation,
$U_{L}=U_{R}^{\dagger}=U_{A}^{\dagger}$, we can diagonalize the source term.
$\Phi^{\prime}\equiv U_{A}\Phi U_{A}$ refers the LSM field in this basis. In
this basis, the source term is proportional to unity, and therefore
$\Phi^{\prime}$ obtains its VEV proportional to unity:
$\langle\Phi^{\prime}\rangle=\bar{\sigma}^{0}\mathbf{1}$, and hence the VEV of
$\Phi$ has an off-diagonal entry that provides $U(1)_{D}$ violation.

$\begin{split}\langle\Phi\rangle=U_{A}^{\dagger}\langle\Phi^{\prime}\rangle
U_{A}^{\dagger}=\bar{\sigma}^{0}\begin{pmatrix}\cos
2\theta_{A}&e^{-i\alpha_{A}}\sin 2\theta_{A}\\\ -e^{i\alpha_{A}}\sin
2\theta_{A}&\cos 2\theta_{A}\end{pmatrix}\,.\end{split}$ (127)

At the LSM potential minimum, we have a linear term of $J$ and this gives a
back-reaction potential to the dark Higgs potential.

$\begin{split}V_{\mathrm{LSM}}(\langle\Phi^{\prime}\rangle)+V_{\mathrm{mix}}(\langle\Phi^{\prime}\rangle,\phi_{D})=-\frac{(c+\mu^{2})^{2}}{4\lambda}-J(\phi_{D})\sqrt{\frac{c+\mu^{2}}{\lambda}}+O(J^{2})\,.\end{split}$
(128)

We define the back-reaction potential as $V_{\mathrm{BR}}(\phi_{D})$, and
expand $\phi_{D}$ around the VEV $v$, $\phi_{D}=v+h$.

$\begin{split}V_{\mathrm{BR}}(\phi_{D})=-fJ(\phi_{D})=-f\sqrt{j_{0}^{2}-4j_{1}j_{2}^{\ast}\left(1+\frac{h}{v}\right)^{2}}\,,\end{split}$
(129)

where $f=\sqrt{\lambda^{-1}(c+\mu^{2})}$ denotes the decay constant. Once we
assume that the shift of the VEV is negligible, the pion mass is given by the
source $J$ with $\phi_{D}=v$, $m_{\pi}^{2}\simeq J(v)/f$. This term tilts the
Higgs potential, and then the true Higgs VEV is shifted from $v$,
$v_{D}=v+\delta v$. We determine $\delta v$ in a perturbative manner of the
source. For simplicity, we take the sizes of $j_{1}$ and $j_{2}$ are the same,
but the signs are opposite to satisfy Eq. 126: $2j_{1}=-2j_{2}\equiv j>0$.
When $j_{0}\simeq j$, the potential is

$\begin{split}V_{\phi_{D}}(\phi_{D})+V_{\mathrm{BR}}(\phi_{D})=\frac{1}{2}m_{h}^{2}h^{2}-fj\sqrt{1+\left(1+\frac{h}{v}\right)^{2}}+\cdots\,,\end{split}$
(130)

and then the shift of the Higgs VEV is given by

$\begin{split}\delta
v=\frac{1}{\sqrt{2}}\frac{fj}{m_{h}^{2}v}+O(j^{2})\simeq\frac{1}{2}\frac{m_{q}\langle\bar{q}q\rangle}{m_{h}^{2}v}\,.\end{split}$
(131)

We use the relation between the source and the pion mass,
$m_{\pi}^{2}\simeq\sqrt{j_{0}^{2}+j^{2}}/f\simeq\sqrt{2}j/f$, and use the
Gell-Mann-Oaks-Renner (GOR) relation, $m_{\pi}^{2}f^{2}\simeq
m_{q}\langle\bar{q}q\rangle$ in the second equation.

When $j_{0}\gg j$, the potential is

$\begin{split}V_{\phi_{D}}(\phi_{D})+V_{\mathrm{BR}}(\phi_{D})&=\frac{1}{2}m_{h}^{2}h^{2}-fj_{0}\sqrt{1+\frac{j^{2}}{j_{0}^{2}}\left(1+\frac{h}{v}\right)^{2}}+\cdots\\\
&=\frac{1}{2}m_{h}^{2}h^{2}-fj_{0}-\frac{fj^{2}}{2j_{0}}\left(1+\frac{h}{v}\right)^{2}+\cdots\end{split}$
(132)

and then the shift of the Higgs VEV is given by

$\begin{split}\delta
v=\frac{f}{m_{h}^{2}v}\frac{j^{2}}{j_{0}}+O(j^{3})\simeq\frac{1}{m_{h}^{2}v}\frac{j^{2}}{m_{\pi}^{2}}\,,\end{split}$
(133)

In the second equation, we use
$m_{\pi}^{2}\simeq\sqrt{j_{0}^{2}+j^{2}}/f\simeq j_{0}/f$.

When $j_{0}\ll j$, the potential is

$\begin{split}V_{\phi_{D}}(\phi_{D})+V_{\mathrm{BR}}(\phi_{D})&=\frac{1}{2}m_{h}^{2}h^{2}-fj\sqrt{\frac{j_{0}^{2}}{j^{2}}+\left(1+\frac{h}{v}\right)^{2}}+\cdots\\\
&\simeq\frac{1}{2}m_{h}^{2}h^{2}-fj\left(1+\frac{h}{v}\right)+\cdots\end{split}$
(134)

and then the shift of the Higgs VEV is given by

$\begin{split}\delta
v=\frac{fj}{m_{h}^{2}v}+O(j^{2})\simeq\frac{m_{q}\langle\bar{q}q\rangle}{m_{h}^{2}v}\,.\end{split}$
(135)

We use $m_{\pi}^{2}\simeq\sqrt{j_{0}^{2}+j^{2}}/f\simeq j/f$ and the GOR
relation, again.

#### $H=JU_{R}^{\dagger}T^{3}U_{L}$

Next, we consider $H=JU_{R}^{\dagger}T^{3}U_{L}$ with $SU(2)$ matrices $U_{L}$
and $U_{R}$, and a real-positive parameter $J>0$,

$\begin{split}U_{L}=\begin{pmatrix}\cos\theta_{L}&-e^{-i\alpha_{L}}\sin\theta_{L}\\\
e^{i\alpha_{L}}\sin\theta_{L}&\cos\theta_{L}\end{pmatrix}\,,\quad
U_{R}=\begin{pmatrix}\cos\theta_{R}&-e^{-i\alpha_{R}}\sin\theta_{R}\\\
e^{i\alpha_{R}}\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}\,,\end{split}$ (136)

We take $\alpha_{L}=\alpha_{R}\equiv\alpha_{V}$ for simplicity,

$\begin{split}JU_{R}^{\dagger}T^{3}U_{L}=\frac{J}{2}\begin{pmatrix}\cos
2\theta_{V}&-e^{-i\alpha_{V}}\sin 2\theta_{V}\\\ -e^{i\alpha_{V}}\sin
2\theta_{V}&-\cos 2\theta_{V}\end{pmatrix}\,.\end{split}$ (137)

Here, we define $2\theta_{V}\equiv\theta_{L}+\theta_{R}$. By comparing with
Eq. 123, we determine parameters as follows.

$\begin{split}j_{0}&=0\,,\qquad j_{3}=J\cos 2\theta_{V}\,,\\\
\frac{\phi_{D}}{v}j_{1}&=\frac{J}{2}e^{-i\alpha_{V}}\sin
2\theta_{V}\,,\qquad\frac{\phi_{D}^{\ast}}{v}j_{2}^{\ast}=\frac{J}{2}e^{i\alpha_{V}}\sin
2\theta_{V}\,.\end{split}$ (138)

The parameters depend on the Higgs field as follows.

$\begin{split}J=\sqrt{j_{3}^{2}+\frac{4|\phi_{D}|^{2}j_{1}j_{2}^{\ast}}{v^{2}}}\,,\quad\tan
2\theta_{V}=\frac{2}{j_{3}}\sqrt{\frac{|\phi_{D}|^{2}j_{1}j_{2}^{\ast}}{v^{2}}}\,,\quad\phi_{D}j_{1}e^{i\alpha_{V}}=\phi_{D}^{\ast}j_{2}e^{-i\alpha_{V}}\,.\end{split}$
(139)

By use of the $SU(2)_{L}\times SU(2)_{R}$ rotation, $U_{L}$ and $U_{R}$, we
can diagonalize the source term. $\Phi^{\prime}\equiv U_{L}\Phi
U_{R}^{\dagger}$ refers the LSM field in this basis. In this basis, the source
term is proportional to $T^{3}$, and therefore $\Phi^{\prime}$ obtains its VEV
proportional to $T^{3}$: as shown in Section A.3, both $\sigma^{3}$ and
$\pi^{3}$ get their VEVs,
$\langle\Phi^{\prime}\rangle=(\bar{\sigma}^{3}+i\bar{\pi}^{3})T^{3}$, and
hence the VEV of $\Phi$ has an off-diagonal entry that provides $U(1)_{D}$
violation.

$\begin{split}\langle\Phi\rangle=U_{L}^{\dagger}\langle\Phi^{\prime}\rangle
U_{R}=(\bar{\sigma}^{3}+i\bar{\pi}^{3})\begin{pmatrix}\cos
2\theta_{V}&-e^{-i\alpha_{V}}\sin 2\theta_{V}\\\ -e^{i\alpha_{V}}\sin
2\theta_{V}&-\cos 2\theta_{V}\end{pmatrix}\,.\end{split}$ (140)

Since the VEV $\bar{\pi}^{3}=f$ does not contain the linear term of $J$, the
pion mass is proportional to $J^{2}$,

$\begin{split}m_{\pi}^{2}=\frac{\lambda_{2}}{2c(2c+\lambda_{2}f^{2})}J^{2}(v)\,,\end{split}$
(141)

where $f=\sqrt{\lambda^{-1}(c+\mu^{2})}$ denotes the decay constant, again. We
assume that the shift of the VEV is negligible, and then the pion mass is
given by the source $J$ with $\phi_{D}=v$. At the LSM potential minimum, we
have a quadratic term of $J$ and this gives a back-reaction potential to the
dark Higgs potential.

$\begin{split}V_{\mathrm{LSM}}(\langle\Phi^{\prime}\rangle)+V_{\mathrm{mix}}(\langle\Phi^{\prime}\rangle,\phi_{D})=-\frac{(c+\mu^{2})^{2}}{4\lambda}-\frac{J^{2}(\phi_{D})}{4c}+O(J^{3})\end{split}$
(142)

Similar to the previous case, we define the back-reaction potential as
$V_{\mathrm{BR}}(\phi_{D})$, and expand $\phi_{D}$ around the VEV $v$,
$\phi_{D}=v+h$.

$\begin{split}V_{\mathrm{BR}}(\phi_{D})=-\frac{J^{2}(\phi_{D})}{4c}=-\frac{1}{4c}\left[j_{3}^{2}+4j_{1}j_{2}^{\ast}\left(1+\frac{h}{v}\right)^{2}\right]\,,\end{split}$
(143)

This gives a shift of the Higgs field as follows up to the second order of
sources.

$\begin{split}\delta
v\simeq\frac{2j_{1}j_{2}^{\ast}}{m_{h}^{2}vc}\,.\end{split}$ (144)

#### $H=U_{R}^{\dagger}(J_{0}T^{0}+J_{3}T^{3})U_{L}$

We consider more generic source term that is characterized by $T^{0}$ and
$T^{3}$. This is a combination of the aforementioned two cases. We consider
$H=U_{R}^{\dagger}(J_{0}T^{0}+J_{3}T^{3})U_{L}$ with $SU(2)$ matrices $U_{L}$
and $U_{R}$, which we use in the previous case, and a real-positive parameter
$J_{0},J_{3}>0$. We take $\alpha_{L}=\alpha_{R}\equiv\alpha_{V}$ for
simplicity, again.

$\begin{split}U_{R}^{\dagger}(J_{0}T^{0}+J_{3}T^{3})U_{L}=\frac{1}{2}\begin{pmatrix}J_{0}\cos
2\theta_{A}+J_{3}\cos 2\theta_{V}&-e^{-i\alpha_{V}}(J_{0}\sin
2\theta_{A}+J_{3}\sin 2\theta_{V})\\\ -e^{i\alpha_{V}}(-J_{0}\sin
2\theta_{A}+J_{3}\sin 2\theta_{V})&J_{0}\cos 2\theta_{A}-J_{3}\cos
2\theta_{V}\end{pmatrix}\,.\end{split}$ (145)

Here we define
$2\theta_{V}=\theta_{L}+\theta_{R}\,,2\theta_{A}=\theta_{L}-\theta_{R}$. By
comparing with Eq. 123, we determine parameters as follows.

$\begin{split}j_{0}&=J_{0}\cos 2\theta_{V}\,,\qquad j_{3}=J_{3}\cos
2\theta_{V}\,,\\\
\frac{\phi_{D}}{v}j_{1}&=\frac{e^{-i\alpha_{V}}}{2}\left[J_{0}\sin
2\theta_{A}+J_{3}\sin
2\theta_{V}\right]\,,\qquad\frac{\phi_{D}^{\ast}}{v}j_{2}^{\ast}=\frac{e^{i\alpha_{V}}}{2}\left[-J_{0}\sin
2\theta_{A}+J_{3}\sin 2\theta_{V}\right]\,.\end{split}$ (146)

The parameters depend on the Higgs field as follows.

$\displaystyle J_{0}^{2}$
$\displaystyle=j_{0}^{2}+\frac{4}{v^{2}}(\phi_{D}j_{1}e^{i\alpha_{V}}-\phi_{D}^{\ast}j_{2}^{\ast}e^{-i\alpha_{V}})^{2}\,,$
$\displaystyle J_{3}^{2}$
$\displaystyle=j_{3}^{2}+\frac{4}{v^{2}}(\phi_{D}j_{1}e^{i\alpha_{V}}+\phi_{D}^{\ast}j_{2}^{\ast}e^{-i\alpha_{V}})^{2}$
$\displaystyle\tan 2\theta_{A}$
$\displaystyle=\frac{\phi_{D}j_{1}e^{i\alpha_{V}}-\phi_{D}^{\ast}j_{2}^{\ast}e^{-i\alpha_{V}}}{2j_{0}v}\,,$
$\displaystyle\tan 2\theta_{V}$
$\displaystyle=\frac{\phi_{D}j_{1}e^{i\alpha_{V}}+\phi_{D}^{\ast}j_{2}^{\ast}e^{-i\alpha_{V}}}{2j_{3}v}\,,$
$\displaystyle e^{i\alpha_{V}}$
$\displaystyle=\frac{\phi_{D}^{\ast}j_{2}^{\ast}}{|\phi_{D}j_{2}|}=\frac{\phi_{D}^{\ast}j_{1}^{\ast}}{|\phi_{D}j_{1}|}\,.$
(147)

By use of the $SU(2)_{L}\times SU(2)_{R}$ rotation, $U_{L}$ and $U_{R}$, we
can diagonalize the source term. $\Phi^{\prime}\equiv U_{L}\Phi
U_{R}^{\dagger}$ refers the LSM field in this basis. As with the previous
case, while the VEVs are diagonal in the mass basis, there are off-diagonal
entries in $\langle\Phi\rangle$.

It is challenging to find an analytic formula that is valid whole parameter
range. In Section A.4, we find the approximate solutions where each of sources
is larger than another and we treat the smaller one as the perturbation. When
$J_{0}$ dominates, the VEVs are given by Eqs. 105 and 106,

$\displaystyle\bar{\sigma}^{0}$ $\displaystyle=f+\frac{J_{0}}{2\lambda
f^{2}}\,,$ $\displaystyle\bar{\sigma}^{3}$
$\displaystyle\simeq\frac{J_{3}}{2c+\lambda_{2}f^{2}}\,,$ (148)
$\displaystyle\bar{\pi}^{0}$ $\displaystyle=0\,,$ $\displaystyle\bar{\pi}^{3}$
$\displaystyle=0\,.$ (149)

At the LSM potential minimum, we have a linear term of $J_{0}$ and this gives
a back-reaction potential to the dark Higgs potential.

$\begin{split}V_{\mathrm{LSM}}(\langle\Phi^{\prime}\rangle)+V_{\mathrm{mix}}(\langle\Phi^{\prime}\rangle,\phi_{D})\simeq-\frac{(c+\mu^{2})^{2}}{4\lambda}-J_{0}(\phi_{D})f\,,\end{split}$
(150)

This back-reaction potential gives the same results with the case of
$H=\frac{J}{2}U_{A}^{2}$.

On the other hand, when $J_{3}$ dominates, the VEVs are given by Eqs. 114 and
115,

$\displaystyle\bar{\sigma}^{0}$
$\displaystyle\simeq\frac{J_{0}2c}{(J_{3})^{2}\lambda_{2}}\left(2c+\lambda_{2}f^{2}\right)\,,$
$\displaystyle\bar{\sigma}^{3}$ $\displaystyle=\frac{J_{3}}{2c}\,,$ (151)
$\displaystyle\bar{\pi}^{0}$ $\displaystyle\simeq-\frac{J_{0}}{J_{3}}f\,,$
$\displaystyle\bar{\pi}^{3}$ $\displaystyle=f-\frac{(J_{3})^{2}}{8c^{2}f}\,.$
(152)

As with the case of $H=JU_{R}^{\dagger}T^{3}U_{L}$, at the LSM potential
minimum, we only have a quadratic term of $J_{3}$ and this gives a back-
reaction potential to the dark Higgs potential.

$\begin{split}V_{\mathrm{LSM}}(\langle\Phi^{\prime}\rangle)+V_{\mathrm{mix}}(\langle\Phi^{\prime}\rangle,h)\simeq-\frac{(c+\mu^{2})^{2}}{4\lambda}-\frac{J_{3}^{2}(\phi_{D})}{4c}\,.\end{split}$
(153)

This back-reaction potential gives the same results with the case of
$H=JR^{\dagger}T^{3}L$.

##  Appendix B Hadron Spectrum

In the previous appendix, we discuss the generic mass spectrum in the LSM
without corrections from QED. In this appendix, we incorporate the QED
correction into the mass spectrum. While the QED corrections usually arise as
the loop corrections, we treat them as effective interactions by regarding the
charge matrices as spurion fields.

###  B.1 Pion Mass Matrix

Let us consider meson spectrum that originates from the universal source
discussed in Section A.2. The meson masses are

$\begin{split}-\mathcal{L}&\supset\frac{1}{2}\sum_{a=1}^{3}m_{\pi}^{2}\pi^{a}\pi^{a}+\frac{1}{2}m_{\eta}^{2}(\pi^{0})^{2}\,,\\\
m_{\eta}^{2}&=2c+\frac{j^{0}}{f}\,,\qquad
m_{\pi}^{2}=\frac{j^{0}}{f}\,.\end{split}$ (154)

Here, we assume that the source $j^{0}$ is positive. When the quark mass basis
differs from its charge basis, we obtain the charge basis by the flavor
rotation. It is expected that the charge basis of mesons is identified with
the mass basis rotated by the same flavor rotation. Let $\Phi$ and
$\hat{\Phi}$ be the LSM fields in the mass basis and in the charge basis,
respectively.

$\begin{split}\hat{\Phi}=U_{L}\Phi
U_{R}^{\dagger}\supset\hat{\pi}^{a}T^{a}=\frac{1}{2}\begin{pmatrix}\hat{\eta}+\hat{\pi}&\hat{\pi}^{+}\\\
\hat{\pi}^{-}&\hat{\eta}-\hat{\pi}\end{pmatrix}\,,\end{split}$ (155)

where the flavor rotation matrices $U_{L}$ and $U_{R}$ that relate the quark
mass basis to its charge bases. In particular, when we take
$U_{L}=U_{R}=U_{V}$ and their phases to be $\alpha_{L}=\alpha_{R}=0$, the
charge neutral pion $\hat{\pi}$ is a linear combination of $\pi^{1}$ and
$\pi^{3}$,

$\begin{split}\begin{pmatrix}\hat{\pi}\\\
\mathrm{Re}(\hat{\pi}^{-})\end{pmatrix}=\begin{pmatrix}\cos 2\theta_{V}&-\sin
2\theta_{V}\\\ \sin 2\theta_{V}&\cos
2\theta_{V}\end{pmatrix}\begin{pmatrix}\pi^{3}\\\
\pi^{1}\end{pmatrix}\,,\end{split}$ (156)

and $\hat{\eta}=\pi^{0}$ and $\mathrm{\hat{\pi}^{-}}=\pi^{2}$. Since the
charge neutral pion has a coupling to dark photons via chiral anomaly, pions
can decay through the (off-shell) dark photons through the mixing.

Once we have $U(1)_{D}$ coupling, we can also add the charge masses for the
dark pions. In general, the masses in the mass basis are given by

$\begin{split}-\mathcal{L}&\supset
4\pi\alpha^{\prime}f_{\pi}^{2}\left[c_{1}\mathrm{tr}(\Pi^{\dagger}q_{L}\Pi
q_{R})+c_{2}\mathrm{tr}(\Pi^{\dagger}q_{L}^{2}\Pi)+c_{3}\mathrm{tr}(\Pi^{\dagger}\Pi
q_{R}^{2})+\mathrm{h.c.}\right]\,,\end{split}$ (157)

Here, $q_{L}$ and $q_{R}$ are the charge matrices that are defined in Eq. 14,
and $\Pi$ is the matrix-form pion fields in the mass basis.

$\begin{split}\Pi\equiv\frac{1}{2}\begin{pmatrix}\pi_{3}&\pi_{1}-i\pi_{2}\\\
\pi_{1}+i\pi_{2}&-\pi_{3}\end{pmatrix}\,.\end{split}$ (158)

Compared to the chiral perturbation theory, $\Pi^{\dagger}\Pi$ is not constant
in the LSM, and then we can add the second and the third terms. Indeed, these
terms are important for giving the correct charge masses. When $L$ and $R$
that relate the quark mass basis to its charge bases are unity, the mass basis
coincides with the charge basis. In this case, the charge matrices are the
same $q_{L}=q_{R}=\hat{q}=\mathrm{diag}(\frac{2}{3},-\frac{1}{3})$ and
$\pi_{1}$ and $\pi_{2}$ are the charged pions, then we obtain

$\displaystyle-\mathcal{L}\supset$
$\displaystyle\frac{c_{1}}{2}4\pi\alpha^{\prime}f_{\pi}^{2}\pi_{3}^{2}+\frac{1}{2}(c_{2}+c_{3})4\pi\alpha^{\prime}f_{\pi}^{2}(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2})=\frac{c_{Q}^{\pi^{\prime}}}{2}4\pi\alpha^{\prime}f_{\pi}^{2}(\pi_{1}^{2}+\pi_{2}^{2})\,,$
(159)

In the last line, we determine the coefficients in order to correctly give
charge masses to the charged pions, $c_{2}=c_{3}=-2c_{1}\equiv
c_{Q}^{\pi^{\prime}}/2$.

Next, we take into account isospin breaking terms. For simplicity, we take the
flavor rotation matrices $U_{L}=U_{R}\equiv U_{V}$, and then the charge
matrices are the same
$q_{L}=q_{R}=U_{V}\mathrm{diag}(\frac{2}{3},-\frac{1}{3})U_{V}^{\dagger}$,
where

$\begin{split}U_{V}=\begin{pmatrix}\cos\theta_{V}&-e^{-i\alpha_{V}}\sin\theta_{V}\\\
e^{i\alpha_{V}}\sin\theta_{V}&\cos\theta_{V}\end{pmatrix}\,.\end{split}$ (160)

In this case, the mass terms of pions are given by

$\begin{split}-\mathcal{L}\supset&-\frac{m_{D}^{2}}{2}(\sin
2\theta_{V}\cos\alpha_{V}\pi_{1}+\sin 2\theta_{V}\sin\alpha_{V}\pi_{2}-\cos
2\theta_{V}\pi_{3})^{2}\\\
&+\frac{m_{D}^{2}}{2}(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2})+\frac{m_{\pi^{3}}^{2}}{2}\pi_{3}^{2}+\frac{m_{\pi^{1}}^{2}}{2}(\pi_{1}^{2}+\pi_{2}^{2})\,,\end{split}$
(161)

where we define
$m_{D}^{2}=4\pi\alpha^{\prime}c_{Q}^{\pi^{\prime}}f_{\pi}^{2}$.
$m_{\pi_{3}}^{2}$ and $m_{\pi_{1,2}}^{2}$ are defined in Eq. 107, which
include the isospin violation. The mass eigenvalues are

$\begin{split}\begin{pmatrix}\pi_{1}^{\prime}\\\ \pi_{2}^{\prime}\\\
\pi_{3}^{\prime}\end{pmatrix}\equiv\begin{pmatrix}c_{\beta_{V}}&&s_{\beta_{V}}\\\
&1&\\\
-s_{\beta_{V}}&&c_{\beta_{V}}\end{pmatrix}\begin{pmatrix}c_{\alpha_{V}}&s_{\alpha_{V}}&\\\
-s_{\alpha_{V}}&c_{\alpha_{V}}&\\\ &&1\end{pmatrix}\begin{pmatrix}\pi_{1}\\\
\pi_{2}\\\ \pi_{3}\end{pmatrix}\,,\end{split}$ (162)

where $c_{\alpha}=\cos\alpha$ and $s_{\alpha}=\sin\alpha$, and the mixing
angle $\beta_{V}$ is defined by

$\begin{split}\tan 2\beta_{V}=\frac{m_{D}^{2}\sin
4\theta_{V}}{m_{\pi^{3}}^{2}-m_{\pi^{1}}^{2}-m_{D}^{2}\cos
4\theta_{V}}\,.\end{split}$ (163)

When there is no $U(1)_{D}$ correction, $m_{D}=0$, $\beta_{V}$ is zero. If
there is no isospin violation in mass basis in advance,
$m_{\pi^{3}}^{2}=m_{\pi^{1}}^{2}$, the degeneracy of the dark pions is
resolved only by the $U(1)_{D}$ correction. This means that the charge basis
and the mass basis coincide with each other, and indeed
$\beta_{V}=-2\theta_{V}$ as $m_{\pi^{3}}^{2}=m_{\pi^{1}}^{2}$. The mass
eigenvalues are

$\displaystyle
m_{\pi^{1}}^{2}+m_{D}^{2}\,,\frac{1}{2}\left(m_{\pi^{3}}^{2}+m_{\pi^{1}}^{2}+m_{D}^{2}\pm\sqrt{(m_{\pi^{3}}^{2}-m_{\pi^{1}}^{2})^{2}+2\cos
4\theta_{V}m_{D}^{2}(m_{\pi^{3}}^{2}-m_{\pi^{1}}^{2})+m_{D}^{4}}\right)\,.$
(164)

In this mass basis, $U(1)_{D}$ gauge interaction of pions is given by

$\begin{split}\mathcal{L}_{\mathrm{LSM}}&\supset
e^{\prime}A^{\prime\mu}\left[\cos(2\theta_{V}+\beta)(\pi_{1}^{\prime}\partial_{\mu}\pi_{2}^{\prime}-\pi_{2}^{\prime}\partial_{\mu}\pi_{1}^{\prime})+\sin(2\theta_{V}+\beta)(\pi_{2}^{\prime}\partial_{\mu}\pi_{3}^{\prime}-\pi_{3}^{\prime}\partial_{\mu}\pi_{2}^{\prime})\right]\,.\end{split}$
(165)

###  B.2 Nucleon Masses

We now consider the mass spectrum of the light dark nucleons. The dominant
nucleon masses arise from the interaction with the LSM fields, while the
global symmetry allows the vector-like mass term that originates from the
quark mass term.

$\begin{split}\mathcal{L}_{B}&\supset-(\xi\overline{N}M_{Q^{\prime}}^{\mathrm{diag}}N+g\overline{N}\Phi^{\dagger}N+\mathrm{h.c.})\,.\end{split}$
(166)

Here, we construct the hadron effective theory after we rotate the quark basis
into their mass basis, and hence the vector-like mass term is proportional to
the diagonalized quark mass matrix. Even though the vector-like mass term is
not dominant contribution, it is important for the nucleon mass difference.
The large-$N_{C}$ scaling of a coupling $g$ is $g=g_{\pi NN}\sim
4\pi\sqrt{N_{C}}$. We assume that the LSM field obtains its VEV to be
diagonalized, $\Phi=fT^{0}+\delta T^{3}$. The nucleon masses are given by Eq.
24. We can also include $U(1)_{D}$ correction to the nucleon mass,

$\begin{split}\mathcal{L}_{B}\supset-\frac{\alpha^{\prime}}{4\pi}\left[\overline{N}(q_{R}^{N})^{{\dagger}2}\Phi^{\dagger}N+\overline{N}\Phi^{\dagger}(q_{L}^{N})^{2}N+\overline{N}(q_{R}^{N})^{{\dagger}}\Phi^{\dagger}q_{L}^{N}N+\mathrm{h.c.}\right]\,.\end{split}$
(167)

The charge matrices are defined in the text, Eq. 27.

We discuss the nucleon mass difference in the SM, proton-neutron mass
difference, by use of the LSM. In the SM, $N_{1}$ and $N_{2}$ correspond to
neutron and proton, respectively. First, we consider the mass difference
originates from isospin-violation by quark masses. In the SM, the mass
difference of quarks is smaller than their sum, hence we use VEVs given by Eq.
105.

$\displaystyle m_{n}-m_{p}-\delta M^{\gamma}$
$\displaystyle=-g\delta=\frac{f_{\pi}^{2}}{2c+\lambda_{2}f_{\pi}^{2}}\frac{m_{n}+m_{p}}{m_{u}+m_{d}}\frac{m_{\pi}^{2}}{f_{\pi}^{2}}(m_{d}-m_{u})$
$\displaystyle\simeq 3.68(m_{d}-m_{u})\,.$ (168)

In the second equality, we rewrite the mass difference in terms of quark
masses and hadron masses by use of the following relations.

$\begin{split}\frac{j^{3}}{j^{0}}=\frac{m_{u}-m_{d}}{m_{u}+m_{d}}\,,\qquad
g=\frac{m_{p}+m_{n}}{f_{\pi}}\,,\qquad j^{0}=m_{\pi}^{2}f_{\pi}\,.\end{split}$
(169)

Since the source terms originate from the quark mass terms, the ratio
$j^{3}/j^{0}$ is determined by the quark mass difference and their sum. The
second relation is obtained from the sum of nucleon masses as quark masses are
negligible. Numerically, we take the SM values of parameters [84]:
$m_{u}=2.16\,\mathrm{MeV}$, $m_{d}=4.67\,\mathrm{MeV}$,
$m_{p}=938.27\,\mathrm{MeV}$, $m_{n}=939.57\,\mathrm{MeV}$,
$f_{\pi}=92\,\mathrm{MeV}$, $\lambda_{2}\simeq(4\pi)^{2}/3$, and $2c\simeq
m_{\eta^{\prime}}^{2}\simeq(957.78\,\mathrm{MeV})^{2}$. The mass difference in
the LSM is not so far from the lattice QCD average of the difference is (See a
review [159])

$\displaystyle m_{n}-m_{p}-\delta M^{\gamma}$ $\displaystyle\simeq
0.95(8)(6)\times(m_{d}-m_{u})~{}[\mu=2~{}\mathrm{GeV}]\,.$ (170)

The electromagnetic correction is [160]

$\displaystyle\delta M^{\gamma}$ $\displaystyle\simeq
1.30(03)(47)~{}[\mathrm{MeV}]\,.$ (171)

##  Appendix C Fitting

In this appendix, we demonstrate the validity of our approximation formulae
Eq. 47 and Eq. 49 by applying the formulae to the specific models. As we
discussed in the text, as for the LHC lifetime frontier, FASER and MATHUSLA,
we apply our formulae to the inelastic DM model [140]. On the other hand, as
for the fixed-target experiments, SeaQuest and E137, we apply them to the dark
vector meson search [158] for SeaQuest, and to the dark photon search [3].

###  C.1 MATHUSLA and FASER

Figure 7:  Our fitting with the results of Ref. [140] for MATHUSLA (left) and
FASER (right). We take the parameters to be $m_{A^{\prime}}=3m_{\chi}$ and
$\alpha^{\prime}=0.1$, which are the same as those in Ref. [140]. The mass
difference parameter, $\Delta$, is taken to be $\Delta=0.1$ (solid),
$\Delta=0.05$ (dashed), and $\Delta=0.03$ (dotted). The green (blue) lines
show the future sensitivities at MATHUSLA (FASER) taken from Ref. [140]. The
light green (cyan) points in the plot are the reference points of MATHUSLA
(FASER) that we have used in the text. The black lines show our approximate
sensitivity given by Eq. 47 and Eq. 49.

We show our fitting in Fig. 7 by use of our approximation formulae, Eq. 47 and
Eq. 49. The MATHUSLA sensitivity plots are shown as green lines in the left
panel, while the FASER plots are shown as blue lines in the right panel, which
are read from Ref. [140]. We take the dark photon mass to be
$m_{A^{\prime}}=3m_{\chi}$ with the dark matter mass $m_{\chi}$ and the
$U(1)_{D}$ coupling to be $\alpha^{\prime}=0.1$ that are same as the
parameters taken in the literature. The on-shell $A^{\prime}$ is produced via
the SM meson decay, the Bremsstrahlung, and Drell-Yan at the LHC, and its
decay $A^{\prime}\to\chi_{1}\chi_{2}$ produces the long-lived particle
$\chi_{2}$. The mass difference of the inelastic dark matter, defined by
$\Delta\equiv(m_{\chi_{2}}-m_{\chi})/m_{\chi}$ with the masses of the excited
state $m_{\chi_{2}}$, is taken to be $\Delta=0.1$ (solid), $\Delta=0.05$
(dashed), and $\Delta=0.03$ (dotted). We assume that the long-lived particles
are produced only via Drell-Yan production at the LHC. We do not take into
account $Z$-resonance for the production, and hence the peaky behavior around
$m_{\chi}\simeq 30\,\mathrm{GeV}$ (corresponding to $m_{A^{\prime}}\simeq
m_{Z}$) are not reproduced in these figures. We take into account the
difference of $m_{A^{\prime}}$-dependence of $N_{A^{\prime}}$ of $\chi_{2}$,
which are scaled as $\epsilon^{2}m_{A^{\prime}}^{-4}$ in forward direction and
$\epsilon^{2}m_{A^{\prime}}^{-2}$ in total, as we mentioned in the text.

We take into account the efficiency due to the energy threshold as discussed
in the text, $A_{\mathrm{eff}}=A^{\mathrm{geo}}A^{\mathrm{prod}}$.
$p_{\mathrm{min}}$ is the larger momentum between $p_{\mathrm{geo}}$ and
$p_{\mathrm{thr}}$; $p_{\mathrm{thr}}=1\,\mathrm{TeV}(0.1/\Delta)$ for FASER
and $p_{\mathrm{thr}}=12\,\mathrm{GeV}(0.1/\Delta)$ for MATHUSLA, while
$p_{\mathrm{geo}}=\sqrt{m_{A^{\prime}}^{2}-4m_{\chi}^{2}}/2\theta$ with a
typical angle $\theta$ where the detector is located: $\theta\simeq 0.5$ for
MATHUSLA and $\theta\lesssim 2\times 10^{-3}$ for FASER. Meanwhile, we take
$p_{\mathrm{max}}$ to be the same as what we have taken in the text:
$p_{\mathrm{max}}=100\,\mathrm{GeV}$ for MATHUSLA and
$p_{\mathrm{max}}=7\,\mathrm{TeV}$ for FASER.

To determine the reference values in Eqs. 47 and 49, we take
$m_{\chi}=2.0\,\mathrm{GeV}$, and we read the boundary values of $\epsilon$ at
$m_{\chi}=2.0\,\mathrm{GeV}$ from the sensitivity plots with $\Delta=0.1$,
which are shown as green and cyan dots in the figure. In the case of MATHUSLA,
we obtain the reference values defined in Eqs. 47 and 49 from this calibration
as follows.

$\begin{split}\frac{r_{\mathrm{min}}}{(c\tau_{\chi}p_{\mathrm{max}}m_{\chi}^{-1})_{0}}\simeq
16.2\,,\qquad\frac{r_{\mathrm{max}}-r_{\mathrm{min}}}{(c\tau_{\chi}p_{\mathrm{min}}m_{\chi}^{-1})_{0}}\simeq
0.083\,.\end{split}$ (172)

Here,
$(c\tau_{\chi}p_{\mathrm{max}}m_{\chi}^{-1})_{0}\,,(c\tau_{\chi}p_{\mathrm{min}}m_{\chi}^{-1})_{0}$
are the reference values $d_{0}(p_{\mathrm{max}})$ and
$d_{0}(p_{\mathrm{min}})$ in the formulae, respectively. On the other hand, in
the case of FASER, we obtain the reference values from this calibration as
follows.

$\begin{split}\frac{r_{\mathrm{min}}}{(c\tau_{\chi}p_{\mathrm{max}}m_{\chi}^{-1})_{0}}\simeq
19.3\,,\qquad\frac{r_{\mathrm{max}}-r_{\mathrm{min}}}{(c\tau_{\chi}p_{\mathrm{min}}m_{\chi}^{-1})_{0}}\simeq
0.0001\,.\end{split}$ (173)

We show the boundaries with our approximation formulae as black lines in the
figure. In both panels, their line types are the same as the sensitivity
plots: $\Delta=0.1$ (solid), $\Delta=0.05$ (dashed), and $\Delta=0.03$
(dotted). We just use the reference points on the sensitivity plot for
$\Delta=0.1$, the other shape of the sensitivity plots is almost reproduced by
our formulae even for other choice of $\Delta$. Since we assume that
$\chi_{2}$ is produced only via the Drell-Yan process, the fitting of lower
sensitivity curves gets worse for $m_{\chi}\simeq 1\,\mathrm{GeV}$, where the
Bremsstrahlung process would be more important.

###  C.2 SeaQuest

Figure 8:  Our fitting with the results of Ref. [158] for SeaQuest: for fixed
$m_{\pi}/f_{\pi}=3$ (left) $m_{\pi}/f_{\pi}=4\pi$ (right). The magenta lines
show their results, and black and red lines show our fitting: magenta thick
lines correspond to the sensitivity lines for visible decay of dark vector
meson, 3-body decay ($V\to\tilde{\pi}\ell^{+}\ell^{-}$) and 2-body decay
($V\to\ell^{+}\ell^{-}$), while magenta thin lines correspond to the
sensitivity lines only for 2-body decay. We take the dark pion mass to be
$m_{\pi}=m_{A^{\prime}}/3$ and the mass of the dark vector meson to be
$m_{V}=0.6m_{A^{\prime}}$. We take the reference points (magenta dots in the
right panel) for our fitting.

We show the validity of our fitting for the SeaQuest sensitivities on the
decay of dark vector mesons in Fig. 8 by applying our approximation formulae,
Eqs. 58 and 59 to the SIMP model. We use the sensitivity plots of the dark
vector mesons decay discussed in Ref. [158]. In the literature, the dark
photon is heavier than the dark hadrons. The on-shell $A^{\prime}$ is produced
through the SM meson decay, Bremsstrahlung, and Drell-Yan at SeaQuest. The
produced dark photon mainly decays into the dark mesons, in particular
$A^{\prime}\to\tilde{\pi}\tilde{\pi}$ and $A^{\prime}\to\tilde{\pi}V$ where
$V$ and $\tilde{\pi}$ collectively denotes vector mesons and pseudo-scalar
mesons, respectively.

The $U(1)_{D}$-neutral dark vector mesons mix with dark photons as with the
$\rho^{0}$-$\gamma$ mixing in the SM, and then the neutral mesons decay into
the SM lepton pair, $\ell^{+}\ell^{-}$, through the dark photon mixing. On the
other hand, the $U(1)_{D}$-charged dark vector mesons decay into the SM
leptons through the off-shell dark photon by emitting $U(1)_{D}$-charged dark
pions. The decay length of the three-body decay tends to be longer than that
of the two-body decay due to three-body phase space factor.

Magenta lines in Fig. 8 show sensitivities of the dark vector meson $V$ decay
at SeaQuest: thick lines correspond to 3-body decay
($V\to\tilde{\pi}\ell^{+}\ell^{-}$) and 2-body decay ($V\to\ell^{+}\ell^{-}$),
while thin lines correspond to the sensitivity lines only for 2-body decay.
The thick sensitivity lines are composed of two parts: upper parts from the
three-body decay $\Gamma(V\to\pi\ell^{+}\ell)$, while lower parts from the
two-body decay $\Gamma(V\to\ell^{+}\ell)$. It is assumed the ratio of the dark
pion mass and the pion decay constant to be $m_{\pi}/f_{\pi}=3$ ($=4\pi$) in
the left (right) panel. In both panels, the dark pion mass is
$m_{\pi}=m_{A^{\prime}}/3$ and the mass of the dark vector meson is
$m_{V}=0.6m_{A^{\prime}}$. The produced numbers of $V$ depend on
$m_{\pi}/f_{\pi}$, and increases as $m_{\pi}/f_{\pi}$ gets larger. In
particular, the branching fraction of $V$ is almost unity when we take
$m_{\pi}/f_{\pi}=4\pi$.

The reference values in Eqs. 58 and 59 are determined by the point at
$m_{A^{\prime}}=0.1\,\mathrm{GeV}$ (magenta dots in the right panel). We use
the plot for $m_{\pi}/f_{\pi}=4\pi$ for the calibration, and hence the
branching fraction to the vector meson is assumed to be unity. The lower
boundary of the three-body decay is hidden since the boundary is overlapped
with the sensitivity to the two-body decay. Therefore, we take the reference
points from the upper boundary of the three-body decay and the lower boundary
of the two-body decay. We obtain the reference values as follows.

$\begin{split}\frac{r_{\mathrm{min}}}{[c\tau_{V}(V\to\tilde{\pi}\ell^{+}\ell^{-})p_{\mathrm{max}}m_{V}^{-1}]_{0}}\simeq
34.6\,,\qquad\frac{r_{\mathrm{max}}-r_{\mathrm{min}}}{[c\tau_{V}(V\to\ell^{+}\ell^{-})p_{\mathrm{min}}m_{V}^{-1}]_{0}}\simeq
0.00001\,.\end{split}$ (174)

Here,
$[c\tau_{V}(V\to\tilde{\pi}\ell^{+}\ell^{-})p_{\mathrm{max}}m_{V}^{-1}]_{0}$
and $[c\tau_{V}(V\to\ell^{+}\ell^{-})p_{\mathrm{min}}m_{V}^{-1}]_{0}$
correspond to the boosted decay lengths referred as to
$d_{0}(p_{\mathrm{max}})$ and $d_{0}(p_{\mathrm{min}})$ in the text,
respectively. The black lines show the boundaries computed by our
approximation formulae, and the boundaries well coincide with the sensitivity
computed by Ref. [158]. Since we do not include the dark photon production
through Drell-Yan, the boundaries are cut at a few GeV. The black-dashed lines
in the left panel also show the boundaries with our formulae. Although the
upper boundary of the sensitivity is not shown below 1 GeV in the literature,
it is expected that our formulae well explain its extrapolation. By use of our
formulae, we also evaluate the upper-boundary of the two-body decay and lower
boundary of the three-body decay, which as shown as red-dashed lines in Fig.
8. The sensitivity of the upper boundaries is also computed by Ref. [158],
which is shown as the thin-magenta lines, and our formula roughly reproduce
their result.

###  C.3 E137

Figure 9:  Our fitting with the results of Ref. [3] for E137. Dashed light
green line shows the boundary of the existing constraint by E137 which is
taken from Ref. [3], while the black solid lines show our fit. Green points on
the boundaries are our reference points for the fitting.

As for the existing constraint on the dark pion decay, we use the constraint
on the visible dark photon decay by E137. We take the reference point at
$m_{A^{\prime}}=0.1\,\mathrm{GeV}$, and read out the kinetic mixing $\epsilon$
on the upper and lower boundaries of the sensitivity area. Fig. 9 shows our
fitting of dark photon visible decay with the constraint by Ref. [3]. The
black lines show our fitting, dashed light green line shows the boundary of
the existing constraint by Ref. [3], and the green points are our reference
points. At these points, we get the reference values as follows.

$\begin{split}\frac{r_{\mathrm{min}}}{[c\tau_{A^{\prime}}(A^{\prime}\to\ell^{+}\ell^{-})p_{\mathrm{max}}m_{A^{\prime}}^{-1}]_{0}}\simeq
8.7\,,\qquad\frac{r_{\mathrm{max}}-r_{\mathrm{min}}}{[c\tau_{A^{\prime}}(A^{\prime}\to\ell^{+}\ell^{-})p_{\mathrm{min}}m_{A^{\prime}}^{-1}]_{0}}\simeq
0.0093\,.\end{split}$ (175)

Our fitting works well for the dark photon visible decay search at E137.

##  Appendix D Production and Geometric Efficiency

Figure 10:  Sensitivity plots for the dark nucleon and dark pion searches at
LHC lifetime frontier, including variation of multiplicity of long-lived
particles: FASER (left) and MATHUSLA (right). The dark nucleon searches: FASER
(blue solid-lines for $m_{N^{\prime}}=8.5\,\mathrm{GeV}$, blue dashed-lines
for $m_{N^{\prime}}=1.1\,\mathrm{GeV}$) and MATHUSLA (green solid-lines for
$m_{N^{\prime}}=8.5\,\mathrm{GeV}$, green dashed-lines for
$m_{N^{\prime}}=1.1\,\mathrm{GeV}$). The dark pion searches: FASER (cyan
solid-lines for $f_{\pi^{\prime}}=0.8\,\mathrm{GeV}$, cyan dashed-lines for
$f_{\pi^{\prime}}=0.1\,\mathrm{GeV}$) and MATHUSLA (yellow solid-lines for
$f_{\pi^{\prime}}=0.8\,\mathrm{GeV}$, yellow dashed-lines for
$f_{\pi^{\prime}}=0.1\,\mathrm{GeV}$). The thick lines are the same as shown
in Fig. 4, while the corresponding thin lines show the sensitivity curves when
the multiplicities of hadrons are similar, in particular $n_{N_{1}}=0.1$ and
$n_{\pi^{\prime}}=0.2$. As in Fig. 4, we also show the existence constraints
and the future sensitivities to the visible dark photon decay (the top and
left-bottom shaded areas and the future sensitivity curves from Belle-II,
LHCb, SHiP, and SeaQuest).

We use the produced number of the long-lived particles given by Eqs. 45 and 54
for the visible decay search at the LHC lifetime frontier. Here, we naïvely
assume that dark nucleons and dark pions are produced when off-shell dark
photons with a fictitious mass of the dark dynamical scale are produced. This
produced number depends on multiplicity of the produced particles, and we
assume $n_{p}=0.04$ and $n_{\pi^{\prime}}=2.0$ in the text, which are the
analogy of the SM particle production at the $J/\psi$ threshold [149]. Since
pions are expected to be more produced than nucleons when they are much
lighter than nucleons, the multiplicity of pions is larger than that of
nucleons In our study, the mass difference of dark nucleons and dark pions can
be smaller than that of the SM hadrons. Therefore, we change the multiplicity
of hadrons: in particular, to be the similar multiplicity, $n_{N_{1}}=0.1$ and
$n_{\pi^{\prime}}=0.2$. We show sensitivity plots in Fig. 10 when the dark
sector hadrons are produced on an almost equal footing.

In this figure, we take $\sin\theta_{V}=5\times 10^{-3}$ and $\Delta_{N}=0.3$
for the visible decay of dark nucleons, while
$m_{\pi^{\prime}}/m_{A^{\prime}}=1.9$ for the visible decay of dark pions. We
commonly use $U(1)_{D}$ coupling to be $\alpha^{\prime}=0.05$
($\alpha^{\prime}=7\times 10^{-3}$) for $n_{g^{\prime}}=1$
($n_{g^{\prime}}=8$). Fig. 10 show the changes of the sensitivity plots of
FASER (left panel) and MATHUSLA (right panel), and both panels include the
visible decays from dark nucleons and dark pions. The color codes are the same
as the previous figures for FASER (MATHUSLA): the blue-solid (green-solid)
lines for $m_{N^{\prime}}=8.5\,\mathrm{GeV}$ and the cyan-solid (yellow-solid)
lines for $f_{\pi^{\prime}}=0.8\,\mathrm{GeV}$, while the blue-dashed (green-
dashed) lines for $m_{N^{\prime}}=1.1\,\mathrm{GeV}$ and the cyan-dashed
(yellow-dashed) lines for $f_{\pi^{\prime}}=0.1\,\mathrm{GeV}$. Thick lines
show the sensitivity curves with Eqs. 45 and 54, while thin lines show the
sensitivity curves with different choice of multiplicities.

Figure 11:  Dark nucleon searches at the LHC lifetime frontier on
$\epsilon$-$m_{A^{\prime}}$ plane: FASER (left columns) and MATHUSLA (right
columns). We choose different values from the parameters taken in Fig. 4: we
take the mass difference to be $\Delta_{N}=0.1$ (top) and 0.3 (bottom), and
the mixing angle to be $\sin\theta_{V}=5\times 10^{-3}$ and
$\sin\theta_{V}=5\times 10^{-2}$. The color codes and the existing constraints
are the same as in Fig. 4.

We show the parameter dependences of dark nucleon sensitivities at the LHC
lifetime frontier in Fig. 11. Existing constraints by dark photon searches and
the direct detection are the same as in Fig. 4. For the larger $\Delta_{N}$,
the decay with on-shell dark photon easily opens, and therefore the
sensitivity to the three-body decay of the nucleons gets narrow in
$m_{A^{\prime}}$ axis. However, it is sensitive to the smaller $\epsilon$
since the lifetime gets shorter. Due to the same reason as what we explained
in the text, the sensitivity range gets shrink once we take tha larger
couplings or the larger DM mass. Fig. 11 shows that some sensitivity curves
disappear when $\sin\theta_{V}=5\times 10^{-2}$ and $\Delta_{N}=0.3$.

Figure 12:  Sensitivity plots and exclusion plots for dark pion searches,
including variation of productions at SeaQuest and E137. We take
$f_{\pi^{\prime}}=0.8\,\mathrm{GeV}$ (left) and
$f_{\pi^{\prime}}=0.1\,\mathrm{GeV}$ (middle) for the sensitivity of SeaQuest,
while $f_{\pi^{\prime}}=0.1\,\mathrm{GeV}$ for the exclusion limit from E137
(right). We take the different values of $m_{\pi^{\prime}}/m_{A^{\prime}}$:
$m_{\pi^{\prime}}/m_{A^{\prime}}=1.9$ , $m_{\pi^{\prime}}/m_{A^{\prime}}=1.6$
, and $m_{\pi^{\prime}}/m_{A^{\prime}}=1.3$ from left-bottom to right-top. For
the SeaQuest plots, we take the production part to be ten times smaller than
that used in the text, which is illustrated as thin lines in the figure. For
the E137 plot, we again take the production part to be ten times smaller than
that used in the text, which is illustrated as dashed-black lines in the
figure.

As for the dark pions at SeaQuest, they are produced via a virtual dark photon
with the fictitious mass of order of the SM $\rho$ mesons, and we use Eq. 57
to estimate their produced number. The proton Bremsstrahlung dominates their
production, and the main contribution comes from the $\rho$ meson resonances
due to the broad width of $\rho$ mesons. As for the dark photon searches at
the SeaQuest [21], as we discussed in the text, the dark photons from meson
decays have less energetic and are produced with a larger angle off from beam-
axis in comparison with the dark photon from proton bremsstrahlung. Therefore,
the geometric efficiency at SeaQuest decreases for the dark photons from meson
decay.

Similarly to the dark photons, the searches for dark pions can be affected by
the geometric efficiency. Even though the production of dark pions is
basically from proton Bremsstrahlung, the geometric efficiency for dark pions
could change since $\rho$ meson dominates their production. In other words,
the invariant mass of final states, $\sqrt{s_{f}}$, is close to the $\rho$
meson mass, $\sqrt{s_{f}}\sim m_{\rho}$, and hence the efficiency could be
reduced in analogy to the efficiency of dark photons from meson decay.

Fig. 12 shows the sensitivities of SeaQuest to the dark pion searches,
including variation of the geometric efficiency at SeaQuest. The different
line-types correspond to the different values of
$m_{\pi^{\prime}}/m_{A^{\prime}}$: $m_{\pi^{\prime}}/m_{A^{\prime}}=1.9$
(solid), $m_{\pi^{\prime}}/m_{A^{\prime}}=1.6$ (dashed), and
$m_{\pi^{\prime}}/m_{A^{\prime}}=1.3$ (dot-dashed). We take different decay
constants, $f_{\pi^{\prime}}=0.8\,\mathrm{GeV}$ (left) and
$f_{\pi^{\prime}}=0.1\,\mathrm{GeV}$ (right). When the geometric efficiency is
ten times smaller than the Bremsstrahlung, the sensitivity areas get smaller
and the sensitivity lines are shown as thin lines in the figures.

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