hep-ph/***

##  1 Introduction

The $B\to K^{(*)}\nu\bar{\nu}$ decays are among the cleanest modes to search
for new physics, due to their well-controlled theoretical uncertainty in the
standard model (SM). The Belle II experiment recently measured the charged
decay mode $B^{+}\to K^{+}\nu\bar{\nu}$ [1]

$\displaystyle{\cal B}(B^{+}\to K^{+}\nu\bar{\nu})_{\tt exp}=(2.3\pm
0.7)\times 10^{-5},$ (1)

which is $2.7\,\sigma$ in excess of the SM prediction [2],

$\displaystyle{\cal B}(B^{+}\to K^{+}\nu\bar{\nu})_{\tt SM}=(4.43\pm
0.31)\times 10^{-6}\;,$ (2)

where the tree-level contribution mediated by $\tau$ leptons has been
subtracted. This result is also about twice as large, but consistent within
errors, with the previous weighted average ${\cal B}(B^{+}\to
K^{+}\nu\bar{\nu})_{\tt exp}^{2021}=(1.1\pm 0.4)\times 10^{-5}$ performed by
the Belle II collaboration [3], and with the new weighted average ${\cal
B}(B^{+}\to K^{+}\nu\bar{\nu})_{\tt exp}^{\tt ave}=(1.3\pm 0.4)\times 10^{-5}$
[1] including the earlier results from Belle [4; 5] and BaBar [6; 7].

The Belle II measurement (new average) suggests an excess that is 3.6–7 (2–4)
times the SM prediction, and has stimulated theoretical exploration of viable
new physics scenarios. These include new contributions to $B^{+}\to
K^{+}\nu\bar{\nu}$ from heavy mediators within the SM effective field theory
(SMEFT) framework or renormalizable models [8; 9; 10; 11; 12; 13; 14; 15; 16;
17; 18], and new decay modes involving new light invisible particles like a
single scalar [19; 20; 21; 22; 23; 24], dark photon [22; 25], sterile
neutrinos [26; 27], or dark matter (DM) pairs [12; 23; 24]. See also earlier
discussions [28; 29; 30; 31; 32; 33] of the slight excess indicated by the
earlier weighted average [3].

For the light DM explanation, some of us identified in Ref. [12] that the
dimension-5 (dim-5) quark scalar current operator,111The operator has been
studied earlier in [34] as an example for using $B\to K+\mathrm{inv}$ as a
probe for dark matter. $\mathcal{O}_{d\phi}^{S,sb}=(\bar{s}b)\,\phi^{2}$, in
the low-energy effective theory ($\phi$LEFT) extended by a light real scalar
DM $\phi$, provides a good fit to the data with a high scale for the Wilson
coefficient (WC) of $|C^{S,sb}_{d\phi}|^{-1}\sim 1.5\times
10^{7}\,\mathrm{GeV}$ for $m_{\phi}=1$ GeV. The field $\phi$ is odd with
respect to a $\mathbb{Z}_{2}$ symmetry. The operator may be generated from the
dim-6 operator in SMEFT extended by $\phi$ ($\phi$SMEFT),

$\displaystyle\mathcal{O}_{qdH\phi^{2}}=(\bar{q}_{L}d_{R}H)\,\phi^{2},$ (3)

where $H$ is the SM Higgs doublet. The focus of this paper is to establish a
viable renormalizable ultraviolet (UV) completion of this operator, which
simultaneously explains the observed DM relic density through the thermal
freeze-out mechanism.

To explain the excess with new decay modes involving a light DM pair, we refer
to the difference between the measurement and the SM value as the “NP window”.
Using the new Belle II measurement together with the SM prediction the NP
window is

$\displaystyle{\cal B}(B^{+}\to K^{+}+\mathrm{inv})_{\tt NP}=(1.86\pm
0.67)\times 10^{-5}$ (4)

assuming three-body decay kinematics with massless neutrinos, which becomes
$(0.86\pm 0.40)\times 10^{-5}$ if the new average experimental value is used.
Usually, the interaction mediating $B^{+}\to K^{+}+\mathrm{inv}$ also leads to
the transitions $B^{0}\to K^{0}+\mathrm{inv}$ and $B\to K^{*}+\mathrm{inv}$,
which may pose strong constraints on the scenarios. Currently, the strongest
experimental constraints on these modes are [4; 5]

$\displaystyle{\cal B}(B^{0}\to K^{0}\nu\bar{\nu})$ $\displaystyle\leq
2.6\times 10^{-5}{\rm~{}(90\%~{}c.l.)},$ (5a) $\displaystyle{\cal B}(B^{+}\to
K^{+*}\nu\bar{\nu})$ $\displaystyle\leq 4.0\times
10^{-5}{\rm~{}(90\%~{}c.l.)},$ (5b) $\displaystyle{\cal B}(B^{0}\to
K^{0*}\nu\bar{\nu})$ $\displaystyle\leq 1.8\times
10^{-5}{\rm~{}(90\%~{}c.l.)},$ (5c)

assuming three-body decay kinematics with massless neutrinos, as predicted by
the SM. Subtracting their SM predictions, these lead to the corresponding NP
bounds for the relevant modes with invisible particles to be [31]: $2.3\times
10^{-5},\,3.1\times 10^{-5}$ and $1.0\times 10^{-5}$, respectively.

The paper is organized as follows. In Section 2 we introduce the
renormalizable UV model which generates the operator in Eq. 3. Its
implications for the $B$ meson decays $B\to K^{(*)}+\mathrm{inv}$ are
discussed in Section 3 and dark matter in Section 4. We discuss constraints
and the available parameter space explaining both the excess in $B^{+}\to
K^{+}+\mathrm{inv}$ and the DM relic abundance in Section 5 and conclude in
Section 6.

##  2 The Model

In order to realize the UV completion of the operator in Eq. 3, we introduce
two heavy vector-like quarks $Q\sim(\mathbf{3},\mathbf{2},1/6)$ and
$D\sim(\mathbf{3},\mathbf{1},-1/3)$ with masses $m_{Q}$ and $m_{D}$,
respectively, in addition to the light scalar field
$\phi\sim(\mathbf{1},\mathbf{1},0)$ to generate the Yukawa interactions
$\bar{q}_{Li}Q_{R}\phi$ and $\bar{d}_{Ri}D_{L}\phi$ with $Q_{R}\equiv
P_{R}Q,D_{L}\equiv P_{L}D$. Here we use lower case letters $q_{L},d_{R},u_{R}$
to represent the SM left-handed quark doublet, right-handed down-type quark
and up-type quark singlets, respectively. The three numbers in parentheses
indicate how they transform with respect to the SM gauge group $SU(3)_{\rm
c}\times SU(2)_{\rm L}\times U(1)_{\rm Y}$. All exotic fields — $\phi$,
$Q_{L,R}$ and $D_{L,R}$ — are required to be odd under an imposed
$\mathbb{Z}_{2}$ symmetry that stabilises the DM particle $\phi$. Taking
$m_{Q,D}>m_{\phi}+m_{q}$, the vector-like quarks $Q$ and $D$ are unstable and
decay into $\phi$ and a SM quark of mass $m_{q}$. Because of the
$\mathbb{Z}_{2}$ symmetry, the vector-like quarks do not mix with SM quarks,
but they mix among themselves after electroweak symmetry breaking. The full
new physics parts of the Lagrangian (kinetic, Yukawa and Higgs potential) are

$\displaystyle\mathcal{L}_{\tt kinetic}^{\tt NP}$
$\displaystyle=\bar{Q}i\not{D}Q-m_{Q}\bar{Q}Q+\bar{D}i\not{D}D-m_{D}\bar{D}D+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}^{2}\phi^{2},$
(6a) $\displaystyle\mathcal{L}_{\tt Yukawa}^{\tt NP}$
$\displaystyle=y_{q}^{p}\bar{q}_{Lp}Q_{R}\phi+y_{d}^{p}\bar{D}_{L}d_{Rp}\phi-
y_{1}\bar{Q}_{L}D_{R}H-y_{2}\bar{Q}_{R}D_{L}H+{\rm h.c.}\;,$ (6b)
$\displaystyle V_{\tt potential}^{\tt NP}$
$\displaystyle=\frac{1}{4}\lambda_{\phi}\phi^{4}+\frac{1}{2}\kappa\,\phi^{2}H^{\dagger}H\;,$
(6c)

where group theory indices are suppressed and the SM quark generations are
denoted by indices $p,r$. The choice of a minus sign with Yukawa couplings
$y_{1,2}$ is for later matching convenience.
$H\sim(\mathbf{1},\mathbf{2},1/2)$ denotes the electroweak Higgs doublet.
Using the freedom to rephase the two heavy vector-like quark fields, we choose
$y_{1}$ and one of the Yukawa couplings $y_{q}^{p}$ to be real without loss of
generality. All other Yukawa couplings generally are complex.

The DM particle $\phi$ couples to the SM Higgs doublet via the Higgs portal
interaction $\phi^{2}H^{\dagger}H$, in addition to its coupling to SM quarks
via the Yukawa interactions. The former is constrained by invisible Higgs
decay with the branching ratio BR($h\to\mathrm{inv})<0.107$ at 95% CL [35]
(see also [36; 37]), which leads to $\kappa\lesssim 0.01$ for light $\phi$
with mass below tens of GeV. In the following phenomenological study, we
neglect the Higgs portal interaction and consequently also its contribution to
the DM mass. Taking it small compared to the bare mass term implies $\kappa\ll
2m_{\phi}^{2}/v^{2}\approx 3\times 10^{-5}(m_{\phi}/{\rm GeV})^{2}$ which is
consistent with radiative corrections from fermion loops.

##  3 $B\to K^{(*)}+\mathrm{inv}$

At energies below the masses $m_{Q}\simeq m_{D}$ of the heavy vector-like
quarks $D$ and $Q$, the interactions of the scalar field $\phi$ with SM quarks
are described to leading order by the following $\phi$SMEFT Lagrangian,

$\displaystyle\mathcal{L}_{\phi\phi qq}^{\tt SMEFT}$
$\displaystyle=C_{qdH\phi^{2}}^{pr}\mathcal{O}_{qdH\phi^{2}}^{pr}+C_{quH\phi^{2}}^{pr}\mathcal{O}_{quH\phi^{2}}^{pr}+\mathrm{h.c.}\;,$
(7)

with the first effective operator given in Eq. 3 and the second,
$\mathcal{O}_{quH\phi^{2}}^{pr}=(\bar{q}_{Lp}u_{Rr}\tilde{H})\phi^{2}$ with
$\tilde{H}=\epsilon H^{*}$.

$d_{R}$$D_{L}$$D_{R}$$Q_{L}$$Q_{R}$$q_{L}$$H$$\phi$$\phi$

$d_{R}$$D_{L}$$d_{R}$$q_{L}$$\phi$$\phi$$H$

$d_{R}$$q_{L}$$Q_{R}$$q_{L}$$\phi$$\phi$$H$

$d_{R}$$D_{L}$$Q_{R}$$q_{L}$$H$$\phi$$\phi$

Figure 1: Feynman diagrams contributing to the matching to the $\phi$SMEFT-
like operator $\mathcal{O}_{qdH\phi^{2}}$ via $t$-channel exchange of the
vector-like fermions $Q$ and $D$. The magenta crosses represent mass
insertions.

The WCs $C_{qdH\phi^{2}}$ and $C_{quH\phi^{2}}$ can be determined by
integrating out the heavy vector-like quarks $Q$ and $D$ at tree level as
shown in Fig. 1. With the aid of the Matchete code [38], we obtain the
matching results,222The matching proceeds through an intermediate step that
uses the two redundant operators, $(\bar{q}_{Lp}i\not{D}q_{Lr})\phi^{2}$ and
$(\bar{d}_{Rp}i\not{D}d_{Rr})\phi^{2}$. These are related to the operators in
Eq. 7 via the equations of motion of the SM quark fields $q_{L}$ and $d_{R}$.

$\displaystyle C_{qdH\phi^{2}}^{pr}$
$\displaystyle=\frac{y_{q}^{p}y_{d}^{r}y_{1}}{m_{Q}m_{D}}+\frac{y_{q}^{p}y_{q}^{x*}(Y_{d})_{xr}}{2m_{Q}^{2}}+\frac{(Y_{d})_{px}y_{d}^{x*}y_{d}^{r}}{2m_{D}^{2}},$
(8a) $\displaystyle C_{quH\phi^{2}}^{pr}$
$\displaystyle=\frac{y_{q}^{p}y_{q}^{x*}(Y_{u})_{xr}}{2m_{Q}^{2}},$ (8b)

where $Y_{u}$ and $Y_{d}$ denote the SM up-quark and down-quark Yukawa
couplings, $\bar{q}_{L}Y_{u}u_{R}\tilde{H}+\bar{q}_{L}Y_{d}d_{R}H$,
respectively. The first three Feynman diagrams in Fig. 1 correspond to the
three terms in $C_{qdH\phi^{2}}^{pr}$, the fourth diagram yields dim-8
operators that are expected to be suppressed by two additional heavy mass
powers and thus will be neglected. The coefficient $C_{quH\phi^{2}}^{pr}$ is
generated by a similar diagram to the third one in Fig. 1 in which the $d_{R}$
and $H$ fields are replaced by $u_{R}$ and $\tilde{H}$ respectively.

To study $B\to K^{(*)}+\mathrm{inv}$ and other low energy processes much lower
than the EW scale, we need to match the $\phi$SMEFT to the $\phi$LEFT by
integrating out the heavy SM states (top, Higgs, $W$ and $Z$) after EW
symmetry breaking. Taking the down-type quark flavour eigenstates to be mass
eigenstates, we obtain four $\phi$LEFT operators

$\displaystyle\mathcal{L}_{\phi\phi qq}^{\tt
LEFT}=\frac{1}{2}C_{d\phi}^{S,ij}(\bar{d}_{i}d_{j})\phi^{2}+\frac{1}{2}C_{d\phi}^{P,ij}(\bar{d}_{i}i\gamma_{5}d_{j})\phi^{2}+\frac{1}{2}C_{u\phi}^{S,ij}(\bar{u}_{i}u_{j})\phi^{2}+\frac{1}{2}C_{u\phi}^{P,ij}(\bar{u}_{i}i\gamma_{5}u_{j})\phi^{2},$
(9)

with their corresponding WCs

$\displaystyle C_{d\phi}^{S,ij}$
$\displaystyle=\frac{(y_{q}^{i}y_{d}^{j}y_{1}+y_{q}^{j*}y_{d}^{i*}y_{1}^{*})v}{\sqrt{2}m_{Q}m_{D}}+\left(\frac{y_{q}^{i}y_{q}^{j*}}{2m_{Q}^{2}}+\frac{y_{d}^{i*}y_{d}^{j}}{2m_{D}^{2}}\right)(m_{d_{i}}+m_{d_{j}}),$
(10a) $\displaystyle iC_{d\phi}^{P,ij}$
$\displaystyle=\frac{(y_{q}^{i}y_{d}^{j}y_{1}-y_{q}^{j*}y_{d}^{i*}y_{1}^{*})v}{\sqrt{2}m_{Q}m_{D}}-\left(\frac{y_{q}^{i}y_{q}^{j*}}{2m_{Q}^{2}}-\frac{y_{d}^{i*}y_{d}^{j}}{2m_{D}^{2}}\right)(m_{d_{i}}-m_{d_{j}}),$
(10b) $\displaystyle C_{u\phi}^{S,ij}$
$\displaystyle=\frac{\tilde{y}_{q}^{i}\tilde{y}_{q}^{j*}}{2m_{Q}^{2}}(m_{u_{i}}+m_{u_{j}}),$
(10c) $\displaystyle iC_{u\phi}^{P,ij}$
$\displaystyle=-\frac{\tilde{y}_{q}^{i}\tilde{y}_{q}^{j*}}{2m_{Q}^{2}}(m_{u_{i}}-m_{u_{j}})\;,$
(10d)

where $\tilde{y}_{q}^{i}\equiv(V_{\tt CKM})_{ix}y_{q}^{x}$ with $V_{\tt CKM}$
being the CKM matrix. In the above we use $i,j$ to represent the quark
flavours in the mass basis. Note that there are no vector operators of the
form $(\bar{q}\gamma^{\mu}q)(\phi\overleftrightarrow{\partial_{\mu}}\phi)$ for
a real scalar $\phi$ due to $(\phi\overleftrightarrow{\partial_{\mu}}\phi)=0$.

The $B\to K\phi\phi$ ($B\to K^{*}\phi\phi$) transition is induced by the
operator $\bar{s}b\phi^{2}$ ($\bar{s}i\gamma_{5}b\phi^{2}$). Their
differential branching ratios have been calculated in [31]333Note that a
factor of $1/2$ needs to be included to account for the double counting of
phase space integration for the identical $\phi$ pair.  and are given by

$\displaystyle\frac{d\Gamma_{B\to K\phi\phi}}{ds_{B}}$
$\displaystyle=\frac{\left|C_{d\phi}^{S,sb}\right|^{2}m_{B}}{512\pi^{3}}f_{0}^{2}\frac{(1-x_{K})^{2}\lambda^{\tfrac{1}{2}}(1,x_{K},s_{B})\sqrt{1-4x_{\phi}/x_{B}}}{(\sqrt{x_{b}}-\sqrt{x_{s}})^{2}},$
(11a) $\displaystyle\frac{d\Gamma_{B\to K^{*}\phi\phi}}{ds_{B}}$
$\displaystyle=\frac{\left|C_{d\phi}^{P,sb}\right|^{2}m_{B}}{512\pi^{3}}A_{0}^{2}\frac{\lambda^{\tfrac{3}{2}}(1,x_{K^{*}},s_{B})\sqrt{1-4x_{\phi}/x_{B}}}{(\sqrt{x_{b}}+\sqrt{x_{s}})^{2}},$
(11b)

in terms of the Källén function
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$. All dimensionful quantities
are normalized relative to the $B$ meson mass: the squared invariant mass of
the DM pair is $q^{2}=s_{B}m_{B}^{2}$ and the mass ratios are
$x_{K^{(*)}}=m_{K^{(*)}}^{2}/m_{B}^{2}$, $x_{\phi}=m_{\phi}^{2}/m_{B}^{2}$,
$x_{b}=m_{b}^{2}/m_{B}^{2}$, and $x_{s}=m_{s}^{2}/m_{B}^{2}$. The $q^{2}$
dependence of the hadronic form factors $f_{0}$ and $A_{0}$ associated with
scalar and pseudo-scalar quark currents is left implicit for notational
simplicity, and we use the recent fitting results from [39] for the later
numerical analysis.

Figure 2: Preferred parameter space to explain the excess in $B^{+}\to
K^{+}+\mathrm{inv}$ via additional decay channels to DM final states,
$B^{+}\to K^{+}\phi\phi$ based on the latest Belle II measurement [1] in red,
with an additional reweighted one to account for the selection efficiency in
blue, and finally based on the new average in purple [1]. The dashed line
indicates the current constraint from $B^{0}\to K^{0}+\mathrm{inv}$ [4; 5] on
$|C^{S,sb}_{d\phi}|$ and the solid orange and green lines the constraints on
$|C^{P,sb}_{d\phi}|$ posed by the searches for $B^{+}\to K^{+*}+\mathrm{inv}$
and $B^{0}\to K^{0*}+\mathrm{inv}$ [4; 5], respectively.

From Eq. 11 it can be seen that the differential branching ratios with final
$K$ and $K^{*}$ states depend on different $\phi$LEFT WCs. Hence, it is
possible to obtain a large enhancement in $B\to K\phi\phi$ without
substantially modifying $B\to K^{*}\phi\phi$. In fact for
$y_{q}^{i}=y_{d}^{i*}$ there is a cancellation in $C_{d\phi}^{P,ij}$ and thus
there are no additional contributions to $B\to K^{*}\phi\phi$. In terms of the
$\phi$LEFT WCs $C_{d\phi}^{S,sb}$, we show the allowed parameter space to
accommodate the Belle II excess in Eq. 4 (the new average number) as the
magenta (purple) region in Fig. 2. It can be seen the excess in $B^{+}\to
K^{+}+\mathrm{inv}$ can be explained for
$C^{S,sb}_{d\phi}\sim(3-8)/(10^{5}\,\rm TeV)$ for a DM mass of $m_{\phi}=1$
GeV, without conflicting with the bound from the neutral kaon mode shown as
the black dashed line. The same plot also shows the constraints on the
$C_{d\phi}^{P,sb}$ from the vector kaon final states as orange and blue
curves, since they are also generated in the model. As can be seen, the
neutral mode puts more a stringent bound on $C_{d\phi}^{P,sb}$ than the
charged mode.

The situation depicted in Eq. 4 would be modified by experimental efficiency
(E.E.) considerations. The inclusion of the Belle II signal selection
efficiency modifies the allowed parameter space. To illustrate the impact this
effect has on our study we include the blue region in the figure. This region
is obtained by scaling the predictions of our model by a ratio of normalized
rates weighted by the reported efficiency $\epsilon$,

$\displaystyle\omega(m)=\frac{\sum_{i}\tilde{\Gamma}_{i,\rm
SM}\epsilon_{i}}{\sum_{i}\tilde{\Gamma}_{i,\rm NP}(m)\epsilon_{i}}.$ (12)

In this factor, $\tilde{\Gamma}_{i,\rm NP}(m)$ ($\tilde{\Gamma}_{i,\rm SM}$)
is the normalized width from the new physics (SM) contribution in the $i$-th
bin [31], and $\epsilon_{i}$ is the experimental efficiency for that bin in
the inclusive tag analysis reported in [1]. It can be seen that the
consideration of efficiency requires a slightly larger $C_{d\phi}^{S,sb}$ for
each fixed DM mass benchmark. Note that the constraints from other $B\to
K^{(*)}+\mathrm{inv}$ decays may also be relaxed due to experimental
efficiency considerations, which depend on the experimental analysis. See Ref.
[40] for a detailed discussion on how the experimental efficiencies could be
incorporated. In Section 5, we combine these results with that of DM analysis
to identify the viable parameter space.

##  4 Dark matter

As invisible Higgs decay constrains the Higgs portal coupling to be small and
thus DM annihilations via the Higgs portal $\phi\phi\to SM$ to be subdominant,
we focus on thermal freeze-out via DM annihilation generated through
$t$-channel exchange of the heavy vector-like quarks, which is described by
the effective operators in Eq. 9. The explanation of the excess $B^{+}\to
K^{+}\phi\phi$ prefers light dark matter, and thus we use chiral perturbation
theory to evaluate the DM annihilation cross section into light mesons. Taking
into account the three light quarks, the QCD Lagrangian extended by the DM-
quark interactions in Eq. 9 takes the form

$\displaystyle{\cal
L}\ni\mathcal{L}_{\mathrm{QCD}}-\left[\overline{q_{R}}(s+ip)q_{L}+\text{ h.c.
}\right]\;,$ (13)

where the scalar and pseudo-scalar sources are

$\displaystyle s=-\frac{1}{2}\begin{pmatrix}C_{u\phi}^{S,uu}&0&0\\\
0&C_{d\phi}^{S,dd}&C_{d\phi}^{S,ds}\\\
0&C_{d\phi}^{S,sd}&C_{d\phi}^{S,ss}\end{pmatrix}\phi^{2},\quad
p=\frac{1}{2}\begin{pmatrix}C_{u\phi}^{P,uu}&0&0\\\
0&C_{d\phi}^{P,dd}&C_{d\phi}^{P,ds}\\\
0&C_{d\phi}^{P,sd}&C_{d\phi}^{P,ss}\end{pmatrix}\phi^{2}\;.$ (14)

The leading order DM-meson interactions appears at ${\cal O}(p^{2})$ in chiral
power counting [41; 42], and thus the relevant part of the effective
Lagrangian in chiral perturbation theory is

$\displaystyle\mathcal{L}_{p^{2}}=\frac{F_{0}^{2}}{4}{\rm
tr}\left[(D_{\mu}U)(D^{\mu}U)^{\dagger}\right]+\frac{F_{0}^{2}}{4}{\rm
tr}\left[\chi U^{\dagger}+U\chi^{\dagger}\right],$ (15)

where $F_{0}\approx 87\,\rm MeV$ is the pion decay constant in the chiral
limit, $\chi=2B(s-ip)$ with $B\approx 2.8\,\rm GeV$, and $U$ is related to the
mesons by

$\displaystyle U=\text{exp}\left[i\frac{\sqrt{2}\Phi}{F_{0}}\right],\quad\Phi$
$\displaystyle=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{pmatrix}.$ (16)

Expanding $U$ to second order in the meson field $\Phi$, the leading order
Lagrangian for DM-meson interactions reads

$\displaystyle\mathcal{L}_{\phi P}\ni-\sqrt{2}BF_{0}\,{\rm
tr}\left[p\Phi\right]-B\,{\rm tr}\left[s\Phi^{2}\right]+{\cal
O}(\phi^{2}P^{3})\;.$ (17)

In the limit $y_{1}v\gg m_{u,d,s}$ and $y_{q}^{i}\simeq y_{d}^{i*}$, which is
preferred by the explanation of the excess in $B^{+}\to K^{+}+\mathrm{inv}$,
the leading contributions originate from the second term and result in

$\displaystyle\mathcal{L}_{\phi P}$
$\displaystyle\ni\frac{B}{2}\phi^{2}\left\\{\left(C_{u\phi}^{S,uu}+C_{d\phi}^{S,dd}\right)\left(\pi^{+}\pi^{-}+\frac{1}{2}\pi^{0}\pi^{0}\right)+\left(C_{u\phi}^{S,uu}+C_{d\phi}^{S,ss}\right)K^{+}K^{-}\right.$
$\displaystyle+(C_{d\phi}^{S,dd}+C_{d\phi}^{S,ss})K^{0}\bar{K}^{0}+\frac{1}{6}\left(C_{u\phi}^{S,uu}+C_{d\phi}^{S,dd}+4C_{d\phi}^{S,ss}\right)\eta^{2}+\frac{1}{\sqrt{3}}\left(C_{u\phi}^{S,uu}-C_{d\phi}^{S,dd}\right)\pi^{0}\eta$
$\displaystyle\left.+\left[C_{d\phi}^{S,ds}\left(\pi^{+}K^{-}-\frac{1}{\sqrt{2}}\pi^{0}\bar{K}^{0}-\frac{1}{\sqrt{6}}\eta\bar{K}^{0}\right)+{\rm
h.c.}\right]\right\\}\;.$ (18)

The WC $C_{u\phi}^{S,uu}$ can be neglected because it is suppressed by small
CKM matrix elements in addition to $m_{u}\ll y_{1}v$. We further neglect the
Yukawa couplings to the down quark, $y_{q,d}^{d}\simeq 0$, which do not enter
in the branching ratio of $B\to K^{(*)}\phi\phi$ and thus the dominant
annihilation channels are into pairs of kaons and $\eta$ mesons,

$\displaystyle\sigma(\phi\phi\to K^{+}K^{-},K^{0}\bar{K}^{0})$
$\displaystyle=\frac{B^{2}|C_{d\phi}^{S,ss}|^{2}}{16\pi
s}\left(\frac{s-4m_{K}^{2}}{s-4m_{\phi}^{2}}\right)^{1/2}\;,$ (19a)
$\displaystyle\sigma(\phi\phi\to\eta\eta)$
$\displaystyle=\frac{B^{2}|C_{d\phi}^{S,ss}|^{2}}{18\pi
s}\left(\frac{s-4m_{K}^{2}}{s-4m_{\phi}^{2}}\right)^{1/2}\;.$ (19b)

The thermal average takes the form [43]

$\displaystyle\langle\sigma
v\rangle=\frac{4x}{K_{2}^{2}(x)}\int_{0}^{\infty}d\epsilon\,\epsilon\sqrt{1+\epsilon}K_{1}(2x\sqrt{1+\epsilon})\sigma,\quad\epsilon\equiv{s-4m_{\phi}^{2}\over
4m_{\phi}^{2}},$ (20)

where $K_{1,2}$ are modified Bessel functions and $x\equiv m_{\phi}/T$ with
$T$ being temperature. Thus the thermally averaged cross sections for DM
annihilation into kaons and $\eta$ mesons are

$\displaystyle\langle\sigma v(\phi\phi\to K^{+}K^{-},K^{0}\bar{K}^{0})\rangle$
$\displaystyle=\frac{B^{2}|C_{d\phi}^{S,ss}|^{2}\eta(x,z_{K})}{64\pi
m_{\phi}^{2}}\;,$ (21a) $\displaystyle\langle\sigma
v(\phi\phi\to\eta\eta)\rangle$
$\displaystyle=\frac{B^{2}|C_{d\phi}^{S,ss}|^{2}\eta(x,z_{\eta})}{72\pi
m_{\phi}^{2}},$ (21b)

in terms of $z_{K,\eta}=m_{K,\eta}^{2}/m_{\phi}^{2}$ and the function

$\displaystyle\eta(x,z)$ $\displaystyle\equiv{4x\over
K_{2}^{2}(x)}\int_{0}^{\infty}d\epsilon\,{\sqrt{\epsilon}\sqrt{1+\epsilon-z}\over\sqrt{1+\epsilon}}K_{1}(2x\sqrt{1+\epsilon})={\cal
O}(1)\;,$ (22)

which takes values between 0.5 and 1.7 for the relevant parameter space.

The relic density of WIMP dark matter is given by

$\Omega_{\phi}=\frac{m_{\phi}\,n_{\phi,\text{fo}}\,s_{0}/s_{\text{fo}}}{\rho_{\rm
crit}}\;,$ (23)

where $n_{\phi,\text{fo}}$ denotes the dark matter number density at freeze-
out, $s_{0,\text{fo}}$ the entropy density today and at freeze-out
respectively, and $\rho_{\rm crit}$ the critical energy density. The required
thermally averaged DM annihilation cross section is approximately given by
[44] $\langle\sigma v\rangle\simeq 2.4\times
10^{-26}\,\frac{\mathrm{cm}^{3}\mathrm{s}^{-1}}{(\hbar c)^{2}c}=2.2\times
10^{-9}\,\mathrm{GeV}^{-2}$, neglecting the slight variation in the required
annihilation cross section between the kaon mass $m_{K}$ and $1$ GeV. A more
precise prediction can be obtained using the results from Figures 4 and 5 in
Ref. [44]. We digitise those two figures to obtain the matching point $x_{*}$
of the two regimes of the freeze-out process as a function of the DM mass and
the thermally averaged cross section at $x_{*}$ which is required to obtain
the correct relic abundance, and use it to predict the required value of the
WC $|C_{d\phi}^{S,ss}|$.

Figure 3: The Wilson coefficient $|C^{S,ss}_{d\phi}|$, for which the observed
DM relic abundance is obtained through annihilation into kaons and $\eta$
mesons, as a function of the DM mass $m_{\phi}$.

While annihilations to final states with more than two mesons are suppressed,
we would have to include annihilations to $K^{*}$ meson pairs were this
channel available. However, we restrict ourselves to DM masses below the
$K^{*}$ threshold, which are also favoured by the explanation of the excess in
$B^{+}\to K^{+}+\mathrm{inv}$. In Fig. 3 we present the WC
$|C_{d\phi}^{S,ss}|$ required to produce the correct DM relic density as a
function of the DM mass $m_{\phi}$. A comparison with the required value of
$C_{d\phi}^{S,sb}\sim(3-8)/(10^{5}\,{\rm TeV})$ for explaining the Belle II
anomaly clearly demonstrates the need for a large ($\sim 10^{3}$) hierarchy
between the strange and bottom couplings.

Finally, we comment on the constraints from direct and indirect detection.
Since the sensitivity of direct detection decreases significantly for sub-GeV
scale DM due to detection threshold issues, direct detection experiments do
not provide any meaningful constraints for the parameter space of interest.
For indirect detection, the main annihilation channel is into photon pairs.
The DM-photon interactions arise at the one-loop order and take the form,
${\cal O}_{\phi\gamma}=\phi^{2}F_{\mu\nu}F^{\mu\nu}$, with the corresponding
WC given by

$\displaystyle C_{\phi\gamma}=\frac{\alpha_{\tt
em}}{36\pi}\left(\frac{5y_{q}^{x}y_{q}^{*x}}{m_{Q}^{2}}+\frac{y_{d}^{x}y_{d}^{*x}}{m_{D}^{2}}\right).$
(24)

The velocity-averaged DM annihilation cross section for
$\phi\phi\to\gamma\gamma$ is estimated to be $\langle\sigma
v\rangle_{\phi\phi\to\gamma\gamma}\approx(2/\pi)m_{\phi}^{2}|C_{\phi}\gamma|^{2}$
in the $s$-wave approximation. The current constraint on $\langle\sigma
v(\phi\phi\to\gamma\gamma)\rangle$ of around ${\cal O}(10^{-28}\,\rm
cm^{3}/s)$ [45], leads to $m_{\phi}|C_{\phi\gamma}|\lesssim 0.004/\rm TeV$.
For $m_{\phi}\sim 0.5\,\rm GeV$ and ${\cal O}(1)$ Yukawa couplings, the
constraint implies $m_{Q}\gtrsim 10\,\rm GeV$, a quite weak bound that can be
easily satisfied.

##  5 Discussion

The decays $B\to X_{s}\gamma$, $B_{s}\to\phi\phi$, $D^{0}\to\pi^{0}+\phi\phi$,
$D^{0}\to\phi\phi$ and $B_{s}-\bar{B}_{s}$ mixing can also be generated in the
model and thus could pose constraints on the allowed parameter space that
solves the Belle II anomaly. In the following, we estimate their effect and
also comment on direct searches for vector-like quarks.

$\boldsymbol{B\to X_{s}\gamma}$: This transition is related to the dipole
operators in the LEFT,
$O_{d\gamma}^{ij}=\bar{d}_{i}\sigma^{\mu\nu}P_{R}d_{j}F_{\mu\nu}$, with the WC
from the one-loop calculation given by

$\displaystyle
C_{d\gamma}^{ij}=\frac{1}{16\pi^{2}}\frac{e}{24}\left[\frac{vy_{q}^{i}y_{d}^{j}y_{1}}{\sqrt{2}m_{D}m_{Q}}I(t)+\frac{5m_{d_{j}}y_{q}^{i}y_{q}^{j*}}{6m_{Q}^{2}}+\frac{m_{d_{i}}y_{d}^{i*}y_{d}^{j}}{3m_{D}^{2}}\right],\quad
I(t)=\frac{5-7t+2t^{2}+3t\ln t}{(1-t)^{2}},$ (25)

with $t\equiv m_{Q}^{2}/m_{D}^{2}$. Denoting the term in square brackets in
Eq. 25 as $\tilde{C}_{d\gamma}^{ij}$, recent results of global fits [46]
result in the constraint $\tilde{C}_{d\gamma}^{ij}\lesssim 260/(10^{5}\,{\rm
TeV})$. The quantity $\tilde{C}_{d\gamma}^{ij}/I(t)$ is roughly the same order
as $C_{d\phi}^{S,sb}$ given in Eq. 10, and since the loop function $I(t)$
ranges between 5 and 2 as $t$ varies between zero and infinity, this implies
that $B\to X_{s}\gamma$ does not restrict our explanation of $B^{+}\to
K^{+}+\mathrm{inv}$, which requires the much smaller value
$C_{d\phi}^{S,sb}\sim(3-8)/(10^{5}\,{\rm TeV})$. Similarly, the related
operator with a gluon field strength tensor is produced in our model with a
coefficient,

$\displaystyle{\cal
O}_{dG}^{ij}=\bar{d}_{i}T^{A}\sigma^{\mu\nu}P_{R}d_{j}G^{A}_{\mu\nu},\quad
C_{dG}^{ij}=-{1\over 16\pi^{2}}{g_{s}\over
4}\left({y_{q}^{i}y_{d}^{j}y_{1}v\over\sqrt{2}m_{D}m_{Q}}+{m_{d_{j}}y_{q}^{j}y_{q}^{j*}\over
6m_{Q}^{2}}+{m_{d_{i}}y_{d}^{i*}y_{d}^{j}\over 6m_{D}^{2}}\right).$ (26)

The same global fits [46] imply a constraint on the term in brackets about ten
times weaker than its counterpart in Eq. 25.

$\boldsymbol{B_{s}(s\bar{b})-\bar{B}_{s}(\bar{s}b)}$ mixing: Naively, we would
expect the generation of dim-6 SMEFT four-quark operators mediating neutral
meson mixing by the Feynman diagrams444Note that the penguin-like diagrams
with a SM gauge boson or Higgs cannot yield such double flavour-changing
operators. shown in Fig. 4.

$q$$Q,D$$q$$q$$Q,D$$q$$\phi$$\phi$

$q$$Q,D$$q$$q$$Q,D$$q$$\phi$$\phi$

Figure 4: Potential leading order Feynman diagrams contributing to neutral
meson mixing.

However, due to the chirality of the quark pairs in each fermion line being
the same, the sum of contributions from the two diagrams vanishes for dim-6
operators. Thus, the leading order SMEFT operators with four quarks which
contribute to neutral meson mixing only arise at dim-8 order, with the
structure $(\bar{q}q)^{2}\otimes\\{D^{2},G^{A},HD,H^{2}\\}$. Their WCs are the
order $y_{\tt Yua}^{6}/(32\pi^{2}m_{Q,D}^{4})$ with $y_{\tt Yua}$ representing
the Yukawa couplings in the model. After electroweak symmetry breaking, the
WCs for the four-quark operators
$(\bar{q}_{p}\Gamma_{1}q_{r})(\bar{q}_{s}\Gamma_{2}q_{t})$ are estimated to be

$\displaystyle C_{qqqq}\sim\frac{y_{\tt
Yua}^{6}}{32\pi^{2}m_{Q,D}^{4}}\frac{v^{2}}{8}\approx 3\times
10^{-8}G_{F}\left(3\,{\rm TeV}\over m_{Q,D}\right)^{4}y_{\tt Yua}^{6}.$ (27)

For the SM case, the WC for the operator
$(s\gamma_{\mu}(1-\gamma_{5})b)(s\gamma^{\mu}(1-\gamma_{5})b)$ is $1.9\times
10^{-6}G_{F}$. We can see the new physics contribution is severely suppressed
relative to the SM once the NP scale is higher than 3 TeV for an ${\cal O}(1)$
coupling constant. From Fig. 2 and Eq. 25, the $B^{+}\to K^{+}+\mathrm{inv}$
excess requires a new physics scale around $50\,\rm TeV$ for ${\cal O}(1)$
coupling constant, and thus the constraints from $B_{s}-\bar{B}_{s}$ mixing
can also be neglected.

$B_{s}\to\mathrm{inv}$: Its branching ratio takes the form

$\displaystyle{\cal
B}(B_{s}\to\phi\phi)=\frac{|C_{d\phi}^{P,sb}|^{2}\tau_{B_{s}}m_{B_{s}}f_{B_{s}}^{2}}{32\pi}\left({m_{B_{s}}\over
m_{b}+m_{s}}\right)^{2}\sqrt{1-4m_{\phi}^{2}/m_{B_{s}}^{2}}.$ (28)

Similar to $B\to K^{*}\phi\phi$ it depends on the pseudo-scalar WC
$C^{P,sb}_{d\phi}$ and thus the current constraint of
$\mathcal{B}(B_{s}\to\mathrm{inv})<5.4\times 10^{-4}$ at 90% c.l. [47] is
easily satisfied with our choice of the Yukawa couplings. For $m_{\phi}=0.6$
GeV and $f_{B_{s}}=225.3(6.6)$ MeV, [48] we find a branching ratio ${\cal
B}(B_{s}\to\phi\phi)\sim 4\times 10^{-5}$ for $|C_{d\phi}^{S,ss}|\sim 0.13/\rm
TeV$ and $\sqrt{r_{d}^{2}+r_{q}^{2}-2r_{d}r_{q}\cos\theta}\sim 10^{-3}$ in Eq.
30 that can simultaneously satisfy the DM relic density and $B^{+}\to
K^{+}+\mathrm{inv}$ requirements. At this level, the rate is about four times
larger than the expected sensitivity at Belle II with 5 ab-1 [49].

$D^{0}\to\mathrm{inv}$ and $D^{0}\to\pi^{0}+\mathrm{inv}$ and decay: These
decay modes have been measured by Belle and BESIII respectively yielding 90%
c.l. upper limits ${\cal B}(D^{0}\to\mathrm{inv})<9.4\times 10^{-5}$ [50] and
${\cal B}(D^{0}\to\pi^{0}\nu\bar{\nu})<2.1\times 10^{-4}$ [51]. These decays
are induced by the operators ${\cal O}_{u\phi}^{S(P),uc}$ with up and charm
quark flavours and could therefore constrain the corresponding WCs. Reference
[52] has recently studied the constraints on ${\cal O}^{S(P),uc}_{u\phi}$ from
these modes (see Eq. 2 and Fig. 5 in that paper). In our notation, those
constraints translate into $|C_{u\phi}^{S(P),uc}|/m_{c}\lesssim 2.06(0.25)/\rm
TeV^{2}$ for $m_{\phi}=0.5\,\rm GeV$ and become weaker for larger $m_{\phi}$.
These bounds are satisfied by the benchmark parameters used below, for which
$|C_{u\phi}^{S(P),uc}|/m_{c}\simeq(0.05/\,{\rm
TeV}^{2})\left({|y_{q}^{s}|\over 2}{3\,{\rm TeV}\over m_{Q}}\right)^{2}$ after
neglecting the Yukawa couplings associated with first and third-generation
quarks.

Direct searches at the LHC: Existing searches for pair-produced vector-like
quarks at the LHC rely on their decays into top (or bottom) quarks associated
with a $Z,W$ or Higgs boson. The $\mathbb{Z}_{2}$ symmetry imposed on our
model implies that their decays into quarks are accompanied instead by the
light dark scalar. As such, the existing bounds do not apply and we leave an
extraction of precise LHC bounds on these particles to a future analysis.
Nevertheless, we expect the bounds to be worse than those obtained by recent
ATLAS [53] and CMS [54] studies that look for pair-produced vector-like quarks
decaying into a final state that contains multi jets with one b-tag, a lepton,
and missing transverse momentum – this produces a mass lower bound near 1.5
TeV.

Figure 5: The allowed parameter space for DM mass $m_{\phi}=0.6\,\rm GeV$,
$m_{Q}=m_{D}=3\,\rm TeV$ and two benchmark CP phases as function of the
coupling ratios $r_{d,q}=|y_{d,q}^{b}|/|y_{d,q}^{s}|$.

The viable parameter space: We are now in a position to discuss the viable
parameter space which explains the excess in $B^{+}\to K^{+}+\mathrm{inv}$ and
provides the correct DM relic density, while satisfying various constraints,
in particular those posed by $B\to K^{*}+\mathrm{inv}$. Firstly, due to the
suppression of the second terms of $C_{d\phi}^{S(P),ij}$ in Eq. 10 by SM quark
masses, we neglect their contribution and focus on the analysis of the first
terms. By decomposing the complex Yukawa couplings as
$y_{q}^{i}\equiv|y_{q}^{i}|e^{-i\alpha_{i}}$ and
$y_{d}^{i}\equiv|y_{d}^{i}|e^{-i\beta_{i}}$, with $\alpha_{i}$ and $\beta_{i}$
being the associated CP phases, the absolute values of the three relevant WCs
become,

$\displaystyle|C_{d\phi}^{S,ss}|$
$\displaystyle\approx\frac{|y_{q}^{s}||y_{d}^{s}|y_{1}v}{\sqrt{2}m_{Q}m_{D}}\left|1+e^{i2(\alpha_{s}+\beta_{s})}\right|,$
(29a) $\displaystyle|C_{d\phi}^{S(P),sb}|$
$\displaystyle\approx\frac{|y_{q}^{s}||y_{d}^{s}|y_{1}v}{\sqrt{2}m_{Q}m_{D}}\left|{|y_{d}^{b}|\over|y_{d}^{s}|}\pm{|y_{q}^{s}|\over|y_{q}^{s}|}e^{i(\alpha_{s}+\alpha_{b}+\beta_{s}+\beta_{b})}\right|.$
(29b)

Next we assume the CP phases $\alpha_{s}+\beta_{s}=0$ and denote
$\alpha_{b}+\beta_{b}\equiv\theta$, $|y_{d}^{b}|/|y_{d}^{s}|\equiv r_{d}$,
$|y_{q}^{b}|/|y_{q}^{s}|\equiv r_{q}$. Then the above WCs obey the approximate
relation,

$\displaystyle|C_{d\phi}^{S(P),sb}|\approx{1\over
2}|C_{d\phi}^{S,ss}|\sqrt{r_{d}^{2}+r_{q}^{2}\pm 2r_{d}r_{q}\cos\theta}.$ (30)

From Fig. 3 we see that for the DM mass in the range $(0.5-0.9)\,\rm GeV$, the
required WC $|C_{d\phi}^{S,ss}|_{\tt DM}\sim 0.13/\rm TeV$ is roughly
independent of that mass. To achieve this value, we choose the masses for the
two heavy quarks to be $m_{Q}=m_{D}=3\,\rm TeV$, the Yukawa couplings related
to the SM quarks to be $|y_{d}^{s}|=|y_{q}^{s}|=2$, and the one inducing heavy
quark mixing to be $y_{1}=1$, leading to a comparable value for DM relic
requirement, $|C_{d\phi}^{S,ss}|\sim 0.16/\rm TeV$. The light-blue (light-
purple) region in Fig. 5 shows the allowed parameter space in $r_{d}-r_{q}$
plane for the CP phase $\theta=0$ ($\theta=\pi/2$) with the DM mass
$m_{\phi}=0.6\,\rm GeV$. It can be seen that the allowed regions require an
${\cal O}(10^{-3})$ hierarchy between the third and second generation Yukawa
couplings, as observed before. This hierarchical structure could be avoided if
the DM is not mainly produced through $K$ and $\eta$ annihilation channels but
includes additional channels like annihilation to pions through couplings to
the down quark or annihilation via the Higgs portal interaction.

##  6 Conclusions

In this paper, we introduce a UV complete model with two vector-like quarks
$Q\sim(\mathbf{3},\mathbf{2},1/6)$ and $D\sim(\mathbf{3},\mathbf{1},-1/3)$
together with a real scalar singlet $\phi$ which are all odd with respect to a
$\mathbb{Z}_{2}$ symmetry. Together they can enhance the branching ratio of
$B\to K+\text{invisible}$ through additional decays to dark matter pairs to
explain the recent Belle II excess and simultaneously generate the observed
dark matter relic density through the thermal freeze-out mechanism via DM pair
annihilation into meson pairs, $\phi\phi\longleftrightarrow KK(\eta\eta)$.
Meanwhile, the model satisfies the strong constraints from $B_{s}-\bar{B}_{s}$
mixing, meson decays and dark matter (in-)direct detection. In particular, the
related constraints from the associated $B\to K^{*}+\text{invisible}$ decays
and invisible $B_{s}$ decays may be easily satisfied by choosing the Yukawa
couplings so that the scalar operator dominates over the pseudo-scalar
operator, which is achieved for similar couplings to left-handed and right-
handed down-type quarks, $y_{d}^{i}\simeq y_{q}^{i*}$.

We are not aware of any dedicated searches for vector-like quarks which decay
to quarks and a DM particle at the LHC, but anticipate lower bounds of order
of $\mathcal{O}(1.5\ \mathrm{TeV})$ based on existing searches for vector-like
quarks that decay with missing energy [53; 54]. Note, however, that this bound
was obtained with the requirement of leptons and additional jets in the final
state, so is not directly applicable to our model. A dedicated search will
provide important additional information and may be able to confirm or rule
out the explanation of the dark matter abundance.

## Acknowledgements

This work was supported in part by Australian Research Council Discovery
Project DP200101470 and in part by the Australian Research Council Centre of
Excellence for Dark Matter Particle Physics (CDM, CE200100008). This work was
also supported in part by Key Laboratory for Particle Physics, Astrophysics
and Cosmology, Ministry of Education, and Shanghai Key Laboratory for Particle
Physics and Cosmology (Grant No. 15DZ2272100) and in part by the NSFC (Grant
Nos. 12375088, 12090064, and 12090060), and by the Guangdong Major Project of
Basic and Applied Basic Research ( Grant No.2020B0301030008) and by the NSFC
(Grant No. NSFC-12305110).

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# The variable 6307Å emission line in the spectrum of Eta Carinae: blueshifted
[S III] $\lambda$6313 from the interacting winds

T. R. Gull11affiliation: Laboratory for Extrasolar Planets and Stellar
Astrophysics, Exploration of the Universe Division, Code 667, Goddard Space
Flight Center, Greenbelt, MD 20771

###### Abstract

The 6307Å emission line in the spectrum of $\eta$ Car (Martin et al., 2006) is
blue-shifted [S iii] $\lambda$6313 emission originating from the outer wind
structures of the massive binary system. We realized the identification while
analyzing multiple forbidden emission lines not normally seen in the spectra
of massive stars. The high spatial and moderate spectral resolutions of
HST/STIS resolve forbidden lines of Fe+, N+, Fe+2, S+2, Ne+2 and Ar+2 into
spatially and velocity-resolved rope-like features originating from
collisionally-excited ions photo-ionized by UV photons or collisions. While
the [Fe ii] emission extends across a velocity range of $\pm$500 km s-1 out to
0$\farcs$7, more highly ionized forbidden emissions ( [N ii], [Fe iii], [S
iii], [Ar iii], and [Ne iii]) range in velocity from $-$500 to $+$200 km s-1,
but spatially extend outward to only 0$\farcs$4\. The [Fe ii] defines the
outer regions of the massive primary wind. The [N ii], [Fe iii] emission
define the the outer wind interaction regions directly photo-ionized by far-UV
radiation. Variations in emission of [S iii] $\lambda$$\lambda$9533, 9071 and
6313 suggest density ranges of 106 \- 1010 cm-3 for electron temperatures
ranging from 8,000 to 13,000° K. Mapping the temporal changes of the emission
structure at critical phases of the 5.54-year period will provide important
diagnostics of the interacting winds.

stars: binaries:spectroscopic, stars: individual: Eta Carinae, stars:winds

##  1 Introduction

Martin et al. (2006) found a variable emission line centered at 6307Å in
multiple spectra of $\eta$ Car, recorded by HST/STIS and by VLT/UVES. They
were unable to identify the origin of the emission line, but demonstrated that
the line was present across the high state (defined by presence of forbidden
lines of doubly-ionized elements, see Damineli et al. (2008) and references
therein) and disappeared during the low state.
Recently Gull et al. (2009), using the same spectra, focused on the spatially-
extended forbidden line emission both from high-ionization (herein defined as
$>$14 eV) and low-ionization (8-13 eV). We found that the forbidden emission
originated from 1) the Weigelt condensations (Weigelt & Ebersberger, 1986), as
narrow lines centered on $-$43 km s-1, 2) the boundaries of the primary wind
($\eta$ Car A) as rope-like [Fe ii], photo-ionized by mid-UV and collisionally
excited at densities around Nc=107cm-3, and 3) the wind interaction region, as
high-ionization emission from [N ii], [Fe iii], [Ar iii], [Ne iii] and [S
iii], photo-ionized by far-UV and collisionally excited for densities, nc
ranging from 105 to 108cm-3. The high-ionization emission lines are present
both for the Weigelt condensations and the wind interaction region during the
5 year high state, but every 5.5 years, disappear during the low state. X-ray
models (Pittard & Corcoran, 2002; Parkin et al., 2009) place the binary
periastron event near the onset of the low state with a three to six month
recovery.
We applied a 3D SPH (smoothed particle hydrodynamics) model (Okazaki et al.,
2008) extended out to 1700 AU (0$\farcs$67) to match the spatial structure
seen in these forbidden lines and realized that the bulk of the high-
ionization emission structure originated from the wind interaction region in
the outer portion of the massive wind structure. We found that portions of the
wind interaction structure, moving ballistically outward, are directly
illuminated by the far-UV radiation of the hot secondary, $\eta$ Car B,
leading to highly-ionized, collisionally excited gas and hence the high-
ionization extended emission.
Further examination of the HST/STIS longslit spectra showed that the
previously unidentified emission at 6307Å is blue-shifted [S iii]
$\lambda$6313 emission from the interacting wind structure. The evidence is
subtle, but convincing. We summarize the observations in Section 2. A
description of how the spectro-images were produced is in Section 3. The
forbidden emission structures are described in Section 4. Discussion in
Section 5 provides insight on the ionization and excitation leading to [S iii]
emission and the potential for monitoring changes with orbital phase,
including mapping temperature with density dependence. We conclude with a
summary in Section 6. Throughout this paper, all wavelengths are in vacuum,
the velocities are heliocentric, directions are compass points (N=north, NNW=
north by northwest and the phase of the binary orbit is referenced to the
X-ray minimum beginning at 1997.9604 (Corcoran, 2005).

##  2 The HST/STIS Observations

Figure 1: HST/STIS aperture positions: Left: Centered on $\eta$ Car. Right:
Centered on Weigelt D. The 2″$\times$2″ Field of View is extracted from an
HST/ACS image recorded in February 2003 through the 550M filter. Weigelt
condensations B, C and D are indicated by the black dots. The projection of
the 52″$\times$0$\farcs$1 aperture is indicated. Note the field is rotated by
69° placing the aperture vertical. North is indicated by the compass.

The spectra discussed here are a portion of the Eta Carinae Treasury
observations accessible through the STScI archives (
http://archive.stsci.edu/prepds/etacar) as reduced by a special reduction tool
developed by K. Ishibashi and K. Davidson. For brevity we focus on two sets of
observations, recorded in July 2002 ($\phi$=0.820) and July 2003
($\phi$=1.001).
Observations were recorded with the HST/STIS moderate dispersion gratings and
CCD detector through the 52″$\times$0$\farcs$1 aperture. We wanted to monitor
the change in both $\eta$ Car and the Weigelt condensations (Weigelt &
Ebersberger, 1986), which drop in excitation during the low state. However the
range in aperture position angle (PA) is limited by the required orientation
of HST solar panels and changes throughout the year. can prevent inclusion of
any one of the three Weigelt condensations within the aperture when centered
on $\eta$ Car (Figure 1). For observations centered around the X-ray minimum,
predicted to be around 1 July, 2003, we scheduled a visit one year earlier,
July 2, 2002 (orbital phase, $\phi$=0.820), at a pre-selected PA=69°, that
would be accessible just before($\phi$=0.995) and after the X-ray minimum
($\phi$=1.001). During all three visits, separate observations centered on
$\eta$ Car and Weigelt D were obtained with the aperture placed as shown in
Figure 1. Additional information on the observations are presented in Martin
et al. (2006) and Gull et al. (2009).

##  3 Spatially-resolved Emission

The HST/STIS spatial-resolution (0$\farcs$1 at H$\alpha$) separates the
spectrum of $\eta$ Car’s core from extended structures, especially the narrow
line emission that originates from the Weigelt condensations (Davidson et al.,
1995), located between 0$\farcs$1 to 0$\farcs$3 in the NW quadrant relative to
$\eta$ Car (see Figure 1). As described by Gull et al. (2009), we found
considerable differences between many broad forbidden emission line profiles
of $\eta$ Car as recorded by the VLT/UVES and the HST/STIS. Extractions with a
0$\farcs$127-high slice of the STIS spectra (five half rows in the reduced
spectro-images) yielded broad wind line profiles for H i, He I and Fe ii lines
that compared favorably with those recorded by VLT/UVES. In contrast, broad
profiles of forbidden lines recorded by VLT/UVES were nearly absent in the
HST/STIS extractions centered on $\eta$ Car. Examination of the HST/STIS long
aperture spectra revealed faint structure in these emission lines extending
out to 0$\farcs$7\. However, the observed emission was highly variable both
with aperture PA and orbital phase, $\phi$. Clearly the forbidden line
emission is spatially extended on scales resolved by HST/STIS but not by
VLT/UVES.
We examined individual lines in more detail and attempted to enhance
visibility of the extended line emission by several reduction procedures.
While various spatial filters were tried, the best results were obtained by
subtraction of measured continuum on a spatial row-by-row basis. We used
spectral plots of the Weigelt condensations (Zethson, 2001) to identify 10 to
20Å intervals with no obvious presence of narrow or broad line emission. At
each position along the aperture, we measured and subtracted the average
continuum in that spectral interval. Examples of the resulting spectro-images
(intensity images with x$=$velocity and y$=$angular size) are presented in
Figure 2.

Figure 2: Spectro-images centered on $\eta$ Car (Columns 1 and 2) and on
Weigelt D (Columns 3 and 4). [Fe II] $\lambda$4815 (Row 1) [Fe III]
$\lambda$4703 (Row 2) [N II] $\lambda$5756 (Row 3) and [S III]
$\lambda\lambda$6313, 9071, 9533 (Rows 4$-$6) Note: Continuum in regions free
of narrow or broad-line emission has been subtracted on a spatial line-by-line
basis to display the extended emission structure. All plots are with a grey
level proportional to $\sqrt{Intensity}$. All observations were recorded with
PA=69°.

These spectro-images provide only a qualitative view of the line profile. We
caution the reader that quantitative measures require much more precision for
the following reasons:

  1. 1.

The STIS utilized a three-axis mechanism to select a grating and to set the
correct tilt angle for the spectral interval of choice. While return to that
grating position is within a few CCD pixels, variations in the tilt are not
fully reproducible. A tilt of 1/20 pixel along the 1024 element row led to
significant photometric variation when attempting to extract spectra at the
0$\farcs$1 spatial resolution.

  2. 2.

The standard calibration for the STIS photometry is properly referenced to
extractions of a stellar spectrum with a 2″-wide aperture, allowing for full
capture of flux from a point source. Apertures with widths comparable to the
diffraction limit of HST sample the point spread function of the telescope,
which changes dynamically even within an orbit.

  3. 3.

Charge transfer inefficiency (CTI) leads to a trail in the direction of
columns and, with on-orbit time, increases. For complex sources like extended
structures, a proper extraction is not available.

These problems complicate attempts to show extended structure in the vicinity
of a bright star, which is exactly the situation with $\eta$ Car. This leads
to the obvious linear striations at the star position in the spectrally-
dispersed (velocity) coordinate. These variations do affect measures of the
extended emission closest to the stellar position. However, for offsets to
Weigelt D, the stellar flux is blocked by the aperture, and quantitative
measures are then possible. We note that the stellar spectrum scattered from
the direction of Weigelt D is very different from the direct spectrum of
$\eta$ Car. The lack of P Cygni absorption in H$\alpha$ across the high state
indicates fully-ionized hydrogen in the region spatially located between
$\eta$ Car and Weigelt D (Gull et al., 2009). On the side of caution, we limit
this discussion to a description of the spatial and velocity structure of the
lines. Even with qualitative descriptions, we gain much insight on the
spatially resolved wind interactions and the source of the 6307Å emission.

##  4 Description of the emission structures

We refer the reader to Figure 2 for the following descriptions of the
forbidden line emission. The first two columns of spectro-images are extracted
from spectra centered on $\eta$ Car in the high state ($\phi$=0.820) and early
in the low state ($\phi$=1.001). Likewise, columns 3 and 4 are centered on
Weigelt D (A more complete summary on variation of the emission structures
with ionization potential and orbital phase is presented by Gull et al.
(2009)).
Four basic structures contribute to these spectro-images:

  1. 1.

The central core of $\eta$ Car, not resolved by HST at 0$\farcs$1, which
contributes the bulk of the continuum and P Cygni wind lines, notably of H i
and Fe ii.

  2. 2.

Weigelt D and other, lesser condensations that contribute many narrow emission
lines centered at $-$40 km s-1.

  3. 3.

Rope-like structures of high-ionization forbidden emission lines with velocity
components extending from $+$200 to $-$500 km s-1.

  4. 4.

Noticeably more-diffuse, rope-like structures of low-ionization forbidden
emission lines, specifically [Fe ii].

Spectro-images of [Fe ii] $\lambda$4815 (Row 1) show narrow rope-like features
extending to 0$\farcs$7 at $-$500 km s-1and other, more diffuse structures
closer to the star extending $\pm$500 km s-1. The narrow emission at $-$40 km
s-1 originates from extended structure WSW of $\eta$ Car, not noted by Weigelt
& Ebersberger (1986), but present throughout the observational period from
1999 to 2004 whenever the STIS aperture sampled this position. The narrow [Fe
ii] emission centered on Weigelt D extends 0$\farcs$5 E and W of Weigelt D,
but at about 0$\farcs$25 E of D, a diffuse emission extends to $-$400 km
s-1and away from $\eta$ Car. During the low state, the outer [Fe ii] emission
drops, becomes more diffuse and is located closer to $\eta$ Car.
The [Fe iii] $\lambda$4703 (Row 2) is interior to the rope-like [Fe ii]
$\lambda$4815\. A series of highly filamentary loops extend from $\eta$ Car to
the east at velocities from $-$40 to $-$500 km s-1. No red-shifted velocity
components are seen at this PA. However, Gull et al. (2009) find that at
PA=$-$28°, observed six times from 1998.0 to 2004.3 ($\phi$=0.000 to 1.122),
red-shifted components extend to $+$200 km s-1 at early phases, but fade late
in the high state. All [Fe iii] $\lambda$4703 disappears during the low state.
The [N ii] $\lambda$5756 (Row 3) has very similar structure to that of [Fe
iii] $\lambda$4703 with higher S/N. A narrow emission line, [Fe ii]
$\lambda$5748, appears at the $-$550 km s-1 position and persists in the low
state, along with weak [N ii] $\lambda$5756\. The structure of [N ii]
$\lambda$5756 extends from $-$40 to $-500$ km s-1 in the spectro-image
centered on Weigelt D during the high state, but also disappears in the low
state.
Three [S iii] lines are shown in Rows 4$-$6 as each is important in accounting
for the 6307Å emission. Unfortunately, the [S iii] $\lambda$6313 (Row 4) was
recorded only at $\phi$=0.820, but the other two lines were observed at both
phases. Most noticeable in the spectro-image of $\eta$ Car at $\phi$=0.820 is
a knot of emission, centered on the stellar position at $-$400 km s-1. The
effective wavelength is 6307Å. That spectral interval was recorded at other
phases, but at other position angles, during the 2003.5 minimum with no [S
iii] $\lambda$6313 present either at the positions of $\eta$ Car or Weigelt D.
The extended structure is less apparent in the spectro-image of $\eta$ Car at
$\phi$=0.820, but is well-defined in the spectro-image centered on Weigelt D.
Narrow lines of [O i] $\lambda$6302, Fe ii $\lambda$$\lambda$6307, 6309 and
6319 contaminate the spectro-image and persist during the minimum.
The [S iii] $\lambda$ 9071 (Row 5) weak emission extends off of $\eta$ Car,
but several narrow lines (N i $\lambda$9063, Fe ii $\lambda$9073) also
contribute to the spectro-image. The high-velocity arc of [S iii] $\lambda$
9071 extends blueward from Weigelt D.
The [S iii] $\lambda$9533 emission (Row 6) is quite similar to that of [Fe ii]
$\lambda$4703 (Row 2) and confirms that [S iii] emission extends to $-$500 km
s-1. The bright emission to the red of [S iii] $\lambda$9533 is H i Pa 8
$\lambda$9548, which originates primarily from the central core.

##  5 Discussion

We associate the low-ionization structure with the massive, slow-moving wind
of $\eta$ Car A. The high-ionization emission is from the interacting wind
region piled up by the fast-moving, less-massive wind of $\eta$ Car B (Pittard
& Corcoran, 2002). The bulk of the interacting wind, by its velocity, appears
to be mostly ionized wind of $\eta$ Car A. The higher velocity side of the
shock is likely less dense and more highly ionized by $\eta$ Car B.
Using the 3D SPH models of Okazaki et al. (2008) and simple geometric models,
we determined that the high-ionization emission originates from a distorted
paraboloidal structure lying in the skirt of the Homunculus. Based upon the
blue-shifted velocities and near symmetry for PAs ranging from $+$22 to
$+$38°, the paraboloid points in our general direction with axis of rotation
projecting onto the sky at PA$\approx-$25°.
Martin et al. (2006) performed a very complete analysis on the unidentified
6307Å line in both the HST/STIS and VLT/UVES spectra, finding very similar
behavior with orbital phase. Their search of possible line identifications
focused primarily on singly-ionized species such as Fe+, V+ and S+, although
they do list [S iii] as a narrow nebular line identified by Zethson (2001) in
the spectrum of the Weigelt condensations. Their candidate of greatest
interest appeared to be Fe iii] $\lambda$6306.43 with unknown atomic data for
the transition.
Most important in their analysis was the tracking of the strength of the
emission throughout the 5.5-year orbit. They found that the line disappeared
during the low state, but might be anti-correlated with Fe ii $\lambda$5529\.
Both suggest a high-ionization source. Nielsen et al. (2007) analyzed the
behavior of the He i absorption, finding an anti-correlation with Fe ii
absorption.
Salient are three facts:

  1. 1.

On the star, both HST/STIS and VLT/UVES see the same emission bump with
similar strengths.

  2. 2.

The emission correlates with high-ionization variations, not the behavior of
the low-ionization emission of Fe ii.

  3. 3.

The extended emission of the extended emission correlated with [S iii]
correlates very well with the extended emission identified with [S iii]
$\lambda\lambda$9071 and 9533.

##  6 Conclusions

We have presented conclusive evidence that the emission line at 6307Å, noted
in the spectra of $\eta$ Car by Martin et al. (2006) is blue-shifted emission
of [S iii] $\lambda$6313 originating from the distorted paraboloidal,
interaction region located between the massive binary members. While the
massive primary, $\eta$ Car A, provides the dominant wind ejecta, 10-3
$M_{\odot}$/y at 500 km s-1, the hotter secondary provides a less massive,
faster wind, 10-5 $M_{\odot}$/y at 3000 km s-1, and far-UV photons that ionize
iron, neon, argon and sulfur to doubly ionized states. Thermal collisions,
mid-UV photons and possible charge exchange excite the doubly-ionized species
to upper states with forbidden transitions leading to forbidden line emission
in regions with densities close to nc. Specifically, [S iii]
$\lambda$$\lambda$9533, 9071 and 6313 have extended spatial structures. The
intensity ratio (Flux ($\lambda$9533) + Flux
($\lambda$9071)/Flux($\lambda$6313) leads to density estimates ranging from
107 \- 108 cm-3on the scale of 0$\farcs$1, the limit of HST/STIS spatial
capabilities. Mapping in these and other doubly-ionized lines will provide
powerful measures for models of wind interactions using various 3-D
hydrodynamical codes.

The observations were accomplished with the NASA/ESA Hubble Space Telescope.
Support for Program numbers 7302, 8036, 8483, 8619, 9083, 9337, 9420, 9973,
10957 and 11273 was provided by NASA directly to the Space Telescope Imaging
Spectrograph Science Team and through grants from the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Incorporated, under NASA contract NAS5-26555. All analysis was
done using STIS IDT software tools on data available through the HST $\eta$
Car Treasury public archive.

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