#  MS-TP-20-42 Xenon-1T excess as a possible signal of a sub-GeV hidden sector
dark matter

Amin Aboubrahima111Email<EMAIL_ADDRESS>Michael Klasena222Email:
<EMAIL_ADDRESS>and Pran Nathb333Email:
<EMAIL_ADDRESS>

aInstitut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,
Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
bDepartment of Physics, Northeastern University, Boston, MA 02115-5000, USA

###### Abstract

We present a particle physics model to explain the observed enhancement in the
Xenon-1T data at an electron recoil energy of 2.5 keV. The model is based on a
$U(1)$ extension of the Standard Model where the dark sector consists of two
essentially mass degenerate Dirac fermions in the sub-GeV region with a small
mass splitting interacting with a dark photon. The dark photon is unstable and
decays before the big bang nucleosynthesis, which leads to the dark matter
constituted of two essentially mass degenerate Dirac fermions. The Xenon-1T
excess is computed via the inelastic exothermic scattering of the heavier dark
fermion from a bound electron in xenon to the lighter dark fermion producing
the observed excess events in the recoil electron energy. The model can be
tested with further data from Xenon-1T and in future experiments such as
SuperCDMS.

##  1 Introduction

Recently the Xenon-1T experiment [1] has analyzed events in the low-energy
region of 1–30 keV of electron recoil energy with an exposure of 0.65 tonne-
years, while claiming a low background rate of $76\pm 2_{\,\mathrm{stat}}$
events/(tonne-year-keV). The experiment observed an excess of recoil electrons
over the background in the 2$-$3 keV range. The collaboration analyzed the
axion couplings to electrons, photons and nucleons, and the neutrino magnetic
moment as possible sources for the signal. However, these models appear to be
in strong tension with stellar constraints [2, 3, 4, 5]. Another possible
source of this excess is traces of tritium in xenon of size $(6.2\pm
2.0)\times 10^{-25}$ mol/mol. The experiment currently can neither confirm nor
exclude such a possibility. Since the publication of the Xenon-1T results, a
variety of models have been proposed which include light sterile neutrinos [6,
7, 8, 9], a goldstino [10], an inflaton [11], string-motivated models [12,
13], boosted dark matter [14, 15, 16], and a variety of other models [17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41].

In this work we discuss the possibility that the observed effect is a signal
from dark matter in the hidden sector. While there are models in the
literature where the hidden sector is used for the explanation of the Xenon-1T
excess, our analysis differs from them in several ways. In our analysis we use
the Stueckelberg extension of the Standard Model where the Stueckelberg sector
consists of an additional $U(1)$ gauge boson interacting via kinetic mixing
with the visible sector. In many previous works a single Dirac fermion is used
which is then split into two Majoranas which are given different masses [32,
30]. In our analysis we consider two Dirac fermions in the hidden sector
carrying $U(1)$ charges with a small mass splitting and interacting with the
dark photon generated by the Stueckelberg mechanism. In the analysis, both the
freeze-in and freeze-out mechanisms operate to generate the desired relic
density. The analysis given here satisfies all the relevant constraints on
kinetic mixing between the hidden sector and the visible sector and on the
dark photon mass from the CRESST 2019 DM-nucleon scattering cross section,
from the neutrino experiment CHARM [42] whose results are reinterpreted as
limits on the dark photon [43] as well as from the Planck relic density
experiment. The outline of the rest of the paper is as follows: In section 2
we discuss the model to explain the Xenon-1T result. In section 3 an analysis
of the dark matter relic density is given. A discussion of the inelastic dark
matter-electron scattering is given in section 4. Event detection rates in the
Xenon-1T detector are discussed in section 5. Constraints on the model and a
fit to the Xenon-1T data is given in section 6. Our conclusions are given in
section 7. Several details of the calculation are given in the Appendix.

##  2 Stueckelberg extension with hidden sector dark fermions

We extend the Standard Model (SM) gauge group by an extra $U(1)_{X}$ under
which the SM is neutral. The extra gauge field $C^{\mu}$ mixes with the SM
$U(1)_{Y}$ hypercharge $B^{\mu}$ via kinetic mixing [44]. Further, we use the
Stueckelberg mechanism [45] to generate mass for the gauge boson of the hidden
sector. The total Lagrangian is then given by

$\mathcal{L}=\mathcal{L}_{\rm SM}+\Delta\mathcal{L},$ (1)

with

$\displaystyle\Delta\mathcal{L}\supset$
$\displaystyle-\frac{1}{4}C_{\mu\nu}C^{\mu\nu}-\frac{\delta}{2}C_{\mu\nu}B^{\mu\nu}+g_{X}J^{\mu}_{X}C_{\mu}-\frac{1}{2}(\partial_{\mu}\sigma+M_{1}C_{\mu}+M_{2}B_{\mu})^{2},$
(2)

where $g_{X}$ is the gauge coupling in the hidden sector, $J_{X}$ is the
hidden sector current and $\sigma$ is a pseudoscalar field which is absorbed
in a gauge-invariant way via the Stueckelberg mechanism to give mass to the
extra neutral gauge boson which we call $\gamma^{\prime}$ (the dark photon).
Further one may introduce matter in the hidden sector which is neutral under
$U(1)_{Y}$ but charged under $U(1)_{X}$ [45, 46]. More generally one may have
both kinetic mixing and mass mixing [47].

The kinetic energy terms in Eqs. (1) and (2) can be diagonalized by a
$GL(2,\mathbb{R})$ transformation

$\displaystyle\left(\begin{matrix}C^{\mu}\cr
B^{\mu}\end{matrix}\right)=\left(\begin{matrix}c_{\delta}&0\cr-
s_{\delta}&1\end{matrix}\right)\left(\begin{matrix}C^{\prime\mu}\cr
B^{\prime\mu}\end{matrix}\right),$ (3)

where $c_{\delta}=1/(1-\delta^{2})^{1/2}$ and
$s_{\delta}=\delta/(1-\delta^{2})^{1/2}$.

In the Standard Model the neutral gauge boson sector arises from the
hypercharge gauge boson $B^{\mu}$ and the third component of the $SU(2)_{L}$
gauge field $A^{\mu}_{a}$ ($a=$ 1$-$3), which leads to a $2\times 2$ mass
squared matrix after spontaneous symmetry breaking. It contains one massless
mode, the photon, and a massive mode, the $Z$-boson. Inclusion of the
Stueckelberg gauge field $C_{\mu}$ enlarges the $2\times 2$ mass squared
matrix of the neutral gauge boson sector in the standard model to a $3\times
3$ mass squared matrix in the Stueckelberg extended model. Thus after
spontaneous electroweak symmetry breaking and the Stueckelberg mass growth,
and on including the $GL(2,\mathbb{R})$ transformation to obtain a diagonal
and a normalized kinetic energy for the gauge bosons, the $3\times 3$ mass
squared matrix of neutral vector bosons in the basis
$(C^{\prime}_{\mu},B^{\prime}_{\mu},A^{3}_{\mu})$ is given by

$\displaystyle\mathcal{M}^{2}_{V}=\left(\begin{matrix}M_{1}^{2}\kappa^{2}+\frac{1}{4}g^{2}_{Y}v_{H}^{2}s^{2}_{\delta}&\kappa\epsilon
M_{1}^{2}-\frac{1}{4}g^{2}_{Y}v_{H}^{2}s_{\delta}&\frac{1}{4}g_{Y}g_{2}v_{H}^{2}s_{\delta}\cr\kappa\epsilon
M_{1}^{2}-\frac{1}{4}g^{2}_{Y}v_{H}^{2}s_{\delta}&\epsilon^{2}M_{1}^{2}+\frac{1}{4}g^{2}_{Y}v_{H}^{2}&-\frac{1}{4}g_{Y}g_{2}v_{H}^{2}\cr\frac{1}{4}g_{Y}g_{2}v_{H}^{2}s_{\delta}&-\frac{1}{4}g_{Y}g_{2}v_{H}^{2}&\frac{1}{4}g^{2}_{2}v_{H}^{2}\cr\end{matrix}\right),$
(4)

where $g_{2}$ is the $SU(2)_{L}$ gauge coupling, $\kappa=(c_{\delta}-\epsilon
s_{\delta})$, $\epsilon=M_{2}/M_{1}$ and $v_{H}$ is the Higgs VEV. The mass-
squared matrix of Eq. (4) has one zero eigenvalue which is the photon, while
the other two eigenvalues are

$M^{2}_{\pm}=\frac{1}{2}\left\\{M^{2}_{0}\pm\sqrt{M_{0}^{4}-M_{1}^{2}v_{H}^{2}\Big{[}(\kappa^{2}+\epsilon^{2})g_{2}^{2}+g^{2}_{Y}c^{2}_{\delta}\Big{]}}~{}\right\\},$
(5)

where
$M^{2}_{0}=(\kappa^{2}+\epsilon^{2})M_{1}^{2}+\dfrac{1}{4}v_{H}^{2}(g_{Y}^{2}c^{2}_{\delta}+g_{2}^{2})$.
Here $M_{-}$ is identified as the $\gamma^{\prime}$ boson mass, while $M_{+}$
as the $Z$ boson. The diagonalization of the mass-squared matrix of Eq. (4)
can be done via two orthogonal transformations, where the first is given by
[47]

$\displaystyle\mathcal{O}=\left(\begin{matrix}1/c_{\delta}&-s_{\delta}/c_{\delta}&0\cr
s_{\delta}/c_{\delta}&1/c_{\delta}&0\cr 0&0&1\cr\end{matrix}\right),$ (6)

which transforms the mass matrix to
$\mathcal{M^{\prime}}^{2}_{V}=\mathcal{O}^{T}\mathcal{M}^{2}_{V}\mathcal{O}$,

$\displaystyle\mathcal{M^{\prime}}^{2}_{V}=\left(\begin{matrix}M_{1}^{2}&M_{1}^{2}\bar{\epsilon}&0\cr
M_{1}^{2}\bar{\epsilon}&M_{1}^{2}\bar{\epsilon}^{2}+\frac{1}{4}g^{2}_{Y}v_{H}^{2}c^{2}_{\delta}&-\frac{1}{4}g_{Y}g_{2}v_{H}^{2}c_{\delta}\cr
0&-\frac{1}{4}g_{Y}g_{2}v_{H}^{2}c_{\delta}&\frac{1}{4}g^{2}_{2}v_{H}^{2}\cr\end{matrix}\right),$
(7)

where $\bar{\epsilon}=\epsilon c_{\delta}-s_{\delta}$. The gauge eigenstates
of $\mathcal{M^{\prime}}^{2}_{V}$ can be rotated into the corresponding mass
eigenstates $(\gamma^{\prime},Z,\gamma)$ using the second transformation such
that
$\mathcal{R}^{T}\mathcal{M^{\prime}}^{2}_{V}\mathcal{R}=\text{diag}(m^{2}_{\gamma^{\prime}},m^{2}_{Z},0)$
where the rotation matrix is given by

$\displaystyle\mathcal{R}=\left(\begin{matrix}\cos\psi\cos\phi-\sin\theta\sin\phi\sin\psi&\sin\psi\cos\phi+\sin\theta\sin\phi\cos\psi&-\cos\theta\sin\phi\cr\cos\psi\sin\phi+\sin\theta\cos\phi\sin\psi&\sin\psi\sin\phi-\sin\theta\cos\phi\cos\psi&\cos\theta\cos\phi\cr-\cos\theta\sin\psi&\cos\theta\cos\psi&\sin\theta\cr\end{matrix}\right).$
(8)

Here the mixing angles are given by

$\tan\phi=\bar{\epsilon},~{}~{}~{}\tan\theta=\frac{g_{Y}}{g_{2}}c_{\delta}\cos\phi,$
(9)

and

$\tan
2\psi\simeq\frac{2\bar{\epsilon}m^{2}_{Z}\sin\theta}{m^{2}_{\gamma^{\prime}}-m^{2}_{Z}+(m^{2}_{\gamma^{\prime}}+m^{2}_{Z}-m^{2}_{W})\bar{\epsilon}^{2}},$
(10)

where $m_{W}=g_{2}v_{H}/2$, $m_{\gamma^{\prime}}\equiv M_{-}$ and $m_{Z}\equiv
M_{+}$. Since the dark photon mixes with the SM gauge bosons, it will couple
with the SM fermions and so

$\mathcal{L}_{\rm
SM}=\frac{g_{2}}{2\cos\theta}\bar{\psi}_{f}\gamma^{\mu}\Big{[}(v_{f}-\gamma_{5}a_{f})Z_{\mu}+(v^{\prime}_{f}-\gamma_{5}a^{\prime}_{f})A^{\gamma^{\prime}}_{\mu}\Big{]}\psi_{f}+e\bar{\psi}_{f}\gamma^{\mu}Q_{f}A_{\mu}\psi_{f},$
(11)

where $f$ runs over all SM fermions and the vector and axial couplings are
given by

$\displaystyle v_{f}$
$\displaystyle=\cos\psi[(1-\bar{\epsilon}\tan\psi\sin\theta)T_{3f}-2\sin^{2}\theta(1-\bar{\epsilon}\csc\theta\tan\psi)Q_{f}],$
(12) $\displaystyle a_{f}$
$\displaystyle=\cos\psi(1-\bar{\epsilon}\tan\psi\sin\theta)T_{3f},$
$\displaystyle v^{\prime}_{f}$
$\displaystyle=-\cos\psi[(\tan\psi+\bar{\epsilon}\sin\theta)T_{3f}-2\sin^{2}\theta(\bar{\epsilon}\csc\theta+\tan\psi)Q_{f}],$
$\displaystyle a^{\prime}_{f}$
$\displaystyle=-\cos\psi(\tan\psi+\bar{\epsilon}\sin\theta)T_{3f}.$

Here $T_{3f}$ is the third component of the isospin and $Q_{f}$ is the
electric charge.

We assume that the hidden sector where $C^{\mu}$ resides contains two mass
degenerate Dirac fermions $D_{1}$ and $D_{2}$ with the common mass $\mu$
which, however, carry different charges $Q_{1}$ and $Q_{2}$ under the
$U(1)_{X}$ gauge group. The interaction Lagrangian for the hidden sector is
then given by

$\displaystyle\mathcal{L}^{\rm int}_{D}=$
$\displaystyle-g^{\gamma^{\prime}}_{X}Q_{1}\bar{D}_{1}\gamma^{\mu}D_{1}A_{\mu}^{\gamma^{\prime}}-g^{\gamma^{\prime}}_{X}Q_{2}\bar{D}_{2}\gamma^{\mu}D_{2}A_{\mu}^{\gamma^{\prime}}-g^{Z}_{X}Q_{1}\bar{D}_{1}\gamma^{\mu}D_{1}Z_{\mu}-g^{Z}_{X}Q_{2}\bar{D}_{2}\gamma^{\mu}D_{2}Z_{\mu},$
(13)

with
$g^{\gamma^{\prime}}_{X}=g_{X}(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})$
and $g^{Z}_{X}=g_{X}(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})$, where
$\mathcal{R}_{ij}$ are elements of the matrix in Eq. (8). To generate
inelastic scattering we need to split the masses of the $D$-fermions. To this
end we add a $U(1)_{X}$ gauge violating mass terms
$\Delta\mu(\bar{D}_{1}D_{2}+\text{h.c.})$ so that the Lagrangian for the
$(D_{1},D_{2})$ mass terms is given by

$\displaystyle\mathcal{L}^{\rm
mass}_{D}=-\mu(\bar{D}_{1}D_{1}+\bar{D}_{2}D_{2})-\Delta\mu(\bar{D}_{1}D_{2}+\bar{D}_{2}D_{1}),$
(14)

We can now go to the mass diagonal basis with Dirac fermions $D_{1}^{\prime}$
with mass $m_{1}=\mu-\Delta\mu$ and $D_{2}^{\prime}$ with mass
$m_{2}=\mu+\Delta\mu$ and we assume $m_{2}>m_{1}$ so $D_{2}^{\prime}$ is the
heavier of the two dark fermions. In this basis, Eq. (13) takes the form

$\displaystyle-\mathcal{L}^{\rm int}_{D}$
$\displaystyle=\frac{1}{2}g^{\gamma^{\prime}}_{X}(Q_{1}+Q_{2})\left(\bar{D}_{1}^{\prime}\gamma^{\mu}D_{1}^{\prime}+\bar{D}_{2}^{\prime}\gamma^{\mu}D_{2}^{\prime}\right)A_{\mu}^{\gamma^{\prime}}$
(15)
$\displaystyle+\frac{1}{2}g^{\gamma^{\prime}}_{X}(Q_{1}-Q_{2})(\bar{D}_{1}^{\prime}\gamma^{\mu}D_{2}^{\prime}+\bar{D}_{2}^{\prime}\gamma^{\mu}D_{1}^{\prime})A_{\mu}^{\gamma^{\prime}}$
$\displaystyle+\frac{1}{2}g^{Z}_{X}(Q_{1}+Q_{2})\left(\bar{D}_{1}^{\prime}\gamma^{\mu}D_{1}^{\prime}+\bar{D}_{2}^{\prime}\gamma^{\mu}D_{2}^{\prime}\right)Z_{\mu}$
$\displaystyle+\frac{1}{2}g^{Z}_{X}(Q_{1}-Q_{2})(\bar{D}_{1}^{\prime}\gamma^{\mu}D_{2}^{\prime}+\bar{D}_{2}^{\prime}\gamma^{\mu}D_{1}^{\prime})Z_{\mu}.$

From Eq. (11) and Eq. (15) we note that the dark photon has couplings with
both the visible sector and the hidden sector. Thus from Eq. (11) we find that
the dark photon couples with quarks and leptons in the visible sector while
from Eq. (15) we find that the dark photon has couplings with
$D_{1}^{\prime},D_{2}^{\prime}$ in the dark sector. These couplings allow for
an inelastic scattering process to occur where a dark fermion hits a bound
electron in a xenon atom producing, in an exothermic process, a recoil
electron with excess energy, i.e.,

$\displaystyle e+D_{2}^{\prime}\to e^{\prime}+D_{1}^{\prime},$ (16)

where the final electron receives an extra boost in energy from the mass
difference $\Delta m=m_{2}-m_{1}=2\Delta\mu$.

##  3 Dark matter relic density

Since the hidden sector matter has feeble couplings with the Standard Model
particles, they are never in thermal equilibrium with the visible sector and
the usual freeze-out analysis for the computation of the relic density does
not apply. However, the hidden sector particles can be produced via the
annihilation of the Standard Model particles into dark photons and dark
fermions via these feeble interactions and the computation of the relic
density in this case is computed using the freeze-in mechanism [48]. Within
the dark sector itself the dark fermions and the dark photons interact via
normal size interactions and are in thermal equilibrium up to a certain
freeze-out temperature $T_{f}$ via the process
$D\bar{D}\to\gamma^{\prime}\gamma^{\prime}$. However, below the dark sector
freeze-out temperature $T_{f}$, the dark fermions and the dark photons
decouple and the dark photons decay to the visible sector much before the big
bang nucleosynthesis, which leaves the dark fermions as the only DM
candidates. In our model we assume that the visible and hidden sectors have
the same temperature. One can consider sectors with different temperatures in
the early universe, but it has been shown that the two sectors will eventually
thermalize and that the effect on the relic density is minimal [49].

For calculating the DM relic density of $D^{\prime}_{1}$ and $D^{\prime}_{2}$,
we assume $m_{1}\simeq m_{2}\simeq m_{D}$ and write only one Boltzmann
equation for the dark fermion. In this limit, we have

$\frac{dY_{D}}{dx}\approx-1.32M_{\rm Pl}\frac{h_{\rm eff}(T)}{g^{1/2}_{\rm
eff}(T)}\frac{m_{D}}{x^{2}}\left(-\langle\sigma v\rangle_{D\bar{D}\to
i\bar{i}}Y_{D}^{\rm eq^{2}}+\langle\sigma
v\rangle_{D\bar{D}\to\gamma^{\prime}\gamma^{\prime}}Y^{2}_{D}\right),$ (17)

where $Y_{D}=n/s$ is the comoving number density (or yield) of DM, $h_{\rm
eff}$ and $g_{\rm eff}$ are the entropy and energy density numbers of degrees
of freedom, $M_{\rm Pl}$ is the reduced Planck mass ($M_{\rm Pl}\sim 2.4\times
10^{16}$ GeV) and $x=m_{D}/T$. The first term on the right-hand-side of Eq.
(17) is the production of dark matter particles via the freeze-in mechanism
and the second term produces DM depletion. Here the thermally averaged cross-
section is given by

$\langle\sigma v\rangle^{D\bar{D}\to
ab}(x)=\frac{x}{8m^{5}_{D}K^{2}_{2}(x)}\int_{4m_{D}^{2}}^{\infty}ds~{}\sigma(s)\sqrt{s}\,(s-4m_{D}^{2})K_{1}\left(\frac{\sqrt{s}}{m_{D}}x\right),$
(18)

while the equilibrium yield is given by

$Y_{D}^{\rm eq}(x)=\frac{45}{4\pi^{4}}\frac{g_{D}}{h_{\rm
eff}(T)}x^{2}K_{2}(x).$ (19)

Here $g_{D}$ is the dark fermion number of degrees of freedom, $K_{1}$ and
$K_{2}$ are the modified second order Bessel functions of degree one and two.
The Boltzmann equation Eq. (17) is solved numerically to determine the yield
at the present time $Y_{\infty}$ which gives us the relic density

$\Omega h^{2}=\frac{m_{D}Y_{\infty}s_{0}h^{2}}{\rho_{c}},$ (20)

where $s_{0}$ is today’s entropy density, $\rho_{c}$ is the critical density
and $h=0.678$ denotes the present Hubble expansion rate in units of 100 km s-1
Mpc-1.

We select three benchmarks with different masses for the dark photon and for
the dark fermions. The benchmarks are listed in Table 1 along with the
couplings, the relic density and the inelastic DM-electron scattering cross-
section. In the analysis, we set $\epsilon=0$, i.e. we assume no mass mixing.

Model |  $m_{D}$ (GeV) |  $m_{\gamma^{\prime}}$ (MeV) |  $g_{X}$ |  $\delta$ |  $\Omega h^{2}$ |  $\bar{\sigma}_{e}$ (cm2)
---|---|---|---|---|---|---
(a) |  1.00 |  55 |  0.040 |  $4.0\times 10^{-5}$ |  0.125 |  $2.80\times 10^{-44}$
(b) |  0.50 |  60 |  0.025 |  $6.0\times 10^{-5}$ |  0.121 |  $1.75\times 10^{-44}$
(c) |  0.30 |  300 |  0.055 |  $7.5\times 10^{-4}$ |  0.116 |  $1.19\times 10^{-44}$
Table 1: Input parameters, the relic density $\Omega h^{2}$ and the inelastic
DM-electron scattering cross section $\bar{\sigma}_{e}$ for the benchmarks
used in this analysis.

The DM relic density satisfies, within theoretical uncertainties, the
experimental value from the Planck Collaboration [50]

$(\Omega h^{2})_{\rm Planck}=0.1198\pm 0.0012.$ (21)

Following the dark freeze-out of DM in the hidden sector, conversion processes
$D^{\prime}_{2}\bar{D}^{\prime}_{2}\longleftrightarrow
D^{\prime}_{1}\bar{D}^{\prime}_{1}$ remain active. The ratio of the number
densities of $D^{\prime}_{1}$ and $D^{\prime}_{2}$ is Boltzmann suppressed,
i.e., $n_{2}/n_{1}\sim\exp(-\Delta m/T_{c})$, where $T_{c}$ is the temperature
below which conversion processes shut off. One can safely assume that
$n_{2}\sim n_{1}$ as long as $T_{c}\gg\Delta m$. Focusing on the process
$D^{\prime}_{2}\bar{D}^{\prime}_{2}\longrightarrow
D^{\prime}_{1}\bar{D}^{\prime}_{1}$, we determine $T_{c}$ for which

$n_{2}\langle\sigma
v\rangle_{D^{\prime}_{2}\bar{D}^{\prime}_{2}\longrightarrow
D^{\prime}_{1}\bar{D}^{\prime}_{1}}/H\sim 1,$ (22)

where $H$ is the Hubble parameter. In Fig. 1 we plot $n\langle\sigma v\rangle$
versus $x$ for benchmarks (a) (left panel) and (c) (right panel). The figure
shows the conversion process (blue) along with two other processes,
$D\bar{D}\to i\bar{i}$ (red) and $D\bar{D}\to\gamma^{\prime}\gamma^{\prime}$
(yellow) and the Hubble parameter (purple). For benchmark (a), one finds that
for the value of the kinetic mixing chosen the DM remains out of equilibrium
with the SM for the entire $x$ range, while increasing the kinetic mixing
causes such a process to enter in equilibrium for a range of $x$ as shown for
benchmark (c). This process, however, decouples before other decoupling
processes.

Figure 1: A plot of $n\langle\sigma v\rangle$ and the Hubble parameter $H(T)$
as a function of $x=m_{D}/T$ for benchmark (a) (left panel) and (c) (right
panel). Dark freeze-out sets in before the DM conversion process shuts off for
model (a). The analysis of model (b) is similar to that of model (a), while
both processes occur nearly at the same temperature for model (c).

As also shown in Fig. 1, dark freeze-out occurs for $x\sim 20$ for all
benchmarks. Turning to the conversion process, for benchmark (a), such a
process shuts off at $x\sim 1.12\times 10^{4}$, which for the DM mass of
benchmark (a) corresponds to $T_{c}\sim 90$ keV. The mass gap of interest here
is $\Delta m\sim 2.8$ keV which makes the number densities of $D^{\prime}_{1}$
and $D^{\prime}_{2}$ almost equal. One can see that the conversion process
shuts off much later than the other ones. This is also observed for benchmark
(b). Unlike (a) and (b), benchmark (c) shows the freeze-out and conversion
process shutting off occur at the same $x\sim 20$ which corresponds to a
temperature much larger than $\Delta m$. Thus for practical purposes
hereafter, we assume that the DM density is divided equally between
$D^{\prime}_{1}$ and $D^{\prime}_{2}$.

After the conversion process terminates, the dark fermion $D_{2}^{\prime}$ can
decay to $D_{1}^{\prime}$. The only decay channel would be $D_{2}^{\prime}\to
D_{1}^{\prime}\bar{\nu}\nu$, which is further suppressed by the dark photon
coupling to the neutrinos which is proportional to the kinetic mixing, and
further the decay is phase-space suppressed because of the small mass gap
$\Delta m\sim\mathcal{O}$(keV). The total 3-body decay width is (including
three neutrino generations)

$\Gamma_{D_{2}^{\prime}\to
D_{1}^{\prime}\nu\bar{\nu}}=\frac{x_{\nu}^{2}}{256\pi^{3}m^{4}_{\gamma^{\prime}}}\left[\frac{f(m_{1},m_{2})}{m_{2}^{3}}+24m_{1}^{3}(m_{1}^{2}+m_{1}m_{2}+m^{2}_{2})\log\left(\frac{m_{2}}{m_{1}}\right)\right],$
(23)

where

$f(m_{1},m_{2})=(m_{2}^{2}-m_{1}^{2})(m_{1}^{6}-2m_{2}m_{1}^{5}-7m_{2}^{2}m_{1}^{4}-20m_{2}^{3}m_{1}^{3}-7m_{2}^{4}m_{1}^{2}-2m_{2}^{5}m_{1}+m_{2}^{6}).$
(24)

Since $m_{1}=m_{2}-\Delta m$ and expanding in $\Delta m$, we get to lowest
order in $\Delta m$

$\Gamma_{D_{2}^{\prime}\to
D_{1}^{\prime}\nu\bar{\nu}}\simeq\frac{x_{\nu}^{2}(\Delta
m)^{5}}{40\pi^{3}m^{4}_{\gamma^{\prime}}},$ (25)

where for a small gauge kinetic mixing, $g_{X}^{\gamma^{\prime}}\approx g_{X}$
and

$x_{\nu}\sim
g_{X}g_{Y}(Q_{1}-Q_{2})\left(\frac{m_{\gamma^{\prime}}}{m_{Z}}\right)\delta.$
(26)

In this work we are interested in $\Delta m\sim 3$ keV and $\delta\sim
10^{-5}$ which results in a decay lifetime of $D_{2}^{\prime}$ order $10^{13}$
years which means that $D_{2}^{\prime}$ is stable over the lifetime of the
universe. Thus, in this model dark matter is constituted of two dark fermions
with essentially degenerate masses which are of order 1 GeV.

##  4 DM-electron scattering cross-section

A DM particle can undergo an elastic scattering with a bound electron in a
xenon atom, but such a scattering can deliver only a few eV to the electron
which is not sufficient to explain the Xenon-1T excess. However, an inelastic
exothermic down-scattering can impart a recoil energy to the electron
equivalent to the mass difference between the incoming and outgoing DM
particles. The model considered here allows for the desired small mass
splitting between the two Dirac fermions $D^{\prime}_{1}$ and $D^{\prime}_{2}$
so that the heavier fermion $D^{\prime}_{2}$ down-scatters to
$D^{\prime}_{1}$. Next, we compute the inelastic scattering cross-section of
the process described $D_{2}^{\prime}(\vec{p}_{1})+e(\vec{p}_{2})\to
D_{1}^{\prime}(\vec{p}_{3})+e^{\prime}(\vec{p}_{4})$. Assuming the dark photon
mass is much greater than the momentum transfer, the averaged matrix element
squared for this process is given by

$\displaystyle\overline{|\mathcal{M}|^{2}}$
$\displaystyle=\frac{2\bar{g}_{X}^{2}g_{2}^{2}}{m^{4}_{\gamma^{\prime}}\cos^{2}\theta}\Bigg{\\{}\frac{1}{2}(a_{f}^{\prime
2}-v_{f}^{\prime
2})\Big{[}(m_{1}-m_{2})^{2}-(t+2m_{1}m_{2})\Big{]}m^{2}_{e}+\frac{1}{4}(v_{f}^{\prime
2}+a_{f}^{\prime 2})\Big{[}(m_{2}^{2}+m_{e}^{2}-u)$
$\displaystyle\times(m_{1}^{2}+m_{e}^{2}-u)+(s-m_{1}^{2}-m_{e}^{2})(s-m_{2}^{2}-m_{e}^{2})-2m_{1}m_{2}(2m_{e}^{2}-t)\Big{]}\Bigg{\\}},$
(27)

where $\bar{g}_{X}=\frac{1}{2}g_{X}(Q_{1}-Q_{2})$ and $s,t,u$ are the
Mandelstam variables. The directional matrix element for free electron-DM
scattering is given by

$\overline{|\mathcal{M}(\vec{q})|^{2}}=\overline{|\mathcal{M}(q)|^{2}}\times|F_{DM}(q)|^{2},$
(28)

where the form factor $F_{DM}(q)$ can be taken as 1 for a small momentum
transfer. The electron-DM scattering differential cross-section is given by

$\displaystyle\frac{d\bar{\sigma}_{e}}{d\Omega}$
$\displaystyle=\frac{1}{64\pi^{2}s}\frac{|\vec{p}_{3}|}{|\vec{p}_{1}|}\overline{|\mathcal{M}(\vec{q})|^{2}}.$
(29)

Keeping the velocity dependence in the Mandelstam variables, we have

$t=-q^{2}\simeq-\Delta m\left(1-\sqrt{\frac{2m_{e}v^{2}}{\Delta
m}}\cos\theta_{\rm
CM}\right),~{}~{}~{}s\sim(m_{2}+m_{e})^{2}\left(1+\frac{\mu_{De}}{m_{2}+m_{e}}v^{2}\right),$
(30)

and integrating over $\theta_{\rm CM}$, the scattering angle in the CM frame,
and $\phi$, we get for the DM-e scattering cross-section

$\bar{\sigma}_{e}\simeq\frac{\bar{g}_{X}^{2}g_{2}^{2}}{16\pi
m^{4}_{\gamma^{\prime}}\cos^{2}\theta}\left(\frac{4\mu^{2}_{De}}{1+\frac{\mu_{De}}{m_{2}+m_{e}}v^{2}}\right)\left[v^{\prime
2}_{f}+(a_{f}^{\prime 2}+v^{\prime 2}_{f})v^{2}\right],$ (31)

where $\mu_{De}=\frac{m_{2}m_{e}}{m_{2}+m_{e}}$ is the dark matter-electron
reduced mass. For $v\sim 10^{-3}$, one can discard the velocity-dependent
piece and get

$\bar{\sigma}_{e}\simeq\frac{\bar{g}_{X}^{2}g_{2}^{2}}{4\pi\cos^{2}\theta}\frac{\mu^{2}_{De}}{m^{4}_{\gamma^{\prime}}}v_{f}^{\prime
2}.$ (32)

The values of the cross-section for the three benchmarks are shown in Table 1.
One finds that the cross-section depends on the gauge coupling in the dark
sector and on kinetic mixing which enters in the expression of
$v^{\prime}_{f}$. Such quantities are constrained by experiments which we
discuss in section 6.

##  5 Detection rate at Xenon-1T

We give here a quantitative analysis of the excess seen in the event rate in
the Xenon-1T experiment arising from $D_{2}^{\prime}$ with mass $m_{2}$
scattering inelastically off an electron into $D_{1}^{\prime}$ with mass
$m_{1}$ delivering a recoil energy $E_{R}$ to the electron. Energy
conservation for this process gives

$\frac{q^{2}}{2m_{2}}-vq\cos\eta=\Delta m-E_{R},$ (33)

where $\eta$ is the angle between the incoming $D_{2}^{\prime}$ momentum and
the momentum transfer $\vec{q}$. Taking $m_{1}\approx m_{2}$, the range of
momentum transfer is given by

$q_{\pm}=\begin{cases}m_{2}v\pm\sqrt{m_{2}^{2}v^{2}-2m_{2}(E_{R}-\Delta
m)},&\text{for }E_{R}>\Delta m\\\ \pm
m_{2}v+\sqrt{m_{2}^{2}v^{2}-2m_{2}(E_{R}-\Delta m)},&\text{for }E_{R}<\Delta
m.\end{cases}$ (34)

The recoil energy can be expressed in terms of the mass difference and in the
limit $\Delta m\ll m_{e}\ll m_{2}$, we have [51]

$E_{R}\simeq\Delta m\left(1-\sqrt{\frac{2m_{e}v^{2}}{\Delta m}}\cos\theta_{\rm
CM}\right),$ (35)

with $q^{2}\simeq 2m_{e}E_{R}$, where $\theta_{\rm CM}$ is the scattering
angle in the CM frame. The velocity-averaged differential cross-section for
inelastic DM scattering is

$\frac{d\langle\sigma v\rangle}{dE_{R}}=\int_{v_{\rm min}}^{v_{\rm
max}}\frac{f(v)}{v}dv\frac{\bar{\sigma}_{e}}{2m_{e}}\int_{q_{-}}^{q_{+}}dq\,a_{0}^{2}qK(E_{R},q),$
(36)

where the Bohr radius $a_{0}=1/(\alpha_{\rm em}m_{e})$ $(\alpha_{\rm em}\simeq
1/137)$ and $K(E_{R},q)$ is the atomic factorization factor (shown in Fig. 2)
and $f(v)$ is the standard Boltzmann velocity distribution after integrating
the angular part. In Eq. (36) the integral on $dq$, i.e.,

$K^{\prime}(E_{R})\equiv\int_{q_{-}}^{q_{+}}dq\,a_{0}^{2}qK(E_{R},q),$ (37)

can be directly evaluated by using Fig. 2.

Figure 2: The atomic ionization factor $K$ summed over all possible atomic
electrons dominated by $n=3$ for Xe at electron recoil energy $E_{R}=2$ keV.
The plot is a function of the momentum transfer $q$, taken from Ref. [52].

From Eq. (35), the range of the electron recoil energy is

$E_{R}^{\pm}\simeq\Delta m\left(1\pm\sqrt{\frac{2m_{e}v^{2}}{\Delta
m}}\right),$ (38)

and the differential cross-section in this range becomes

$\frac{d\langle\sigma
v\rangle}{dE_{R}}=\frac{\bar{\sigma}_{e}}{2m_{e}}K^{\prime}(E_{R})\int_{0}^{v_{\rm
max}}\frac{f(v)}{v}dv~{}\Theta(E_{R}-E_{R}^{-})\Theta(E_{R}^{+}-E_{R}),$ (39)

where for $E_{R}^{+}-E_{R}^{-}\ll E_{R}^{\pm}$, and where
$\Theta(E_{R}-E_{R}^{-})\Theta(E_{R}^{+}-E_{R})\simeq(E_{R}^{+}-E_{R}^{-})\delta(E_{R}-\Delta
m)$. Thus we have

$\frac{d\langle\sigma v\rangle}{dE_{R}}=\sqrt{\frac{2\Delta
m}{m_{e}}}\bar{\sigma}_{e}K^{\prime}(E_{R})\delta(E_{R}-\Delta
m)\int_{0}^{v_{\rm max}}f(v)dv.$ (40)

Note that for $v_{\rm max}=\sqrt{2\Delta m/m_{e}}\gg v_{0}$ (the most probable
velocity), we get $\int_{0}^{v_{\rm max}}f(v)dv\simeq 1$. In practice, the
electron recoil energy is not manifested as a Dirac delta function but rather
smeared by the detector resolution. This can be modeled by [1]

$\sigma_{r}=a\sqrt{E_{R}}+b~{}E_{R},$ (41)

with $a=(0.310\pm 0.004)\sqrt{\text{keV}}$ and $b=0.0037\pm 0.0003$. We assume
the resolution function is a Gaussian of the form

$R_{S}(E,E_{R})=\frac{1}{\sqrt{2\pi}\sigma_{r}}\exp\left[-\frac{(E-E_{R})^{2}}{\sigma^{2}_{r}}\right]~{}\alpha(E),$
(42)

where $\alpha(E)$ is the efficiency given in Fig. 2 of Ref. [1] which we take
to be 0.8 for our purposes. As a result, the DM detection rate is

$\displaystyle\frac{dR}{dE}$ $\displaystyle=n_{\rm
Xe}\frac{\rho_{2}}{m_{2}}\int\frac{d\langle\sigma
v\rangle}{dE_{R}}R_{S}(E,E_{R})dE_{R}$ $\displaystyle=n_{\rm
Xe}~{}\rho_{2}\sqrt{\frac{2\Delta
m}{m_{e}}}\frac{\bar{\sigma}_{e}}{m_{2}}K^{\prime}(\Delta m)R_{S}(E,\Delta
m),$ (43)

where $n_{\rm Xe}\simeq 4.2\times 10^{27}/$ton is the number of xenon atoms in
the detector and $\rho_{2}\simeq 0.15$ GeV/cm3 assuming that $D_{2}^{\prime}$
makes half the amount of the observed relic density. At the recoil energy of
interest, $E_{R}\simeq\Delta m\simeq 2.5$ keV and $K^{\prime}(\Delta m)\simeq
19.4$. The event detection rate becomes

$\frac{dR}{dE}\simeq(1.5\times
10^{45}~{}\text{GeV/cm}^{2})\frac{\bar{\sigma}_{e}}{m_{2}}R_{S}(E,\Delta m),$
(44)

which has units of $(\text{t}\cdot\text{yr}\cdot\text{keV})^{-1}$.

##  6 Constraints and fit to Xenon-1T data

Using Eq. (44) we attempt to fit the theory predictions of the model based on
the benchmarks (a), (b) and (c) of Table 1 to the Xenon-1T data. But before
doing so, let us discuss the stringent experimental constraints that must be
satisfied.

Those constraints are summarized in Fig. 3. Here CRESST-III (2019) [53] gives
the most sensitive limits on the DM-nucleon scattering for the light mass
range, 0.1$-$10 GeV. The DM-nucleon scattering cross-section against a nucleus
with mass number $A$ and proton number $Z$ can be written as

$\sigma_{N}=\frac{g_{X}^{2}(Q_{1}+Q_{2})^{2}g_{2}^{2}v_{f}^{\prime
2}}{16\pi\cos^{2}\theta}\frac{\mu^{2}_{DN}}{m^{4}_{\gamma^{\prime}}}\left(\frac{Z}{A}\right)^{2},$
(45)

where $\mu_{DN}$ is the DM-nucleon reduced mass and $Z/A\sim 0.5$.

Figure 3: Top panels: Exclusion limits for benchmarks (a), (b) and (c) of
Table 1 in the kinetic mixing-dark photon mass plane specific to our model
with constraints from the CRESST 2019 DM-nucleon scattering cross-section and
the projected SuperCDMS limit. Also shown is the dark photon constraint from
CHARM. The green patches show the regions where the DM relic density is
satisfied and the grey stripes indicate the region producing the fit to the
Xenon-1T excess. Bottom panels: Constraints from the Planck experiment on the
DM annihilation to two $e^{+}e^{-}$ pair.

Dark photon experiments [54] such as CHARM set limits on visible and invisible
decays of the dark photon. These limits are shown in the kinetic mixing-dark
photon mass plane in the top panels of Fig. 3 for benchmarks (a) (left), (b)
(center) and (c) (right) recast to our model parameters. Also shown are the
SuperCDMS [55] sensitivity projections for benchmarks (a) and (b) while (c)
remains out of reach of SuperCDMS. In the bottom panels, we present the recast
limits from the Planck experiment [56, 57] for the same benchmarks in the
gauge coupling $g_{X}$-dark photon mass plane. These limits pertain to the
annihilation of DM to two new vector bosons followed by their decay to two
$e^{+}e^{-}$ pairs. The benchmarks of Table 1 satisfy all of the above
constraints. There is a larger parameter space than the benchmarks of Table 1
where the experimental limits can be evaded while satisfying the relic density
and producing the correct fit to the Xenon-1T excess. We exhibit those regions
in the top panels of Fig. 3. The green patches show the parts of the parameter
space giving the correct DM relic density for a specific choice of the dark
coupling $g_{X}$ and the grey stripes indicate the regions producing the
correct fit to the Xenon-1T excess.

Figure 4: The event rate at Xenon-1T plotted over the background for
benchmarks (purple line) vs. the electron recoil energy for models (a) (blue
line), (b) (red line) and (c) (yellow line) of Table 1.

We exhibit in Fig. 4 the Xenon-1T data points (SR1) for the electron recoil
events in the detector along with the background-only hypothesis (B0). The
event rates for the three benchmarks in our model are plotted over the
background showing a clear enhancement near 2.5$-$2.8 keV as expected. Lighter
DM particles give a larger event rate since the latter is dependent on
$\bar{\sigma}_{e}/m_{2}$. Thus more data in the future which can measure the
height of the peak more accurately can determine more precisely the allowed
range of the dark matter mass.

##  7 Conclusion

In this work we have investigated the Xenon-1T signal as arising from sub-GeV
hidden sector dark matter. Specifically we consider a $U(1)$ Stueckelberg
extension of the Standard Model with two dark Dirac fermions degenerate in
mass charged under the $U(1)$ gauge group. The fermion mass degeneracy is
broken by a small mass mixing term which removes the degeneracy and produces
two Dirac fermions $D_{1}^{\prime},D_{2}^{\prime}$ where $D_{2}^{\prime}$ is
heavier than $D_{1}^{\prime}$ with a mass splitting size $\sim$ 3 keV. The
observed effect is explained by the exothermic inelastic scattering
$eD_{2}^{\prime}\to eD_{1}^{\prime}$. The scattering occurs via exchange of a
dark photon which is the massive gauge boson of the hidden sector. The
coupling of the dark photon to the electron arises from gauge kinetic mixing
of the $U(1)$ gauge boson of the hidden sector with the gauge boson of the
$U(1)_{Y}$ hypercharge. In the work here we have given a detailed analysis of
the dark matter relic density constituted of dark fermions while the dark
photons decay before the BBN time. As noted above there are two dark fermions
in the system $D_{1}^{\prime}$ and $D_{2}^{\prime}$ where $D_{2}^{\prime}$ has
a mass slightly greater than that of $D_{1}^{\prime}$ and decays to
$D_{1}^{\prime}$ via the channel $D^{\prime}_{2}\to
D_{1}^{\prime}+\nu\bar{\nu}$. However, the lifetime for this decay is larger
than the age of the universe and so for all practical purposes dark matter is
constituted of two dark fermions $D_{2}^{\prime}$ and $D_{1}^{\prime}$ in
essentially equal amount. In the analysis of the relic density one encounters
stringent constraints on kinetic mixing and on the dark photon mass from
CRESST 2019 DM-nucleon scattering cross section, and from the CHARM
experiment. We have translated these constraints for our analysis for the
model points considered in Table 1 and show that the model parameters of Table
1 are consistent with these constraints (Fig. 3). We note in passing that
these constraints, i.e., on the kinetic mixing and on the dark photon mass,
are projected to become more stringent from future data from SuperCDMS as
shown in Fig. 3 putting further constraints on the allowed parameter space of
GeV size dark matter models. Further, the Planck experiment gives constraints
on the $U(1)_{X}$ gauge coupling and the dark photon mass from the dark matter
annihilation to $4e$ which arise in our case from the annihilation channel
$D^{\prime}\bar{D}^{\prime}\to\gamma^{\prime}\gamma^{\prime}\to 4e$. Here
again we show that our model is consistent with these constraints as exhibited
in Fig. 3.

In Fig. 4 we showed that the models listed in Table 1, which satisfy all the
known constraints on kinetic mixing, on the gauge coupling of the $U(1)_{X}$
of the dark sector, on the dark photon mass, and generate relic density of
dark matter consistent with the Planck experiment, can explain the Xenon-1T
excess. We noted that the size of the peak for the excess events is model-
dependent and more data in the future from the Xenon-1T collaboration will
help delineate the nature of the dark sector more accurately. Further checks
on the model can also come from additional data from the direct detection
experiments that focus on the low mass region of dark matter in the GeV
region.

Acknowledgments: The research of AA and MK was supported by the BMBF under
contract 05H18PMCC1, while the research of PN was supported in part by the NSF
Grant PHY-1913328.

## Appendix

In this appendix we give further details of the analysis presented in the main
body of the paper. Thus in appendix A we discuss the generation of the mass
term for the $D_{1}$ and $D_{2}$ quarks and of the mixing term involving
$\bar{D}_{1}D_{2}+\bar{D}_{2}D_{1}$ from a Higgs mechanism. In appendix B we
give the cross-sections for the annihilation of $D\bar{D}$ into
$q\bar{q},\ell\bar{\ell}$, $\nu\bar{\nu}$ and $\gamma^{\prime}\gamma^{\prime}$
and the cross section for the conversion process $D_{2}\bar{D}_{2}\to
D_{1}\bar{D}_{1}$. In appendix C details of the partial decay widths of the
dark photon $\gamma^{\prime}$ are given, i.e.,
$\gamma^{\prime}\to\ell\bar{\ell},q\bar{q},\nu\bar{\nu}$.

##  Appendix A Generation of $\bar{D}_{1}D_{2}+\bar{D}_{2}D_{1}$ term from
spontaneous symmetry breaking

We now show that the term $\Delta\mu(\bar{D}_{1}D_{2}+\bar{D}_{2}D_{1})$ can
be produced via spontaneous breaking which also gives mass to the $U(1)_{X}$
gauge boson. Thus consider a $U(1)_{X}$ gauge field coupled to a complex
scalar field charged under the $U(1)_{X}$ with charge $Q_{\phi}$, and further
that the complex scalar couples to the Dirac fermions $D_{1},D_{2}$.
Specifically we consider the Lagrangian

$\displaystyle\mathcal{L}_{\phi}=-|(\partial_{\mu}\phi-
ig_{X}Q_{\phi}C_{\mu}\phi)|^{2}-V(\phi\phi^{*})-\lambda\frac{H^{c}H}{\Lambda}(\bar{D}_{1}D_{1}+\bar{D}_{2}D_{2})-(\lambda^{\prime}\bar{D}_{1}D_{2}\phi+\text{h.c.}).$
(46)

The Lagrangian above is invariant under $U(1)$ gauge transformations when

$\displaystyle-Q_{1}+Q_{2}+Q_{\phi}=0,$ (47)

where $Q_{1}$ and $Q_{2}$ are the $U(1)_{X}$ charges of $D_{1}$ and $D_{2}$.
The potential $V(\phi)$ gives a VEV $\phi_{0}=\langle\phi\rangle$ and after
spontaneous breaking the dark photon and the dark fermions get masses as
follows

$\displaystyle\mathcal{L}_{\rm
m}=-\frac{1}{2}m^{2}_{\gamma^{\prime}}A^{\mu^{\prime}}A^{\prime}_{\mu}-\mu(\bar{D}_{1}D_{1}+\bar{D}_{2}D_{2})-\Delta\mu(\bar{D}_{1}D_{2}+\bar{D}_{2}D_{1}),$
(48)

where

$\displaystyle m_{\gamma^{\prime}}$
$\displaystyle=\sqrt{2}g_{X}Q_{\phi}\phi_{0},$ $\displaystyle\mu$
$\displaystyle=\lambda\frac{v^{2}}{\Lambda},$ $\displaystyle\Delta\mu$
$\displaystyle=\lambda^{\prime}\phi_{0},$ (49)

where $v\sim 250$ GeV is the standard model Higgs VEV. For $g_{X}\sim
Q_{\phi}\sim 1$, an $m_{\gamma^{\prime}}$ in the range 50$-$300 MeV requires
$\phi_{0}\sim 100$ MeV. A Dirac fermion mass $\mu\sim 1$ GeV, requires the
cutoff scale $\Lambda\sim 100$ TeV. The cutoff scale could have a low scale
string origin. Further, $\Delta\mu\sim 2$ keV requires $\lambda^{\prime}\sim
10^{-5}$. Such a small $\lambda^{\prime}$ could also have a low scale string
origin. Thus quite remarkably if we assume that ‘flavor changing’ $\Delta\mu$
term arises from a higher dimensional operator such as
$(H^{c}H/\Lambda^{2})(\bar{D}_{1}D_{2}\phi+\bar{D}_{2}D_{1}\phi^{*})$, then
after spontaneous breaking it produces a $\lambda^{\prime}\sim
v^{2}/\Lambda^{2}\sim 10^{-5}$ which is precisely the size we want to generate
$\Delta\mu\sim 2$ keV.

##  Appendix B Relevant cross-sections

We present here the relevant cross-sections needed for the computation of the
dark matter relic density. In the computations of these cross-sections, we
have assumed $m_{D^{\prime}_{1}}\approx m_{D^{\prime}_{2}}\approx m_{D}$ as
the mass difference between their masses is tiny and has no substantial effect
on the size of the cross sections computed below.

  1. 1.

$D\bar{D}\to Z/\gamma^{\prime}\to q\bar{q}$:
The total cross-section for the process $D\bar{D}\to Z/\gamma^{\prime}\to
q\bar{q}$ is given by

$\displaystyle\sigma^{D\bar{D}\to q\bar{q}}(s)=$
$\displaystyle\frac{c^{2}_{X}g_{X}^{2}g_{2}^{2}}{8\pi
s\cos^{2}\theta}\sqrt{\frac{s-4m^{2}_{q}}{s-4m^{2}_{D}}}\Bigg{[}\frac{(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})^{2}(\alpha^{2}\eta_{q}T^{2}_{3q}-2\alpha\beta\kappa_{q}Q_{q}T_{3q}+2\beta^{2}Q_{q}^{2}\kappa_{q})}{(s-m_{Z}^{2})^{2}+m^{2}_{Z}\Gamma_{Z}^{2}}$
$\displaystyle+\frac{(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})^{2}(\alpha^{\prime
2}\eta_{q}T^{2}_{3q}-2\alpha^{\prime}\beta^{\prime}\kappa_{q}Q_{q}T_{3q}+2\beta^{\prime
2}Q_{q}^{2}\kappa_{q}))}{(s-m^{2}_{\gamma^{\prime}})^{2}+m^{2}_{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}^{2}}$
$\displaystyle-2(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})\Big{\\{}Q_{q}\beta(2\beta^{\prime}Q_{q}-\alpha^{\prime}T_{3q})\kappa_{q}$
$\displaystyle+\alpha
T_{3q}(\alpha^{\prime}T_{3q}\eta_{q}-\beta^{\prime}Q_{q}\kappa_{q})\Big{\\}}\times\frac{(s-m^{2}_{Z})(s-m^{2}_{\gamma^{\prime}})+\Gamma_{Z}\Gamma_{\gamma^{\prime}}m_{Z}m_{\gamma^{\prime}}}{[(s-m_{Z}^{2})^{2}+m^{2}_{Z}\Gamma_{Z}^{2}][(s-m^{2}_{\gamma^{\prime}})^{2}+m^{2}_{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}^{2}]}\Bigg{]},$
(50)

where $m_{q}$, $m_{Z}$ and $m_{\gamma^{\prime}}$ are the quark, $Z$ and
$\gamma^{\prime}$ masses, respectively, and $T_{3q}=1/2(-1/2)$ and
$Q_{q}=2/3(-1/3)$ for up-(down)-type quarks, and with

$\displaystyle\kappa_{q}$
$\displaystyle=(s+2m^{2}_{D})(s+2m^{2}_{q}),~{}~{}~{}\eta_{q}=(s+2m^{2}_{D})(s-m^{2}_{q}),$
(51) $\displaystyle\alpha$
$\displaystyle=\cos\psi-\bar{\epsilon}\sin\theta\sin\psi,~{}~{}~{}\beta=\sin^{2}\theta\cos\psi-\bar{\epsilon}\sin\theta\sin\psi,$
$\displaystyle\alpha^{\prime}$
$\displaystyle=\sin\psi+\bar{\epsilon}\sin\theta\cos\psi,~{}~{}~{}\beta^{\prime}=\sin^{2}\theta\sin\psi+\bar{\epsilon}\sin\theta\cos\psi.$

  2. 2.

$D\bar{D}\to Z/\gamma^{\prime}\to\ell\bar{\ell}$:
The total cross-section for the process $D\bar{D}\to
Z/\gamma^{\prime}\to\ell\bar{\ell}$ is given by

$\displaystyle\sigma^{D\bar{D}\to\ell\bar{\ell}}(s)=$
$\displaystyle\frac{c^{2}_{X}g_{X}^{2}g_{2}^{2}}{96\pi
s\cos^{2}\theta}\sqrt{\frac{s-4m^{2}_{\ell}}{s-4m^{2}_{D}}}\Bigg{[}\frac{(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})^{2}(\alpha^{2}\eta_{\ell}-4\alpha\beta\kappa_{\ell}+8\beta^{2}\kappa_{\ell})}{(s-m_{Z}^{2})^{2}+m^{2}_{Z}\Gamma_{Z}^{2}}$
$\displaystyle+\frac{(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})^{2}(\alpha^{\prime
2}\eta_{\ell}-4\alpha^{\prime}\beta^{\prime}\kappa_{\ell}+8\beta^{\prime
2}\kappa_{\ell})}{(s-m_{\gamma^{\prime}}^{2})^{2}+m^{2}_{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}^{2}}$
$\displaystyle+2(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})\Big{\\{}2\beta(\alpha^{\prime}-4\beta^{\prime})\kappa_{\ell}$
$\displaystyle-\alpha(\alpha^{\prime}\eta_{\ell}-2\beta^{\prime}\kappa_{\ell})\Big{\\}}\times\frac{(s-m^{2}_{Z})(s-m^{2}_{Z^{\prime}})+\Gamma_{Z}\Gamma_{\gamma^{\prime}}m_{Z}m_{\gamma^{\prime}}}{[(s-m_{Z}^{2})^{2}+m^{2}_{Z}\Gamma_{Z}^{2}][(s-m^{2}_{\gamma^{\prime}})^{2}+m^{2}_{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}^{2}]}\Bigg{]},$
(52)

where

$\displaystyle\kappa_{\ell}$
$\displaystyle=(s+2m^{2}_{D})(s+2m^{2}_{\ell}),~{}~{}~{}\eta_{\ell}=(s+2m^{2}_{D})(s-m^{2}_{\ell}).$
(53)

  3. 3.

$D\bar{D}\to Z/\gamma^{\prime}\to\nu\bar{\nu}$:
The total cross-section for the process $D\bar{D}\to
Z/\gamma^{\prime}\to\nu\bar{\nu}$ is given by

$\displaystyle\sigma^{D\bar{D}\to\nu\bar{\nu}}(s)$
$\displaystyle=\frac{c^{2}_{X}g_{X}^{2}g^{2}_{2}}{32\pi\cos^{2}\theta}\frac{(s+2m^{2}_{D})s^{1/2}}{\sqrt{s-4m^{2}_{D}}}\Bigg{\\{}\frac{\alpha^{\prime
2}(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})^{2}}{(s-m_{\gamma^{\prime}}^{2})^{2}+m^{2}_{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}^{2}}+\frac{\alpha^{2}(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})^{2}}{(s-m_{Z}^{2})^{2}+m^{2}_{Z}\Gamma_{Z}^{2}}$
$\displaystyle-2\alpha\alpha^{\prime}(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})(\mathcal{R}_{12}-s_{\delta}\mathcal{R}_{22})$
$\displaystyle\times\frac{(s-m^{2}_{Z})(s-m^{2}_{\gamma^{\prime}})+\Gamma_{Z}\Gamma_{\gamma^{\prime}}m_{Z}m_{\gamma^{\prime}}}{[(s-m_{Z}^{2})^{2}+m^{2}_{Z}\Gamma_{Z}^{2}][(s-m^{2}_{\gamma^{\prime}})^{2}+m^{2}_{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}^{2}]}\Bigg{\\}}.$
(54)

In all of the above, the coefficient $c_{X}$ is defined as

$c_{X}=\begin{cases}\frac{1}{2}(Q_{1}+Q_{2}),&\text{for
}\bar{D}^{\prime}_{1}D^{\prime}_{1}/\bar{D}^{\prime}_{2}D^{\prime}_{2}\\\
\frac{1}{2}(Q_{1}-Q_{2}),&\text{for
}\bar{D}^{\prime}_{1}D^{\prime}_{2}/\bar{D}^{\prime}_{2}D^{\prime}_{1}.\end{cases}$
(55)

  4. 4.

$D^{\prime}_{i}\bar{D}^{\prime}_{j}\longleftrightarrow\gamma^{\prime}\gamma^{\prime}$:
The total cross-section for the process
$D^{\prime}_{i}\bar{D}^{\prime}_{j}\to\gamma^{\prime}\gamma^{\prime}$ is

$\sigma^{D^{\prime}_{i}\bar{D}^{\prime}_{j}\to\gamma^{\prime}\gamma^{\prime}}(s)=\sum_{ij}c_{ij}^{4}\sigma_{0}(s),$
(56)

where

$\displaystyle\sigma_{0}(s)=\frac{g_{X}^{4}(\mathcal{R}_{11}-s_{\delta}\mathcal{R}_{21})^{4}}{8\pi
s(s-4m^{2}_{D})}$
$\displaystyle\Bigg{\\{}-\frac{\sqrt{(s-4m^{2}_{\gamma^{\prime}})(s-4m^{2}_{D})}}{m^{4}_{\gamma^{\prime}}+m^{2}_{D}(s-4m^{2}_{\gamma^{\prime}})}[2m^{4}_{\gamma^{\prime}}+m^{2}_{D}(s+4m^{2}_{D})]$
$\displaystyle+\frac{\log
A}{s-2m^{2}_{\gamma^{\prime}}}(s^{2}+4m^{2}_{D}s+4m^{4}_{\gamma^{\prime}}-8m^{4}_{D}-8m_{D}^{2}m^{2}_{\gamma^{\prime}})\Bigg{\\}},$
(57)

and

$A=\frac{s-2m^{2}_{\gamma^{\prime}}+\sqrt{(s-4m^{2}_{\gamma^{\prime}})(s-4m^{2}_{D})}}{s-2m^{2}_{\gamma^{\prime}}-\sqrt{(s-4m^{2}_{\gamma^{\prime}})(s-4m^{2}_{D})}},$
(58)

with

$c_{ij}=\begin{cases}\frac{1}{2}(Q_{1}+Q_{2}),&\text{for }i=j=1,2\\\
\frac{1}{2}\sqrt{(Q_{1}+Q_{2})(Q_{1}-Q_{2})},&\text{for }i,j=\\{1,2\\},i\neq
j.\end{cases}$ (59)

The reverse processes are given by

$9(s-4m^{2}_{\gamma^{\prime}})\sigma^{\gamma^{\prime}\gamma^{\prime}\to
D^{\prime}_{i}\bar{D}^{\prime}_{j}}(s)=8(s-4m^{2}_{D})\sigma^{D^{\prime}_{i}\bar{D}^{\prime}_{j}\to\gamma^{\prime}\gamma^{\prime}}(s).$
(60)

  5. 5.

The conversion process $D_{2}^{\prime}\bar{D_{2}^{\prime}}\longrightarrow
D_{1}^{\prime}\bar{D_{1}^{\prime}}$:

$\sigma^{D_{2}^{\prime}\bar{D_{2}^{\prime}}\longrightarrow
D_{1}^{\prime}\bar{D_{1}^{\prime}}}\simeq\frac{g_{X}^{4}(Q_{1}+Q_{2})^{4}}{4\pi}\frac{m^{2}_{D}}{m^{4}_{\gamma^{\prime}}}\frac{(1-5r+7r^{2})}{(1-r)^{2}},$
(61)

where $r=m^{2}_{\gamma^{\prime}}/4m^{2}_{D}$.

##  Appendix C Decay widths for the processes
$\gamma^{\prime}\to\ell\bar{\ell},q\bar{q},\nu\bar{\nu}$

  1. 1.

The decay width of $\gamma^{\prime}$ to leptons is given by

$\displaystyle\Gamma_{\gamma^{\prime}\to\ell\bar{\ell}}=\frac{g_{2}^{2}}{24\pi
m_{\gamma^{\prime}}\cos^{2}\theta}\sqrt{1-\left(\frac{2m_{\ell}}{m_{\gamma^{\prime}}}\right)^{2}}$
$\displaystyle\Bigg{[}\frac{1}{4}\alpha^{\prime
2}(m^{2}_{\gamma^{\prime}}-m^{2}_{\ell})-\alpha^{\prime}\beta^{\prime}(m^{2}_{\gamma^{\prime}}+2m^{2}_{\ell})$
$\displaystyle+2\beta^{\prime
2}(m^{2}_{\gamma^{\prime}}+2m^{2}_{\ell})\Bigg{]}.$ (62)

  2. 2.

The decay width of $\gamma^{\prime}$ to quarks is given by

$\displaystyle\Gamma_{\gamma^{\prime}\to q\bar{q}}=\frac{g_{2}^{2}}{8\pi
m_{\gamma^{\prime}}\cos^{2}\theta}\sqrt{1-\left(\frac{2m_{q}}{m_{\gamma^{\prime}}}\right)^{2}}$
$\displaystyle[\alpha^{\prime
2}(m^{2}_{\gamma^{\prime}}-m^{2}_{q})T_{3q}^{2}-2\alpha^{\prime}\beta^{\prime}(m^{2}_{\gamma^{\prime}}+2m^{2}_{q})Q_{q}T_{3q}$
$\displaystyle+2\beta^{\prime
2}(m^{2}_{\gamma^{\prime}}+2m^{2}_{q})Q_{q}^{2}].$ (63)

  3. 3.

The decay width of $\gamma^{\prime}$ to neutrinos is given by

$\Gamma_{\gamma^{\prime}\to\nu\bar{\nu}}=\frac{g_{2}^{2}}{32\pi\cos^{2}\theta}m_{\gamma^{\prime}}\alpha^{\prime
2}.$ (64)

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Hadron Molecules Revisted
R.S. Longacrea
aBrookhaven National Laboratory, Upton, NY 11973, USA

###### Abstract

Hadron Molecules are particles made out of hadrons that are held together by
self interactions. In this report we discuss seven such molecules and their
self interactions. The $f_{0}(980)$, $a_{0}(980)$, $f_{1}(1400)$, $\Delta
N(2150)$ and $\pi_{1}(1400)$ molecular structure is given. We predict that two
more states the $K\overline{K}K(1500)$ and $a_{1}(1400)$ should be found.

##  1 Introduction

The first two molecular states $f_{0}(980)$ and $a_{0}(980)$ are the
isosinglet and the isotriplet states of the $K$$\overline{K}$ bound system[1].
This binding requires a quark-spin hyperfine interaction in the over all $q$
$q$ $\overline{q}$ $\overline{q}$ system. We will see that this binding is
different from the particle exchange mechanisms that bind the rest of
molecules of this report. The exchanges of Ref.[1] that bind the
$K$$\overline{K}$ system are quark exchanges where the quark-spin hyperfine
interaction leads to an attractive potential. This attractive potential can
only make states if the mesons of the fall apart mode $q$ $\overline{q}$ \-
$q$ $\overline{q}$ are below threshold. Thus we have states of
$K$$\overline{K}$ lying just below threshold in scalar channel($0^{++}$). This
is somewhat like the deuteron in the $p$ $n$ system.

The work of Ref.[1] has one flaw with regards to long-range color van der
Waals-type forces[2]. It has been pointed out that confining potentials of the
type used in Ref.[1] leads to a long-range power-law residual potential
between color singlets of $r^{-2}$ between two mesons. Ref.[1] calculates the
potential between mesons to be given by

$V_{vdW}=-{{20MeV}\over{r^{2}}}.$ (1)

This is to be compared with the coulomb force between charged mesons

$V_{coulomb}=-{{1.5MeV}\over{r}}.$ (2)

The r in equation 1 and equation 2 is given in fm. Let us compare the
potential strength between them by setting them equal

$V_{vdW}=V_{coulomb}=-{{20MeV}\over{r^{2}}}=-{{1.5MeV}\over{r}}.$ (3)

This occurs at the distance of 13.3 fm which is a very large distance. The
size of the scalar bound states are much smaller 2 fm at best. This long range
van der Waals force implies that gluons are massless(like the photon) and can
travel to the edge of the universe in a virtual state(like photons in EM-
fields). Gluons can only try to travel to the edge if they are in a color
singlet state (glueballs or glueloops). This would lead to an exponential
cutoff

$V_{r}=-{{e^{-\mu r}}\over{r}},$ (4)

where $\mu$ is the glueball mass. Sine mesons are much lighter than glueballs,
and the pion is the lightest, it is the longest carrier of the strong force.
Meson exchanges will be the binding force of the other hadron molecules of
this report.

The report is organized in the following manner:

Sec. 1 is the introduction to $f_{0}(980)$ and $a_{0}(980)$. Sec. 2 looks at
particle exchange calculations and the generation of $f_{1}(1400)$. Sec. 3
consider a dibaryon state $\Delta N(2150)$ and the similarity to
$f_{1}(1400)$. This similarity predicts the $a_{1}(1400)$ state. Sec. 4 an
exotic meson state $1^{-+}$ which is seen in $\eta$ $\pi$ p-wave scattering
$\pi_{1}(1400)$ is explained. Sec. 5 predicts another molecular state
$K\overline{K}K(1500)$ which should be found. Sec. 6 is the summary and
discussion.

##  2 $K$$\overline{K}$$\pi$ as an interacting system

In order to bind $K$$\overline{K}$$\pi$ together, we need to develop a
dynamical theory that uses particle exchange mechanism not the quark exchange
that led to the van der Walls forces[2]. It is standard to break up the
scattering of a three particle system into a sum of isobar spectator
scatterings[3]. To complete this task we develop a unitary isobar model which
has long-range particle exchange forces. We will assume that the only
interaction among the three particles are isobars decaying into two particles
where one of these particle exchange with the spectator forming another
isobar. This one-particle-exchange(OPE) occurs in the $K^{*}$$\overline{K}$,
$\overline{K^{*}}$$K$ and $a_{0}$$\pi$ isobar systems. See Figure 1(a) for the
OPE mechanism(note $a_{0}$ is the $\delta$ isobar which is its older name). We
choose as our dynamical framework the Blankenbecler-Sugar formalism[4] which
yields set of coupled integral equations for amplitudes $X(K^{*}\rightarrow
K^{*}$), $X(K^{*}\rightarrow\overline{K^{*}}$), $X(K^{*}\rightarrow a_{0}$),
$X(\overline{K^{*}}\rightarrow K^{*}$),
$X(\overline{K^{*}}\rightarrow\overline{K^{*}}$),
$X(\overline{K^{*}}\rightarrow a_{0}$), $X(a_{0}\rightarrow K^{*}$),
$X(a_{0}\rightarrow\overline{K^{*}}$), and $X(a_{0}\rightarrow a_{0}$). These
amplitudes describe the isobar quasi-two-body processes
$K^{*}\overline{K}\rightarrow K^{*}\overline{K}$,
$K^{*}\overline{K}\rightarrow\overline{K^{*}}K$, $K^{*}\overline{K}\rightarrow
a_{0}\pi$, etc., whose solution are Lorentz invariant, and satisfy two- and
three-body unitarity, and cluster properties. In operator formalism these
equations have the structure(a schematic representation is show in Figure
1(b)),

$X_{ba}=B_{ba}(W_{E})+B_{bc}(W_{E})G_{c}(W_{E})X_{ca}(W_{E});a,b,c=K^{*},\overline{K^{*}},a_{0}.$
(5)

In equation 5, $W_{E}$ is the overall c.m. energy of the three-particle
system. The index $c$ which is summed over represent each isobar. Integration
is over solid angles where each total angular momentum is projected out as an
individual set of coupled equations. The c.m. momentum of the spectator which
has the same magnitude as the isobar is integrated from zero to $\infty$. Thus
all effective masses of isobars are probed starting at the kinematic limit
down to negative $\infty$. All details are given in Ref.[5]. $G_{c}(W_{E})$ is
the propagator of isobar $c$ and $W_{E}$ determines the kinematic limit where
c.m. momentum of the spectator and the isobar is zero. In Figure 2 we show the
Breit-Wigner shape of the $K^{*}$ propagator $G_{K^{*}}(2.0)$ as a function of
isobar mass($K\pi$). In Figure 3 we show the Breit-Wigner shape of the $a_{0}$
propagator $G_{a_{0}}(2.04)$ as a function of isobar mass($K\overline{K}$).

Figure 1:  (a) Long-range one-particle-exchange(OPE) mechanism. Isobars
$K^{*}$, $\overline{K^{*}}$, or $\delta$($a_{0}$ newer name) plus
$\overline{K}$, $K$ or $\pi$ which absorbs the exchange particle which has
decayed from the isobar forming another isobar. (b) Unitary sum of OPE
diagrams in terms of coupled integral equations.

Figure 2:  The absolute value squared of the imaginary part divided by the
propagator for $K\pi$ propagation of the $K^{*}$ which is an $I={{1}\over{2}}$
and $J=1$ mode. This is equal to the square of the T-matrix scattering of this
$K\pi$ mode.

Figure 3:  The absolute value squared of the imaginary part divided by the
propagator for $K\overline{K}$ propagation of the $a_{0}$ which is an $I=1$
and $J=0$ mode. This is equal to the square of the T-matrix scattering of this
$K\overline{K}$ mode.

Equation 5 can be rewriting as

$\sum_{k}(\delta_{ik}-M_{ik})X_{kj}=B_{ij};i,j,k=K^{*},\overline{K^{*}},a_{0}.$
(6)

This Fredholm integral equation leads to a Fredholm determinant as a function
of $W_{E}$ for each partial wave or total $J$ projection. We have solved this
Fredholm determinant for two $J^{PC}$ states $0^{-+}$ and $1^{++}$. The
results of this analysis is shown in Figure 4. We see no binding effect in the
$0^{-+}$ determinant, while in the $1^{++}$ channel there is a large effect
around $1.40$ GeV. At the energy of $1.40$ GeV the $K^{*}$ and
$\overline{K^{*}}$ the peaks of Figure 2 are just coming into play. Since for
$1^{++}$ they are in a s-wave they have maximum effect. In the $0^{-+}$ these
peak are suppressed by a p-wave barrier. Around the mass $1.40$ GeV one can
form a picture of the system being $K$$\overline{K}$($a_{0}$) molecule at the
center of gravity with a light pion revolving in a p-wave orbit. The momentum
of the pion is such that at each half-revolution a $K^{*}$ or a
$\overline{K^{*}}$ is formed(see Figure 5). The phase shift and production
cross sections of this molecular state is explored in detail in Ref.[5].

Figure 4: The value of 1 over the Fredholm determinant squared for $J^{PC}$ =
$1^{++}$ and $J^{PC}$ = $0^{-+}$ as a function of $K$$\overline{K}$$\pi$
mass(smooth curves).

Figure 5: The meson system mainly resonates in the s-wave
$K^{*}$$\overline{K}$ and $K$$\overline{K^{*}}$ mode with a pion rotating in a
p-wave about a $K$ $\overline{K}$ system which forms a isospin triplet. The
pion moves back and forth forming $K^{*}$ and $\overline{K^{*}}$ states with
one $K$ or $\overline{K}$.

Figure 6: The dibaryon system mainly resonates in the s-wave $\Delta$ $N$ mode
with a pion rotating in a p-wave about a spin aligned $N$ $N$ system which
forms a isospin singlet. The pion moves back and forth forming $\Delta$ states
with one nucleon and then the other.

##  3 Two more molecular states

###  3.1 Dibaryon state $\Delta N(2150)$ as a molecule.

The dibaryon state interacts in three two-body scattering channels. Its mass
is 2.15 GeV and has a strong interaction resonance decay width of 100 MeV. It
interacts in the $N$$N$ d-wave spin anti-aligned[6], $d$$\pi$ p-wave spin
aligned[7], and $\Delta$$N$ s-wave spin aligned[8]. The dibaryon system mainly
resonates in the s-wave $\Delta$$N$ mode with a pion rotating in a p-wave
about a spin aligned $N$$N$ system which forms a isospin singlet. The pion
moves back and forth forming $\Delta$ states with one nucleon and then the
other(see Figure 6). All three isospin states of the pion can be achieved in
this resonance. Thus we can have $\pi^{+}$$d$, $\pi^{0}$$d$, and $\pi^{-}$$d$
states. If the pion is absorbed by any of the nucleons it under goes a spin
flip producing a d-wave $N$$N$ system. The resonance decays into $N$$N$,
$\pi$$d$, or $\pi$$N$$N$. In the last section we saw a meson system that had
an analogous orbiting pion in a p-wave mode about a $K\overline{K}$ in a
s-wave[5]. Both systems have a similar lifetime or width of $\sim$ .100
GeV[9].

###  3.2 $a_{1}(1400)$ state is predicted

Unlike the $f_{1}(1400)$ the $\Delta N(2150)$ has an isosinglet at the center
of motion. The $K$$\overline{K}$ isosinglet state of Sec. 1 could form the
center of motion for an isotriplet molecular state $a_{1}(1400)$. The set of
integral equation would be the same as in the $f_{1}(1400)$ case making a
similar Fredholm determinant. Like for $\Delta N(2150)$ which had a $d\pi$
decay mode, one would expect that there would be a $f_{0}(980)$ $\pi$ decay
mode. We can calculate the branching ration of $f_{1}(1400)$ to $a_{0}$$\pi$
from the Dalitz plot calculated using equation 20 of Ref.[5]. The ratio in the
plot going into $a_{0}$$\pi$ is 22%. The reason this mode is so small is
because $\sqrt{Imag(D_{a_{0}})}\over{|D_{a_{0}}|}$ is much smaller than
$\sqrt{Imag(D_{K^{*}})}\over{|D_{K^{*}}|}$[5]. Where as the ratios of
$Imag(D_{a_{0}})\over{|D_{a_{0}}|}$ and $Imag(D_{K^{*}})\over{|D_{K^{*}}|}$
are one at resonance(see Figure 2 and 3). For the $\Delta N(2150)$ the $d\pi$
branching ratio is 25%[9]. We should expect that the branching of
$a_{1}(1400)\rightarrow f_{0}\pi$ should be the same as
$f_{1}(1400)\rightarrow a_{0}\pi$. Dr. Suh-Urk Chung has claimed such a state
has been observed[10].

##  4 Exotic state $J^{PC}$ = $1^{-+}$ $\pi_{1}(1400)$ as a molecule.

In Sec. 2 we explained the $f_{1}$(1420) seen in $\overline{K}K\pi$[5].
Following the same approach we can demonstrate the possibility that the
$\pi_{1}$(1400) is a $\overline{K}K\pi\pi$ molecule, where the
$\overline{K}K\pi$ in a relative s-wave with the other $\pi$ orbiting them in
a p-wave. Since the $\overline{K}K\pi$ is resonating as the $\eta$(1295), it
is possible that the offshell $\overline{K}K\pi(\eta)$ would couple to the
ground state $\eta$, thus creating a $\eta\pi$ p-wave decay mode.

As was done in Ref.[5], we need to arrange a set of Born terms connecting all
of the possible intermediate isobar states of the $\overline{K}K\pi\pi$ system
($\eta$(1295)$\pi$, $a_{0}$(980)$\rho$(770), $K_{1}$(1270)$\overline{K}$ or
$\overline{K_{1}}(1270)$$K$). We assume that the only interaction among the
particles occurs through one-particle exchange (OPE), thus connecting the
above isobar states (Figure 7). In order to completely derive the dynamics one
would have to develop a true four-body scattering mechanism with OPE Born
terms connecting two- and three-body isobar states. We can take a short cut
and use the three-body formalism developed in Ref. [5], if we note that the
set of diagrams (Figure 8) could be summed using a true four-body formalism,
and be replaced by the Born term of Figure 9. Here the $a_{0}$(980) is treated
as a stable particle and the $\pi\pi$ p-wave phase shift ($\rho_{med}$) is
assumed to be modified by the sum of terms in Figure 8. With this assumption,
then binding can occur if we use the $N/D$ propagators for the
$\eta$(1295)(see Figure 10) and $\rho_{med}$(see Figure 11). In Figure 11 we
also show the unaltered p-wave phase shift ($\rho$). Figure 12 shows the final
state enhancement times the $\eta$(1295) $\pi$ p-wave kinematics. The bump is
driven by the collision on the Dalitz plot of the $\eta$(1295) Breit-Wigner
(Figure 10) and the rapid increase of the $\pi\pi$ p-wave phase shift (Figure
11).

We have suggested the possibility that the $\pi_{1}$(1400) is a final state
interaction for the $K\bar{K}\pi$ system in a s-wave orbiting by a $\pi$ in a
p-wave. The $\eta\pi$ decay mode is generated by the off shell appearance of
the $\eta$ from the $K\bar{K}\pi$ system ($0^{-+}$). Our model thus predicts
that a strong $J^{PC}=1^{-+}$ should be seen in the $K\bar{K}\pi\pi$ system at
around 1.4 GeV/c2. If the $\pi_{1}$(1400) is only seen in the $\eta\pi$
channel then its hard to understand three facts about its production. First,
that the force between the $\eta$ and $\pi$ in a p-wave should be repulsive
(QCD) [11]. This is not a problem if the $\eta\pi$ is a minor decay mode.
Second, why should the production be so small compared to the $a_{2}$ which
has only a 14% branching to $\eta\pi$? One would think it should be produced
in unnatural parity exchange not natural. Again this is not a problem if minor
decay mode.

Figure 7: One-particle-exchange (OPE) Born terms for $\overline{K}K\pi\pi$
system.

Figure 8: The set of infinite terms where all $K$ and $\overline{K}$ exchanges
are summed.

Figure 9: The Born that is used in the three-body effective analysis, where
the $\pi\pi$ p-wave is altered by the sum of terms in Figure 8.

Figure 10: The absolute value squared of the imaginary part of the $\eta$
(1290) propagator divided by the complete propagator, thus forming the square
of the T-matrix scattering amplitude.

Figure 11: The absolute value squared of the imaginary part of the $\pi\pi$
p-wave phase shift: the solid line is the modified phase shift; the dashed
line is the original vacuum phase shift which is the $\rho$ meson.

Figure 12: The value of 1 over the Fredholm determinate squared times the
kinematics of p-wave $\pi\eta$(1290).

Finally, it is reasonable to think that the largest decay amplitude would be
the modes that have an $a_{0}$(980) in the final state. However in Ref.[5] the
same conclusion was initially drawn, except when one puts in all the numerical
factors the $a_{0}$ modes become suppressed. The explanation comes from the
very powerful attraction of the kaons in the $a_{0}$ mode. The isobar decay
amplitude is proportional to $\sqrt{N}/D$ both $N$ and $D$ are large numbers
while the ratio is near one at the threshold(see last section and Ref.[5]).
Thus the decay amplitude becomes proportional to $1/\sqrt{N}$ . We predict
that the major mode could be $\pi\pi$ p-wave having no $\rho$ peak (work
above) forming a $K\pi\pi$ or a $\overline{K}\pi\pi~{}J^{p}=1^{+}$ plus a
$\overline{K}$ or $K$ with overall $G$-parity minus. The $K\pi\pi$ should more
or less be a phase space distribution.

##  5 $K\overline{K}K(1500)$ state is predicted

We saw in Sec.1 and Sec. 2 that the $K$$\overline{K}$ system had attraction
through the $a_{0}(980)$ resonance. It seems only natural to investigate the
possibility that a three-K molecule might exist. This is only worthwhile if we
consider only exotic quantum numbers. The only exotic quantum number which can
be be obtained is the isotopic spin. Thus a set of coupled equations for the
$K$$\overline{K}$$K$ system in a overall s-wave with isotopic spin of
$3\over{2}$ is created[5]. The Fredholm determinate squared times of the
equations is shown in Figure 13.

Isopin spin $3\over{2}$ implies there are four states
$K^{+}$$\overline{K^{0}}$$K^{+}$,$K^{+}$$K^{-}$$K^{+}$,
$K^{0}$$\overline{K^{0}}$$K^{0}$, and $K^{0}$$K^{-}$$K^{0}$. The
$K^{+}$$\overline{K^{0}}$$K^{+}$ is double charged. There would also be a
$K^{-}$$K^{0}$$K^{-}$ which is the anti-matter state of the
$K^{+}$$\overline{K^{0}}$$K^{+}$. These states are unique to this type of
binding mechanism.

##  6 Summary and Discussion

In this report we have discussed seven possible hadron molecular states. These
states are particles made out of hadrons that are held together by self
interactions. The seven molecules and their self interactions are explored.
The $f_{0}(980)$, $a_{0}(980)$ relied quark exchange forces which made states
of $K$$\overline{K}$ lying just below threshold in scalar channel($0^{++}$).
This is somewhat like the deuteron in the $p$ $n$ system. The $f_{1}(1400)$,
$\Delta N(2150)$ and $\pi_{1}(1400)$ molecular structure are held together by
long range particle exchange mechanisms not the quark exchange that led to the
van der Walls forces[2]. These exchange mechanisms also predicts that two more
states the $K\overline{K}K(1500)$ and $a_{1}(1400)$ should be found. For the
$a_{1}(1400)$ the set of integral equation would be the same as in the
$f_{1}(1400)$ case making a similar Fredholm determinant. Like for $\Delta
N(2150)$ which had a $d\pi$ decay mode, one would expect that there would be a
$f_{0}(980)$ $\pi$ decay mode. Dr. Suh-Urk Chung has claimed such a state has
been observed[10].

Figure 13: The value of 1 over the Fredholm determinant squared for $J^{P}$ =
$0^{-}$ as a function of $K$$\overline{K}$$K$ mass(smooth curves).

##  7 Acknowledgments

This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.

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