# Constraints on energy scales from dark matter decay in a gauged $B-L$ model

Guillermo Gambini  Instituto de Física Gleb Wataghin, UNICAMP, Campinas, SP,
Brasil  McGill University, Department of Physics, 3600 University Street,
Montréal, QC H3A 2T8, Canada  Pedro C. de Holanda  Instituto de Física Gleb
Wataghin, UNICAMP, Campinas, SP, Brasil  Saulo Carneiro  Instituto de Física,
Universidade Federal da Bahia, 40210-340, Salvador, BA, Brasil

###### Abstract

Popular extensions of the standard model of particle physics feature new
fields and symmetries which could, for example, dynamically generate neutrino
masses from $B-L$ spontaneous symmetry breaking. If a new light scalar that
decays into dark radiation appears in the spectrum of the theory, it could
significantly modify the cosmological observables. In this case, cold dark
matter could have a stable and a decaying component and limits on its decay
rate $\Gamma_{\rm dcdm}$ can be used to put constraints on the new energy
scales of a given model. We illustrate this idea using a gauged $B-L$ model
where the dark radiation is in the form of light neutrinos.

##  I Introduction

One of the driving forces of intensive research nowadays is the nature of the
dark sector and its interactions with the content of the standard model of
particle physics (SM). This sector is embedded in the cosmological standard
model ($\Lambda$CDM) in the form of dark matter (DM) and dark energy. While
the latter accounts for the accelerated expansion of the universe and its
effects manifest in the form of a cosmological constant $\Lambda$, the former
is most likely a new yet-unknown form of matter. The large amount of
astronomical and cosmological observations are compatible with a cold relic
whose self-interaction could solve, for example, small-scale problems
Kaplinghat:2015aga ; Tulin:2017ara ; McDermott:2017vyk ; Cline:2020gon , while
its interactions with SM particles could explain anomalies in terrestrial
experiments Lindner:2020kko ; Ge:2021cjz ; Ge:2021snv ; Cline:2019snp ;
Cline:2020mdt , as well as discrepancies between low-redshift and cosmic
microwave background (CMB) determinations of the Hubble parameter $H_{0}$ and
the matter perturbation amplitude $\sigma_{8}$ Sam ; Alcaniz:2019kah ;
Blinov:2020uvz ; Abellan:2020pmw ; Escudero:2021rfi .

The resolution of these $H_{0}$ and $\sigma_{8}$ tensions Riess:2011yx ;
Heymans:2013fya ; Planck:2015lwi ; Riess:2016jrr  could also indicate the
presence of a new neutrino species Carneiro:2018xwq , primordial black holes
Nesseris:2019fwr , decaying dark matter Enqvist:2015ara  and more (see
DiValentino:2021izs  for a review). In particular, a natural (decaying) DM
candidate is the Majoron which appears in models where the $B-L$ symmetry is
broken by gravitational effects Akhmedov:1992et  or spontaneously broken by
the vacuum expectation value (VEV) of a new complex singlet scalar
Chikashige:1980ui . Neutrino masses could be a consequence of this spontaneous
symmetry breaking (SSB) at high scales if neutrinos are Majorana particles.
Besides that, tiny Dirac neutrino masses can be achieved by coupling new
right-handed neutral fermions and active neutrinos to new scalar doublets with
VEVs $v_{\rm D}\ll v_{\rm SM}\approx$ 246 GeV Davidson:2009ha . This dynamical
generation of Dirac neutrino masses can be extended to Majorana neutrino
masses by introducing new scalar singlets with larger VEVs $v_{\rm SM}\ll
v_{\rm M}$ Montero:2011jk . In this work, we put constraints on these Dirac
and Majorana VEVs from the cosmic evolution and decay of a massive Majoron DM
candidate that arises in gauged $B-L$ models and is able to constitute all or
a fraction of the DM respecting limits from the cosmic microwave background
(CMB) and the large-scale structure (LSS) of the universe.

In section 2 we briefly describe the gauged $B-L$ model we use to illustrate
the constraints cosmology puts on these new scales $v_{D}$ and $v_{M}$.
Constraints from late-time neutrino production and observational limits are
presented in sections 3 and 4, respectively. We conclude in section 5.

##  II A $B$–$L$ model with exotic charges

The model we use to study decaying DM is based on the gauge group $SU(3)_{\rm
C}\otimes SU(2)_{\rm L}\otimes U(1)_{\rm Y^{\prime}}\otimes U(1)_{\rm B-L}$,
where C, L, Y′, and $B$–$L$ stand for color, left chirality, a new charge
different from SM hypercharge, and baryon minus lepton number, respectively.
Taking $B$–$L$ as a gauge symmetry, many models arise from the set of
solutions to anomaly equations Costa:2019zzy . In Ref.montero2009gauging , a
model with three right-handed neutrinos with $B$–$L=5,-4,-4$ charges is
proposed. Its representation content is that of the SM slightly extended by
these new particles and six new scalars: two doublets $\Phi_{1,2}$ and four
singlets $\phi_{1,2,3,X}$,

$SU(2)_{\rm L}\otimes U(1)_{\rm Y^{\prime}}\otimes U(1)_{\rm
B-L}\xrightarrow[]{\left<\phi\right>}SU(2)_{\rm L}\otimes U(1)_{\rm
Y}\xrightarrow[]{\left<H,\Phi\right>}U_{\rm em}.$ (1)

All of these new scalars have unique $B$–$L$ quantum numbers
sanchez2014complex .

In this model, the most general renormalizable and gauge-invariant scalar
potential reads

$\displaystyle V_{B-L}$ $\displaystyle=$
$\displaystyle-\mu^{2}_{H}H^{\dagger}H+\lambda_{H}|H^{\dagger}H|^{2}$
$\displaystyle-\mu_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{11}|\Phi_{1}^{\dagger}\Phi_{1}|^{2}-\mu_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}+\lambda_{22}|\Phi_{2}^{\dagger}\Phi_{2}|^{2}-\mu_{s\alpha}^{2}|\phi_{\alpha}|^{2}+\lambda_{s\alpha}|\phi_{\alpha}^{*}\phi_{\alpha}|^{2}$
$\displaystyle+\lambda_{12}|\Phi_{1}|^{2}|\Phi_{2}|^{2}+\Lambda_{H\gamma}|H|^{2}|\Phi_{\gamma}|^{2}+\Lambda_{Hs\alpha}|H|^{2}|\phi_{\alpha}|^{2}+\Lambda^{{}^{\prime}}_{\gamma\alpha}|\Phi_{\gamma}|^{2}|\phi_{\alpha}|^{2}+\Delta_{\alpha\beta}(\phi_{\alpha}^{*}\phi_{\alpha})(\phi_{\beta}^{*}\phi_{\beta})$
$\displaystyle+\lambda_{12}^{{}^{\prime}}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\Lambda_{H\gamma}^{{}^{\prime}}(H^{\dagger}\Phi_{\gamma})(\Phi_{\gamma}^{\dagger}H)$
$\displaystyle+[\beta_{123}\phi_{1}\phi_{2}(\phi_{3}^{*})^{2}+\Phi^{\dagger}_{1}\Phi_{2}(\beta_{13}\phi_{1}\phi_{3}^{*}+\beta_{23}\phi_{2}^{*}\phi_{3})-i\kappa_{H1X}\Phi_{1}^{T}\tau_{2}H\phi_{X}$
$\displaystyle-i\kappa_{H2X}(\Phi_{2}^{T}\tau_{2}H)(\phi_{X}^{*})^{2}+\beta_{X}(\phi_{X}^{*}\phi_{1})(\phi_{2}\phi_{3})+\beta_{3X}(\phi_{X}^{*}\phi_{3}^{3})+H.c.],$

where $\gamma=1,2$, $\alpha,\beta=1,2,3,X$, and $\alpha>\beta$ in the
$\Delta_{\alpha\beta}$ terms. The quadratic potential reads
$V_{2}=\frac{1}{2}\bm{\varphi}^{\tau}M\bm{\varphi}$, where the scalar mass
matrix can be written as $M=\partial^{2}V_{B-L}/\partial\bm{\varphi}^{2}$.
Being particularly interested in the CP-odd sector, we take
$\bm{\varphi}=\\{\text{Im}_{\varphi}\\}$ with
$\varphi^{0}=\frac{1}{\sqrt{2}}(v_{\varphi}+\text{Re}_{\varphi}+\text{Im}_{\varphi})$
and $\varphi=H,\Phi_{1},\Phi_{2},\phi_{1},\phi_{2},\phi_{3},\phi_{X}$. In
order to find the squared masses, we proceed to find the roots of the
eigenvalue equation $\det(\text{M}-\lambda\,\text{I}_{7\times 7})=0$. This is
a hard task since the large number of free parameters forbids useful
analytical expressions. For this reason, we depart from the ‘general model’
approach (where all couplings are independent) and proceed to simplify the
model assuming similar terms in $V_{B-L}$ may have similar couplings. Namely,

$\displaystyle\lambda_{11}=\lambda_{12}=\lambda_{s1}=\lambda_{s3}=\lambda_{sX},$
$\displaystyle\Lambda_{H1}=\Lambda_{H1}=\Lambda_{H2}=\Lambda_{Hs1}=\Lambda_{Hs3}=\Lambda_{HsX}=\Lambda_{H1}^{\prime}=\Lambda_{H2}^{\prime},$
$\displaystyle\Lambda_{11}^{\prime}=\Lambda_{13}^{\prime}=\Lambda_{1X}^{\prime}=\Lambda_{21}^{\prime}=\Lambda_{23}^{\prime}=\Lambda_{2X}^{\prime}=\lambda_{12}=\lambda_{12}^{\prime}=\Delta_{13}=\Delta_{1X}=\Delta_{3X},$
$\displaystyle\Lambda_{12}^{\prime}=\Lambda_{22}^{\prime}=\Delta_{12}=\Delta_{23}=\Delta_{2X}.$
(2)

We leave all other parameters free except for the VEVs of the new scalars
which we make $v_{\rm D}$ for $\left<\Phi_{i}\right>$ and $v_{\rm M}$ for
$\left<\phi_{j}\right>$, where $i=1,2$, $j=1,2,3,X$, and $v_{\rm D}\ll v_{\rm
SM}\approx 246\text{ GeV}\ll v_{\rm M}$. This choice was made in
Ref.sanchez2014complex  in order to obtain significant phenomenological
results and here we follow their work closely. In that paper, the masses in
the CP-odd scalar sector were found in the limit where $v_{\rm D}/v_{\rm
M}\rightarrow 0$,

$m_{I_{1}}^{2}=0,\hskip 7.22743ptm_{I_{2}}^{2}=0,\hskip
7.22743ptm_{I_{3}}^{2}=\mathcal{O}(v_{\rm D}/v_{\rm M}),$ (3)
$m_{I_{4}}^{2}=\frac{1}{2}\left[\Lambda_{Hs2}v_{\rm
SM}^{2}+(\Lambda_{12}^{\prime}+\lambda_{22}^{\prime}-2\Lambda_{Hs2})v_{\rm
D}^{2}+(\Delta_{12}+\Delta_{23}+\Delta_{2X})v_{\rm
M}^{2}-2\mu_{s2}^{2}\right],$ (4) $m_{I_{5,6}}^{2}=\frac{1}{4}v_{\rm
M}\left[(1+\sqrt{2})v_{\rm SM}-2\beta_{13}v_{\rm
M}\mp\sqrt{4\beta_{13}^{2}v_{\rm M}^{2}+(3-2\sqrt{2})v_{\rm SM}^{2}}\right],$
(5) $m_{I_{7}}^{2}=-5\beta_{3X}v_{\rm M}^{2},$ (6)

where $I_{i}$ ($i=1...7$) are linearly-independent eigenvectors of M. Note
that even if an analytical expression for $m_{I_{3}}^{2}$ is not shown in
Eq.(3), apart from the two Goldstone bosons, the remaining particles are much
heavier than $I_{3}$ unless their masses are fine-tuned not to be so.

A method to obtain approximate expressions for eigenvalues and eigenvectors is
Rayleigh-Schrödinger perturbation theory lancaster1985theory ;
mccartin2009rayleigh  where the mass matrix is expanded in terms of a small
parameter Alvarez-Salazar:2019cxw . In our case, $\zeta\equiv v_{\rm D}/v_{\rm
M}\ll$1 is the small parameter that allows us to obtain the zeroth-order
eigenvector (see Appendix B),

$I_{3}^{(0)}=\frac{1}{\sqrt{10}}\text{Im}_{\phi_{3}}+\frac{3}{\sqrt{10}}\text{Im}_{\phi_{X}}.$
(7)

For the mass of $I_{3}$, we have found a better approximation using a
different method (see Appendix A). We defined three small parameters and found
the lightest root of the characteristic polynomial of the scalar mass matrix.
The mass of the lightest CP-odd scalar $I_{3}$ has been found to be a function
of the two SSB scales and two dimensionless parameters,
$m^{2}_{I_{3}}=m^{2}_{I_{3}}(v_{\rm D},v_{\rm M},\beta_{13},\beta_{3X})$.
Namely,

$m_{I_{3}}^{2}\simeq\frac{37\,v_{\rm SM}\,v_{\rm M}^{2}v_{\rm
D}^{2}\,\beta_{13}\,\beta_{3X}}{5\sqrt{2}(1+\sqrt{2})v_{\rm
M}^{3}\beta_{13}\beta_{3X}-v_{\rm SM}\left(11v_{\rm D}^{2}\beta_{13}+10v_{\rm
M}^{2}\beta_{3X}\right)},$ (8)

where $\beta_{13}<0$ and $\beta_{3X}<0$ because of the positivity of other two
masses. Numerically, as $v_{\rm SM}\approx 246$ GeV and $v_{\rm SM}\ll v_{\rm
M}$, for $\beta_{13}\sim\mathcal{O}$(1), we have

$m_{I_{3}}\approx 0.73\sqrt{\frac{v_{\rm D}}{1\text{ MeV}}}\sqrt{\frac{v_{\rm
D}}{v_{\rm M}}}\text{ GeV}.$ (9)

The unstable $I_{3}$ pseudoscalar decays mainly into active neutrinos, with
decaying rate schechter1982neutrino

$\Gamma_{I_{3}\rightarrow\nu\nu}\approx\frac{m_{I_{3}}}{16\pi}\frac{\sum_{i}m^{2}_{\nu_{i}}}{v_{\rm
M}^{2}},$ (10)

where $v_{\rm M}$ is the SSB scale of the $B-L$ symmetry. Since neutrino
masses are unknown, we take the sum of these squared masses as a free
parameter (see Appendix C). In this way, it is convenient to rewrite Eq.(9)
and Eq.(10) like

$\frac{m_{I_{3}}}{\text{eV}}\approx 23.1\left(\frac{v_{\rm
D}}{\text{KeV}}\right)\left(\frac{\text{1000 TeV}}{v_{\rm M}}\right)^{1/2},$
(11)

$\frac{\text{Gyr}}{\Gamma_{I_{3}\to\nu\nu}}=\frac{\tau_{I_{3}}}{\text{Gyr}}\approx
9.1\left(\frac{5\times 10^{-3}\text{ eV}^{2}}{\sum
m^{2}_{\nu}}\right)\left(\frac{\text{KeV}}{v_{\rm D}}\right)\left(\frac{v_{\rm
M}}{\text{1000 TeV}}\right)^{5/2}.$ (12)

##  III Late-time neutrino production

Including the $I_{3}$ particle into the dynamics of the different components
in the Universe, we can write the following equations:

$\displaystyle\dot{\rho}_{I_{3}}+3H\rho_{I_{3}}$ $\displaystyle=$
$\displaystyle-\Gamma\rho_{I_{3}},$ (13)
$\displaystyle\dot{\rho}_{\nu}+4H\rho_{\nu}$ $\displaystyle=$
$\displaystyle+\Gamma\rho_{I_{3}},$ (14)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}$ $\displaystyle=$ $\displaystyle 0,$
(15)

where $\rho_{I3}$ is the density of the unstable particle $I_{3}$, which
produces an increase on the neutrino density $\rho_{\nu}$, and
$\Gamma=\Gamma(I_{3}\rightarrow\nu\nu)$. The stable cold dark matter is
denoted by $\rho_{m}$.

Considering the I3 freeze-out temperature TD $\gtrsim m_{t}\approx$ 173 GeV,
the $I_{3}$ abundance can be written as Kolb:1990vq ; lattanzi2007decaying

$\Omega_{I_{3},0}\,h^{2}=\frac{m_{I_{3}}}{1.25\text{
KeV}}\exp\left(-t_{0}/\tau\right),$ (16)

where $\tau_{I_{3}}=1/\Gamma_{I_{3}\rightarrow\nu\nu}$ is the $I_{3}$ lifetime
and $t_{0}$ is the age of the Universe.

Figure 1: Relative densities for the case $v_{\rm D}=1$ MeV and $v_{\rm M}=1$
TeV.

As we can see, both the lifetime and abundance of $I_{3}$ depend strongly on
$v_{\rm D}$ and $v_{\rm M}$, so we will use these observables to put limits on
these VEVs. As a first step, let’s analyze the example $v_{\rm D}=1$ MeV and
$v_{\rm M}=1$ TeV from sanchez2014complex . These values can give valid
neutrino masses by means of the type-I seesaw mechanism and also the correct
relic abundance for a stable DM candidate when a $\mathbb{Z}_{2}$ symmetry is
imposed on the $\phi_{2}$ singlet scalar. Using these values, we get

$\tau_{I_{3}}\approx 0.1\text{ year}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Omega_{I_{3},0}\,h^{2}\approx
10^{3}\exp\left(-t_{0}/(0.1\text{ yr})\right).$ (17)

So, in this case $I_{3}$ represents a negligible fraction of the total dark
matter in the universe today. This is fine if there is another DM candidate
that can account for the cold dark matter (CDM) relic density
$\Omega_{cdm,0}$. However, we can calculate the amount of produced neutrinos
from the decays of $I_{3}$ and check whether this is in conflict with
observations. This calculation is done numerically by integrating equations
(13) to (15) with initial conditions for $\rho_{I_{3}}$ derived from eq. (16)
and $\rho_{\nu,i}=0$ for the neutrinos produced from $I_{3}$ decays.

In Fig. 1 the fractional abundances are shown in comparison with the standard
cosmological scenario. Due to the large abundance of $I_{3}$ at early times,
there is a period of CDM dominance driven by $I_{3}$ around $t\sim 100$ s.
When the decay starts to take place, the energetic content is populated by the
relativistic neutrinos, which would be the main ingredient in the Universe
until the CDM starts to preponderate again. Such scenario would affect
dramatically the cosmological observables, and would hardly be acceptable by
the latest perturbation data. Therefore, since some choices for the model
parameters can affect the cosmology, we present in the following some weak
limits on such parameters.

##  IV Observational limits

If $I_{3}$ is a long-lived particle ($\tau_{I_{3}}\gg t_{0}$) we have, from
Eqs. (11) and (12),

$\displaystyle\Omega_{I_{3}}$ $\displaystyle\approx$
$\displaystyle\frac{m_{I_{3}}}{1.25\text{ KeV}}h^{-2},$ (18)
$\displaystyle\approx$ $\displaystyle\frac{9\times
10^{-3}}{h^{2}}\left(\frac{v_{\rm D}}{\text{KeV}}\right)\left(\frac{\text{1000
TeV}}{v_{\rm M}}\right)^{1/2},$ $\displaystyle\approx$
$\displaystyle\frac{9\times
10^{-3}}{h^{2}}\left(\frac{\text{Gyr}}{\tau_{I_{3}}}\right)\left(\frac{v_{\rm
M}}{\text{1000 TeV}}\right)^{2},$

assuming $\sum m_{\nu}^{2}=5\times 10^{-3}\text{ eV}^{2}$. This gives us an
upper bound for the energy scale ($v_{\rm M}$) of the $B-L$ spontaneous
symmetry breaking, since the relative matter density is below 1,

$v_{\rm M}\lesssim 6.9\,h\left(\frac{\tau_{I_{3}}}{\rm Gyr}\right)^{1/2}\times
10^{3}\ {\rm TeV}.$ (19)

For instance, if $\tau_{I_{3}}=100$ Gyr, using $h=0.674$ we get

$v_{\rm M}\lesssim 46.6\times 10^{3}\ {\rm TeV}.$ (20)

For lower values of $\tau_{I_{3}}$, we can make the following estimation. If
at $t=\tau_{I_{3}}$ all $I_{3}$ is converted immediately to neutrinos, then
the energy density would decrease with an extra scaling factor $a$ in
comparison with the cold dark matter. Assuming a matter dominated universe,
such event happens at

$a\approx\left(\frac{3H_{0}\tau_{I_{3}}}{2}\right)^{2/3}.$

Then,

$\Omega_{\nu}h^{2}\approx(H_{0}\tau_{I_{3}})^{2/3}\frac{m_{I_{3}}}{1.25\text{
KeV}}\approx(H_{0}\tau_{I_{3}})^{2/3}\,10^{-7}\left(\frac{1\text{
Gyr}}{\tau_{I_{3}}}\right)\left(\frac{v_{\rm M}}{1\text{ TeV}}\right)^{2}.$

Replacing $H_{0}\approx 0.1\,h$ Gyr-1:

$\Omega_{\nu}h^{2}\approx(0.1\,h)^{2/3}10^{-7}\left(\frac{1\text{
Gyr}}{\tau_{I_{3}}}\right)^{1/3}\left(\frac{v_{\rm M}}{1\text{
TeV}}\right)^{2}.$

For $\tau_{I_{3}}\approx 0.1$ yr, we would have $v_{\rm M}<76$ TeV, since
$\Omega_{\nu}\lesssim 1$. These numbers help to restrict the space of
parameters, however we would expect a much stronger restriction when analysing
other cosmological observables.

Figure 2: Value of scale factor at matter-radiation equality. For reference,
$a_{eq}\sim 3\times 10^{-4}$ is the $\Lambda$CDM prediction.

For instance, we can calculate the scale factor when the radiation and matter
densities are equal. A strong production of radiation (matter) would increase
(decrease) the value of the scale factor, affecting the growth of structures
or the CMB anisotropies. One way to estimate the limit on the amount of
radiation produced by the decay of $I_{3}$ would be to translate the limits on
the number of neutrino families given by Planck:2018vyg , i.e.
$N_{eff}=2.99\pm 0.17$, to a $3\sigma$ interval for the time of equality
between radiation and non-relativistic species,

$a_{eq.}=(3.0\pm 0.2)\times 10^{-4}.$ (21)

In Fig. 2 we show the scale factor at the equality. The safe choice would be
to stay on the left of both blue and green lines. There is a fine-tuned region
where the blue and green lines are almost indistinguishable, and it would
produce the ‘correct’ scale factor.

A fraction $f_{\rm dcdm}$ of the initial dark matter could be allowed to decay
into dark radiation. Its decay rate $\Gamma_{\rm dcdm}$ has been constrained
by a global analysis of both CMB and low redshift datasets Audren:2014bca ;
Poulin:2016nat . Using these limits, it is possible to put constraints on the
new energy scales of our model. Assuming an initial dark matter relative
density $\Omega^{\rm ini}_{\rm dm}$ with an initial stable component
$\Omega_{\rm sdm}=(1-f_{\rm dcdm})\Omega^{\rm ini}_{\rm sdm}$, we have

$\displaystyle\Omega_{\rm dm}$ $\displaystyle=$ $\displaystyle\Omega_{\rm
sdm}+\Omega_{\rm dcdm}$ (22) $\displaystyle=$ $\displaystyle\left[(1-f_{\rm
dcdm})+f_{\rm dcdm}\,e^{-\Gamma_{\rm dcdm}t}\right]\Omega^{\rm ini}_{\rm dm}.$

Ref.Poulin:2016nat  showed that if $\Gamma_{\rm dcdm}<H_{0}\sim 0.07$ Gyr-1,
then $f_{\rm dcdm}\,\Gamma_{\rm dcdm}<\mathcal{O}(10^{-3}){\text{ Gyr}}^{-1}$,
where the exact value of the upper bound depends on the dataset. On the
contrary, if $\Gamma_{\rm dcdm}>H_{0}$, then $f_{\rm dcdm}\lesssim 0.04$. We
used the most conservative of these bounds to map the parameter space of our
model $v_{\rm M}$ vs. $v_{\rm D}$, looking for the allowed regions for fixed
values of $f_{\rm dcdm}$. Results are shown in Figs. 3 and 4 where white
regions indicate the values of $\Gamma_{I_{3}}$ compatible with these bounds,
blue regions are excluded at 95$\%$ C.L., and green regions are excluded
because $\Gamma_{\rm dcdm}>H_{0}$ and $0.04\leq f_{\rm dcdm}$. Also, black
curves indicate different results for the present value of the DM relative
density

$\Omega_{dm}=\frac{m_{I_{3}}}{1.25\text{ KeV}}\,h^{-2}\left(\frac{1-f_{\rm
dcdm}}{f_{\rm dcdm}}+e^{-\Gamma_{I_{3}}t}\right),$ (23)

and colored curves represent the different values the lifetime of $I_{3}$ can
take (in Gyr). For example, below the blue dotted curve (13.8 Gyr),
$\tau_{I_{3}}<t_{0}$ (the age of the universe). The panels on the left and
right sides differ in the values of the mass of the lightest (heaviest)
neutrino. For instance, in the normal hierarchy, the lightest neutrino is
$\nu_{1}$ and the present limits on its mass are $0\leq m_{\nu_{1}}\lesssim
30$ meV, whereas in the inverted hierarchy the heaviest neutrino is $\nu_{2}$
and $49.9$ meV $\lesssim m_{\nu_{2}}\lesssim 52$ meV. These masses can affect
the sum of squared-masses according to the following expression (see Appendix
C),

$\sum_{i=1}^{3}m_{\nu_{i}}^{2}=\bigg{\\{}\begin{array}[]{cc}3\,m_{\nu_{1}}^{2}+\Delta
m_{\rm Sun}^{2}+\Delta m_{\rm atm}^{2}&(\text{NH}),\\\
3\,m_{\nu_{2}}^{2}-\Delta m_{\rm Sun}^{2}+\Delta m_{\rm
atm}^{2}&(\text{IH}).\end{array}$ (24)

We calculated the values of $\sum m_{\nu_{i}}^{2}$ using Eq.(24), the central
values of $\Delta m_{\rm Sun}^{2}$ and $\Delta m_{\rm atm}^{2}$ from
esteban2020fate , and $\sum m_{\nu}<$120 meV pdg .

Figure 3: Normal hierarchy: DM relative density (solid black) and $I_{3}$
lifetime in Gyr (dotted) curves. Blue regions are ruled-out because $I_{3}$
represents too-large of a fraction of the cold dark matter and its mean
lifetime is too short, even if it is larger than the present age of the
universe. Green regions are excluded because $\Gamma(I_{3}\to\nu\nu)>H_{0}$
and $f_{\rm dcdm}$ used in these plots are larger than 0.04.

Figure 4: Like Fig.3, but for neutrino mass inverted hierarchy.

##  V Conclusions

A dark matter model should predict its right abundance today, but this is not
enough for a successful model. The model analysed here has some regions on the
parameter space that produce the right amount of dark matter, but also present
some discrepancies with the standard cosmological model through a very high
production of radiation, which would distort the CMB anisotropies, for
instance.

The total dark matter of the universe could be multi-component. The lifetime
of its decaying component is strongly constrained when the produced dark
radiation is formed by neutrinos, gravitational waves or other new
relativistic degrees of freedom. In the model presented here, the unstable
dark matter candidate $I_{3}$ decays mainly into neutrinos, allowing us to use
the sum of their squared-masses and the fraction $f_{\rm dcdm}$ before decays
begin ($\Omega^{\rm ini}_{\rm dcdm}=f_{\rm dcdm}\,\Omega_{\rm dm}^{\rm ini}$)
as free parameters to constraint the new energy scales $v_{\rm D}$ and $v_{\rm
M}$. The latter is the VEV responsible for $B-L$ spontaneous symmetry breaking
and the former is responsible for dynamically generating Dirac neutrino
masses.

Our results show that, even if taking the most conservative limits on
$\Gamma_{\rm dcdm}$, a large area of the parameter space is ruled-out when the
decaying DM candidate $I_{3}$ represents a significant amount of the initial
CDM and its total decay rate is lower than the present Hubble parameter, i.e
$\Gamma(I_{3}\rightarrow\nu\nu)<H_{0}$, or when the decay rate of $I_{3}$ is
larger than the present value of the Hubble rate and $f_{\rm dcdm}>0.04$. The
strongest bounds come from the present value of the total (stable plus
decaying) DM relative density. These are dominant when the initial fraction of
decaying dark matter is of a few percent.

## Acknowledgements

We thank Flavia Sobreira and Cássio Pigozzo for helpful discussions. GG
ackowledges support from CNPq grant No.141699/2016-7 (Brasil), NSERC (Natural
Sciences and Engineering Research Council, Canada), McGill Space Institute,
and McGill Graduate $\&$ Postdoctoral Studies. PH is partially supported by
CNPq (grant no. 310952/2018-2) and FAPESP (grant no. 2014/19164-6). SC is
partially supported by CNPq with grant $\\#$ 311584/2020-9.

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##  Appendix A An approximate expression for the mass of $I_{3}$

In order to calculate the mass of the dark matter candidate in this model, it
is useful to make the following definitions

$\zeta\equiv\frac{v_{\rm D}}{v_{\rm M}},\hskip 7.22743pt\eta\equiv\frac{v_{\rm
D}}{v_{H}},\hskip 7.22743pt\chi\equiv\frac{v_{H}}{v_{\rm M}},$ (25)

where $v_{\rm D}\ll v_{H}\approx v_{SM}\approx 246$ GeV $\ll v_{\rm M}$. From
Eq.(II) the mass matrix for CP-odd scalars $M_{I}$ reads

$M_{I}^{2}=\frac{v_{SM}^{2}}{2\chi^{3}}\ (1-\eta^{2})^{2}\ m_{I}^{2},$ (26)

where $m_{I}^{2}/\chi$ is given by

$\begin{pmatrix}(1+\sqrt{2})\frac{\zeta^{2}}{\chi}&\sqrt{2}\
\zeta&\zeta&0&0&0&(-2+\sqrt{2})\ \zeta^{2}\\\ \sqrt{2}\
\zeta&\sqrt{2}\chi-\beta_{13}&\beta_{13}&\beta_{13}\zeta&0&-\beta_{13}\zeta&\sqrt{2}\zeta\chi\\\
\zeta&\beta_{13}&\chi-\beta_{13}&-\beta_{13}\zeta&0&\beta_{13}\zeta&-\sqrt{2}\zeta\chi\\\
0&\beta_{13}\zeta&-\beta_{13}\zeta&-\beta_{13}\zeta^{2}&0&\beta_{13}\zeta^{2}&0\\\
0&0&0&0&\frac{2\kappa\chi^{2}}{(1-\eta^{2})^{2}}&0&0\\\
0&-\beta_{13}\zeta&\beta_{13}\zeta&\beta_{13}\zeta^{2}&0&-9\beta_{3X}-\beta_{13}\zeta^{2}&3\beta_{3X}\\\
(-2+\sqrt{2})\
\zeta^{2}&\sqrt{2}\zeta\chi&-2\zeta\chi&0&0&3\beta_{3X}&(4+\sqrt{2})\zeta^{2}\chi-\beta_{3X}\end{pmatrix}$
(27)

Since the role of $\eta$ is small, we can safely take $\eta\rightarrow 0$
independently of $\zeta$ and $\chi$. From the $(5,5)$ element of this matrix,
we can see one eigenvalue is equal to $2\kappa\chi^{2}$ and
det($M_{I}^{2}$)=0, so at least another one is zero. Solving for their
eigenvalues $\lambda$, we get the following eigenvectors in the $\\{$ Im(H),
Im($\Phi_{1}$), Im($\Phi_{2}$), Im($\phi_{1}$), Im($\phi_{2}$),
Im($\phi_{3}$), Im($\phi_{X}$) $\\}$ basis

$\displaystyle\\{6\chi,-9\zeta,0,-8,0,1,3\\}\text{ and
}\\{-\chi,\zeta,\zeta,0,0,0,0\\}\hskip 7.22743pt$ for
$\displaystyle\lambda=0,$ (28) $\displaystyle\\{0,0,0,0,1,0,0\\}$ for
$\displaystyle\lambda=2\kappa\chi^{2},$

up to normalization factors. In Ref.[29] all masses were found (in the limit
$\zeta\rightarrow 0$) in the CP-odd scalar sector except for $m_{I_{3}}$,
which was given a numerical estimate of $\sim\mathcal{O}(\zeta^{1/2})$ in GeV.
Here we find an approximate value for this mass, which is the mass of the
unstable DM candidate $I_{3}$. The equation for the eigenvalues of $m_{I}^{2}$
is

$\lambda^{2}(\lambda-2\kappa\chi^{3})p(\lambda)=0,$ (29)

where $p(\lambda)=\sum_{i=0}^{4}c_{i}\,\lambda^{i}$ with $c_{4}$=1. Expanding
and comparing coefficients with det($\lambda\,\text{I}_{7\times
7}-m_{I}^{2}$), we get (to lowest order in the new parameters defined in
Eq.(25)),

$\displaystyle c_{0}$ $\displaystyle\approx$ $\displaystyle
74\sqrt{2}\beta_{13}\beta_{3X}\zeta^{2}\chi^{6},$ $\displaystyle c_{1}$
$\displaystyle\approx$ $\displaystyle
10\sqrt{2}\chi^{5}\beta_{3X}+11\sqrt{2}\zeta^{2}\chi^{5}\beta_{13}-10(1+\sqrt{2})\chi^{4}\beta_{13}\beta_{3X},$
$\displaystyle c_{2}$ $\displaystyle\approx$
$\displaystyle\sqrt{2}\chi^{4}-10(1+\sqrt{2})\chi^{3}\beta_{3X}-(1+\sqrt{2})\chi^{3}\beta_{13}+20\chi^{2}\beta_{13}\beta_{3X},$
$\displaystyle c_{3}$ $\displaystyle\approx$
$\displaystyle-(1+\sqrt{2})\chi^{2}+10\chi\beta_{3X}+2\chi\beta_{13}.$

As mentioned above, since $m_{I_{3}}$ is expected to be small, we can expand
$p(\lambda)$ around zero $p(\lambda)\approx p(0)+p^{\prime}(0)\,\lambda$ and
solve $p(\lambda)=0$ for $\lambda$, i.e.

$\lambda\approx-\frac{p(0)}{p^{\prime}(0)}=-\frac{c_{0}}{c_{1}}=\frac{74\sqrt{2}\beta_{13}\beta_{3X}\zeta^{2}\chi^{2}}{10\beta_{3X}((1+\sqrt{2})\beta_{13}-\sqrt{2}\chi)-11\sqrt{2}\beta_{13}\zeta^{2}\chi},$
(30)

which gives us

$m_{I_{3}}^{2}\simeq\frac{v_{SM}^{2}}{2\chi^{3}}\lambda=\frac{37\,v_{SM}\,v_{\rm
M}^{2}v_{\rm D}^{2}\,\beta_{13}\,\beta_{3X}}{5\sqrt{2}(1+\sqrt{2})v_{\rm
M}^{3}\beta_{13}\beta_{3X}-v_{SM}\left(11v_{\rm D}^{2}\beta_{13}+10v_{\rm
M}^{2}\beta_{3X}\right)}.$ (31)

##  Appendix B Rayleigh-Schrodinger perturbation theory

Here we find approximate expressions for $I_{3}$ and its mass using Rayleigh-
Schrodinger perturbation theory [30, 31, 32]. First of all, let’s write
$\tilde{m}_{I}^{2}\equiv m_{I}^{2}/\chi$ as

$\tilde{m}_{I}^{2}=\tilde{m}^{2}_{0}+\zeta\tilde{m}^{2}_{1}+\zeta^{2}\tilde{m}^{2}_{2},$
(32)

where

$\tilde{m}^{2}_{0}=\begin{pmatrix}0&0&0&0&0&0&0\\\
0&\sqrt{2}\chi-\beta_{13}&\beta_{13}&0&0&0&0\\\
0&\beta_{13}&\chi-\beta_{13}&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&2\kappa\chi^{2}&0&0&0\\\ 0&0&0&0&0&-9\beta_{3X}&3\beta_{3X}\\\
0&0&0&0&0&3\beta_{3X}&-\beta_{3X}\end{pmatrix}$ (33)

$\tilde{m}^{2}_{1}=\begin{pmatrix}0&\sqrt{2}&1&0&0&0&0\\\
\sqrt{2}&0&0&\beta_{13}&0&-\beta_{13}&\sqrt{2}\,\chi\\\
1&0&0&-\beta_{13}&0&\beta_{13}&-2\,\chi\\\ 0&\beta_{13}&-\beta_{13}&0&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&-\beta_{13}&\beta_{13}&0&0&0&0\\\
0&\sqrt{2}\,\chi&-2\,\chi&0&0&0&0\end{pmatrix}$ (34)

$\tilde{m}^{2}_{2}=\begin{pmatrix}\frac{1+\sqrt{2}}{\chi}&0&0&0&0&0&-2+\sqrt{2}\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&-\beta_{13}&0&\beta_{13}&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&\beta_{13}&0&-\beta_{13}&0\\\
-2+\sqrt{2}&0&0&0&0&0&(4+\sqrt{2})\chi\end{pmatrix}$ (35)

The first unperturbed eigenvalues and eigenvectors come from
$\tilde{m}^{2}_{0}$ and they are (before normalization)

$\displaystyle\lambda^{(0)}_{1}$ $\displaystyle=$ $\displaystyle
0,\,\lambda^{(0)}_{2}=0,\,\lambda^{(0)}_{3}=0,\,\lambda^{(0)}_{4}=-10\,\beta_{3X},\,\lambda^{(0)}_{5}=2\kappa\chi^{2},$
$\displaystyle\lambda^{(0)}_{6,7}$ $\displaystyle=$
$\displaystyle\left[-2\beta_{13}+(1+\sqrt{2})\chi\mp\sqrt{4\beta_{13}^{2}+(3-2\sqrt{2})\chi^{2}}\right]/2,$
(36)

$\displaystyle I^{(0)}_{1}$ $\displaystyle=$
$\displaystyle(1,0,0,0,0,0,0),\,I^{(0)}_{2}=(0,0,0,1,0,0,0),\,I^{(0)}_{3}=(0,0,0,0,0,1,3),$
$\displaystyle I^{(0)}_{4}$ $\displaystyle=$
$\displaystyle(0,0,0,0,0,-3,1),\,I^{(0)}_{5}=(0,0,0,0,1,0,0),\,I^{(0)}_{6}=(0,r^{-},1,0,0,0,0),$
$\displaystyle I^{(0)}_{7}$ $\displaystyle=$
$\displaystyle(0,r^{+},1,0,0,0,0),$ (37)

where
$2\beta_{13}\,r^{\pm}=(\sqrt{2}-1)\chi\pm\sqrt{4\beta_{13}^{2}+(3-2\sqrt{2})\chi^{2}}$,
which is in agreement with [29]. The first order corrections are obtained
using

$\lambda^{(1)}_{i}=\left<I^{(0)}_{i}|\tilde{m}_{1}^{2}|I^{(0)}_{i}\right>\hskip
7.22743pt\text{and}\hskip
7.22743ptI^{(1)}_{i}=-\left(\tilde{m}_{0}^{2}-\lambda^{(0)}_{i}\text{I}_{7\times
7}\right)^{\text{PS}}\left(\tilde{m}_{1}^{2}-\lambda^{(1)}_{i}\text{I}_{7\times
7}\right)I^{(0)}_{i},$ (38)

where I7×7 is the identity matrix and PS stands for pseudoinverse. For $i=3$
we have $\lambda^{(0)}_{3}=0$, so

$I^{(1)}_{3}=-\tilde{m}_{0}^{\text{PS}}\tilde{m}_{1}^{2}I^{(0)}_{3}=(0,3t-8\beta_{13},-6t+8\sqrt{2}\beta_{13},0,0,0,0),$
(39)

where $\sqrt{2}\,t=(2+\sqrt{2})\beta_{13}-2\chi$. Next,

$\lambda_{3}^{(2)}=\left<I^{(0)}_{3}|\tilde{m}_{0}^{2}|I^{(0)}_{3}\right>+\left<I^{(0)}_{3}|\tilde{m}_{1}^{2}|I^{(1)}_{3}\right>\hskip
289.07999pt$
$\lambda_{3}^{(2)}=\frac{1}{10}\left(9(4+\sqrt{2})\chi-\beta_{13}\right)+\frac{1}{3\sqrt{10}N_{3}}\bigg{[}(6t-8\sqrt{2}\beta_{13})(6\chi-\beta_{13})+(3t-8\beta_{13})(3\sqrt{2}\chi-\beta_{13})\bigg{]},$
(40)

with the normalization factor
$N_{3}=[90\chi^{2}+6(2-7\sqrt{2})\beta_{13}\chi+(87-54\sqrt{2})\beta_{13}^{2}]^{1/2}$.

Now, we can compute the second order contribution

$I^{(2)}_{3}=-(\tilde{m}_{0}^{2})^{\text{PS}}\left[(\tilde{m}_{2}^{2}-\lambda_{3}^{(2)})I_{3}^{(0)}+\tilde{m}_{1}^{2}I_{3}^{(1)}\right]=(0,0,0,0,0,-3,1)/\sqrt{10}.$
(41)

Finally, $I_{3}$ and its mass can be approximated to

$I_{3}\simeq I^{(0)}_{3}+\zeta\,I^{(1)}_{3}+\zeta^{2}\,I^{(2)}_{3}\hskip
14.45377pt\text{and}\hskip
14.45377ptm_{I_{3}}^{2}\simeq\frac{v_{SM}^{2}}{2\chi^{2}}\zeta^{2}\lambda^{(2)}_{3}\approx\frac{1}{2}v_{\rm
D}^{2}\lambda_{3}^{(2)}.$ (42)

##  Appendix C Sum of neutrino squared masses

When neutrino masses obey a ‘normal hierarchy’ (NH), then $\Delta m_{\rm
atm}^{2}=\Delta m_{31}^{2}$. When the hierarchy is inverted (IH), $\Delta
m_{\rm atm}^{2}=\Delta m_{32}^{2}$. On the other hand, $\Delta m_{\rm
Sun}^{2}=\Delta m_{21}^{2}$ irrespective of the hierarchy. Taking this into
consideration, we can write the sum of squared masses depending on one
neutrino mass only,

$\sum_{i=1}^{3}m_{\nu_{i}}^{2}=\bigg{\\{}\begin{array}[]{cc}3\,m_{\nu_{1}}^{2}+\Delta
m_{\rm Sun}^{2}+\Delta m_{\rm atm}^{2}&(\text{NH})\\\
3\,m_{\nu_{2}}^{2}-\Delta m_{\rm Sun}^{2}+\Delta m_{\rm
atm}^{2}&(\text{IH})\end{array}$ (43)

The sum of neutrino masses is bounded from above $\sum m_{\nu}<$ 120 meV [40].
Using this limit, $\Delta m_{\rm Sun}^{2}$, and $\Delta m_{\rm atm}^{2}$, we
get (see Fig.(5))

$\displaystyle 0\leq$ $\displaystyle m_{\nu_{1}}$ $\displaystyle\lesssim 30\
{\rm meV}\hskip 7.22743pt\text{(Normal hierarchy)},$ $\displaystyle 49.9\ {\rm
meV}\lesssim$ $\displaystyle m_{\nu_{2}}$ $\displaystyle\lesssim 52.3\ {\rm
meV}\hskip 7.22743pt\text{(Inverted hierarchy)}.$ (44)

Therefore, upper and lower bounds for $\sum m_{\nu_{i}}^{2}$ are approximately
(see Fig.(6))

$\displaystyle 2500\ {\rm meV^{2}}\lesssim$ $\displaystyle\sum
m_{\nu_{i}}^{2}$ $\displaystyle\lesssim 5200\ {\rm meV^{2}}\hskip
7.22743pt\text{(Normal hierarchy)},$ $\displaystyle 4900\ {\rm
meV^{2}}\lesssim$ $\displaystyle\sum m_{\nu_{i}}^{2}$ $\displaystyle\lesssim
5600\ {\rm meV^{2}}\hskip 7.22743pt\text{(Inverted hierarchy)}.$ (45)

Figure 5: Upper and lower bounds on $m_{\nu_{1}}$ and $m_{\nu_{2}}$ for the
normal and inverted hierarchy, respectively.

Figure 6: Constraints on the sum of neutrino squared masses.


# The Dirac equation on metrics of Eguchi-Hanson type

Zhuohua Cai† †School of Mathematics and Information Science, Guangxi
University, Nanning, Guangxi 530004, PR China<EMAIL_ADDRESS>and
Xiao Zhang♭ ♭Guangxi Center for Mathematical Research, Guangxi University,
Nanning, Guangxi 530004, PR China  ♭Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, PR China  ♭School of
Mathematical Sciences, University of Chinese Academy of Sciences, Beijing
100049, PR China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We investigate parallel spinors on the Eguchi-Hanson metrics and find the
space of complex parallel spinors are complex 2-dimensional. For the metrics
of Eguchi-Hanson type with the zero scalar curvature, we separate variables
for the harmonic spinors and obtain the solutions explicitly.

PASC numbers: 03.60.Pm, 04.20.Gz, 04.60.-m
Key words: Parallel spinors, Dirac equation, Eguchi-Hanson metric

##  1\. Introduction

Eguchi-Hanson metrics were constructed in [1, 2], which are complete
4-dimensional Ricci flat, anti-self-dual ALE Riemannian metrics on $R_{\geq
0}\times P^{3}$. They are one type of gravitational instantons, and play
important roles in the Euclidean approach of gravitational quantization [3].
Since then, gravitational instantons including the Eguchi-Hanson metrics
attract much attention both in geometry and in general relativity.

In order to investigate the physical behaviour of spin-$\frac{1}{2}$ particles
in general relativity, Chandrasekhar solved the Dirac equation in Kerr
spacetimes by separating variables [4]. We refer to [5, 6] and references
therein for further developments of the the Dirac equation in various
spacetimes. In [7], Sucu and Ünal solved the Dirac equation on Eguchi-Hanson
and Bianchi $VII_{0}$ gravitational instanton metrics. In particular, the
solutions are obtained as the product of two hypergeometric functions for the
Eguchi-Hanson gravitational instanton metric.

As it actually relates to manifold’s scalar curvature, it is interesting to
study the Dirac equation on much more general metrics of Eguchi-Hanson type
with the zero scalar curvature. This type of metrics were constructed by
LeBrun using the method of algebraic geometry [8] and by the second author
solving an ordinary differential equation [9] on $R_{\geq 0}\times
S^{3}/Z_{d}$ $(d>2)$, and they provides the counter-examples of Hawking and
Pope’s generalized positive action conjecture.

In this note we first introduce the Eguchi-Hanson metrics, the metrics of
Eguchi-Hanson type with the zero scalar curvature as well as the spin
connections on them. Then we investigate the parallel spinors and the harmonic
spinors. As the existence of a nontrivial parallel spinor on a Riemannian spin
manifold implies that the manifold is Ricci-flat, parallel spinors only exist
on the Eguchi-Hanson metrics, and we find the space of complex parallel
spinors are complex 2-dimensional. For the metrics of Eguchi-Hanson type with
the zero scalar curvature, we separate variables for the harmonic spinors and
obtain the solutions explicitly.

##  2\. Metrics of Eguchi-Hanson Type

Metrics of Eguchi-Hanson type are given by

$g=f^{-2}\mathrm{d}r^{2}+r^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+f^{2}\sigma_{3}^{2}\right),$
(2.1)

where $f$ is a real function of $r$, $\sigma_{1}$, $\sigma_{2}$ and
$\sigma_{3}$ are the Cartan-Maurer forms for $SU(2)\cong S^{3}$

$\displaystyle\sigma_{1}=\dfrac{1}{2}\left(\sin\psi\mathrm{d}\theta-\sin\theta\cos\psi\mathrm{d}\phi\right),$
$\displaystyle\sigma_{2}=\dfrac{1}{2}\left(-\cos\psi\mathrm{d}\theta-\sin\theta\sin\psi\mathrm{d}\phi\right),$
$\displaystyle\sigma_{3}=\dfrac{1}{2}\left(\mathrm{d}\psi+\cos\theta\mathrm{d}\phi\right)$

with three Euler angles $\theta$, $\phi$, $\psi$. The frame and coframe are

$e^{1}=f^{-1}\mathrm{d}r,\quad e^{2}=r\sigma_{1},\quad e^{3}=r\sigma_{2},\quad
e^{4}=rf\sigma_{3},$ (2.2)

and

$\displaystyle e_{1}=f\partial_{r},\quad
e_{2}=\dfrac{2}{r}\left(\sin\psi\partial_{\theta}-\dfrac{\cos\psi}{\sin\theta}\partial_{\phi}+\dfrac{\cos\theta\cos\psi}{\sin\theta}\partial_{\psi}\right),$
(2.3) $\displaystyle
e_{3}=\dfrac{2}{r}\left(-\cos\psi\partial_{\theta}-\dfrac{\sin\psi}{\sin\theta}\partial_{\phi}+\dfrac{\cos\theta\sin\psi}{\sin\theta}\partial_{\psi}\right),\quad
e_{4}=\dfrac{2}{rf}\partial_{\psi}$

respectively.

Let $\\{\omega^{i}_{j}\\}$ be the connection 1-forms given by
$\mathrm{d}e^{i}=-\omega^{i}_{j}\wedge e^{j}$. Then [9]

$\displaystyle\omega^{2}_{1}=\dfrac{f}{r}e^{2},\quad\omega^{3}_{1}=\dfrac{f}{r}e^{3},\quad\omega^{4}_{1}=\left(\dfrac{f}{r}+f^{\prime}\right)e^{4},$
(2.4)
$\displaystyle\omega^{3}_{4}=\dfrac{f}{r}e^{2},\quad\omega^{4}_{2}=\dfrac{f}{r}e^{3},\quad\omega^{2}_{3}=\left(\dfrac{2}{rf}-\dfrac{f}{r}\right)e^{4}.$

Let $B>0$ be certain positive constant. Taking

$\displaystyle f=\sqrt{1-\dfrac{B}{r^{4}}},$

and

$\displaystyle r\geq\sqrt[4]{B},\quad 0\leq\theta\leq\pi,\quad 0\leq\phi\leq
2\pi,\quad 0\leq\psi\leq 2\pi,$

metric (2.1) is the Eguchi-Hanson metric [1, 2]

$g=\left(1-\dfrac{B}{r^{4}}\right)^{-1}\mathrm{d}r^{2}+r^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+\left(1-\dfrac{B}{r^{4}}\right)\sigma_{3}^{2}\right),$
(2.5)

which is Ricci-flat and geodesically complete. Topologically, it is

$\displaystyle R_{\geq 0}\times SU(2)/Z_{2}\cong R_{\geq 0}\times SO(3)\cong
R_{\geq 0}\times P_{3}.$

Let $d>2$ be a positive integer. Taking

$\displaystyle f=\sqrt{1-\dfrac{2A}{r^{2}}-\dfrac{B}{r^{4}}},\quad
A=-\dfrac{d-2}{2}\sqrt{\dfrac{B}{d-1}},\quad r_{0}=\sqrt[4]{\dfrac{B}{d-1}}$

and

$\displaystyle r\geq r_{0},\quad 0\leq\theta\leq\pi,\quad 0\leq\phi\leq
2\pi,\quad 0\leq\psi\leq\frac{4\pi}{d},$

metric (2.1) is

$g=\left(1-\dfrac{2A}{r^{2}}-\dfrac{B}{r^{4}}\right)^{-1}\mathrm{d}r^{2}+r^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+\left(1-\dfrac{2A}{r^{2}}-\dfrac{B}{r^{4}}\right)\sigma_{3}^{2}\right),$
(2.6)

which is scalar-flat and geodesically complete, constructed in [8, 9].
Topologically, it is

$\displaystyle R_{\geq 0}\times SU(2)/Z_{d}\cong R_{\geq 0}\times
S^{3}/Z_{d}.$

##  3\. Spin Connection and Parallel Spinors

We refer to [10] for an introductory knowledge of spin geometry. It is clear
that the underline manifolds of both metric (2.5) and metric (2.6) are spin
and their complex spinor bundles are complex 4-dimensional equipped with spin
connection

$\nabla_{e_{k}}\Phi=e_{k}\Phi+\dfrac{1}{4}\sum_{i,j=1}^{4}g\left(\nabla_{e_{k}}e_{i},e_{j}\right)e_{i}\cdot
e_{j}\cdot\Phi$

for spinor $\Phi=\left(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4}\right)^{t}$. Using
(2.4), we obtain

$\displaystyle\nabla_{e_{1}}\Phi=e_{1}\Phi,$ (3.1)
$\displaystyle\nabla_{e_{2}}\Phi=e_{2}\Phi+\dfrac{f}{2r}\left(e_{1}\cdot
e_{2}-e_{3}\cdot e_{4}\right)\cdot\Phi,$
$\displaystyle\nabla_{e_{3}}\Phi=e_{3}\Phi+\dfrac{f}{2r}\left(e_{1}\cdot
e_{3}+e_{2}\cdot e_{4}\right)\cdot\Phi,$
$\displaystyle\nabla_{e_{4}}\Phi=e_{4}\Phi+\dfrac{1}{2}\left(\left(\dfrac{f}{r}+f^{\prime}\right)e_{1}\cdot
e_{4}+\left(\dfrac{f}{r}-\dfrac{2}{rf}\right)e_{2}\cdot
e_{3}\right)\cdot\Phi.$

Throughout the paper, we fix the following representations for $e_{i}$ in
$Spin(4)$.

$\displaystyle e_{1}\mapsto\begin{pmatrix}0&0&\eqmakebox[c]{1}&0\\\
0&0&0&\eqmakebox[c]{1}\\\ \eqmakebox[c]{-1}&0&0&0\\\
0&\eqmakebox[c]{-1}&0&0\\\ \end{pmatrix},\qquad
e_{2}\mapsto\begin{pmatrix}0&0&0&\eqmakebox[c]{i}\\\ 0&0&\eqmakebox[c]{i}&0\\\
0&\eqmakebox[c]{i}&0&0\\\ \eqmakebox[c]{i}&0&0&0\\\ \end{pmatrix},$ (3.2)
$\displaystyle e_{3}\mapsto\begin{pmatrix}0&0&0&\eqmakebox[c]{-1}\\\
0&0&\eqmakebox[c]{1}&0\\\ 0&\eqmakebox[c]{-1}&0&0\\\ \eqmakebox[c]{1}&0&0&0\\\
\end{pmatrix},\qquad e_{4}\mapsto\begin{pmatrix}0&0&\eqmakebox[c]{i}&0\\\
0&0&0&\eqmakebox[c]{-i}\\\ \eqmakebox[c]{i}&0&0&0\\\
0&\eqmakebox[c]{-i}&0&0\\\ \end{pmatrix}.$

Spinor $\Phi$ is parallel if $\nabla_{X}\Phi=0$ for any tangent vector $X$.
Any Riemannian metric carried a parallel spinor must be Ricci-flat, see, e.g.
[10]. Thus it is interesting to investigate the parallel spinors for the
Eguchi-Hanson metric.

######  Theorem 3.1.

Parallel spinors for the Eguchi-Hanson metric (2.5) are complex constant
spinors which form a complex 2-dimensional space.

Proof: Under (3.2), we have

$\displaystyle\nabla_{e_{1}}\Phi=e_{1}\begin{pmatrix}\Phi_{1}\\\ \Phi_{2}\\\
\Phi_{3}\\\ \Phi_{4}\\\ \end{pmatrix}=0,$ (3.3)
$\displaystyle\nabla_{e_{2}}\Phi=e_{2}\begin{pmatrix}\Phi_{1}\\\ \Phi_{2}\\\
\Phi_{3}\\\ \Phi_{4}\\\ \end{pmatrix}+\dfrac{f}{2r}\begin{pmatrix}0\\\ 0\\\
-2\mathrm{i}\Phi_{4}\\\ -2\mathrm{i}\Phi_{3}\\\ \end{pmatrix}=0,$ (3.4)
$\displaystyle\nabla_{e_{3}}\Phi=e_{3}\begin{pmatrix}\Phi_{1}\\\ \Phi_{2}\\\
\Phi_{3}\\\ \Phi_{4}\\\ \end{pmatrix}+\dfrac{f}{2r}\begin{pmatrix}0\\\ 0\\\
2\Phi_{4}\\\ -2\Phi_{3}\\\ \end{pmatrix}=0,$ (3.5)
$\displaystyle\nabla_{e_{4}}\Phi=e_{4}\begin{pmatrix}\Phi_{1}\\\ \Phi_{2}\\\
\Phi_{3}\\\ \Phi_{4}\\\
\end{pmatrix}+\left(\dfrac{1}{rf}+\dfrac{f^{\prime}}{2}\right)\begin{pmatrix}0\\\
0\\\ -\mathrm{i}\Phi_{3}\\\ \mathrm{i}\Phi_{4}\\\ \end{pmatrix}=0.$ (3.6)

Firstly, (3.3) implies

$\displaystyle\partial_{r}\Phi=0.$

Secondly, (3.4) gives

$\displaystyle\left(\sin\psi\partial_{\theta}-\dfrac{\cos\psi}{\sin\theta}\partial_{\phi}+\dfrac{\cos\theta\cos\psi}{\sin\theta}\partial_{\psi}\right)\begin{pmatrix}\Phi_{3}\\\
\Phi_{4}\end{pmatrix}=\dfrac{f}{r}\begin{pmatrix}\mathrm{i}\Phi_{4}\\\
\mathrm{i}\Phi_{3}\end{pmatrix}.$

Since the left hand side is independent on $r$, but the right hand side
contains function $f(r)$, it must be

$\displaystyle\Phi_{3}=\Phi_{4}=0.$

Now (3.6) gives

$\displaystyle\partial_{\psi}\Phi=0.$

Thus (3.4), (3.5) give

$\displaystyle\dfrac{2}{r}\left(\sin\psi\partial_{\theta}-\dfrac{\cos\psi}{\sin\theta}\partial_{\phi}\right)\begin{pmatrix}\Phi_{1}\\\
\Phi_{2}\end{pmatrix}$ $\displaystyle=0,$
$\displaystyle\dfrac{2}{r}\left(-\cos\psi\partial_{\theta}-\dfrac{\sin\psi}{\sin\theta}\partial_{\phi}\right)\begin{pmatrix}\Phi_{1}\\\
\Phi_{2}\end{pmatrix}$ $\displaystyle=0.$

Therefore

$\displaystyle\partial_{\theta}\begin{pmatrix}\Phi_{1}\\\
\Phi_{2}\end{pmatrix}=\partial_{\phi}\begin{pmatrix}\Phi_{1}\\\
\Phi_{2}\end{pmatrix}=0.$

So $\Phi_{1}$ and $\Phi_{2}$ are arbitrary complex constants and the proof of
the theorem is complete. Q.E.D.

##  4\. Harmonic Spinors

Harmonic spinors are the solutions of the Dirac equation

$\displaystyle\mathcal{D}=\sum_{k=1}^{4}e_{k}\cdot\nabla_{e_{k}}=0.$

We shall find the explicit solutions of harmonic spinors on metric (2.6) of
Eguchi-Hanson type with zero scalar curvature ($d>2$). In the case $d=2$,
harmonic spinors on Eguchi-Hanson metric (2.5) were solved by Sucu and Ünal in
[7], and the solutions can be obtained in terms of hypergeometric functions.

######  Theorem 4.1.

Let positive integer $d>2$. There exists harmonic spinors

$\displaystyle\Phi=\mathrm{e}^{\mathrm{i}\left(n+\frac{1}{2}\right)\phi}\begin{pmatrix}C_{1}\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m+\frac{1}{2}\right)\psi}u_{1}(r)v_{1}(\theta)\\\
C_{2}\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m-\frac{1}{2}\right)\psi}u_{2}(r)v_{2}(\theta)\\\
C_{3}\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m+\frac{1}{2}\right)\psi}u_{3}(r)v_{3}(\theta)\\\
C_{4}\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m-\frac{1}{2}\right)\psi}u_{4}(r)v_{4}(\theta)\end{pmatrix},$

on metric (2.6) of Eguchi-Hanson type with zero scalar curvature, where $m$,
$n$ are integers, $C_{i}$ are complex constants,

$\displaystyle\begin{matrix}\left\\{\begin{aligned}
u_{1}(r)&=\Big{(}r^{2}-r_{0}^{2}\Big{)}^{\frac{1-d-dm}{2d}}\Big{(}r^{2}+(d-1)r_{0}^{2}\Big{)}^{-\frac{1-d-dm}{2d}-\frac{d\left(1+2m\right)}{4}},\\\
u_{2}(r)&=\Big{(}r^{2}-r_{0}^{2}\Big{)}^{\frac{1-d+dm}{2d}}\Big{(}r^{2}+(d-1)r_{0}^{2}\Big{)}^{-\frac{1-d+dm}{2d}-\frac{d\left(1-2m\right)}{4}},\\\
u_{3}(r)&=\dfrac{1}{r}\Big{(}r^{2}-r_{0}^{2}\Big{)}^{-\frac{1}{2}-\frac{1-d-dm}{2d}}\Big{(}r^{2}+(d-1)r_{0}^{2}\Big{)}^{-\frac{1}{2}+\frac{1-d-dm}{2d}+\frac{d\left(1+2m\right)}{4}},\\\
u_{4}(r)&=\dfrac{1}{r}\Big{(}r^{2}-r_{0}^{2}\Big{)}^{-\frac{1}{2}-\frac{1-d+dm}{2d}}\Big{(}r^{2}+(d-1)r_{0}^{2}\Big{)}^{-\frac{1}{2}+\frac{1-d+dm}{2d}+\frac{d\left(1-2m\right)}{4}},\end{aligned}\right.\end{matrix}$
(4.1) $\displaystyle\begin{matrix}\left\\{\begin{aligned}
v_{1}(\theta)&=v_{3}(\theta)=\left(\sin\dfrac{\theta}{2}\right)^{n+\frac{1}{2}-\frac{d}{2}\left(m+\frac{1}{2}\right)}\left(\cos\dfrac{\theta}{2}\right)^{-n-\frac{1}{2}-\frac{d}{2}\left(m+\frac{1}{2}\right)},\\\
v_{2}(\theta)&=v_{4}(\theta)=\left(\sin\dfrac{\theta}{2}\right)^{-n-\frac{1}{2}+\frac{d}{2}\left(m-\frac{1}{2}\right)}\left(\cos\dfrac{\theta}{2}\right)^{n+\frac{1}{2}+\frac{d}{2}\left(m-\frac{1}{2}\right)}.\end{aligned}\right.\end{matrix}$
(4.2)

Proof: Denote

$\displaystyle F_{1}=\dfrac{f}{r}+\dfrac{f^{\prime}}{2}-\dfrac{1}{rf},\quad
F_{2}=\dfrac{2f}{r}+\dfrac{f^{\prime}}{2}+\dfrac{1}{rf}.$

Under (3.2), harmonic spinors satisfy

$\displaystyle\left(f\partial_{r}-\dfrac{2\mathrm{i}}{rf}\partial_{\psi}+F_{1}\right)\Phi_{1}-\dfrac{2}{r}\mathrm{e}^{\mathrm{i}\psi}\left(\partial_{\theta}-\mathrm{i}\csc\theta\partial_{\phi}+\mathrm{i}\cot\theta\partial_{\psi}\right)\Phi_{2}=0,$
(4.3)
$\displaystyle\left(f\partial_{r}+\dfrac{2\mathrm{i}}{rf}\partial_{\psi}+F_{1}\right)\Phi_{2}+\dfrac{2}{r}\mathrm{e}^{-\mathrm{i}\psi}\left(\partial_{\theta}+\mathrm{i}\csc\theta\partial_{\phi}-\mathrm{i}\cot\theta\partial_{\psi}\right)\Phi_{1}=0,$
$\displaystyle\left(f\partial_{r}+\dfrac{2\mathrm{i}}{rf}\partial_{\psi}+F_{2}\right)\Phi_{3}+\dfrac{2}{r}\mathrm{e}^{\mathrm{i}\psi}\left(\partial_{\theta}-\mathrm{i}\csc\theta\partial_{\phi}+\mathrm{i}\cot\theta\partial_{\psi}\right)\Phi_{4}=0,$
$\displaystyle\left(f\partial_{r}-\dfrac{2\mathrm{i}}{rf}\partial_{\psi}+F_{2}\right)\Phi_{4}-\dfrac{2}{r}\mathrm{e}^{-\mathrm{i}\psi}\left(\partial_{\theta}+\mathrm{i}\csc\theta\partial_{\phi}-\mathrm{i}\cot\theta\partial_{\psi}\right)\Phi_{3}=0.$

Separating variables

$\begin{pmatrix}\Phi_{1}\\\ \Phi_{2}\\\ \Phi_{3}\\\ \Phi_{4}\\\
\end{pmatrix}=\mathrm{e}^{\mathrm{i}\left(n+\frac{1}{2}\right)\phi}\begin{pmatrix}\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m+\frac{1}{2}\right)\psi}h_{1}(r,\theta)\\\
\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m-\frac{1}{2}\right)\psi}h_{2}(r,\theta)\\\
\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m+\frac{1}{2}\right)\psi}h_{3}(r,\theta)\\\
\mathrm{e}^{\mathrm{i}\frac{d}{2}\left(m-\frac{1}{2}\right)\psi}h_{4}(r,\theta)\end{pmatrix},$
(4.4)

where $m$, $n$ are integers, the Dirac equation gives

$e^{\mathrm{i}\left(\frac{d}{2}-1\right)\psi}\vec{L}(r,\theta)=\vec{R}(r,\theta),$
(4.5)

where

$\displaystyle\begin{aligned}
\vec{L}(r,\theta)&=\begin{pmatrix}\dfrac{rf}{2}\left(\partial_{r}+\dfrac{d\left(2m+1\right)}{2rf^{2}}+\dfrac{1}{r}+\dfrac{f^{\prime}}{2f}-\dfrac{1}{rf^{2}}\right)h_{1}\\\
\left(\partial_{\theta}-\left(n+\dfrac{1}{2}\right)\csc\theta+\dfrac{d}{2}\left(m+\dfrac{1}{2}\right)\cot\theta\right)h_{1}\\\
-\dfrac{rf}{2}\left(\partial_{r}-\dfrac{d\left(2m+1\right)}{2rf^{2}}+\dfrac{2}{r}+\dfrac{f^{\prime}}{2f}+\dfrac{1}{rf^{2}}\right)h_{3}\\\
\left(\partial_{\theta}-\left(n+\dfrac{1}{2}\right)\csc\theta+\dfrac{d}{2}\left(m+\dfrac{1}{2}\right)\cot\theta\right)h_{3}\end{pmatrix},\\\
\vec{R}(r,\theta)&=\begin{pmatrix}\left(\partial_{\theta}+\left(n+\dfrac{1}{2}\right)\csc\theta-\dfrac{d}{2}\left(m-\dfrac{1}{2}\right)\cot\theta\right)h_{2}\\\
-\dfrac{rf}{2}\left(\partial_{r}-\dfrac{d\left(2m-1\right)}{2rf^{2}}+\dfrac{1}{r}+\dfrac{f^{\prime}}{2f}-\dfrac{1}{rf^{2}}\right)h_{2}\\\
\left(\partial_{\theta}+\left(n+\dfrac{1}{2}\right)\csc\theta-\dfrac{d}{2}\left(m-\dfrac{1}{2}\right)\cot\theta\right)h_{4}\\\
\dfrac{rf}{2}\left(\partial_{r}+\dfrac{d\left(2m-1\right)}{2rf^{2}}+\dfrac{2}{r}+\dfrac{f^{\prime}}{2f}+\dfrac{1}{rf^{2}}\right)h_{4}\end{pmatrix}.\end{aligned}$

Note that $\vec{L}(r,\theta)$ and $\vec{R}(r,\theta)$ are independent with
$\psi$, and $d\geq 3$, we obtain

$\displaystyle\vec{L}(r,\theta)=\vec{R}(r,\theta)=0.$ (4.6)

Thus, for $i=1,3$, $j=2,4$, we have

$\displaystyle\partial_{\theta}h_{i}$
$\displaystyle=\left(\left(n+\dfrac{1}{2}\right)\csc\theta-\dfrac{d}{2}\left(m+\dfrac{1}{2}\right)\cot\theta\right)h_{i},$
$\displaystyle\partial_{\theta}h_{j}$
$\displaystyle=\left(-\left(n+\dfrac{1}{2}\right)\csc\theta+\dfrac{d}{2}\left(m-\dfrac{1}{2}\right)\cot\theta\right)h_{j}.$

Solving these ordinary differential equations, we obtain

$\displaystyle h_{i}$
$\displaystyle=u_{i}(r)\left(\sin\dfrac{\theta}{2}\right)^{n+\frac{1}{2}-\frac{d}{2}\left(m+\frac{1}{2}\right)}\left(\cos\dfrac{\theta}{2}\right)^{-n-\frac{1}{2}-\frac{d}{2}\left(m+\frac{1}{2}\right)},$
$\displaystyle h_{j}$
$\displaystyle=u_{j}(r)\left(\sin\dfrac{\theta}{2}\right)^{-n-\frac{1}{2}+\frac{d}{2}\left(m-\frac{1}{2}\right)}\left(\cos\dfrac{\theta}{2}\right)^{n+\frac{1}{2}+\frac{d}{2}\left(m-\frac{1}{2}\right)}.$

Substituting them into (4.6), we have

$\displaystyle u_{1}^{\prime}$
$\displaystyle=\dfrac{1}{r^{4}-2Ar^{2}-B}\left(Ar-d\left(m+\dfrac{1}{2}\right)r^{3}\right)u_{1},$
$\displaystyle u_{2}^{\prime}$
$\displaystyle=\dfrac{1}{r^{4}-2Ar^{2}-B}\left(Ar+d\left(m-\dfrac{1}{2}\right)r^{3}\right)u_{2},$
$\displaystyle u_{3}^{\prime}$
$\displaystyle=\dfrac{1}{r\left(r^{4}-2Ar^{2}-B\right)}\left(3Ar^{2}+B+\left(d\left(m+\dfrac{1}{2}\right)-3\right)r^{4}\right)u_{3},$
$\displaystyle u_{4}^{\prime}$
$\displaystyle=\dfrac{1}{r\left(r^{4}-2Ar^{2}-B\right)}\left(3Ar^{2}+B-\left(d\left(m-\dfrac{1}{2}\right)+3\right)r^{4}\right)u_{4}.$

Solving these ordinary differential equations, we obtain (4.1) and (4.2). This
gives the proof of theorem. Q.E.D.

Finally we study the singular sets of harmonic spinors constructed in Theorem
4.1. The singularity occurs when at least one of the power indices of
$r-r_{0}$ is negative in (4.1) or at least one of the power indices of
$\sin\dfrac{\theta}{2}$ or $\cos\dfrac{\theta}{2}$ is negative in (4.2). The
relations between polar coordinates $\\{r,\theta,\psi,\phi\\}$ and Cartesian
coordinates $\\{x_{0},x_{1},x_{2},x_{3}\\}$ of the metric (2.6) are [9]

$\displaystyle x_{1}$
$\displaystyle=r\cos\dfrac{\theta}{2}\cos\dfrac{\psi+\phi}{2},\qquad
x_{2}=r\cos\dfrac{\theta}{2}\sin\dfrac{\psi+\phi}{2},$ $\displaystyle x_{3}$
$\displaystyle=r\sin\dfrac{\theta}{2}\cos\dfrac{\psi-\phi}{2},\qquad
x_{0}=r\sin\dfrac{\theta}{2}\sin\dfrac{\psi-\phi}{2}.$

Since $r\geq r_{0}$, we obtain

$\displaystyle\sin\frac{\theta}{2}=0\Longleftrightarrow
x_{3}=x_{0}=0,\quad\cos\frac{\theta}{2}=0\Longleftrightarrow x_{1}=x_{2}=0.$
(4.7)

Now we denote

$\displaystyle a_{m}^{\pm}$ $\displaystyle=\dfrac{1-d\pm dm}{2d},$
$\displaystyle a_{mn}^{\pm}$
$\displaystyle=n+\frac{1}{2}+\frac{d}{2}\Big{(}-m\pm\frac{1}{2}\Big{)},$
$\displaystyle b_{mn}^{\pm}$
$\displaystyle=n+\frac{1}{2}+\frac{d}{2}\Big{(}m\pm\frac{1}{2}\Big{)}.$

It is clearly

$\displaystyle a_{mn}^{+}-a_{mn}^{-}=\frac{d}{2},\quad
b_{mn}^{+}-b_{mn}^{-}=\frac{d}{2}.$ (4.8)

######  Proposition 4.1.

The harmonic spinor $\Phi$, constructed in Theorem 4.1, as well as its norm
$|\Phi|$ are singular at $r=r_{0}$. Moreover, it holds that

  * (1)

If $-\frac{d}{2}<a_{mn}^{-}<0$, $-\frac{d}{2}<b_{mn}^{-}<0$, $v_{1}$, $v_{2}$,
$v_{3}$, $v_{4}$ and $|\Phi|$ are singular either at $x_{1}x_{2}$-plane, or at
$x_{3}x_{0}$-plane;

  * (2)

If $-\frac{d}{2}<a_{mn}^{-}<0$, $b_{mn}^{-}\geq 0$, either $v_{1}$, $v_{2}$,
$v_{3}$, $v_{4}$, $|\Phi|$ are singular at $x_{1}x_{2}$-plane, or $v_{1}$,
$v_{3}$, $|\Phi|$ are singular at $x_{3}x_{0}$-plane;

  * (3)

If $-\frac{d}{2}<a_{mn}^{-}<0$, $b_{mn}^{-}\leq-\frac{d}{2}$, either $v_{1}$,
$v_{2}$, $v_{3}$, $v_{4}$, $|\Phi|$ are singular at $x_{1}x_{2}$-plane, or
$v_{2}$, $v_{4}$, $|\Phi|$ are singular at $x_{3}x_{0}$-plane;

  * (4)

If $a_{mn}^{-}\geq 0$, $-\frac{d}{2}<b_{mn}^{-}<0$, either $v_{2}$, $v_{4}$,
$|\Phi|$ are singular at $x_{1}x_{2}$-plane, or $v_{1}$, $v_{2}$, $v_{3}$,
$v_{4}$, $|\Phi|$ are singular at $x_{3}x_{0}$-plane;

  * (5)

If $a_{mn}^{-}\geq 0$, $b_{mn}^{-}\geq 0$, $v_{2}$, $v_{4}$, $|\Phi|$ are
singular at $x_{1}x_{2}$-plane, or $v_{1}$, $v_{3}$, $|\Phi|$ are singular at
$x_{3}x_{0}$-plane,

  * (6)

If $a_{mn}^{-}\geq 0$, $b_{mn}^{-}\leq-\frac{d}{2}$, $v_{2}$, $v_{4}$,
$|\Phi|$ are singular either at $x_{1}x_{2}$-plane or at $x_{3}x_{0}$-plane;

  * (7)

If $a_{mn}^{-}\leq-\frac{d}{2}$, $-\frac{d}{2}<b_{mn}^{-}<0$, either $v_{1}$,
$v_{3}$, $|\Phi|$ are singular at $x_{1}x_{2}$-plane, or $v_{1}$, $v_{2}$,
$v_{3}$, $v_{4}$, $|\Phi|$ are singular at $x_{3}x_{0}$-plane;

  * (8)

If $a_{mn}^{-}\leq-\frac{d}{2}$, $b_{mn}^{-}\geq 0$, $v_{1}$, $v_{3}$,
$|\Phi|$ are singular either at $x_{1}x_{2}$-plane, or at $x_{3}x_{0}$-plane;

  * (9)

If $a_{mn}^{-}\leq-\frac{d}{2}$, $b_{mn}^{-}\leq-\frac{d}{2}$, either $v_{1}$,
$v_{3}$, $|\Phi|$ are singular at $x_{1}x_{2}$-plane, or $v_{2}$, $v_{4}$,
$|\Phi|$ are singular at $x_{3}x_{0}$-plane.

Proof: The norm of harmonic spinors in Theorem 4.1 is

$\displaystyle|\Phi|^{2}=\sum_{i=1}^{4}|\Phi_{i}|^{2}=\sum_{i=1}^{4}|C_{i}|^{2}u_{i}^{2}v_{i}^{2}.$

If any $u_{i}$ or $v_{i}$ goes to $\infty$ at some subset of the space, the
harmonic spinors and their norm become singular there.

The power orders of $r-r_{0}$ in $u_{i}$ are

$\displaystyle a_{m}^{-},\quad
a_{m}^{+},\quad-\frac{1}{2}-a_{m}^{-},\quad-\frac{1}{2}-a_{m}^{+}$

respectively. Note that $m$, $d$ are integers and $d>2$. Thus

$\displaystyle m=0$ $\displaystyle\Longrightarrow
a_{m}^{\pm}<0,\quad-\frac{1}{2}-a_{m}^{\pm}<0,$ $\displaystyle m\leq-1$
$\displaystyle\Longrightarrow a_{m}^{-}>0,\quad-\frac{1}{2}-a_{m}^{+}>0,\quad
a_{m}^{+}<0,\quad-\frac{1}{2}-a_{m}^{-}<0,$ $\displaystyle m\geq 1$
$\displaystyle\Longrightarrow a_{m}^{-}<0,\quad-\frac{1}{2}-a_{m}^{+}<0,\quad
a_{m}^{+}>0,\quad-\frac{1}{2}-a_{m}^{-}>0.$

This indicates that $r_{0}$ is the singular point of the harmonic spinor
$\Phi$ as well as its norm $|\Phi|$.

From (4.2), we obtain

$\displaystyle v_{1}(\theta)$
$\displaystyle=v_{3}(\theta)=\left(\sin\frac{\theta}{2}\right)^{a^{-}_{mn}}\left(\cos\frac{\theta}{2}\right)^{-b^{-}_{mn}-\frac{d}{2}},$
$\displaystyle v_{2}(\theta)$
$\displaystyle=v_{4}(\theta)=\left(\sin\frac{\theta}{2}\right)^{-a^{-}_{mn}-\frac{d}{2}}\left(\cos\frac{\theta}{2}\right)^{b^{-}_{mn}}.$

Therefore the rest parts of the theorem follow from the above formulas, (4.7)
and the fact that $\sin\frac{\theta}{2}$, $\cos\frac{\theta}{2}$ can not be
both zero. Q.E.D.

Acknowledgement. The authors would like to thank referee’s valuable
suggestions. This work is supported by Chinese NSF grants 11731001, the
special foundations for Guangxi Ba Gui Scholars and Junwu Scholars of Guangxi
University.

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