Download Can the Strongly Interacting Dark Matter Be a Heating Source of

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Progress of Theoretical Physics, Vol. 87, No.3, March 1992
Can the Strongly Interacting Dark Matter
Be a Heating Source of Jupiter?
Masahiro KAWASAKI, Hitoshi MURAYAMA and Tsutomu YANAGIDA
Department of Physics, Tohoku University, Sendai 980
(Received October 28, 1991)
We show that the strongly·interacting massive particle (SIMP) with mass of 3x106 -107 GeV is
astrophysically interesting as a dark matter candidate in" the galactic halo. The annihilation of
SIMPs inside Jupiter naturally explains the intrinsic heat flux irrespective of details of the planetary
models. We discuss its effect in all Jovian planets as well as in the Sun and the Earth. We also
comment that such a SIMP is accommodated in a class of hadronic axion models.
The dark matter (for a recent review, see, e.g., Ref. 1)) giving a large fraction of
the gravitational mass of the galaxy may be composed of elementary particles.
These particles must be so weakly-interacting or so few and heavy that they are not
conspicuous. Many dark-matter candidates have been proposed!) and the possibilities of their experimental detection have been intensively studied. 2 ) Among them the
strongly-interacting dark-matters 3 ) are very interesting since they may allow the
direct detection in the dark-matter search experiments on the Earth. In fact, only
small regions have been left open for the strongly-interacting massive particles h
(SIMPs), one with mass of 105 -10 7 GeV and the other with mass above 1010 GeV. 3 )
Furthermore, the recent experiment by Caldwell et a1. 4 ) has excluded some portion of
the window. However the SIMP with mass of 3 X 106-10 7 Ge V is still allowed. *)
In this paper we point out that for the window, SIMPs are trapped in the planets
and can annihilate in their centers producing possible heating energies. We indeed
find that our model may explain the mysteriously large intrinsic heat flux of Jupiter 5 )
within the three standard deviations irrespective of the detailed parameters of the
planet as well as the annihilation cross section. However for other Jovian planets,6),7)
one must require that the SIMP's annihilation has not yet reached the steady state,
since otherwise it overproduces the heating energy. In this case the predicted heat
fluxes depend on details of the densities and temperatures of the planet cores and the
annihilation cross section. We also examine SIMP's annihilation effects in the Earth
and the Sun. The final comment is devoted to stressing that this stable SIMP is
naturally accommodated in a class of hadronic axion models.
Our basic assumption is that the dark halo of our galaxy 'is a cloud of the SIMPs
h with the mass density Ph=OA GeV fcm 3 near the solar system. 8 ) The mean velocity
of h has been estimated as vh=300km/sec. 8 ) For the SIMPs h to be trapped inside
*) The authors of Ref. 3) argue that this window is only applicable to the particle with cosmic
asymmetry, since their annihilation inside the Earth will produce too many neutrinos which can be detected
at the proton· decay detectors. However, it is not the case if the annihilation cross section 'is very small as
we will discuss later.
686
M. Kawasaki, H. Murayama and T. Yanagida
Jupiter, we have a condition,3)
> 5" 10-27cm2( 107mh
GeV ) '
(1)
(Jp~ X
where (Jp is the cross section on protons. For the window mh:::::: 10 10 Ge V Eq. (1) leads
to a large cross section (Jp <10- 23 cm2 which has been already ruled out in the previous
analysis. 3) The window of mh=3X10 6 -107 GeV with a coherent cross section,3),4)
(2)
meets the trapping condition (1) and hence we will, in this paper, restrict our analysis
to this window. Namely, we assume that the SIMP interacts with a proton by
coherent interaction with the cross section (2). This magnitude of (Jp nicely coincides
with that of the ordinary strong-interaction cross section, suggesting the SIMP h to
be a kind of stable heavy hadrons. We must assume that the low-energy interaction
of h with nuclei is repulsive, since otherwise the h makes bound states with the
ordinary nuclei and the annihilation probability of h is drastically reduced (we
assume no number-asymmetry of h).
With the cross section (2), most of h's coming on planets are trapped in all Jovian
planets as well as in the Earth and the Sun. The trapping number rate is given by
=
(R)2 '
3.5 X 10 20 sec -) m7 -) P0.4V300 10 10 cm
(3)
where R is the radius of the planet, PO.4=Ph/O.4 GeV cm- 3, V300=Vh/300 km sec-) and
m7= mh/107 Ge V. Our crucial observation is that the annihilation of all the trapped
h's in the core produces a universal heat flux for all planets
(4)
which is close to the observed intrinsic heat flux of Jupiter,5)
(5)
Let us discuss first how the accumulated SIMPs distribute inside the planets. After
trapped, the SIMPs lose their energy by multiple scattering off nuclei and will be
rapidly thermalized. Therefore we assume that the SIMP distribution is isothermal
with the central temperature Tc of the planet. As the SIMPs are accumulated in the
center of the planet, their annihilation rate increases.*) Finally the SIMPs willreach
a steady state where the annihilation rate balances with .the. trapping rate Ntr ,
producing the heat flux (4). The number of the SIMPs Nss in the ~teady state i~ given
by
*)
SIMPs
We have checked that the cross section on protons
will fall down into the center of the planets.
(Jp
in our window is small enough that the trapped
Can the StronglY Interacting Dark Matter Be a Heating Source of Jupiter?
687
"
((Jahnv
)-1/2 " "
1/2
=2.4X1037 m7- 5 / 4 10 20GeV 2
(P0.4V300)
P
x ( 10 g
~m 3
)-3/4(
)3/4( R )
K 10 cm '
T.
10 4
10
(6)
where pc is the central density of the planet, (J!;n the annihilation cross section of hand
GN the Newton constant.*) For the SIMPs to reach the "isothermal steady state",
this number Nss should be less than the total number accumulated until the present (= .
NtrTe , Te=4.5 Gyr). This condition requires the annihilation cross section to be
greater than the critical one which is given by
21
2
h)
")-1
( (Jan
V cr = 2.3 X 10- Ge V- m7 -1/2( PO.4 V300
P
X ( 10 g
~m 3
K)3/2( 10 Rcm )-2
)-3/2( T.
10 4
10
(7)
However, if the SIMP is relatively heavy, the density of the accumulated SIMPs
inside the planet eventually exceeds the core density of the planet, and then the SIMPs
become self-gravitating and will start to collapse. Since the SIMPs are assumed to
be able to annihilate with each other, the annihilation rate increases during the
collapse, and will balance with the trapping rate somewhere unless the annihilation
cross section is extremely small.**) Thus here again a steady state can be formed.
However the detailed distribution may be very different from that of the "isothermal
steady state".
The criterion of the gravitational collapse was given in Ref. 9),
(8)
where Mh is the total mass of SIMPs accumulated, and
(9)
is the radius of the isothermal SIMP gas inside the planet. The critical lines in the
parameter space (mh, (J!;nV) will be given by the following equations. If the SIMPs
have reached the "isothermal steady state" (((J!;nV) > ((J!;nV)cr), the total mass of SIMPs
is just Mh = mhNss , and the collapse occurs if
m7>7.5X10
-3( 10 20GeV
(J!;n V
)2/5
2
(P0.4V300)
-2/5
*) We assume that the SIMP his self·annihilating.
If the SIMP h has particle anti·particle distinction,
2! should be replaced by 4.
**) The "annihilation cross section can be as small as 0(10- 12°) cm 2 so that the annihilation rate can
balance with the trapping rate, in the extreme case of the SIMPs forming a Fermi·degenerate gas.
688
M. Kawasaki, H. Murayama and T. Yanagida
x(
Pc
10 g cm. 3
)1!5(
Tc )3/5(
R
104 K
10 10 cm
)-4/5
(10)
Otherwise «IJi:nV)<(6~nV)cr), the total mass of SIMPs is Mh=mhNtrTe, and the collapse
occurs if
(11)
We will call this case "collapse" hereafter. As discussed above, we assume that the
SIMPs' annihilations produce the universal heat flux Eq. (4) after the gravitational
collapse. For the region of the "non-steady state" the expected heat flux is given by
- 1 (.,':T
F -2f
lvtrTe
I/2
X
m7
2
)2[ GNPCmh
J3/ (
3kT
c
V
( 10 (JhGeV
20
)(
2
2
h ) 2(mhc2 )
4TCR
O'anV
)3/2(
P
10 g ~m
3
T.
10 4
)-3/2(
K
R )2
10 10 cm
(12)
Before applying the above discussion to each planet, we have to examine what
value of the annihilation cross section (O'~nV) is allowed to avoid the constraints from
the proton decay detectors on the Earth. The neutrino event rate at the proton decay
detectors is estimated following Ref. 3), with the energy fraction ~ of the neutrinos
with energy between 1 GeV and 10 TeV:
{I
.
N v ~109E
c;mln,
((O'~nV)}
h
)
O'anV cr
(13)
for the world's working time ~4 kton yr. Since the energy fraction can be never
larger than unity, we conservatively require that the annihilation cross section
satisfies
(14)
to escape the detection (Nv:S 1). At the first glance one may not anticipate this small
annihilation cross section. However, we will show later that this can be justified for
a class of strongly-interacting particles. The consequence of this small annihilation
cross section is that the "isothermal steady state" is not possible in all planets (see Eq.
(7», and only either the "non-steady state" or steady state after the gravitational
collapse is possible inside the planets. The criterion which separates the "non-steady
state"and "collapse" is given by Eq. (11).
Now we discuss the situations in each planet. . The observed heat flux of Saturn
is (2.01±0.14) X 103 erg cm-2 sec-I 6) which is roughly close to that of Jupiter, while that
of Neptune is one order of magnitude smaller «5.25±1.48) X 102 erg cm- 2 sec-I) and
only the upper bound is known for Uranus « (0.75±1.75) X 102 erg cm- 2 sec- I).7) Here
we find that Jupiter and Saturn have abnormally large intrinsic heat fluxes. The
fluxes amount to two order of magnitudes larger than that of the Earth (64 erg cm- 2
sec-I), and hence an explanation seems to be required besides the geophysical effects
(e.g., the radiochemial activity).
Can the Strongly Interacting Dark Matter Be a Heating Source of Jupiter?
689
The difference between Saturn and Jupiter is hard to explain in our model, but
also even in any geophysical models,ll) since the planetary parameters are very
similar for these two giant planets. Furthermore, the measurement of the intrinsic
heat flux from Saturn is disturbed by the existence of the rings as noted in 6). The
discrepancy may be some error due to this disturbance. Thus we do not regard the
discrepancy serious in this paper.
For Neptune and Uranus, their observed heat fluxes are much lower than the
universal steady-state value, and hence the SIMPs must be in a "non-steady" state.
As noted before, the SIMPs can never be in an "isothermal steady state" with the
small annihilation cross section in Eq. (14). The issue is, therefore, whether the
SIMPs form a "non-steady state" inside Neptune and Uranus.
In sum, the h's accumulated in Jupiter and Saturn have to be in a self-gravitating
situation producing the heat flux in Eq. (4), whilethey have to be in "non-steady state"
inside Neptune and Uranus as well as the Earth. Since the detailed parameters of the
planet cores are still highly uncertain, we cannot draw a definite conclusion on
whether the situation discussed here can be realized. Instead we show, in Table I, the
planetary parameters required in our scenario. We also quote the parameters of a
recent modePO) together. As seen from the table, the model parameters for Jupiter,
Saturn and the Earth seem to be consistent with our scenario, where the SIMPs inside
the first two planets are in "collapse" while they are in "non-steady state" inside the
Earth. In fact, if one tentatively employes Pc=10g cm- 3 and .Tc=104 K for the Earth,
the produced heat flux is 0(10- 1°) smaller than the observed one, thus there is no
problem with the overproduction of the heat inside the Earth. There are potential
problems on Neptune and Uranus, where the planetary· parameters given in the
modePO) do not seem to support our scenario. However, the models of Neptune and
Uranus seem more uncertain compared to those of Jupiter and Saturn, due to the
larger ratio of the rocks composed of heavy elements. Moreovor, one must reexamine the dynamics of the planets including the SIMP accumulation and annihilation in their cores. In this point of view, the required parameter regions given in
Table I should be taken as the constraints in the future planetary model building.
The Sun also traps the h's. With the annihilation cross section in Eq. (14), SIMPs
will form the "non-steady state" inside the Sun and the gravitational collapse does not
occur due to the high temperature. The study on the high-energy neutrino from the
WIMP annihilation in the Sun has rejected some dark matter particles of mh
;S 0(10 2 ) GeV. 12 ) However, our SIMP h is much heavier, with the flux of 0(10- 5 )
Table 1. The regions of the planetary parameters required in our scenario, for
the case mh=3XI0 6 GeV. Only the parameter Pc- 113 T c matters, since it is
the combination which appears in Eq. (11). The unit is in (10 g cm- 3t1!3(10'
K). The parameter regions for each planet are· obtained from the require·
ment that the SIMPs are in a self·gravitating situation inside Jupiter and
Saturn, while they are in "non-steady state" inside the other planets. Also
the parameters from a model lOl are shown for comparison.
Pc- 113Tc
our scenario
model
Jupiter
Saturn
Neptune
Uranus
Earth
<19.
<14.
>4.1
>4.4
>0.70
2.0
1.6
0.91
0.84
1.0
690
M. Kawasaki, H. Murayama and T. Yanagida
smaller, and hence we believe that the search is irrelevant to our case.
The final comment concerns a particle· physics model for the SIMP h. One can
easily .construct a class of hadronic axion models/ 3 ) in which a new left·handed
fermion QL and a boson cP are introduced. Both fields are assigned as SU(2)L x U(l)y
trivial representations, but QL as· an SU(3)c octet. All usual quarks, leptons and
Higgs bosons do not carry the Peccei-Quinn charge, while the new fields QL and cP are
supposed to transform under the U(1)PQ as QL -> eiaQL and cP-> e- 2ia cP. The QL
acquires a heavy Majorana mass term mQQLQL, when the Peccei-Quinn symmetry is
spontaneously broken by <cP>*O, through a Yukawa coupling of cP, .£ y=gyQLQLcP
+h.c. The heavy Majorana octet quark Q (== QL + QRC ) is completely stable, since it
cannot mix with the usual triplet quarks as long as the color SU(3)c is kept unbroken.
We assume that the lightest Q-hadron is a bound state of aQ and a gluon, and the
SIMP h is nothing but this stable Q-hadron.
The characteristics of this particle-physics model are listed as follows.
(1) The SIMP h is electromagnetically neutral and hence can be a candidate for the
dark matter. 9 )
(2) The h is a Majorana fermion, since the heavy quark Q is a Majorana octet.
Therefore, we have no number asymmetry of h.
(3) The h has an ordinary strong-interaction cross section on nuclei. Since h does
not carry isospin, it interacts with nuclei by one scalar-boson exchange (e.g.,
6(600)) which gives coherent cross sections. Since the sign of the Yukawa
coupling can differ between that to nuclei and to h, this scalar-boson exchange
can give repulsive interactions between them.
(4) The mass of h is given by mh -:: =. mQ= gy<cP>, while<cP> should be in the axion
window <cP>=10 Io -10 I2 GeV.*) To obtain the mass mh=3 X 106 -10 7 GeV we need
gy -:::=.10- 3 -10- 6 • However, we do not consider it is very unnatural, since even in
the standard model a small Yukawa coupling ~4 X 10-6 is already required to get
the small electron mass.
(5) The cosmological abundance of the h particles is calculated as
(15)
(16)
where D is the dilution factor -:: =. 27 and Xf -I-:::=. 20. N is the effective degrees of
freedom at the period of the annihilation, N-:::=.50. Here Q h is the fraction of
critical density of the h's, ho is the reduced Hubble constant ho=0.5-1 and 6~n is
the annihilation cross section of QQ-> gg and qq.**) We obtain ·Qhho2=104-105
For the hadronic axion model, there are left two windows, one with <9'»""10 7 GeV and the other with
<9'»=10 10 -10 12 GeV. However, the former window is not applicable to our specific model. See Ref. 14).
**) The annihilation cross section of Q is computable by the perturbative QeD in the early hot universe.
On the other hand, the annihilation cross section of h at very low energies depends on details of long-range
interaction between h's, and hence the cross sections O~n and O;n are completely different in general.
*)
Can the Strongly Interacting Dark Matter Be a Heating Source of Jupiter?
691
for the range of mh=3 X 166 -10 7 GeV. For the SIMP h to be a viable candidate
for the dark halo, an extra entropy production should be invoked after the
decoupling of h. Such an entropy may be supplied by the first-order QeD phase
transition*) or the Weinberg-Salam phase transition.**)
(6) The annihilation cross section of h at very low energies depends on the long-
range potential between h's. Since we have assumed that the E-exchange dominates the long-range part of the strong interaction, it gives rise to an attractive
potential betweeen h's. However at smaller distances, the axial vector exchange
can dominate, which gives a repulsive core. Thus two h's become first a loosely
bound state,and only after tunneling through the repulsive core potential, they
can become a single QQ bound state emitting an even number of pions. Thus the
annihilation is suppressed by the tunneling probability FT, as (]!:n ~ FT· (]~n. The
FT is estimated in the WKB approximation as
(17)
Taking the potential height of, say, Vo;(:l MeV and range lo~O.l fm, we can
understand that the annihilation cross section of the h's at very low energies is
strongly reduced from that of the Q's, i.e., (]!:n:S 0(10- 15 ) x (]~n, as required in our
scenario (see Eq. (14». Notice that this enormous suppression of the tunneling
probability FT is achieved because of the large mass of the SIMP h.
(7) After forming the single QQ bound state, they will eventually annihilate into gg.
The high-energy gluon can convert into a pair of heavy quarks like top, bottom
or charm, which can decay semileptonically (with branching ratio ~ 30%) producing high-energy neutrinos with Ell ~ 106 Ge V. The conversion rate of one gluon
into a pair of heavy quarks with mass mq is
Q's(4mQ 2) I
67Z"
mQ2
ogm/'
(18)
which gives the rough estimate of the neutrino flux at the surface of the Earth
(19)
These high-energy neutrinos will produce multi-ring events in proton decay
detectors, which have been not yet analyzed. However with the small annihilation cross section (l4), the expected event rate is less than 0(10- 8 ) kton- 1 yr-\
which is unobservable.
Summarizing the discussion on the particle-physics model, the hypothesis of the
SIMP h being a bound state of an octet fermion Q and a gluon meets all conditions
for our scenario if the required entropy production is supplied in the period between
*) The deconfinement transition in pure gauge QeD is now known to be of the first-order. 15 )
However,
the introduction of the dynamical quarks will make the notion of the confinement obscure, and the order of
the chiral symmetry phase transition is still not known. 16 )
**) The first-order phase transition even with a large Higgs boson mass is consistent with the recent LEP
data if one introduces two Higgs doublets (see 17».
692
M. Kawasaki, H. Murayama and T. Yanagida
T ~ mQ and the nucleosynthesis. Therefore, the existence of a particle-physics model
for the SIMP h makes the present proposal of explaining the intrinsic heat flux of
Jupiter a very interesting scenario.
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