暗黒物質由来のスペクトル

第４回計画研究B02若若⼿手研究会＠早稲⽥田⼤大学 8/31-‐‑‒9/1
暗⿊黒物質由来のスペクトル
神戸大学 M2 橋本　隆
1. 
2. 
3. 
4. 
暗黒物質探索手法エネルギースペクトル角度スペクトルまとめ
1. 暗黒物質探索手法暗⿊黒物質探索索⼿手法
sues for the detection via WIMP-nucleus elastic scatterings. Direct
e• direct
evidence of the existence of the dark matter in the halo of
WIMPと原⼦子核との弾性散乱を検出し、暗⿊黒物質特有の信号を得る
on, the
framework of the direct detection focusing on the
–  theoretical
エネルギースペクトルの季節変動
ed[33].–  到来⽅方向の異異⽅方性
到来⽅方向異異⽅方性(数倍)
季節変動(数％)
座
白鳥
ctrum
od is to detect the
座
鳥
白
energy deposition
on the
DM Nucleus
θ
nucleus via WIMP-nucleus
hus the calculation of the expected energy spectrum is the first step
Because we are on the Earth, the relative velocity between the Earth
xy aﬀects the expected energy spectrum. The solar system rotates in
後 rotates around
of 8 kpc from the center of the galaxy, and the Earth
ity of the Earth relative to the galactic halo is written as
vE (= |vE |) = vsun + vorb cos θorb sin 2πy
≃ 244 + 15 sin(2πy) [km/sec],
前
cosθ
(2.1)
vsun: 銀河ハローに対する太陽の相対速度度
vobs: 太陽の周りを回る地球の公転速度度
θobs: 銀河⾯面と地球の公転⾯面がなす⾓角
y: 3⽉月2⽇日から経過した時間を年年で表したもの
y of the Sun relative to the galactic halo, vorb is the orbital rotation
3
2. エネルギースペクトル n0 3
n0
3
n ≡
dn
, (2
dn
=
f
(v,
v
)d
v,
dn
=
f
(v,
v
)d
v,
(2.2)
E
: The schematic view of the motionkEof the Sun and the Earth in our 0galaxy.
k !
0
!
!
!
!
!
!
v
2π
+1
v
esc
esc
vesc
+1 of the Sun and v
ure
2.1.1: The schematic view2π
of the motion
the
escEarth in our galaxy.
2 vE )v 2 dv ,
n
≡
dn
,
k
=
dφ
d(cos
θ)
f
(v,
0
where
k
is
the
constan
dn
,
k
=
dφ
d(cos
θ)
f
(v,
v
)v
dvnormalize
, as
density
of WIMPs, dn, with the velocity of v ∼ v + dv isEwritten
WIMPの数密度度分布dnは
WIMPの数密度度
0 of WIMPs,0 dn, with 0−1
−1of v ∼0 v + dv is0 written as
umber density
the velocity
n0
the
escape
velocity
of the
galaxy
3
dn
=
f
(v,
v
)d
v,
(2.2)
n
e
normalize
constant,
n
is
the
total
number
of
WIMPs
in
our
galaxy,
E
0
3
0
lize constant, n0 isdnthe
of WIMPs in our (2.2)
galaxy, vesc isvesc
=k total
f (v, vEnumber
)d v,
! vesc ! 90
! !2π k !! +1
! Lewin,
v
angle
taking
WIMP
velo
! azimuth
J.D.
Smith/Astroparticle
Physics
6 (19%)
87-11
locity
of
the
galaxy,
θ
is
the
vEWIMP
and
WIMP
velocity,
and
φ
v
2π
+1angle between
vesc esc PE
esc
n
: WIMPの全数密度度
2
the
galaxy,
θ
is
the
angle
between
v
and
velocity,
and
φ
is
o
2
E
≡ n0 ≡ dn , dn , k =k = dφ
d(cosθ)θ) f (v,fv(v,
,
dφ
d(cos
, dv
E )v
E )vvdv
ｋ: 規格化定数
0
0 vector
−1 as a zenith.
0the
0
0 and
0
velocity
distribution
da
eWIMP
taking
WIMP
velocity
Assuming
a Boltzmann
distributio
distribution,
is the local
Galactic
escape
velocity;
dndistribution,
is thenof
thethe
particle
velocity
vector
asuesc
a −1
zenith.
Assuming
a Boltzmann
s the normalize
constant,
isthe
the total
totalwithin
number
of of
WIMPs
galaxy,
vesc is v
0 velocities
with
relative
d3v
about
v.in our
ormalize
constant,
n0 nis
number
WIMPs
in
our
galaxy,
esc is
istribution
of
the
dark
matter
f
(v,
v
)
is
written
as
WIMPは銀河の重⼒力力に束縛され、熱平衡状態になっていると考られる。
E
on
of
the
dark
matter
f
(v,
v
)
is
written
as
e velocity of the galaxy,
θ is theMaxwellian
angleE between
vE matter
and WIMP velocity,
and φ is
We θassume
dark
ty よって WIMPの速度度分布はボルツマン分布に従うと考えられており
of the galaxy,
is thea angle
between
v and velocity
WIMP distribution:
velocity, and φ is
E
2 /v 2
ngle taking WIMP velocity vector as a zenith.
Assuming
a Boltzmann distribution,
−(v+v
)
E
2
2
0 ,
2 /v
2
f
(v,
v
)
=
e
−(v+v
)
E
E
king
WIMP velocity
as
a zenith.
a Boltzmann distribution,(2.3)
0Assuming
f(Y,Q)vvector
(v,
) == ee--(“+“E)
, as
ty distribution of f
the
darkEmatter
f (v, vE ) is /uo;
written
vo: 分散速度度
(2
where
v0 is the velocity dispersio
ibution of the dark matter f (v,−(v+v
vE ) is
written
as
2
2
E ) /v0and if the galaxy rotation velocity is flat,
he v
velocity
dispersion
our
(v,of
v=E )00,
= e galaxy
,
(2.3)
then, for fuesc
=∞ とした時
esc
ty dispersion
of our galaxy−(v+v
and if)2 /vthe
galaxy
rotation
velocity
is flat, it
2 becomes rotation velocity. The no
E
0 ,
f (v,
e constant
(2.3) or not
tion
The
normalize
is derived
caseisvflat,
E) =
esc is
is the velocity.
velocity dispersion
ofvour
galaxy
and if the kgalaxy
rotation in
velocity
it infinite
k = klJconstant
= (m;)3’2;
city. The normalize
k is derived in case vesc is infinite or not as
otation velocity. The normalize constant k is derived in case v
is infinite or not as
esc
2 3/2
elocity
dispersion
of
our
galaxy
and
if
the
galaxy
rotation
velocity
is flat,give
it
k
whereas the
distribution
truncated ’ at 1v +(vVE
j==uesc
would
k =same
k20 =
(πv
)
∞)
(2
vescが有限であるとすると
esc
0
3/2
%
k=
k0"
) 02$
(v
=
∞)
k=
= k(πv
=
(πv
)3/2 k is derived (v
= esc
∞)
(2.4)or not as(2.4)"
0#
esc
0
velocity. "
The
normalize
constant
in
case
v
is
infinite
% %2
# " #$v $vesc 2 v 2 2 vesc
2 esc
−v
/v
esc -&&
0
esc −vesc2/v022 e
vosc
&=
k = k1 = k0 (2e
k k==kk11 =
=kk0 k=kt=ka
−5esc
̸= (2.5)
∞),
v
2 (v
0verf
esc: 脱出速度度
escerf esc −2
e
=
̸
∞), (v
erf
−v
/v
esc
1/2
esc
1/2
0
2
3/2
vLO
v0
k1 = k0 erf k = k0 =v−
(vescesc≠=∞)
∞),
0 (πv
001/2
)π v0πe
(v
(2.4)(2.5)
v0
" v#0
$ π
%
&
2 & x −t22
& x −t
2
v
(x) ≡2 π1/2 x0 e−t dt is thevesc
error function.
2
2
esc
2
−v
--+
co.
Derivations
of
these
and
subsequent
results
are g
so
k,
+
ko
as
vex
esc /v0
2
≡k−t
e
dt
is
the
error
function.
=
k
=
k
erf
−
e
(v
=
̸
∞),
(2.5)
2
1/2
1
0
esc
0
where
erf(x)
≡
e
dt
is
t
π
1/2
v0days
π600per
0
e dt event
is the
0target
π 1/2 h/k,
km
s-’ is(see Appendix B), we obtain
= 0.9965.
23Okms-‘,
~esc=mass
iﬀerential
rateerror
R pervfunction.
.
---e
lr’J2
lJ0 1,
, σ depends on the momentum transfer. In this section, we discuss the cros
WIMPの計数率率率
mentum transfer σ . The cross section taking account of the momentum transfer
pends
on the
momentum
this section,
discussaccount
the cross of
section
ro momentum
transfertransfer.
σ0 . TheIn cross
sectionwetaking
the momentum
he
error
function.
0 latter section 2.3. 運動量量移⾏行行が0と考えσ=σ
(定数)とした
discussed in the
The total event rate 0R
is derived by in
per
target
mass
per 2.3.
daysTheis total
sedWIMPの微分係数率率率は
in
the latter
section
on (2.6) as
) as
dR =
event rate R is derived by integrating
!
[tru]=[counts/kg/day] NA
!
N
0
N
0
[dru]
σvdn,
(2.6)
R
=
σ
vdn
[tru] [dru]=[counts/keV/kg/day]
0
R
=
σ
vdn
(2.7)
0
A
A
A
N : アボガドロ数
A
26
σ: WIMPと原⼦子核の散乱断⾯面積
rate
R0 for
vE=∞の時の全計数率率率は
0for
and
=0∞
isnumber
described
as the
rate
R=0A
vvEesc=mass
and
vesc = ∞
is described
as
rntal(6.02×10
),
is
the
of
target
nucleus,
v=
vevent
=0、v
E
esc
2 the
NA ρtarget,
D 2 N σ ρis the WIMP-nucleus cross section.
y relative
R0 = to
σ0 v 0 A D
(2.8)
1/2
R
σ0 v 0
A MD1/2
MD: WIMPの質量量
π0 =
" π #"
#
"
#
A MD ρ
MN: 標的原⼦子核の質量量
361
σ0
v0
"
#
"
#
D# "
=
,
(2.9)
[tru]
(=0.932A)
361 0.3 GeV/c
σ0 2 /cm3
ρDkm/s
v0
MD M
1 pb
220
28
N
=
,
2
3
MD MN 1 pb
0.3 GeV/c /cm
220 km/s
he mass
density of the dark matter, MD is the mass of the dark matter, MN (=
規格化
of the
target
nucleus.
The dark
Equation
(2.9) is
by σ0 of
= the
1 pb,dark matte
ρeDmass
is the
mass
density
of the
matter,
Mnormalized
D is the mass
, ρD = 0.3 GeV/c2 /cm3 .
) is the mass of the target nucleus. The Equation (2.9) is normalized by σ0
2 a mass
3 MN caused by an elastic scattering of the
the0.3
nucleus
with
0energy
km/s,ERρDof =
GeV/c
$ /cm . %
with the kinematic energy E = 12 MD v 2 in the laboratory system is
recoil energy ER of the nucleus with a mass MN caused by an elastic scatteri
%
ER = Er (1 − cos θ) $/2, 1
(2.10)
2
atter with the kinematic energy E = 2 MD v in the laboratory system is
r=
4MD MN
,
(2.11)
2 Er
3.
c= /cm
. dEρRD = 0.3 GeV/c
$ 1
%
220 km/s,
Emin /cm
2
! vmax
dark matter with the
kinematic
energy E = 2 MD v in the laborato
2
obtained as
1
v
The
recoil energy
of theM
nucleus
with
mass
by an elasticofscattering
of the
0 aby
N caused
nucleus
with =
aERmass
an%Melastic
the
N caused
dR(0, ∞) scattering
R
dR(v),
$
=
e
,
E r
E0 r2 % energy
v 2 = 12 MD v2 in dE
k matter with$the 1kinematic
the laboratory
system
is
vmin E
!
"
2/2)のWIMPに⾓角度度θ(重⼼心
v v)Er
kis dR(0,
∞) θ) /2,
R
EdR(0,
=
(1
−
cos
c energy
E = 2 MD v in the laboratory
system
実験室系で標的原⼦子核が運動エネルギーE(=M
R
D
=
e
−
e
,
dE
k
dE
E r
! #
$
#
$"
dR(v
, ∞)
R π the
v
v
+v E
v(2.10)
−v
he 系)で散乱した時の反跳エネルギーは
minimum kinetic energyEof
WIMP
capable
to
give
energy
to
=
Er
(1
−
cos
θ)
/2,
R
=
erf
−R
erf
4M
dE
E r D
4 M
v N
v
v
!
"
ここで、
E
=
Er
(1
−
cos
θ)
/2,
(2.10)
ra,=
dR(v
v )
k dR(v , ∞)
R ,
R
the maximum
kinetic energy of WIMP
laboratory
system,
E. 0 is
=
− 2 e
4MD MN in dE
k
dEM )E r
(M
+
D
N
r
=
,
(2.11)
2shows the normalized energy spectrum.
v02
Figure
2.1.2
D + MN ) using Equation (2.6), (2.2),
= v2 E. The energy
is(Mobtained
4MD Mspectrum
N
where
θ is the scattering
angle
in the center of mass system,
r is the
r=
, center
(2.11)
ereエネルギースペクトルは
θ is the
scattering angle in2 the
of mass system, r is the reduced
mass. Assuming
1
(MD + M
N)
!
v
the
scattering
is
isotropic
in
the
center
of
mass system,
cosflat
θ distribu
max
scattering
cos0.9θ distribution
become
and ER
dR is isotropic
R0 kin
1 of mass system,
0 the center
3
the center
of mass
system,
r isvE
the
reduced
mass.
Assuming
=in the
f (v,
)d v.
(2.12)
dR
0.8
oin
becomes
flat
0
≤
E
≤
Er
region.
Then
the
energy
spectrum
is
written
as
R
also
becomes
flat
in
the
0
≤
E
≤
Er
region.
Then
the
energy
spectr
dER
dER
E0 r k vmin v
R
0.7
es center of mass system, cos θ distribution
become
flat and ER
! Emax
0.6
! ofEmax
dR the specific
1 formula
dR(0, ∞)of Equation
R0 −E /E r (2.12),
gration
energy spectra are
dR
=
e
,
(2.13)
=
dR(E)
dR
1 as
0.5
is
written
dER region.
E0 r Then the
R ≤ Er
dERenergy Espectrum
Er
dE
!
"
=0.4 R
dR(E)
dR(0, vesc )
k0 dR(0, ∞) −E /E r
R0 −v /v min!
=
e
−
e
vmax
R v02 0.3(2.14)Emin Er
1 , dE
dER
dER
E0 r
!k1 E1/2
$
#
$" dR(v),
=
! vmax 2
max ! #
dR(vE , ∞)
R0 π
v0 1 vmin + vE
vrmin − vE v 2
dR
E
0.2
v0
vmin
=
erf
− erf 0
,
(2.15)1
dR(E)
dER =
E0 r 4 vE
v0
v0
=0.1
dR(v),
!
dE
Er R0 −v /v "
2
dR(vR
k0Emin
dR(vE , ∞)
E , vesc )
E
r
v
0
ere Emin = E
energy of WIMP
capable
give the energy ER to
vto
= R /r is the
−
e kinetic
.
min
0(2.16)
! R vminimum
29
0
1
2
3
4
5
6
7
8
9
10
dER
k1
dE
E
r
0
2
max
E (normalized energy)
1
v
r
arget
Emaxenergy
is the
maximum
kinetic energy of WIMP in a laboratory Esystem,
E0 is
0
.2 showsnuclei,
the normalized
spectrum.
=
dR(v),
2
where Eminr 2= Ev0R /rv 2is the minimum
kinetic energy of WIMP capable
WIMPと標的原子核(F)における反跳エネルギー
fined as E0 = 12E
M0D v0 =
E.
The
energy
spectrum
is obtained using Equation (2.6), (2.2),
vmin
v2
Figure 2.1.2: The normalized energy spectrum of nucleus caused by an elastic scatterin
の規格化されたエネルギースペクトル
F is assumed for the
target.
a as
target
nuclei, Emax is the maximum
kinetic
energy of WIMP in a
1
d (2.3)
!
v02 vmax 1to give the energy ER to
mum kinetic energy of
1dRWIMP
2R0 kcapable
エネルギースペクトル
0
R
esc
R
0
R
0
R
2
esc
0
2
esc
0
1
0
min
E
R
E
0
E
0
2
0
2 /v 2
−vesc
0
0
2
0
0
rate)
19
min
E
0
E0r dR
(normalized event rate)
R0 dE
R
0
1/2
2 /v 2
−vesc
0
0
R
0
esc
E
0
−ER /E0 r
1
E
R
−ER /E0 r
0
用いて次のように表すことができる。
(p)
e to the Spin-Independent(SI)
and(p)
Spin-Dependent(SD) interactions, respectively,
fTG = 1 − q=u,d,s fTq ,
2
diagrams of each interaction are shown in Figure 2.2.1.
λ2 J(J + 1) or n(neutro
! ”odd”(n)
SDnucleon;
SD µχ−N
where
refers an unpaired
p(proton)
(n) subscript
σχ−N = σχ−p 2
fTG = 1 − q=u,d,s fTq .
µχ−p
0.75
l
l
s
s =
the
orbital
and
spin
nucleon
g-factor;
g
=
1,
g
=
0,
g
=
5.586
and
g
p
n
p
n
n Independent (SI) cross section
2
表
2.1
にいくつかの標的核種に対する
λ
J(J
+
1)
の値をまとめる。
(n)WIMPがニュートラリーノであった場合、散乱にはSpin Independent(SI)と
2 J(J + 1)/a
2 J(J +
SI
2 µ A2 and
the
spin
J,
measured
the
calculated
Λ
(≡
λ
,
therefore
σ
∝
µ
is
practically
derived.
Thus,
the
mag
odd
χ−N
χ−N
Tq
section
of SI interaction is written as
Spin dependent(SD)が考えられる。それぞれのニュートラリーノと原⼦子核
原子核中の陽子の寄与
SI ,
is の弾性散乱の散乱断⾯面積は
written
using nucleus.
neutralino-proton
SI
cross
section;
σ
for
somebytarget
For
experimental
use,
the
SD cross 2section is wr
χ−p
2
4µ
元素
J
自然存在比 (%) λ J(J + 1)
散乱断⾯面積
SI
σχ−N
=
χ−N
[Zfp + (A − Z)fn ]2 ,
1
(2.18)
H
1/2SD 2
100
0.750
2π
2
µ
µ
対する SD 反応の散乱断面積は、陽子に対する
SI
反応の散乱断面積
σ
を
χ−N 2
7
χ−p χ−N λ J(J + 1)
SI
SI
SD
Li SD
3/2
92.5
0.244
σ
=
σ
A
.
(2.22)
σ
=
σ
.
A are theχ−N
atomic number
mass
number
of
a
target,
respectively.
µ
is
the
χ−p and
χ−N
すことができる。
χ−N 19 χ−p 2
µ2χ−p
F 1/2 µχ−p 100 0.75
0.647
of the neutralino mass2 MD2 and a target mass MN , 23
Na 3/2
100
0.0041
SD
SD µχ−N λ J(J + 1)
σχ−N
= σχ−p
(2.22)
σ
: 陽
子との散乱断面積 2
127 SI-interacting χ-‐p
0.75 MA
a target
large
isσNSD
eﬀective
I 5/2
100 neu0.007
SDµwith
SD for
χ−p SD
DM
Valueselement
of
σ
/σ
and
/σ
for
some
target
nucleus
are shown
µ
=
.
(2.19)
χ−N
χ−p M + Mχ−N
χ−n
ここで
χ−N
133
Cs 7/2
100 cross
0.052
D
N
2
the SI cross λsections
normalized
by neutralino-proton
SI
標的核種に対する
J(J + 1) の値をまとめる。
Figure 2.2.4, respectively.
e proton-neutralino and neutron-neutralino SI couplings, respectively. They are
s.
原子核中の中性子の寄与
原子核中の陽子の寄与
(p)
(n)
e sum of the neutralino-quarks SI couplings, fTq and fTq as
λ2(%)
J(J λ+2 J(J
1) + 1)unpair
元素
J Isotope
自然存在比 (%)J λ2 J(JAbundance(%)
+ 1)
元素
J µmag
自然存在比
1
1 H100 (p) α1/2
#
#
H
1/2
fp
α3q 3 He 1/22.793
20.750
1.3 × 10−4 0.750 0.928
3q
(p)
100
p
=
f
+
f
(2.20)
7
T
TG
29
Li M
3/2
92.5 q Mq
0.244
732
27
Mq Si 1/23.256 4.7
0.063
p
Li
3/2
92.5
0.244
p
q=u,d,s
q=c,t,b
19
F 1/2
100
0.647
73
0.065
#
11 B (n) α3q
α3q Ge 9/22.689 7.8
fn
2 (n) #80.1
23
3/2
0.112
p
Na 3/2
100
0.0041
=
fTq
+ fTG
, 129 Xe 1/2
(2.21)
26.4
0.124
270.007
Mq
15 N
127
n
q
I M
5/2
100 M1/2
q=u,d,s
q=c,t,b
0.4
−0.283 21.2 0.087 0.055
p
131
Xe 3/2
133
Cs 7/2
100
0.052
19 F
183
1/2
100 and
0.647 0.003
p
W
1/22.629
14.3 The
and
M
are
the
mass
of
the
proton,
the neutron
quarks,
respectively.
n
q
23 (p)
(p)
(n)
(n)
100 表
2.218
0.041
p
ue of the fTq原子核中の中性子の寄与
, fTq , Na
fTG and3/2
fTG are calculated
from
mass
of quarksλ2and
2.1:the
標的核種に対する
J(J + 1) の値 [?]。
9
10
19
8
10
SD
SD
/ σDM-p
σDM-N
SI
SI
/ σDM-p
σDM-N
中村輝石博士論文2014年3月京都大
F
73
Ge
128
Xe
102
107
6
10
10
7
Li
19
F
5
23
10
Na
127
I
2
10
1
3
102
10
MD [GeV/c2]
3
10
MD [GeV/c2]
SI SD SI SD
oss sections normalized by neutralino-proton SI cross
section,
/σ/σ
as aasfunction
Figure
2.2.3:σχ−N
σχ−N
a function of neutralino mass. Red, black, blue and gre
χ−p
χ−p
19
73
1287
19
s. Black, red and blue line shows target diﬀerence
as F,
Ge and as Xe.
isotope
diﬀerence
Li, F, 23 Na and 127 I, respectively.
SD
DM-N
ction of SD interaction is written as
"2 J + 1
32 2 2 !
SD
σχ−N = GF µχ−N ap ⟨Sp ⟩ + an ⟨Sn ⟩
,
π
J
SD
/ σDM-n
•  XeはSIに対する散乱断⾯面積が⼤大きい
•  NEWAGEで⽤用いているFはSDに対する散乱断⾯面積が⼤大きい
Dependent
(SD) cross section
3
10
29
Si
73
Ge
(2.23)
129
131
Xe
F (qr·nr)dr
)=3
exp − n ,
(2.29)
ρ(r)exp(iq
19
qr
2
e ’scatteringenergy
centres’,
ρ(r),
asSI
with
Fr)dr
target
is factors
shown of
in some
Figure
2.3.2,
F of
(qr
= n ρ(r)exp(iq
·the
n ) interaction
!spectra
SI
form
isotopes
e rn is an eﬀective
eﬀective
neutralino∞nuclear radius. The shows
4π
!! ∞
=
rssin(qr)ρ(r)dr.
19 F tar
is
taken
into
account.
1/3
4π
energy
spectra
of
SI
interaction
with
r
≃
1.14A
fm,
and
≃
0.9
fm
representing
the
nuclear
skin
thickness.
Figure
2.3.1
n
with
this form qfactor
as
0
=
r sin(qr)ρ(r)dr.
F (qrn ) =
フォームファクター
実際はσは定数ではなく、以下のような補正を受ける
! ∞nucleon, the nuclei
q: 運動量量移⾏行行
n, since neutralino would
interact
with any 4π
of the
2
19
y spectra
SI
F target
is shown
inand
Figure
2.3.2,
wheresince
the
form
factor
σ(qrnof
)=
σinteraction
F approximated
(qrn ),withrn: 原⼦子核の⼤大きさ[fm]
(2.28)
0be
can
as
a
thin
shell,
this
assumption
is
called
the w
For
the
SD
interaction,
neutralino
=
r
sin(qr)ρ(r)dr.
For
the
SI
interaction,
since
neutralino
would
interact
with
any
of
the
nucleon,
the
nucle
a solid sphere. In this case, the form factor is written using Bessel
q 0
en into account.
can be
approximated
as
a thin
and
tha
2
this
case,
the
form
factor
is
written
using
Bessel
function
j0 =shell,
sin(x)/x
approximated
as a solid sphere. In this
case,
the
form
factor
is
written
using
Besse
−nSIの場合
xbe
cos(x))/x
as
SDの場合
s: 原子核の厚み= fm
a zero momentum
transfer. 0.9 In
the firstthis
Born
(plane
wave)
2
case,
the
form
factornucleon,
is written
Be
r the
interaction,
since
interact
with the
surplus
theusing
nuclei
"neutralino
# aswould
nction
jSI
= (sin(x) −
xsince
cos(x))/x
i (x)
2
or
theSD
interaction,
neutralino
would
interact
with
any
of
the
nucleon,
the
j1 (qrntransformation
)
(qs)
given
by
Fourier
of assumption
the density
distribution
Fis(qr
)(2.29)
= the
j0 (qr
n#
n ),
F
(qr
)
=
3
exp
−
, this
n
"
e approximated as
a
thin
shell,
and
called
odd-group
model. In
2
2
be approximatedqras
case,
the
form
factor
is
written
n a solid sphere.
j1In
(qrthis
)
(qs)
Fusing
(qrn
n
F (qrn ) = q ρ(r)exp(iq · r)dr
isneutralino
takenofinto
account.
the SI form factors
of
some
isotopes assince
a 0function
thewould
recoilinteract
energy E
R . Expected
For the SD interaction,
with
the surpl
F2
F2
F (qr
3
exp −j0 = sin(x)/x
,
(2.29
n) =
ase, the form factor is written
using
Bessel
function
as
2
1/3
qr
2
tion
=where
(sin(x)
cos(x))/x
rn−≃x1.0A
fm. as
Figure
2.3.3 shows
the2.3.1
SD form factors of some
and jsi (x)
≃ 0.9
representing
the nuclear
skinnthickness.
Figure
! fm
1/3
where "
rn ≃ 1.0A# fm. Figure 2.3.3 shows t
1/3
s
of
some
isotopes
as
a
function
of
the
recoil
energy
E
Expected
of
the fm,
recoil
spectra
SD interaction
with
R . nuclear
FE(qr
)Expected
=(qr
j0 (qr
(2.30)
R .nrepresenting
2 skinofthickness.
here
and·energy
s ≃ 0.9
fm
the
Figure
2.3.
n ),energy
ρ(r)exp(iq
r)dr
n ≃ 1.14A
n ) r=
j
)
(qs)
1
n
of
the
recoil
energy
E
.
Expected
energy
sp
R
19 F target is shown
1
F
(qr
)
=
3
exp
−
,
1
action
with
in
Figure
2.3.2,
where
the
form
factor
nisotopes
in! Figure
where
the form
is taken
ows the SI form
factors2.3.4,
of some
as
a factor
function
of 2the into
recoilaccount.
energy ER . Expecte
qr
∞
n
1/3
in
Figure
2.3.4,
where
the
formasfactor
is tak
4π fm. Figure 2.3.3 shows
rn ≃ 1.0A
the SD form factors of some isotopes
a function
19
ergy =
spectra
of SI interaction
with F target is shown in Figure 2.3.2, where the form facto
r sin(qr)ρ(r)dr.
10
19 F target is shown
1/3
q
10
recoil
energy
E
.
Expected
energy
spectra
of
SD
interaction
with
0
R
e
r
≃
1.14A
fm,
and
s
≃
0.9
fm
representing
the
nuclear
skin
thickness. Figu
n into
n,
since
neutralino
would interact with the surplus nucleon, the nuclei
taken
account.
36
10
2.3.4,
where
the form
taken the
into
account.
thin
shell,
andfactors
this
assumption
called
model.
In recoil energy ER . Ex
saure
the
SI
form
offactor
someisisisotopes
asodd-group
a function
of the
For the SDwould
interaction,
sincewith
neutralino
would
with
thenuclei
surplus nucleon, the nucle
10
eutralino
interact
any of
the interact
nucleon,
the
-1
19
F
-1
74
19
F
7
Ge
Li
132
127
Xe
I
-2
-2
is spectra
written
using
Bessel
functionwith
j0 =19sin(x)/x
as is shown in Figure 2.3.2, where the form
gy
of
SI
interaction
F
target
10
n be approximated
as a the
thin form
shell, factor
and thisisassumption
is called the odd-group model. I
phere.
In this case,
36 written using Bessel
ken
into
account.
10
is case,
the
form
using Bessel function
j0 = (2.30)
sin(x)/x as
F (qrfactor
) = j0is
(qrwritten
),
2
n
n
10
)/x as
中村輝石博士
or the SD interaction,
since neutralino
would interact with the surplus nucleon, th
"
#
論文2014年3月
Figure 2.3.3
shows
the
SD
form
factors
of
some
a50 function
10 0
F
(qr
)
j0 (qrn10
), as
(2.30
2
0
100 150 200 250 300 350 400
n =isotopes
50 100 150 200 250 300 350 400
j1 (qr
(qs)
recoil
energy
[keV]
n)
recoil
energy [keV]
京都大
-3
-3
-4
-5
-4
SD反応で予想させるエネルギースペクトル
19
SD, F, M =100 [GeV], σ=1 [pb]
rate [counts/keV/kg/day]
2
R
F (E )
Form Factor [SD (keV)]
1
-1
10
-2
10
-3
10
-4
10
-5
10
-6
D
2
1.8
赤：6月青：12月
1.6
1.4
1.2
10
1
-7
10
-8
10 0
2
]
0.8
100 200 300 400 500 600 700 800 900 1000
ER [keV]
F原子核の場合のフォームファクター
19
event rate [counts/keV/kg/day]
SD, F, σ=1 [pb], average
0.6
0.4
0.2
10
19
19
19
1
19
F MD =10GeV
F MD =50GeV
0
0
20
40
60
80
100
120
140
160 180 200
recoil energy [keV]
F MD =100GeV
F MD =200GeV
-1
10
ニュートラリーノの質量が100GeVの場合の季節変動 SDによる反応を仮定し、フォームファクタを考慮に入れ
ている
-2
10
-3
10 0
20
40
60
80
100
120
140 160 180 200
recoil energy [keV]
F原子核に開けるSD反応で予想されるエネルギースペクトル σSDχ-‐p = 1 pbとした
3. 角度スペクトル D
nce the Cygnus 400
direction Dis varying per hour and
per day,
the systematic
400
recoil energy [keV]
recoil energy [keV]
到来⽅方向異異⽅方性
1.6
ily and seasonal 350environmental changes will 3 be 350canceled.
18
16
1.4
2.5
14
ry 実験室系におけるWIMPとの弾性散乱による⾓角度度スペクトルは
system, expected
angular
spectrum
of
recoil
nucleus is written as
250
250
300
12
300
!
"
10
2
2
200
200
d R
1 R0
(vE cos θ − v1.5min )
8
≃
exp −
,
150
150
2
dER d cos
θ
2 E0 r
v0 1
6
100
D. Spergel Phys. 4Rev. D 37 (1988) 1353.
2
座
白鳥
1.2
1
Nucleus0.8
(2.31)
0.6
DM θ
100
0.4
rmed by the2 WIMP-wind
direction and the 0.5recoil
nuclei direction[36]. Fig50
50
0.2
19 F for the SD
0
0
0
0
0
e0.2実験室系におけるWIMPとの弾性散乱による⾓角度度スペクトルは、標
expected
the
is
0.4 0.6 0.8 1 angle-energy
-1 -0.8 -0.6 -0.4distribution,
-0.2
0
0.2 0.4 0.6 where
0.8
1
-1 target
-0.8 -0.6 -0.4 -0.2
0 0.2
0.4 0.6 0.8 1
cosθ
cosθ
cosθ
0.1
rate [count/keV/kg/day/cos θ]
D
rate [count/keV/kg/day/cos θ]
的原⼦子核F、SDによる弾性散乱、M
SD = 1 pb. Figure
D=100GeV、σ= 1pbとした場合
100
GeV,
and
σ
2.4.5 shows the expected cos θ distriχ−p σ=1 [pb]
SD, F, M =100 [GeV],
0.1
y selecting the events
with recoil energy of 1000.09
− 120 keV in Figure 2.4.4.
0.09
3
2.5
recoil energy [keV]
400
350
300
250
2
200
1.5
150
1
80-100keV
0.08
1.4
0.07
41
1.2
0.06
0.05
0.8
0.04
0
-1 -0.8
0.4 0.6
0.8 -0.61 -0.4 -0.2
cosθ
0.07
0.06
0.05
0.03
0.4
0.2
0.01
0
0.02
00.2 0.4 0.6 0.8 1 0
-1 -0.8 -0.6 -0.4 -0.2
cosθ
0.01
0
0.2
0.4
0.6
0.8
1
cosθ
色分布は[counts/keV/kg/day/cosθ]での計数率
0.1
θ]
0.08
0.04
0.03
50
100-120keV
0.6
0.02
0
100~120keV部
分を抽出
1
100
0.5
0.2
1.6
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
cosθ
SD, F, MD=50 [GeV], σ=1 [pb]
180
18
160
16
400
recoil energy [keV]
400
recoil energy [keV]
200
SD, F, MD=100 [GeV], σ=1 [pb]
350
3
300
140
14
120
12
100
10
80
8
60
6
40
4
20
2
2.5
250
0.4 0.6
0.8 1
cosθ
0.5
0.45
20-40keV
0.4
0.35
0.3
0.25
1.2
1
200
0.8
1.5
150
150
1
0
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
cosθ
0.1
0.09
0.4
50
0.2
0.5
50
80-100keV
0.08
0.07
0.06
0.05
0.6
100
0
0
-1
rate [count/keV/kg/day/cos θ]
0.2
1.4
300
250
200
rate [count/keV/kg/day/cos θ]
rate [count/keV/kg/day/cos θ]
0
1.6
350
2
100
0
-1 -0.8 -0.6 -0.4 -0.2
recoil energy [keV]
SD, F, MD=10 [GeV], σ=1 [pb]
0.09
0.03
0.1
0.02
0.02
0.05
0.01
0.01
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
cosθ
0
0.6
100-120keV
0.05
0.03
0.8 1
cosθ
0
0.06
0.15
0.4 0.6
0.8 1
cosθ
0.07
0.04
0.2
0.2 0.4 0.6
0.08
0.04
0
0
0.1
0.2
0
-1 -0.8 -0.6 -0.4 -0.2
-0.8 -0.6 -0.4 -0.2
0
-1
-0.8 -0.6 -0.4 -0.2
0.2
0.4
0.8
1
cosθ
•  MD=100GeVの場合、100~∼120keVでのcosθ分布を⾒見見るとの異異⽅方性を観
測できる
•  MD=10GeVの場合、20~∼40keVでのcosθ分布を⾒見見る必要がある
まとめ
•  WIMPとの原⼦子核反跳イベントからは特有のスペクトルを得ることができる
–  季節変動
–  到来⽅方向異異⽅方性
•  WIMPがニュートラリーノであったとき、SIな散乱の場合Aが⼤大きいほど感
度度が⾼高く、SDの場合λJ(J+1)が⼤大きいほうが感度度が⾼高い
•  どのように実験を進めていくか考えるには、得られるスペクトルを予想す
ることは⾮非常に重要
–  エネルギースペクトルに関して: J. D. Lewin and P. F. Smith Astropart. Phys. 6 (1996) 87 –  ⾓角度度スペクトルに関して: D. Spergel Phys. Rev. D 37 (1988)

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