暗物质与标准模型费米子耦合的有效场论 Ref: Beltran, et al., PRD 80, 043509(2009)
(一)湮灭方面
一、Dirac fermionic WIMP
设暗物质粒子 和 是 Dirac 旋量,
f 和 f 是标准模型中的费米子
�
�
f
f
p
p’ k’
k
Mandelstam 变量
2 2
2 2
2 2
2 2 f
s p p p p m
k k k k m
2 2 2
2 2 2
2
2
f
f
t k p k p m m
k p k p m m
2 2 2
2 2 2
2
2
f
f
u k p k p m m
k p k p m m
2 22 2 fs t u m m
质心系下,
cm 2 2 2p kE E E E
22 22cm 2 2 2p ks p p E E E E
2
22 2 24 4 fs k k E m k
2 cos4
sp k E p k p k
2cm 2 4 4
4
sv v E m m
m
p pp
2 2 2 216 4 4 fs m s m p k
2
2 2 2
cm
41 1 1
2 2 16 4 42 4
f
p p
s m
E E v v s s mE
k
2
0
2cos sin
3d
,
0cos sin 0d
2 2 2 2 2 2 2cos c2 o24
s2 2f f f
st k p k p m m E m m m m
p k p k
2 2 2 2 2 2 2co2 s4
s2 c2 2of f f
su k p k p m m E m m m m
p k p k
质心系下的非相对论近似,
v 是两暗物质粒子间的相对速度,
两暗物质粒子的速度分别是2
v和
2
v,能量均为
21
2 2
vE m m
,
22
2 2 22 2cm
14 4
2 22s p p E
vvE m m m m
2
2 22 2 2 2 2
2 2 2 2 2
4 4 /4 4 4
4 4 4ff f
v m ms m m m v m
s m m m v m v
2
22 2 2
2
2 2
2 2 2
0
22 2
2 2
44 4 /
4
14 4 /
2 4 4 /
2 1 /4 1 /
ff
f
fv
f
f
s mv v m m
s m
m m vv m m
vm m
m m
2
2
222 2 2 2 2 2 2
0
2
2 2
1 1
4 4 4
1
4 16
1
v
mv
m m v m m m v
v
m m
s
1. Scalar 耦合:2SG
ff
** † 0
† 0
spins spins
,
u k v k u k v k
v k u k v k u k
u p u p p m v p v p p m
2SG
i u k v k v p u pi
2*2
spins spins
2
spins
2
2
22 2 2 2 2
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr tr
4 2
14 16 64
4 2
S
S
S
Sf f
Sf f
Gu k v k v p u p u k v k v p u p
Gu k v k v k u k v p u p u p v p
Gv k v k u k u k u p u p v p v p
Gk m k m p m p m
Gs m m s m m
2
2 2 2 2 24 162
Sf f
Gs m m s m m
2
2spinsCM cm
22
2 2 2 2 2
2 2
2 22 2 2 2
2 2
1 1
2 2 42 4
1 4 4 162
16 44
41 14 16
16 4 4 2
S
p p
fS
f f
f Sf f
d
d E E v v E
sm
Gs m m s m m
s sm
s m Gs m m s m m
s s m
2
k
22CM 2 2 2 2 2
2
2 222
2
414 16
16 2 4
4 441
16 2 4
fSS f
ffS
s mGs m m s m m
s s m
s m s ms mG
s m s
f
2 222CM
2
2 2 22 2 2 2 2 2 2 2 2 2
2 22 2
2 2 22 2
2 22 2
4 441
16 2 4
1 12 1 / 4 4 4 4
16 2 4 164 1 /
1 12 1 /
16 2 4 164 1 /
ffSS
Sf f
f
Sf
f
s m s ms mGv v
s m s
G v vm m m m v m m m v m
m mm m
G v vm m
m mm m
2 2 2 2 2 2
2 22 2 2 2 2 2
2 2
22 2 2 2 2 2
22 2 2 2 2 2
4 4
12 1 / 1 /
16 2 4 1 /
12 1 / 1 /
16 2
1 11 / 1 /
2 2 4
f
Sf f
f
Sf f
Sf f
m v m m m v
G vm m m m m v
m m
Gm m m m m v
Gm m m m m v
2. Pseudoscalar 耦合:5
52PG
f f
55
2PG
i u k v ki v p u p
2*2 5 5
5 5spins spins
25 5
5 5spins
25 5
5 5
25 5
5 5
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr tr
4 2
P
P
P
Pf f
Gu k v k v p u p u k v k v p u p
Gu k v k v k u k v p u p u p v p
Gu k u k v k v k v p v p u p u p
Gk m k m p m p m
22 2 2 2
2 22 2
116 16 16 16
4 2
14
4 2 2
Pf f
P P
Gm m k k p p k k m p p m
G Gs s
2
2spinsCM cm
22
2
2 2
22
2 2
1 1
2 2 42 4
1 42
16 44
41 1
16 4 2 4
P
p p
fP
fP
d
d E E v v E
sm
Gs
s sm
s mGs
s m
k
22CM
2
41
16 2 4fP
P
s mGs
s m
22CM
2
22
2
41
16 2 4
41
16 2 4
fPP
fP
s mGv v
s m
s mGv s
s m
s
22CM
2
2 22 2 2 2 2
2 2
22 2 2 2 2 2
2 2
22 2 2 2 2 2
2 2
41
16 2 4
12 1 / 4
16 2 4 1 /
1 18 1 / 2 1 /
16 2 1 /
1 1 11 / 2 1 /
2 2 8 1 /
fPP
Pf
f
Pf f
f
Pf f
f
s mGv v s
s m
G vm m m m v
m m
Gm m m m m m v
m m
Gm m m m m m v
m m
2
2
2 222 2 2 2 2
2 2
3 2 /11 /
2 2 8 1 /
fPf
f
m mGm m m m v
m m
3. Vector 耦合:2VG
f f
2VG
i u k k p u pi v v
2*2
spins spins
2
spins
2
2
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr tr
4 2
vVv
vVv
vVv
vVf f v
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p v k u k u p v p
Gu k u k v k v k v p v p u p u p
Gk m k m p m p m
22 2 2 2
1 2
22 2 2 2
22 2 2 2
164 32 32 32 32
4 2
164 cos 16 4
4 2
16 cos 42
Vf f
Vf
Vf
Gm m k k m k p k p k p k p p p m
Gm m s s
Gm m s s
p k
p k
2
2spinsCM cm
22
2 2 2 22
2
2222 2
2 2
1 1
2 2 42 4
1 1 4 16 cos 464 2
4
16 cos414
64 2 4
V
p p
fV
f
fVf
d
d E E v v E
sm
Gm m s s
s sm
s mGm m s
s m s
k
p k
p k
222CM 2 2
2CM
2 2222 2
2
2 222
2 22
16 cos414 s
32 2 4
4 3218 2
32 2 4 3
3241 4 4
8 216 2 4 3
fV VV f
fVf
ffV
f
s md Gd m
d s m s
s mGm m s
s m s
s sm m
s mGm m s
s m s
inm s d
p k
p k
2 2222 2
2
2 2 222 2
2
4 4414
16 2 4 3
412 4
12 2 4
ffVf
f fVf
s m s ms mGm m s
s m s
s m m mGs m m
s m s
2 222CM 2 2
2
2 2 22 2 2 2 2 2 2 2 2 2 2 2 2
2 22 2
2 22 2
2
4 4414
16 2 4 3
1 1 12 1 / 4 4 4 4
16 2 3 4 164 1 /
12 1 /
16 2 4 1 /
ffVV f
Vf f f
f
Vf
f
s m s ms mGv v m m s
s m s
G v vm m m v m m v m m m m m v
m mm m
G vm m
m
2 2 2 2 2 2 2 2
2
2 22 2 2 2 2 2 2 2 2 2 2 2
2 2
2 222 2 2 2 2 2 2
2 2
18 4 1 /
3
1 18 4 2 1 / 2 1 / 1 /
16 2 34 1 /
2 /18 2 / 1 /
16 2 1 /
f f
Vf f f f
f
fVf f
f
m m m m m v m vm
G vm m m m m m m m m v m v
m m
m mGm m m m m m v
m m
2 2 2 2 2 2
2 222 2 2 2 2 2 2 2 2 2 2
2 2
2 222 2 2 2 2 2 2
2 2
21 / 4 /
3
2 /1 28 2 / 1 / 1 / 4 /
16 2 31 /
2 /1 1 11 / 2 / 4 /
2 2 8 1 / 12
f f
fVf f f f
f
fVf f f
f
m m m m m v
m mGm m m m m m m m m m v
m m
m mGm m m m m v m m
m m
2 2
22 2 2 222 2 2 2 2 2
2 2
14 7 / 2 /1 11 / 2 /
2 2 24 1 /f fV
f ff
v
m m m mGm m m m m v
m m
4. Axial Vector 耦合:5
52AG
f f
55
2AG
i u k v k v ui p p
2*2 5 5
5 5spins spins
25 5
5 5spins
25 5
5 5
25
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr
4 2
vAv
vAv
vAv
Af
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p v k u k u p v p
Gu k u k v k v k v p v p u p u p
Gk m k
55 5
22 2 2 2
222 2 2 2 2
222 2 2 2 2
tr
164 32 32 32 32
4 2
1128 64 cos 16 4
4 2
32 16 cos 42
vf v
Af f
Af f
Af f
m p m p m
Gm m k k m k p k p k p k p p p m
Gm m m m s s
Gm m m m s s
p k
p k
2
2spinsCM cm
22
22 2 2 2 22
2
22 22 22 2
2 2
1 1
2 2 42 4
1 1 4 32 16 cos 464 2
4
32 16 cos414
64 4 2
A
p p
fA
f f
ff Af
d
d E E v v E
sm
Gm m m m s s
s sm
m ms m Gm m s
s m s
k
p k
p k
22 22 2CM 2 2
2CM
2 22 2 222 2
2
2 22 2 22
2
32 16 cos414 s
32 4 2
4 64 3218 2
32 2 4 3
324 641 4 4
32 2 4
ffA AA f
f fAf
ff fA
m ms md Gd m
d s m s
s m m mGm m s
s m s s
s sm m
s m m mG
s m s
inm s d
p k
p k
2 2
2 2 2 2222 2
2
2 2 222 2
2
8 23
96 4 4414
16 2 4 3
414 28
12 2 4
f
f ffAf
f fAf
m m ss
m m s m s ms mGm m s
s m s
s m m mGs m m
s m s
2 2 2 222CM 2 2
2
2 2 22 2 2 2 2 2 2 2 2 2 2 2 2
2 22 2
22
96 4 4414
16 2 4 3
1 1 12 1 / 96 4 4 4
16 2 3 4 164 1 /
12 1 /
16 2
f ffAA f
Af f f
f
Af
m m s m s ms mGv v m m s
s m s
G v vm m m m m v m m v m m m v
m mm m
Gm
f
22 2 2 2 2 2
2 2
2 22 2 2 2 2 2 2 2 2 2
2 2
2 222 2 2 2 2 2 2 2
2 2
14 4 7 /
34 1 /
1 28 1 / 1 / 4 7 /
16 2 31 /
/1 1 21 / / 4 7 /
2 2 8 1 / 3
f f
f
Af f f f f
f
fAf f f
f
vm m m m m v
m m
G vm m m m m m m m m v
m m
m mGm m m m m m m v
m m
22 2 2 222 2 2 2 2 2
2 2
2 2 2 222 2 2 2
2 2
8 19 / 14 /1 11 / /
2 2 24 1 /
8 / 19 14 /1 11 / 1
2 2 24 1 /
f fAf f
f
f fAf f
f
m m m mGm m m m m v
m m
m m m mGm m m v
m m
5. Tensor 耦合:2TG
f f
,2
i
** † 0
† † † † † 0
† 0
2
2
2
2
iu k v k u k v k
iv k u k
iv k u k
iv k u k
v k u k v k u k
2TG
i u k v k v p pi u
2*2
spins spins
2
spins
2
2
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr tr
4 2
T
T
T
Tf f
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p v k u k u p v p
Gu k u k v k v k v p v p u p u p
Gk m k m p
2
22 2
222 2 2 2
1 1 1 1 1tr , , tr , ,
4 2 2 2 2 2
1384 128 256 256
4 2
1256 512 cos 64
4 2
Tf f
Tf
Tf f
m p m
Gk m k m p m p m
Gm m k k p p k p k p k p k p
Gm m m m s
p k
2
22 2 2 264 128 cos 162
Tf f
Gm m m m s
p k
2
2spinsCM cm
22
22 2 2 22
2
22 22 22 2
2 2
1 1
2 2 42 4
1 1 4 64 128 cos 1664 2
4
64 128 cos4116
64 4 2
T
p p
fT
f f
ff Tf
d
d E E v v E
sm
Gm m m m s
s sm
m ms m Gm m
s m s
k
p k
p k
22 22 2CM 2 2
2CM
2 22 2 222 2
2
2 2 2 22 2
2
64 128 cos4116 sin
32 4 2
4 384 256132
32 4 2 3
192 12841 4 4
16 4 2
ffT TT f
f fTf
f ff T
m ms md Gd m
d s m s
s m m mGm m
s m s
s sm m m m
s m G
s m
m d
p k
p k
2 2
2 2 2 22 22 2
2
2 2 222 2
2
163
192 8 4 44116
16 4 2 3
412 40
6 4 2
f
f ff Tf
f fTf
m ms
m m s m s ms m Gm m
s m s
s m m mGs m m
s m s
2 2 2 22 2CM 2 2
2
22 22 2 2 2 2 2 2 2 2 2 2 2
2 22 2
2
192 8 4 44116
16 4 2 3
1 1 12 1 / 192 8 4 4 16
16 2 3 4 164 1 /
12 1 /
16
f ff TT f
Tf f f
f
f
m m s m s ms m Gv v m m
s m s
Gv vm m m m m v m m m v m m
m mm m
m
f
222 2 2 2 2 2 2 2 2
2 2
222 2 2 2 2 2 2 2
2 2
2 22 2 2 2 2 2
2 2
1 116 2 12 8 1 /
2 3 34 1 /
1 42 1 / 16 2 2 5 /
16 2 34 1 /
1 832 2 1 / 4 2 1
16 2 31 /
Tf f f
f
Tf f f
f
Tf f f
f
Gvm m m m v m v m m
m m
Gvm m m m m m m v
m m
G vm m m m m m
m m
2 2 2 2 2 2
2 222 2 2 2 2 2 2 2 2
2 2
22 2 2 222 2 2 2 2 2
2 2
/ 2 5 /
1 2 /1 1 11 / 4 1 2 / 2 5 /
2 2 2 1 / 3
7 8 / 10 /1 11 / 4 1 2 /
2 2 6 1 /
f f
fTf f f
f
f fTf f
f
m m m m m v
m mGm m m m m v m m v
m m
m m m mGm m m m m v
m m
6. Scalar-Pseudoscalar 耦合: 52SPG
fi f
52SPG
i u k i v k v pi u p
2*2
5 5spins spins
2
5 5spins
2
5 5spins
2
5 5
1 1
4 4 2
1
4 2
1
4 2
1tr
4 2
SP
SP
SP
SP
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p v k u k u p v p
Gu k v k v k u k v p u p u p v p
Gu k u k v k v k
2
5 5
2
2 2
tr
1tr tr
4 2
42
SPf f
SP
v p v p u p u p
Gk m k m p m p m
Gs m s
2
2spinsCM cm
22
2 22
2
222
2 2
1 1
2 2 42 4
1 1 4 464 2
4
414
64 4 2
SP
p p
fSP
f SP
d
d E E v v E
sm
Gs m s
s sm
s m Gs m
s m
k
22
CM 22
CM
414
16 4 2fSP SP
SP
s md Gd s
d s mm
22CM 2
2
2 22 2 2 2
2 2
2
2 2 2 2
414
16 4 2
12 1 /
16 2 4 1 /
11 /
8 2
f SPSP
SPf
f
SPf
s m Gv v s m
s m
G vm m m v
m m
Gm m m v
7. Pseudoscalar-Scalar 耦合: 52PSG
i ff
52PSG
i u k v k v p ii u p
2*2
5 5spins spins
2
5 5spins
2
5 5spins
2
1 1
4 4 2
1
4 2
1
4 2
1tr
4 2
PS
PS
PS
PS
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p v k u k u p v p
Gu k v k v k u k v p u p u p v p
Gu k u k v k v k
5 5
2
5 5
2
2 2
tr
1tr tr
4 2
42
PSf f
PSf
v p v p u p u p
Gk m k m p m p m
Gs m s
2
2spinsCM cm
22
2 22
2
222
2 2
1 1
2 2 42 4
1 1 4 464 2
4
414
64 4 2
PS
p p
fPS
f
f PSf
d
d E E v v E
sm
Gs m s
s sm
s m Gs m
s m
k
22
CM 22
CM
414
16 4 2fPS PS
PS f
s md Gd s
d s m
m
22CM 2
2
2 22 2 2 2 2 2
2 2
2 2 22 2 2 2 2 2 2
2 2
2
2 2 2
414
16 4 2
12 1 / 1 4 4
16 8 1 /2
12 1 / 4
16 2 1 /2
11
16 2
f PSPS f
PSf f
f
fPSf f
f
PSf f
s m Gv v s m
s m
G vm m m m m v
m m
m mGm m m m m v v
m m
Gm m m
2 2
2 2
2
2 2 2 2 2
2 2
3/ 8
1 /
1 31 / 1
2 8 1 /2
f
PSf f
f
m vm m
Gm m m m v
m m
8. Vector-Axial Vector 耦合: 52VAG
f f
52
VAGi u k v k v p u pi
2*2
5 5spins spins
2
5 5spins
2
5 5
1 1
4 4 2
1
4 2
1tr tr
4 2
1
4 2
vVAv
vVAv
vVAv
VA
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v p v p u p u p
G
2
5 5
222 2 2 2 2
tr tr
4 4 16 cos 162
vf f v
VAf f
k m k m p m p m
Gs m s m s m m
p k
2
2spinsCM cm
2222 2 2 2 2
2 2
2222 2 2 2
2 2
1 1
2 2 42 4
41 14 4 16 cos 16
64 4 2
41 14 4 16 cos 16
64 4 2
VA
p p
f VA
1
f f
f VAf f
d
d E E v v E
s m Gs m s m s m m
s s m
s m Gs m m m m
s m s s
k
p k
p k
22
2CM 2 2 2 22
CM
222 22 2 2 2
2
222
2
41 14 4 16 cos 16 sin
32 4 2
41 16 1 14 4 16
16 4 32
414 4
16 4 2
fVA VAVA f f
f VAf f
f VAf
s md Gd s m m
d s m s s
s m Gs m m m m
s m s s
s m Gs m
s m
1m m d
p k
p k
2 2 2 2 2
22 2 22 2
2
1 164 4
3
412 2 8
12 4 2
f f
f fVAf
m s m s m m ms s
s m m mGs m m
s m s
22 2 2CM 2 2
2
2 22 2 2 2 2 2 2
2 2
2 2 22
2 2 2 2
2 2
412 2 8
12 4 2
1 12 1 / 1 6 6
12 28 1 /2
312 1 / 6
12 4 1 /2
f fVAVA f
VAf f
f
fVAf f
f
s m m mGv v s m m
s m s
G vm m m m m m v
m m
m m vGm m m m
m m
f
2 2 2
2 2 22 2 2 2 2
2 2
1
2
7 2 /11 / 1
24 1 /2
f
fVAf f
f
m m v
m mGm m m m v
m m
9. Axial Vector-Vector 耦合: 52
AVGf f
52
AVGi u k v k v p u pi
2*2
5 5spins spins
2
5 5spins
2
5 5
1 1
4 4 2
1
4 2
1tr tr
4 2
1
4 2
vAVv
vAVv
vAVv
AV
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v p v p u p u p
G
2
5 5
222 2 2 2 2
tr tr
4 4 16 cos 162
vf f v
AVf f
k m k m p m p m
Gs m s m s m m
p k
2
2spinsCM cm
2222 2 2 2 2
2 2
22 2 222 2
2 2
1 1
2 2 42 4
41 14 4 16 cos 16
64 4 2
41 14 4 16 cos 16
64 4 2
AV
p p
f AVf f
f fAVf
d
d E E v v E
s m Gs m s m s m m
s s m
s m m mGs m m
s m s s
k
p k
p k
22 2 2
2CM 2 22
CM
22 2 22 22 2
2
222
2
41 14 4 16 cos 16 sin
32 4 2
41 16 14 4 16
16 4 32
414
16 4 2
f fAV AVAV f
f fAVf
f AVf
s m m md Gd s m m
d s m s s
s m m mGs m m
s m s s
s m Gs m
s m
d
p k
p k
2 22 2 2
22 2 22 2
2
14 4 4 16
3
412 2 8
12 4 2
ff
f fAVf
m mm s m s m
s s
s m m mGs m m
s m s
22 2 2CM 2 2
2
2 2 22 2 2 2 2 2 2 2 2
22 2
2
2 2 2 2
412 2 8
12 4 2
1 12 1 / 1 4 2 2 8 1
12 4 48 1 /2
1 11 /
6 22
f fAVAV f
AVf f
f
AVf f
s m m mGv v s m m
s m s
G v vm m m m v m m m m
mm m
Gm m m m
f
2v
10. Alternative Tensor 耦合:2TG
f f
2TG
i u ki v k v p
u p
2*2
spins spins
2
spins
2
1 1
4 4 2
1
4 2
1tr tr
4 2
T
T
T
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v
2
2
1tr tr
4 2
1 1 1 1tr , , tr , ,
4 2 2 22
1
4 2
Tf f
Tf f v
T
p v p u p u p
Gk m k m p m p m
Gk m k m p m p m
G
1
2
222 2 2 2
222 2 2 2
2 1024 1024 cos 128
64 512 cos 5122
f f
Tf f
m m m m s
Gm m s m m
p k
p k
2
2spinsCM cm
2222 2 2 2
2 2
22 2 222 2
2 2
1 1
2 2 42 4
41 164 512 cos 512
64 4 2
41 51264 cos 512
64 4 2
T
p p
f Tf f
f fTf
d
d E E v v E
s m Gm m s m m
s s m
s m m mGm m
s m s s
k
p k
p k
22 2 22CM 2 2
2CM
2 2 2 22 22 2
2
2 22
2
41 51264 cos 512 sin
32 4 2
41 51264 512
16 4 32
4164
16 42
f fT TfT
f fTf
fTf
s m m md Gd m m
d s m s s
s m m mGm m
s m s s
s mGm m
s m
d
p k
p k
2 22 2 2
2 2 2 22 2
2
324 4 512
3
422 32
3 42
ff
f fTf
m ms m s m
s s
s m m mGs m m
s m s
2 2 2 2CM 2 2
2
2 2 22 2 2 2 2 2 2 2 2
22 2
2 22 2
2 2
422 32
3 42
2 12 1 / 1 4 2 32 1
3 48 1 /2
22 1 / 1
3 8 1 /2
f fTfT
T
4f f f
f
Tf
f
s m m mGv v s m m
s m s
G vm m m m v m m m m
mm m
G vm m
m m
v
2 2 2 2 2
2 2 2 2
2 2 2 2 2 2 2
2 2
2 2 22 2 2 2 2
2 2
6 2
322 1 / 6 2
3 4 1 /2
7 8 /41 / 6
3 4 1 /2
f f
fTf f f
f
fTf f
f
m m m m v
m m vGm m m m m m v
m m
m mGm m m m v
m m
11. 左手-左手(L-L)耦合: 5 51 12LLG
f f
5 51 12LLi
Gi u k v k v p u p
* *† 05 5
†† † 0 �5 5
† 05 5
5
1 1
1 1
1 1
1
u k v k u k v k
v k u k v k u k
v k u k v k u k
v k u k
2*2
5 5 5spins spins
2
5 5 5spins
2
5 5
1 11 1 1 1
4 4 2
11 1 1 1
4 2
1tr 1 1 tr
4 2
vLLv
vLLv
LLv
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v
5
5
5 5
2
5 5 5 5
222
222
1 1
1tr 1 1 tr 1 1
4 2
12 8 64 cos 128 cos
4 2
4 8 cos 16 cos2
v
vLLf f v
LL
LL
p v p u p u p
Gk m k m p m p m
Gs s
Gs s
p k p k
p k p k
2
2spinsCM cm
2222
2 2
2 22
2 2
1 1
2 2 42 4
41 14 8 cos 16 cos
64 4 2
41 168 cos cos
16 42
LL
p p
f LL
fLL
d
d E E v v E
s m Gs s
s s m
s mGs
s m s
k
p k p k
p k p k
2 22CM
2CM
2 22 2
2
2 22 2
2
2 2
41 168 cos cos sin
8 42
41 16
8 4 32
41 14 4
8 4 32
41
3 42
fLL LLLL
fLL
fLLf
fLL
s md Gd s
d s m s
s mGs
s m s
s mGs s m s m
s m s
s mG
s
d
p k p k
p k
2 2
2 22
4 ff
m ms m m
m s
2 2 2 2CM 2 2
2
2 2 22 2 2 2 2 2 2 2 2
22 2
2 22 2
2 2
414
3 42
1 12 1 / 1 4 4 1
3 48 1 /2
12 1 / 1
3 8 1 /2
f fLLLL f
LL
4f f f
f
LLf
f
s m m mGv v s m m
s m s
G v vm m m m v m m m m
mm m
G vm m
m m
2 2 2 2
2 22 2 2 2 2 2
2 2
2 2 2 4 42 2 2 2
2 2
2
2
13
4
31 12 1 / 3
3 48 1 /2
11 10 / 2 /21 / 3
3 8 1 /2
21
2
f
LLf f
f
f fLLf
f
LL
m m m v
mGm m m m m v
m m
m m m mGm m m v
m m
Gm
2 2 4 4
2 2 2
2 2
11 10 / 2 // 1
24 1 /f f
f
f
m m m mm m v
m m
12. 右手-右手(R-R)耦合: 5 51 12RRG
f f
5 51 12RRi
Gi u k v k v p u p
2*2
5 5 5spins spins
2
5 5 5spins
2
5 5
1 11 1 1 1
4 4 2
11 1 1 1
4 2
1tr 1 1 tr
4 2
vRRv
vRRv
RRv
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v
5
5
5 5
2
5 5 5 5
222
222
1 1
1tr 1 1 tr 1 1
4 2
12 8 64 cos 128 cos
4 2
4 8 cos 16 cos2
v
vRRf f v
RR
RR
p v p u p u p
Gk m k m p m p m
Gs s
Gs s
p k p k
p k p k
2
2spinsCM cm
2222
2 2
2 22
2 2
1 1
2 2 42 4
41 14 8 cos 16 cos
64 4 2
41 168 cos cos
16 42
RR
p p
f RR
fRR
d
d E E v v E
s m Gs s
s s m
s mGs
s m s
k
p k p k
p k p k
2 22CM
2CM
2 22 2
2
2 22 2
2
2 2
41 168 cos cos sin
8 42
41 16
4 4 32
41 14 4
4 4 32
41
3 42
fRR RRRR
fRR
fRRf
fRR
s md Gd s
d s m s
s mGs
s m s
s mGs s m s m
s m s
s mG
s
d
p k p k
p k
2 2
2 22
4 ff
m ms m m
m s
13. 左手-右手(L-R)耦合: 5 51 12LRG
f f
5 51 12LRi
Gi u k v k v p u p
2*2
5 5 5spins spins
2
5 5 5 5spins
2
5 5
1 11 1 1 1
4 4 2
11 1 1 1
4 2
1tr 1 1 tr
4 2
LR
vLRv
LRv
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v
5
5 5
2
5 5 5 5
222
222
1 1
1tr 1 1 tr 1 1
4 2
12 8 64 cos 128 cos
4 2
4 8 cos 16 cos2
v
vLRf f v
LR
LR
p v p u p u p
Gk m k m p m p m
Gs s
Gs s
p k p k
p k p k
2
2spinsCM cm
2222
2 2
2 22
2 2
1 1
2 2 42 4
41 14 8 cos 16 cos
64 4 2
41 168 cos cos
16 42
LR
p p
f LR
fLR
d
d E E v v E
s m Gs s
s s m
s mGs
s m s
k
p k p k
p k p k
2 22CM
2CM
2 22 2
2
2 22 2
2
2 2
41 168 cos cos sin
8 42
41 16
4 4 32
41 14 4
4 4 32
41
3 42
fLR RRLR
fLR
fLRf
fLR
s md Gd s
d s m s
s mGs
s m s
s mGs s m s m
s m s
s mG
s
d
p k p k
p k
2 2
2 22
4 ff
m ms m m
m s
14. 右手-左手(R-L)耦合: 5 51 12RLG
f f
5 51 12RLi
Gi u k v k v p u p
2*2
5 5 5 5spins spins
2
5 5 5 5spins
2
5 5
1 11 1 1 1
4 4 2
11 1 1 1
4 2
1tr 1 1 tr
4 2
RL
vRLv
RLv
Gu k v k v p u p u k v k v p u p
Gu k v k v p u p u p v p v k u k
Gu k u k v k v k v
5 5
2
5 5 5 5
222
222
1 1
1tr 1 1 tr 1 1
4 2
12 8 64 cos 128 cos
4 2
4 8 cos 16 cos2
v
vRLf f v
RL
RL
p v p u p u p
Gk m k m p m p m
Gs s
Gs s
p k p k
p k p k
2
2spinsCM cm
2222
2 2
2 22
2 2
1 1
2 2 42 4
41 14 8 cos 16 cos
64 4 2
41 168 cos cos
16 42
RL
p p
f RL
fRL
d
d E E v v E
s m Gs s
s s m
s mGs
s m s
k
p k p k
p k p k
2 22CM
2CM
2 22 2
2
2 22 2
2
2 2
41 168 cos cos sin
8 42
41 16
4 4 32
41 14 4
4 4 32
41
3 42
fRL RRRL
fRL
fRLf
fRL
s md Gd s
d s m s
s mGs
s m s
s mGs s m s m
s m s
s mG
s
d
p k p k
p k
2 2
2 22
4 ff
m ms m m
m s
Maxwell 速度分布律
(此处所用的记号 与上面不同!) v3/22 2
3 23/20
2exp 4 exp 4
2 24
2
mv mv kTd v v dv
kT kT mmkT
3/2 2
exp2 2
m mf v
kT kT
v
3/2 22 3
0exp 4
2 2
m mvQ Q
kT kT
v d v
3/2 22 4
0
34 exp
2 2
m mvv v
kT kT m
3 kT
d v
Velocity dispersion 2 3kTv v
m
对暗物质,可取 270 km/sv [Jungman et al. Phys. Rep. 267, (1996) 195-373]
设两暗物质粒子的速度分别为 和 ,则它们之间相对速度平方的热平均值 1v 2v
2 2 2 2 2 21 2 1 1 2 2 1 1 2 22 2 2v v v v v v v v v v v 22v ,
即是单粒子 Velocity dispersion v 平方的两倍,故可取
2 22 11 2 270 km/2 2 1.458 10s cv v v 5 2 2m /s
Dirac 费米子型暗物质给出的遗迹密度
设 (计上标准模型中所有费米子的贡献),则 2annv a bv
32 9 1 0
* * pl
39 1 0
pl *
1.0665207 10 GeV2.75 K / 3
1.0665207 10 GeV2.75 K 3 /
f
/s f f
f
f f
xTh
g T g T M a b x
xT
M g T a b x
f
其中 /f fx m T , fT 是冻结温度,Planck 质量 19pl 1.2209 10 GeVM
0 2.72 K25 0.00T (Ref: Mather et al. ApJ (1999) 512:511-520)
*g 随温度关系如下图所示(Coleman & Roos, PRD 68, 027702 (2003) 中 Fig. 1,夸克-强子转变温度取在 200 MeV)
10
100
1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 10000
g *
T (GeV)
The quark-hadron transitionis chosen to occur at 200 MeV
g* (Coleman)
与 freeze out 相关的 fx 可由方程
pl
3*
6 /45ln 2
8 2
f
f
f
gm M a b xx c c
x g
解得,其中 c 是一个量级为 1 的数。
1
2c 因上式的对数依赖关系, c 的数值对结果的影响不大,通常可取 。
二、Complex Scalar WIMP
设暗物质粒子和* 是复标量场玻色子,
f 和 f 是标准模型中的费米子
1. Scalar 耦合:*
2SF
ff
2Si
Fi u k v k
2*2
spins spins
2
spins
2
2
22
22
2
2
tr2
tr2
4 22
2 42
S
S
S
Sf f
Sf
Sf
Fu k v k u k v k
Fu k v k v k u k
Fu k u k v k v k
Fk m k m
Fk k m
Fs m
2
2spinsCM cm
2 22
2 2
1
2 2 2 4
41 12 4
16 4 4 2
S
p p
f Sf
d
d E E v v E
s m Fs m
s s m
k
2 2
CM 22
2 2 2
2
2 2 2
2
412 4
16 4 2
4 412
16 42
4 41
8 42
f SS f
f fS
f fS
s m Fs m
s s m
s m s mF
s m s
s m s mF
s m s
2 22CM
2
2 2 22 2 2 2 2 2
2 22 2
2 22 2 2 2 2 2 2
2 2
2
4 412
16 2 4
1 12 2 1 / 4 4
16 2 4 164 1 /
1 12 2 1 / 1 / /
16 2 44 1 /
12 2 1
16 2
f fSS
Sf f
f
Sf f f
f
S
s m s mFv v
s m s
F v vm m m m v m
m mm m
F vm m m m m m v
m m
F
3/22 2 2 2 2 2 2
23/22 2 2 2 2 2 2
2 2 2 22
2 2 2
1/ 1 / 1 2 /
4
1 12 1 / 1 / 1 2 /
8 2 4
1 11 2 1 1 2
8 42
f f f
Sf f f
f f fS
m m m m m m v
Fm m m m m m v
m m mFv
m m m
2. Vector 耦合: * * *
2 2V VF F
i f f i f f
2SF
i pi p u k v k
2*2
spins spins
2
spins
2
2
222 2
2
2
tr2
tr2
2 4 16 cos2
V
V
V
Vf f
V
Fp p u k v k p p u k v k
Fp p p p u k v k v k u k
Fp p p p u k u k v k v k
Fk m p p k m p p
Fs m s
p k
2
2spinsCM cm
2 222 2
2 2
22 22
2 2
1
2 2 2 4
41 12 4 16 cos
16 4 4 2
16 cos414
32 4 2
V
p p
f V
f V
d
d E E v v E
s m Fs m s
s s m
s m Fs m
s m s
k
p k
p k
22 2CM 2
2CM
2 2222
2
2 222
22
22
16 cos414 s
16 4 2
4 3212 8
16 2 4 3
1641 4 4
48 2 4 3
41
8 2
fV VV
fV
ffV
fV
s md Fd s m
d s m s
s mFs m
s m s
s sm m
s mFs m
s m s
s mF
in d
p k
p k
2 2
22
2 22 2
2
2 2 2 22 2
2
4 44
4 3
2 4 241
8 4 32
412 2 8
12 42
f
ffV
f fVf
s m s ms m
s m s
s m s ms mF
s m s
s m m mFs m m
s m s
2 222CM
2
2 2 22 2 2 2 2 2 2 2
2 22 2
2 22 2 2 2 2
2 2
2 4 241
8 2 4 3
1 2 12 1 / 4 2
8 2 3 4 164 1 /
1 12 1 / 2
8 2 34 1 /
1
8
ffVV
Vf f
f
Vf f
f
s m s ms mFv v
s m s
F v vm m m v m m m v
m mm m
F vm m m m v
m m
22 2 2 2 2
2
2 2 2 2 2 2
2 2 22 2
2 2
21 / 2
2 3
1 21 / 2 /
8 32
11 2
12 2
Vf f
Vf f
f fV
Fm m m m v
Fm m m m m v
m mFm v
m m
3. Scalar-Pseudoscalar 耦合:*SP
52
Ffi f
SP5
2
Fi u vi k i k
2*2 SP
5 5spins spins
2SP
5 5spins
2SP
5 5
2SP
5 5
2SP
2
2
tr2
tr2
22
f f
Fu k i v k u k i v k
Fu k v k v k u k
Fu k u k v k v k
Fk m k m
Fs
2SP
2spinsCM cm
2 2SP
2 2
22SP
2 2
1
2 2 2 4
41 12
16 4 4 2
41 1
8 4 2 4
p p
f
f
d
d E E v v E
s m Fs
s s m
s mF
s m
k
2 2CM SPSP 2
CM
41
8 42fSP
s md Fd
d s m
22CM SPSP 2
2 22 2SP
2 2
2 2 2
2 2 2
2 2 2
2 2 2
41
8 2 4
12 1 /
8 2 4 1 /
11 2
8 4 1 /2
11 2
8 4 1 /2
f
f
f
fSP
f
fSP
f
s mFv v
s m
F vm m
m m
mF v
m m m
mF v
m m m
4. Vector-Axial vector 耦合: * * *VA5 5
2 2VF F
i f f i f f
VA5
2
Fi p u k v ki p
2*2 VA
5 5spins spins
2VA
5 5spins
2VA
5 5
2VA
5 5
22 2VA
2
2
tr2
tr2
2 42
f f
f
Fp p u k v k p p u k v k
Fp p p p u k v k v k u k
Fp p p p u k u k v k v k
Fk m p p k m p p
Fs m m
22 216 cos 16 fs m m
p k 2
2VA2
spinsCM cm
2 222 2 2 2VA
2 2
22 22 22 2VA
2 2
2 22VA
2 2
1
2 2 2 4
41 12 4 16 cos 16
16 4 4 2
16 16 cos41 12 4
16 4 4 2
414
32 4 2
p p
ff f
fff
ff
d
d E E v v E
s m Fs m m s m m
s s m
m ms m Fs m m
s m s
s m Fs m m
s m
2
k
p k
p k
22 22
16 16 cosfm m
s
p k
22 22 2CM 2 2VA VAVA 2
CM
2 22 2 222 2VA
2
2 2 222 2VA
2
16 16 cos414 s
16 4 2
4 32 3212 8
16 2 4 3
324 321 4
2 816 2 4
fff
f ff
f ff
m ms md Fd s m m
d s m s
s m m mFs m m
s m s s
sm
s m m mFs m m
s m s
in d
p k
p k
2 2
2 2 2 2 2 2 222VA
2
2 22 2
2
2 2 2 22 2
2
43
4 16 4 441
8 2 4 3
2 4 441
8 4 32
414 16
12 42
f
f f ff
ffVA
f fVAf
sm
s
s m m s m m s m s ms mF
s m s s
s m s ms mF
s m s
s m m mFs m m
s m s
2 222CM VAVA 2
2 2 22 2 2 2 2 2 2 2VA
2 22 2
2 22 2 2 2 2 2VA
2 2
22VA
2 4 441
8 2 4 3
1 2 12 1 / 4 4
8 2 3 4 164 1 /
1 22 1 / 1 /
8 2 34 1 /
1 41 /
8 2 3
ff
f f
f
f f
f
f
s m s ms mFv v
s m s
F v vm m m v m m m v
m mm m
F vm m m m m v
m m
Fm m
2 2 2 2 2
3/22 22 2
2
1 /
11
6 2
f
fVA
m m m v
mFm v
m
三、Complex Vector WIMP
设暗物质粒子 X 和*X 是复矢量场玻色子,
f 和 f 是标准模型中的费米子
*2
i iv
i X
p pp p g
M
2 2 2
2 2 2 2
2 2 2
2
2
42 1
1
14 8 1 /
4
11
4 4
f f
X X f X
X X
X
s m m vv
s M M m M
M
v
s M v
Ms
1. Scalar 耦合:, *
eff2
S f
f
KX X ff
, *
2S fK
i p p u k vi k
2*2 , * *
spins , spins ,
2
, * *
spins ,
2
,
2 2
1 1
9 9 2
1
9 2
1tr
9 2
1
S f j i j ivv
i j i j
S f i iv j jv
i j
S f
X X
Kp p u k v k p p u k v k
Kp p p p u k v k v k u k
K p pp pg g u k u k v k v k
M M
2
,
2 2
2 22 2,
2 4
2 2
,
4
2
, 2
tr9 2
14 tr
9 2
12 tr
9 2
12 12 3 4
9 2
S ff f
X X
S ff f
X X
S ff f
X
S f ff
K p pp pg g k m k m
M M
K p pp pk m k m
M M
K p pk m k m
M
K mm
2 22 3
2 4 2 4
2
, 3 2 2 2 4 2 2 2 44
1 1
4
14 12 16 48
18 2
f
X X X X
S ff X X f X f X
X
ms s s
M M M M
Ks m M s M m M s m M
M
2,
2spins ,CM cm
22, 3 2 2 2 4 2 2 2
2 2 4
22 2, 2 2 2 4 2 2
2 4 2
1 1
2 2 92 4
41 1 14 12 16 48
16 4 4 18 2
414 12 16 48
1152 4 2
S f
i jp p
f S ff X X f X f X
X X
f S f ff X X f X
X X
d
d E E v v E
s m Ks m M s M m M s m M
s s M M
s m K m Ms m M s M m M
M s M
4
k
4X
s
,,ann
CM
22 2 4, 2 2 2 4 2 2
4 2
2 2 2 4, 2 2 2 4 2 2
4 2
414 12 16 48 s
576 4 2
414 12 16 48
288 42
S fS f
f
f S f f Xf f X X f X
f X X
S f f f Xf f X X f X
fX X
dc d
d
s m K m Mc s m M s M m M
M s M s
K s m m Mc s m M s M m M
M s M s
in d
2 2 2 4, 2 2 2 4 2 2
,ann 4 2
2 2 22, 2 2 2 2 2 2 2 2 4 2 2 2 2
4 2 2 2
414 12 16 48
288 42
12 1 1 4 4 4 12 16 12 1
288 8 1 /2
S f f f XS f f X X f X
fX X
S f ff X X f X X X X f X
X X f X
K s m m Mv c v s m M s M m M
M s M s
K m vc M M v m M M M v M m M
M M m M
f Xm M
2
2 2 2, 2 2 2 2 2 2 2
4 2 2 2
2 2 2 2 22, 2 2 2 2 2 2 2
4 2 2 2
4
12 1 1 12 4
288 8 1 /2
1212 1 12 4
288 8 1 /2
1
12
f
S f ff X f X X f X
fX X f X
X f XS f ff X f X X f X
fX X f X
v
K m vc M m M M m M v
M M m M
M m M vK mc M m M M m M v
M M m M
2 2 2 2
, 22 2 2
11 11 1
24 122S f f f f
ff X X X
K m m mc v
M M M
2. Vector 耦合: , *eff
2V f
f
KX i X f f
, *
2V f
v
Ki p p p p u ki v
k
2*2 , * *
spins , spins ,
2
, * *
spins ,
2
,
2
1 1
9 9 2
1
9 2
1
9 2
V f j i j iv
i j i j
V f i i j jv
i j
V f
X
Kp p p p u k v k p p p p u k v k
Kp p p p p p p p u k v k v k u k
K p pp p p p g
M
2
2 2
,
4
2 222, 4 3 2 2
4 2 4 2
tr
12 tr
9 2
4 cos 16 cos1 1 22 7 12 48 cos
9 42
X
V ff f
X
V fX
X X X X
p pg u k u k v k v k
M
K p pk m p p k m p p
M
Ks s s M s
M M M M
p k p kp k
2 22
2, 4 3 2 24 2 4 2
4 cos 16 cos2 1 27 12
9 42V f
XX X X X
Ks s s M s
M M M M
48 cos
p k p kp k
2,
2spins ,CM cm
2 2222, 4 3 2 2
2 2 4 2 4 2
2,
2 2
1 1
2 2 92 4
4 cos 16 cos41 1 2 1 27 12
16 4 4 9 42
41
288 4 2
V f
i jp p
f V fX
X X X X X
f V f
X
d
d E E v v E
s m Ks s s M s
s s M M M M M
s m K
s M
48 cos
k
p k p kp k
2 2 22
3 2 24 2 4 2
4 cos 16 cos cos1 27 12 48
4 XX X X X
s s s MM M M M s
p k p k p k
,,ann
CM
2 2 222, 3 2 2
2 4 2 4 2
2 2 22, 3 2
2 4 2
4 cos 16 cos cos41 1 27 12 48
144 4 42
4 41 1 27
72 4 42
V fV f
f
f V ff X
f X X X X X
f V ff
X X X
dc d
d
s m Kc s s s M
ss M M M M M
s m Kc s s
s M M M
p k p k p k
p k
sin d
2 2 2 2
24 2
2 2 2 2 2 2 22, 3 2 2
2 4 2 4 2
2,
2
1612 48
33 3
4 4 4 4 4 441 1 27 12
72 4 4 12 32
41
72 4 2
Xf X X
X f X f X ff V ff X
f X X X X X
f V ff
X
s MsM M
s M s m s M s m s M s ms m Kc s s s M
ss M M M M M
s m Kc
s M
p k p k
2
4
4 2 3 4 2 2 2 2 2 2 2 6 4 2 2
22, 3 2 2 4 6 2 2 2 2 4
2 4
1
4
1 48 28 4 4 4 4 48 4 4 4
3 3
41 13 8 28 48 4 4 4 12
72 4 122
1
72
f X
X X X f X X f X X X f
f V ff X X X X f X X
f X X
f
M s
s M s M s s M s m s M s M s m s M s M s M s m
s m Kc s s M s M s M s M s m s M s M
s M M s
sc
22, 2 2 2 4 2 2 2 2 4
2 4
22, 2 2 2 2 4
2 4
2 2 2 2, 2 2
4
4 13 4 4 12 4 4 4 12
4 122
41 12 4 4 12
72 4 62
2 414 4
432 2
f V fX X X X f X X
f X X
f V ff f X X X
f X X
f XV ff f X
X
m Ks s M s M s M s M s m s M s M
s M M s
s m Kc s m s M s M s M
s M M s
s m s M sKc s m s M
M
412 X
f
M
s
2 2 2 2 2 42,
,ann 4 2
2 2 2 2,
4 2 22 2
22 2 2 2 2 2 2 2 2 2 2 2 2 2
4 2 4 1241
432 42
1 12 1 1 1
432 4 48 1 /2
4 4 4 2 4 4 4
X f X XV f fV f
fX X
V f ff
fX X Xf X
X X X X X f X X X X X
s M s m s M s MK s mv c v
M s M s
K m v vc
M M Mm M
M M v M M M v m M M v M M M v
2 4
2 2, 2 2 2 4
4 2
2 2 2, 2 2
2 2
12
1 12 1 4 2 12
432 42
11 2
36 2
X
V f ff X f X
fX X
V f f ff X
f X X
M
K mc v M m M
M M
K m mc M v
M M
3. Tensor 耦合: , * *eff
2T f
f
Ki X X X X f f
, * *
2T f
v
Ki i p ki p p p u
k v
2*2 , * * * *
spins , spins ,
2
, * * * *
spins ,
1 1
9 9 2
1
9 2
1
9
T f j i j i j i j iv
i j i j
T f j i j i j i j iv
i j
Kp p p p u k v k p p p p u k v k
Kp p p p u k v k p p p p v k u k
2
, * * * * * * * *
spins ,
2
,
2 2 2
2
1
9 2
T f i i j j i i j j i i j j i i j jv v
i j
T f
X X X
Kp p p p p p p p p p p p p p p p
u k v k v k u k
K p p p pp pg g g g
M M M
2 2 2 2
2
,
2 2 2
tr
1
9 2
X X X X
T f
X X X
p p p p p p p p p pg g g g
M M M M M
u k u k v k v k
K p p p pp pg g g g
M M M
2X
2 2 2 2
22 2, 2
4 2 4
1 1tr , ,
2 2
cos1 12 4 16 8 16
9 2
X X X X
f f
T f f f
X X X
p p p p p p p p p pg g g g
M M M M
k m k m
K m ms
M M M
p k
2XM
222
2 2
2 22 2 2, 2 2
4 2 4 2 2
cos64 32
cos cos8 14 2 4 16 8
9 2
fX X
T f f ff
X X X X X
s mM M
K m ms s m
M M M M M
p k
p k p k
2,
2spins ,CM cm
2 222 2 2, 2 2
2 2 4 2 4 2 2
2 2,
2 4 2
1 1
2 2 92 4
cos cos41 1 8 14 2 4 16 8
16 4 4 9 2
41
72 42
T f
i jp p
f T f f ff
X X X X X X
T f ff
X X
d
d E E v v E
s m K m ms s
s s M M M M M M
K s mm
M s M
k
p k p k
m
4 22
2 22 2 4 2 24 cos 2 4 16 cos 8 X fXX X f X
M mMM s M m M
s s
p k p k
,,ann
CM
2 2 4 222 2, 2 2 4 2 2
4 2
2 22 2, 2 2 4 2 2
4 2
414 cos 2 4 16 cos 8 sin
36 42
41 42 4 16
18 4 32
T fT f
f
T f f X fXf f X X f X
f X X
T f ff f X X f X
X X
dc d
d
K s m M mMc m M s M m M
M s M s s
K s mc m M s M m M
M s M
p k p k
p k
d
4 222 2
2 2 4 22, 2 2 2 2 4 2 2 2 2
4 2
2 2, 2 2 2 2
4 2
83
41 14 4 2 4 4 4 8
18 4 12 32
4112 4 4 2
216 42
X fX
f
T f f X fXf f X X f X f X X f
fX X
T f ff f X X f
X X
M mM
s s
K s m M mMc m M s s M s m M m M s M s m
M s M s s
K s mc m M s s M s m
M s M
p k
4 224 2 2 2 2
2 2 2 4, 2 2 2 2 2 4
4 2
4 48 4 4 4 96
418 4 16 8 160
216 42
X fXX f X X f
f
T f f f Xf f X f X X
fX X
M mMM m M s M s m
s s
K s m m Mc s m M s m M M
M s M s
2 2 2 4, 2 2 2 2 2 4
,ann 4 2
2 2 22, 2 2 2 2 2 2 2 2 2 2 4 2 2
4 2 2 2
418 4 16 8 160
216 42
12 1 1 4 8 4 4 16 8 40 1
216 8 1 /2
T f f f XT f f X f X X
fX X
T f ff X X f X X X f X X
X X f X
K s m m Mv c v s m M s m M M
M s M s
K m vc M M v m M M M v m M M
M M m M
f Xm M
2
2 2 2, 4 2 2 2 2 2 2
4 2 2 2
2 2 2 22, 2
2 2 22 2
4
12 1 1 24 24 18 12
216 8 1 /2
12 1 1 24 24 12 18
216 8 1 /2
f
T f ff X f X f X X
fX X f X
T f f f ff
X X Xf X
v
K m vc M m M m M M v
M M m M
K m m mvc v
M M Mm M
2 2 2 2, 2
2 2 2
11 8 1 5 6
36 2
f
T f f f ff
f X X X
K m m mc v
M M M
4. Scalar-Pseudoscalar 耦合:, *
eff 52
SP f
f
KX X fi f
, *5
2SP fK
i p p u k ii v k
2*2 , * *
5 5spins , spins ,
2
, * *5 5
spins ,
2
,
2 2
1 1
9 9 2
1
9 2
1Tr
9 2
SP f j i j i
i j i j
SP f i i j j
i j
SP f
X X
Kp p u k i v k p p u k i v k
Kp p p p u k v k v k u k
K p pp pg g u
M M
5 5
2 2
,5 54
2
, 3 24 2
12 Tr
9 2
2 1 13
9 42
SP ff f
X
SP f
X X
k u k v k v k
K p pk m k m
M
Ks s s
M M
2,
2spins ,CM cm
22, 3 2
2 2 4 2
2 2, 2 2 4
2 4 2
1 1
2 2 92 4
41 1 2 1 13
16 4 4 9 42
41 13
288 4 42
SP f
i jp p
f SP f
X X X
SP f fX X
X X
d
d E E v v E
s m Ks s
s s M M M
K s ms M s M
M s M
k
s
,,ann
CM
2 2, 2 2 4
4 2
2 2, 2 2 4
4 2
2 2, 2 2
4 2
41 13 sin
144 4 42
41 12 6
144 4 22
414
288 42
SP fSP f
f
SP f ff X
f X X
SP f ff X
fX X
SP f ff X
X X
dc d
d
K s mc s M
M s M
K s mc s M s M
M s M
K s mc s M s
M s M
X
X
s M d
412 Xf
M
2 2, 2 2 4
,ann 4 2
2 2 22, 2 2 2 2 2 2 2 4
4 2 2 2
2 2 2,
4 2 2
414 12
288 42
12 1 1 4 4 4 12
288 8 1 /2
12 1 1
288 8 1 /2
SP f fSP f X X
fX X
SP f ff X X X X
fX X f X
SP f ff
X X f
K s mv c v s M s M
M s M
K m vc M M v M M
M M m M
K m vc
M M m
X XM v M
4 4 2
2
2 2, 2
2 2 2
12 4
1 31 3 1
36 8 1 /2
X Xf X
SP f ff
f X f X
M M vM
K mc v
M m M
5. Vector-Axial Vector 耦合: , *eff 5
2VA f
f
KX i X f f
, *5
2VA f
v
Ki pi p p p u k v k
2*2 , * *
5 5spins , spins ,
2
, * *5 5
spins ,
2
,
1 1
9 9 2
1
9 2
1
9 2
VA f j iv
i j i j
VA f i i j jv
i j
VA f
Kp p p p u k v k p p p p u k v k
Kp p p p p p p p u k v k v k u k
Kp p p p g
5 52 2
2 2
,5 54
22 2 2, 4 3 2
4 2 4 2 4
tr
12 tr
9 2
4 cos 11 1 22 7 8
9 42
X X
VA ff f
X
VA f f f
X X X X X
p pp pg u k u k v k v k
M M
K p pk m p p k m p p
M
K m ms s s
M M M M M
p k
2
22 2 22
2 22 2 22, 4 3 2 2 2
4 2 4 2 4 2
6 cos28 12 48 cos 48
4 cos 16 cos2 1 27 8 28 12 48 cos 48
9 42
f X fX
VA f f ff X f X
X X X X X X
m M s m MM
K m ms s s m M s m
M M M M M M
p kp k
p k p kp k
2
2 2
X
M
2,
2spins ,CM cm
22,
2 2
2 22 224 3 2 2 2 2
4 2 4 2 4 2
1 1
2 2 92 4
41 1 2
4 916 4 2
4 cos 16 cos1 27 8 28 12 48 cos 48
4
VA f
i jp p
f VA f
X
f ff X f
X X X X X X
d
d E E v v E
s m K
s s M
m ms s s m M s m M
M M M M M M
k
p k p kp k
2
22,
2 2
2 22 2 2 223 2 2 2
4 2 4 2 4 2
41
288 4 2
4 cos 16 cos1 2 487 8 28 12 cos 48
4
X
f VA f
X
f f f Xf X
X X X X X X
s m K
s M
m m m Ms s s m M
s sM M M M M M
p k p kp k
,,ann
CM
22,
2
2 22 2 2 223 2 2 2
4 2 4 2 4 2
41
144 4 2
4 cos 16 cos1 2 487 8 28 12 cos 48 sin
4
VA fVA f
f
f VA ff
f X
f f f Xf X
X X X X X X
dc d
d
s m Kc
s M
m m m Ms s s m M d
s sM M M M M M
c
p k p kp k
2 2 2 2 22 2 2 2 22 2, 3 2 2 2
2 4 2 4 2 4 2
22,
2
34 2
4 4 161 1 2 487 8 28 12 48
72 34 4 3 32
41
72 4 2
1 2
4
f VA f f f f Xf f X
f X X X X X X X
f VA ff
f X
f
X X
s m K m m m Ms s s m M
s ss M M M M M M M
s m Kc
s M
ms
M M
p k p kp k
2 2 2 2 2 22 2 2 22 2 2
4 2 4 2
22,
2
4 2 2 3 4 2 2
4
4 4 4 4 4 47 8 28 12 48
12 3
41
72 4 2
14 2 28 32 4
31
4
X f X f X ff f Xf X
X X X X
f VA ff
f X
X f X X f X
X
s M s m s M s m s M s mm m Ms s m M
s sM M M M
s m Kc
s M
s M m s M M m s M
M s
2 2 2
2 2 2 4 2 6 4 2 2 2 6
22,
2
4 2 2 3 4 2 2 2 4 2 6 2 64
4
44 4 112 48 4 4 4 192
3
41
72 4 2
1 14 2 28 32 112 48 192
34
f
X X f X f X X X f f X
f VA ff
f X
X f X X f X f X f XX
s m s
M s M s m M m M s M s M s m m M
s m Kc
s M
s M m s M M m s M m s M s m MM s
2 2 2 2 4
22, 2 2 2 2 4 2 2 2 2 4
2 4
2 2 2 2 2 42,
4 2
4 4 4 12
41 1 14 4 4 12 4 4 4 12
72 34 42
4 4 4 1241
432 42
1
X f X X
f VA ff X f X X X f X X
f X X
X f X XVA f ff
fX X
s M s m s M s M
s m Kc s M s m s M s M s M s m s M s M
s M M s
s M s m s M s MK s mc
sM s M
2 2 2 2 4, 2 2
4
4 4 124 4
432 2
f X XVA ff f X
fX
s m s M s MKc s m s M
sM
2 2 2 2 4, 2 2
,ann 4
2 2, 2 2 2 2 4
4 2
2 2 2 2,
4 2 22 2
4 4 1214 4
432 2
41 14 4 4 12
432 42
1 12 1 1 1
432 4 48 1 /2
f X XVA fVA f f X
fX
VA f ff X f X X
fX X
VA f ff
X X Xf X
s m s M s MKv c v s m s M
M s
K s mc v s M s m s M s M
M s M s
K m v vc
M M Mm M
2 2 2 2 2 2 2 2
22 2 2 2 2 2 2 4
2 2, 2 2 2 2 4
4 2 2
3/22 2, 2 2
2
4 4 4 4
4 4 4 12
1 12 1 4 4 12
432 42
11
18 2
X X X X X ff
X X X X X X
VA f ff X X f X
fX X X
VA f ff X
f X
M M v M M M v m
M M v M M M v M
K mc M v M m M
M M M
K mc M v
M
6. Alternative Vector 耦合: , *eff
2V f
vf
KX i X f f
, *
2V fK
i p p pi p u k v k
2*2 , * *
spins , spins ,
2
, * *
spins ,
,
1 1
9 9 2
1
9 2
1
9 2
V f j i j i
i j i j
V f i i j j
i j
V f
Kp p p p u k v k p p p p u k v k
Kp p p p p p p p u k v k v k u k
K
2
2 2
2
,
2 2
2
,
tr
1tr
9 2
1
9 2
X X
V ff f
X X
V f
p p p pp p p p g g u k u k v k v k
M M
K p p p pp p p p g g k m k m
M M
K
2223 2 2 2 2
2 2 2
2 222, 3 2 2 2 2 2
2 2 2
cos1 12 2 4 8 16 8 32 cos 32
2
cos1 14 8 16 32 16 64 cos 64
9 2
ff X f X
X X X
V f ff X f X
X X X
ms s m M s m M
M M M
K ms s m M s m M
M M M
p kp k
p kp k
2
2,
2spins ,CM cm
2 22 22, 3 2 2 2
2 2 2 2 2
2
2 2
1 1
2 2 92 4
cos41 1 1 14 8 16 32 16 64 cos 64
4 916 4 2
41
576 4
V f
i jp p
f V f ff X f
X X X X
f V
X
d
d E E v v E
s m K ms s m M s m
s s M M M M
s m K
s M
2 2XM
k
p kp k
2 2 22 2 2, 2 2 2
2 2 2
cos cos14 8 16 32 16 64 64
2
f f f Xf X
X X X
m ms s m M
s sM M M
p k p k M
,,ann
CM
2 2 22 2 2 2, 2 2 2
2 2 2 2
2 2,
2
cos cos41 14 8 16 32 16 64 64 sin
288 4 2
41 1
144 42
V fV f
f
f V f f f Xf f X
f X X X X
V f ff
X
dc d
d
s m K m m Mc s s m M d
s ss M M M M
K s mc
s M
p k p k
2 2 2 22 2 22 2 2
2 2 2
2 2 2 2 22 2 4, 2 2 2 2 2 4 2
2 2
4 8 16 32 16 64 6433
414 8 16 32 16 64 64
3 3144 42
f f Xf X
f X X X
V f f f Xf f X f X X X
X X
m m Ms s m M
s sM M M
K s m m Mc s m M s m M M M
s sM s M
p k p k
p k p k
2 2 2, 3 2 2 2 2 2 2 2 4 2 2 2 4
2 2
2 2, 3 2 2 2 2 2 4
2 2
4 41 1 14 8 4 4 32 16 4 4 64
3 3144 42
41 14 8 32 16 64
144 42
f
V f f Xf f X X f f X X X f f X
fX X
V f ff f X f X X f
X X
K s m Mc s m M s s s M s m m M s M s s M s m m M
sM s M
K s mc s m M s m M s M s m
sM s M
22 4 2 2
2 22 2 2, 2 2 2 2 2
2 2
2 22, 2 2
2 2
2
, 22
14 4
3
41 1 14 4 4 4 4
3144 42
41 44 2
3144 42
14
108 2
X X ff
V f ff X f X X f
fX X
V f ff X f
fX X
V ff f
X
M s M s m
K s mc s s M m s M s M s m
sM s M
K s mc s M s m
sM s M
Kc s m
M
2 2
24 2
4X f
Xf
s M s ms M
s
2 2 2, 2 2
,ann 2
22 2 22,
2 2
2 2 2 2, 2 2 2
2 2 22 2
4 214 4
108 2
4 241
108 42
1 12 1 1 1 4
108 4 48 1 /2
X fV fV f f X
fX
X fV f ff
fX X
V f ff X X
X X Xf X
s M s mKv c v s m s M
M s
s M s mK s mc v
M s M s
K m v vc M M v
M M Mm M
22 2 2 2
2 2 2, 4 4
2 2 2
2 2 2, 2 4
2 2
4 4 2
11 2
108 2
11 2
108 2
2X X X f
f
V f f ff X
fX X X
V f f ff X
f X X
M M M v m
K m mc M v
M M M
K m mc M v
M M
7. Alternative Vector-Axial Vector 耦合: , *eff 5
2VA f
f
KX i X f f
, *5
2VA fK
i p p p pi u kk v
2*2 , * *
5 5spins , spins ,
2
, * *5 5
spins ,
1 1
9 9 2
1
9 2
1
VA f j i j i
i j i j
VA f i i j j
i j
Kp p p p u k v k p p p p u k v k
Kp p p p p p p p u k v k v k u k
9
,5 52 2
2
,5 52 2
tr2
1tr
9 2
VA f
X X
VA ff f
X X
p p p p g g u k u k v k v kM M
K p p p pp p p p g g k m k m
M M
2K p p p p
2 222, 3 2 2 2 2 2
2 2 2
2 222, 3 2 2 2
2 2 2
cos1 1 12 4 2 8 24 8 32 cos 64
9 22
cos1 18 4 16 48 16 64 cos
9 2
VA f ff X f X
X X X
VA f ff X
X X X
K ms s m M s m M
M M M
K ms s m M s
M M M
p kp k
p kp k
2 2128 f Xm M
2,
2spins ,CM cm
2 22 22, 3 2 2 2
2 2 2 2 2
2
2 2
1 1
2 2 92 4
cos41 1 1 18 4 16 48 16 64 cos 128
16 4 4 9 2
41
576 4
VA f
i jp p
f VA f ff X f
X X X X
f
X
d
d E E v v E
s m K ms s m M s m
s s M M M M
s m
s M
2 2XM
k
p kp k
2 2 22 2 2, 2 2 2
2 2 2
cos cos18 4 16 48 16 64 128
2VA f f f X
f XX X X
K m m Ms s m M
M M M s s
p k p k
,,ann
CM
2 2 22 2 2 2, 2 2 2
2 2 2 2
2 2,
2 2
cos cos41 18 4 16 48 16 64 128 sin
288 4 2
41 1
144 42
VA fVA f
f
f VA f f f Xf f X
f X X X X
VA f ff
X X
dc d
d
s m K m m Mc s s m M
s ss M M M M
K s mc
s M M
p k p k
d
2 2 2 22 2 22 2 2
2 2
2 22 2, 3 2 2 2 2 2 4 2 4 2
2 2
,
2
8 4 16 48 16 64 12833
41 1 168 4 48 16 128 4
3144 42
1
144 2
f f Xf X
f X X
VA f ff X f f X X f X X
fX X
VA f
X
m m Ms s m M
s sM M
K s mc s M m s m M s M s m M s M
sM s M
K
M
p k p k
p k
2 222 2 2 2 2 2 2 2 2 2 2
2
2 22, 2 2 2 2 2 2 2 2
2 2
,
2
4 1 14 4 4 4 4 32 4 4 4
34
41 14 3 12 12 96 4 4
3144 42
1
144 2
ff X X X f X f X X X
f X
VA f ff X X f f X X f
fX X
VA f
X
s mc s M s M s s M m s s M m M s M s M s m
ss M
K s mc s M s M s m s m M s M s m
sM s M
K
M
f
2
2 2 2 2 2 2 2
2 2 2, 2 2 2 2
2
44 4 4 28
3
14 4 4 28
108 2
f f X X f f Xf
VA f f Xf f X X f
fX
c s m s M s M m s m Ms
K m Mc s m s M s M m
sM
2 2 2, 2 2 2 2
,ann 2
2 2 2 2, 2 2 2
2 2
2 2 2,
2 2
14 4 4 28
108 2
414 4 28
108 42
12 1 1
108 8 12
VA f f XVA f f X X f
fX
VA f f f Xf X X f
fX X
VA f ff
X X f
K m Mv c v s m s M s M m
M s
K s m m Mc v s M s M m
M s M s
K m vc
M M m
22 2 2 2 2 2 2 2 2 2 2
22 2
2 2, 2 2 2 2 2 2 2
2 2 2
2 2, 2 2
2
14 4 4 4 28 1
4 4/
1 12 1 4 4 28
108 42
11
18 2
X X X X X X f f Xf XX
VA f ff X X f f X X
fX X X
VA f ff f
f X
vM M v M M M v M m m M
MM
K mc M M m m M M v
M M M
K mc m v
M
8. Alternative Tensor 耦合: , * *eff
2T f
f
Ki X X X X f f
, * *
2T f
v
Ki i p p p p ui k
v k
2*2 , * * * *
spins , spins ,
2
, * * * *
1 1
9 9 2
1
9 2
T f j i j i j i j iv
i j i j
T f j i j i j i j iv
Ki p p p p u k v k i p p p p u k v k
Kp p p p p p p p u
spins ,
2
, * * * * * * * *
spins ,
2
,
1
9 2
1
9 2
i j
T f i i j j i i j j i i j j i i j jv v
i j
T f
k v k v k u k
Kp p p p p p p p p p p p p p p p
u k v k v k u k
Kg
2 2 2 2 2 2 2
2
,
tr
1
9 2
X X X X X X X
T f
p p p p p p p p p p p pp p p pg g g g g g g
M M M M M M M M
u k u k v k v k
K
2X
2 2 2 2 2 2 2
1 1tr , ,
2 2
X X X X X X X
f f
p p p p p p p p p p p pp p p pg g g g g g g g
M M M M M M M M
k m k m
2X
2 2 2
, 2 22 4 2
2 2 2
, 2 22 4 2
cos cos1 12 16 64 32 256 256
9 2
cos cos32 14 2 16 16
9 2
T ff
X X X
T ff
X X X
Ks s m
M M M
Ks s m
M M M
p k p k
p k p k
2,
2spins ,CM cm
2 2 22, 2 2
2 2 2 4 2
2 22,
2 2 2 4
1 1
2 2 92 4
cos cos41 1 32 14 2 16 16
16 4 4 9 2
cos41 14 2
18 4 2
T f
i jp p
f T ff
X X X X
f T f
X X X
d
d E E v v E
s m Ks s m
s s M M M M
s m Ks
s M M M
k
p k p k
p k
2 2
2
cos16 16 f
X
m
M s s
p k
,,ann
CM
2 2 22 2,
2 2 4 2
2 2 2 2 22,
2 2 4 2
cos cos41 14 2 16 16 sin
9 4 2
42 14 2 16
9 4 3 32
T fT f
f
f T f ff
f X X X X
T f ff
X X X X
dc d
d
s m K mc s d
ss M M M M s
K s mc s
s M M M M
p k p k
p k p k
2
2 2 2 2 22, 2 2 2 2
2 2 2
2 2, 2 2 2 2 2 2 2
2 2 2
16
42 14 2 16 16
39 4 32
42 1 1 14 4 2 4 4 16
39 4 122
f
f
T f ff X f X
fX X X
T f ff X f X X f
X X X
m
ss
K s mc s M s m M
sM s M M
K s mc s s s M s m M s s M s m m
sM s M M
p k p k
f
2
2 2, 2 2 2 2 4 2 2 2 2 4
4 2
2 22, 2 2 4 2 4 2 2
4 2
,
4
42 112 4 4 24 4 4 4 192
129 42
42 112 24 192 4 4
129 42
2
9
Xf
T f ff X X f X X X f
fX X
T f ff X X f X X f
fX X
T f
X
M
K s mc M s s s M s m M s M s M s m
sM s M
K s mc M s M s m M s M s m
sM s M
K
M
f Xm M
2 23 2 2 2 2 2 4 2 4
2
2 2 2 4, 2 2 2 2 2 4
4 2
4 14 32 8 128
1242
414 32 8 128
54 42
ff X f f X X f X
f X
T f f f Xf X f f X X
fX X
s mc s M m s m M s M s m M
ss M
K s m m Mc s M m s m M M
sM s M
2 2 2 4, 2 2 2 2 2 4
,ann 4 2
2 2 2 22, 2 2 2 2 2 2 2 2 2 2 4
4 2 2 2
414 32 8 128
54 42
12 1 1 4 4 4 32 8 32 1
54 8 1 /2
T f f f XT f X f f X X
fX X
T f fX X X f X X f X X
X X f X
K s m m Mv c v s M m s m M M
M s M s
K m v vM M v M m M M v m M M
M M m M
2 2
2 2 22, 4 2 2 4 2
4 2 22 2
2 2 22,
2 22 2
4
12 1 1 24 48 12 1
54 8 1 /2
12 1 1 24 1 2 12
54 8 1 /2
f Xf
T f f fX f X X
fX X Xf X
T f f f
X Xf X
m M
K m mvM m M M v
M M Mm M
K m mv
M Mm M
22
2
2 2 2 2 2 2, 2 2
2 2 2 2 2
2 2 2 2 2 4 4,
2 2 2 2
1
1 2 /11 8 1 2 4 1
9 1 /2
5 6 / 4 /11 8 1 2
9 1 /2
f
f X
T f f f f X f
f X X f X X
T f f f f X f X
X X f X
mv
M
K m m m M mv v
M M m M M
K m m m M m Mv
M M m M
2
f
设 ,则 2
0
ii
i
v a v
1 21 2
1 2 1 2
/ 2/ 21 2
1 20/ /
01 2 1 2
E E T iE E T ii
iiE E T E E T
i
d d e a v d d e a vv
d d e d d e
非相对论近似下,
3 3
3 32 2 2 2
d d
E md
p p, 2 2 2 2 24 4 1s m m v m v / 4 , 1/22 2
1 2 2 1 / 4 2 / 4E E s m v m mv ;
在替换 下,22 p pp 1 23 3 3 3
1 2 1d d d d p p p p ,而 2 mvp ;并注意到 22
10
2 1 !!
2n ay
n n
ny e dy
a a
;故
1 2
2
2
22
2
1 2
2
2
3 32 /41 2
3 3 /43 31 2
3 3 3 3 /42 /
/ 2/ 2 2
41 2 1 23 3
3 3 /4 3 /41 2 2
3
1 2
//
1 2
3 /42
2
1
2 2 2 2
2 2 2 2
T iE E T i i
i
i iE E TT
x i x
i ix
m mv
mv T
mv Tm mv
v v
v
d dd dm m
d d d
e vd d e a v e v
a ad d e ee
e v ea a
e
dm m
d d d
d d
p pp p
p p p p
p p p
p p
2
2
2
2
2
2
2
2 2
2 2
2 /4
3 /4 2 /42
2 2 /4 2 2 /4 2
2 3 2
3
2 2
2 /4 2 /4 22
2 2
2
4
4
4 /
4 /
2 2 14 4
i x i
ix x
ix x y
i i ix x y
yi i
i iy
v
v v
i v i v i
v v
i
v m e va
e m e
e e x ea a a
dvv
d d
e e x
vv
dvv dvv dyy
dvv dvv dyy
dyy
d
e
ie
a ax y xey
p
1 1
1 1
!!2 1 !!42
2 1 !! 2
2
2 2 1 !!
ii
i i
i
i i
ia
x
ia
x
于是
1 2
1 2
/ 21 2
/0 01 2
2 2 1 !!E E T i i
i
i iE E Ti i
d d e a v iv a
xd d e
pl
pl 03 3
* *
2 2 1 !!/45 45
ln 2 ln 28 82 2
i ifi
if
f f
gm M a i xgm M v
x c c c cx g x g
02 2
vdc d
x
x
00 02 2 2
12
0 00 0
10
0 00
2 2 1 !!
1 1
2 2 1 !! 2 2 1 !!1
2 2 1 !!
12 2 1 !!
1
f f f
f
f
i
i ixi
x x x xf
iii i
i ixi i x
i
i i iif fi
ii f
ia
vd xc dx c dx
x x
xc a i x dx c a i
i
ia
x i xc a i c
i x
32 9 1 0
pl *0
1.0665207 10 GeV2 2 1 !!2.75 K
1
f
i
f i ii f
xTh
iM g T a
i x
v将 展开至 4O v ,有 2pl 0 1 2
3*
6 / 60 /45ln 2
8 2
f f
f
f
gm M a a x a xx c c
x g
,
3
2 9 1 0
2pl * 0 1 1
1.0665207 10 GeV2.75 K 3 / 20 /
f
f f f
xTh
M g T a a x a x
。
(二)散射方面
一、Dirac fermionic WIMP
设暗物质粒子 和 是 Dirac 旋量,
f 和 f 是标准模型中的费米子
�
f
p p’
k’k
�
f
考虑质心系下的弹性散射,如下图所示,
�^
p’
�
�
f
f
k’
p=m v z� �
^k=-m v zf f
E m= + m v2
� �
E m= + m v2
ff
� �-12
-12 f f
m v =m v� � f f
�������
| | | |v v=� � f’ | | | |v v= ’f
p . p=E m v’ - cos2 2 2� � � �
k . k=E m v’ - cos2 2 2
�f f f
v v << c, , ��f�
质心系下的非相对论近似,
v 是两暗物质粒子间的相对速度,
两暗物质粒子的速度分别是2
v和
2
v,能量均为
21
2 2
vE m m
,
v c 时,2 2 2, 2
f fp p E m k E mk
22 2cm
242s p p E E m , 2 24 ~s m v
2 22 2 2 2 2 2 2 2 2 21 1,
4 4f fm v m v m v m v m v p k
1. Scalar 耦合:,
int2
S fGff
,
2S fG
i u p u p ui k u k
2*2 ,
spins spins
2
spins
2
2
22 2 2
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr tr
4 2
116
4 2
S f
S
S
Sf f
Sf
Gu p u p u k u k u p u p u k u k
Gu p u p u p u p u k u k u k u k
Gu p u p u p u p u k u k u k u k
Gp m p m k m k m
Gm m k k p p m k
2
22 2 2 2 2 2 2 2
22 2
116
4 2
162
f
Sf f f f
Sf
k m p p
Gm m m m m m m m
Gm m
(1)对于核子 N(n,p),拉氏量相应项,
2S N
N N
G ,
对应着
22 , 2 2
spins
116
4 2S N
N
Gm m ,其中
, ,
, ,, , , ,
N NS N S q q
q q
N NN NS q q S q Q
q u t s q c b tq q
mG G f
m
m mG f G f
m m
,
而, ,
21
27N N
Q qq u d s
f f
,
依照 Beltran 所引 J. R. Ellis, A. Ferstl, and K. A. Olive, Phys. Lett. B 481, 304(2000),form factor Nqf 的值为
0.020 0.004puf , , , 0.026 0.005p
df 0.118 0.062psf
0.014 0.003nuf , , 。 0.036 0.008p
df 0.118 0.062psf
2
2spinsCM cm
2
2spins
2
2 2spins
2, 2 2
2 2
2 2 2,
22
1 1
42 2 2 4
1 1
42 42 2
1 1
44 16
116
24 16
24
N
p k p k
N N
NNN N N
N
S NN
N
N S N
N
d
d E E v v E
m v
m mmm m v v
m
m m
Gm m
m m
m m G
m m
k
故
2 2 2
,CM2 2
N S NN
N
m m G
m m
(2)对于原子数为 A,原子序数为Z 的原子核与 散射,
2
2 , ,2 2
spins
116
4 2 2S p S n
A
G Gm m Z A Z
(SI 通用)
2
2spinsCM cm
2
2 2spins
2
, ,2 22 2
22 2, ,
22
1 1
42 2 2 4
1 1
44 16
116
2 24 16
2 24
A
p k p k
A
S p S nA
A
S pA S n
A
d
d E E v v E
m m
G Gm m Z A Z
m m
Gm m GZ A Z
m m
k
故
22 2, ,CM
2 2 2S pA S n
A
A
Gm m GZ A Z
m m
2. Pseudoscalar 耦合:,
int 5 52
P fGf f
,5 5
2P fG
i u p u p u k ui k
2*2 ,
5 5 5 5spins spins
2
,5 5 5 5
spins
2
,5 5 5
2
,
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr
4 2
P f
P f
P f
P f
Gu p u p u k u k u p u p u k u k
Gu p u p u p u p u k u k u k u k
Gu p u p u p u p u k u k u k u k
Gp
5
5 5 5 5
2
, 2 2 2 2 2
2
, 2 2
2
2
tr
12 2 8 32
4 2
4 42
4
~
f f
P ff f
P ff
m p m k m k m
Gs m m s m m
Gs m s m
s m
v
故 CM 2~N v , CM 2~A v
3. Vector 耦合:,
int2
V fGf f
,
2V fG
i u p u p u k ui k
2*2 ,
spins spins
2
,
spins
2
,
2
,
1 1
4 4 2
1
4 2
1tr tr
4 2
1tr
4 2
V f vv
V f vv
V f vv
V f
Gu p u p u k u k u p u p u k u k
Gu p u p u p u p u k u k u k u k
Gu p u p u p u p u k u k u k u k
Gp
22, 2 2 2 2 2
22, 2 2 2 2 2
tr
12 2 8 32 cos 64
4 2
4 16 cos 322
vf f v
V ff f
V ff f
m p m k m k m
Gs m m s m m
Gs m m s m m
p k
p k
(1)对于核子 N(n,p),拉氏量相应项,
2V N
N N
G ,
对应着 2
22 , 2 2 2 2
spins
14 16 cos 32
4 2V N
N N
Gs m m s m m
2 p k ,其中
, , , , ,2 , 2V p V u V d V n V u V dG G G G G G , ,
2
2spinsCM cm
2
2spins
2
2 2spins
22, 2 2 2 2
2 2
,22
1 1
42 2 2 4
1 1
42 42 2
1 1
44 16
14 16 cos 32
24 16
1
264
N
p k p k
N N
NNN N N
N
V NN N
N
V N
N
d
d E E v v E
m v
m mmm m v v
m
m m
Gs m m s m m
m m
G
m m
2
k
p k
2
22 2 2 2 24 16 cos 32N Ns m m s m m
p k
22,CM 2 2 2 2 2
2
CM
22 2, 2 2 2 2 2
2
2
, 4 2 2 2 2 22
14 16 cos 32 sin
232
1 164 32
3216
1 16 1 116 16
3 4 4216
N V NN N N
N
V NN N
N
V NN
N
d Gd s m m s
d m m
Gs m m s m m
m m
Gm m m m m v
m m
m m d
p k
p k
2 2 2 2
22 2,
2
32
2
N
N V N
N
m v m m
m m G
m m
(2)对于原子数为 A,原子序数为Z 的原子核与 散射,
2
22 , ,2 2 2 2 2
spins
14 16 cos 32
4 2 2V p V n
N N
G Gs m m s m m Z A Z
p k
2
2spinsCM cm
2
2 2spins
22 , ,2 2 2 2 2
2 2
2 22 ,2 2 2 2 2
22
1 1
42 2 2 4
1 1
44 16
14 16 cos 32
2 24 16
4 16 cos 32264
A
p k p k
A
V p V nN N
A
V pAN N
A
d
d E E v v E
m m
G Gs m m s m m Z A Z
m m
Gm ms m m s m m Z A
m m
k
p k
p k
2
,
2V nG
Z
故
22 2, ,CM
2 2 2V pA V n
A
A
Gm m GZ A Z
m m
(结果与 scalar 耦合的情况一样,除了耦合常数的联系方式不同)
3. Axial-Vector 耦合:,
int 5 52
A fGf f
,5 5
2A fG
i u p u p u k u ki
2*2 ,
5 5 5 5spins spins
2
,5 5 5 5
spins
2
,5 5 5
1 1
4 4 2
1
4 2
1tr tr
4 2
A f vv
A f vv
A f vv
Gu p u p u k u k u p u p u k u k
Gu p u p u p u p u k u k u k u k
Gu p u p u p u p u k u k u k u k
5
2
,5 5 5 5
22, 2 2 2
22, 2 2 2
1tr tr
4 2
12 2 8 32 cos
4 2
4 16 cos2
A f vf f v
A ff
A ff
Gp m p m k m k m
Gs m m s
Gs m m s
p k
p k
(1)对于核子 N(n,p),拉氏量相应项,
5 52A N
N N
G ,
对应着 2
22 , 2 2 2
spins
14 16 co
4 2A N
N
Gs m m s
s
p k ,其中
, ,, ,
NA N A q q
q u d s
G G
,
0.78 0.02, 0.48 0.02, 0.15 0.02p n p n p nu d d u s s (Beltran et al. 采用)
0.842 0.012, 0.427 0.013, 0.085 0.018p n p n p nu d d u s s (Belanger et al. micrOMEGAs 2.2, arXiv:0803.2360 提到)
2
2spinsCM cm
2
2spins
2
22spins
22, 2 2 2
22
1 1
42 2 2 4
1 1
42 42 2
1 1
464
14 16 co
264
N
p k p k
N N
NNN N N
N
A NN
N
d
d E E v v E
m v
m mmm m v v
m
m m
Gs m m s
m m
s
k
p k
22,CM 2 2 2
2
CM
22 2, 2 2 2
2
2
, 4 2 2 2 2 2 2 22
4
14 16 cos si
232
1 164
3216
1 16 1 116 16
3 4 4216
2
N A NN f
N
A Nf
N
A NN
N
d Gd s m m s
d m m
Gs m m s
m m
Gm m m m m v m v
m m
m m
n d
p k
p k
22 2,
2 2N A N
N
m G
m m
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