（一）湮灭方面 - yzhxxzxy.github.ioyzhxxzxy.github.io/doc/1012_Eff_DM.pdf · G v m m mm mm mmmv mv mm G mm mmm mm mv mm

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 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 v k u k u p v 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




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




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 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 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 p v p v k u k


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




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 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 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 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 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 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，


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，


*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


i j


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 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



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



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


i j i j


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

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


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




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




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


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




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


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

Documents

（一）湮灭方面 - yzhxxzxy.github.ioyzhxxzxy.github.io/doc/1012_Eff_DM.pdf · G v m m mm mm mmmv mv mm G mm mmm mm mv mm