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Page 1: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 2: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 3: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 4: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 5: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 6: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

Ekin ∼ O(10 keV )ER ∼ O(1 − 10 keV )

Page 7: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 8: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 9: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

χ̄χq̄q

χ̄γµχq̄γµq

Page 10: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

χ̄σµνγ5χFµν

Λ

Page 11: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

q

mN∼ q

mχ∼ 10−3

q

m0,

q

Λ

σ ∼ 10−44cm2

σ ∼ 10−38cm2

σW ∼ 10−36cm2

v ∼ 10−3

Page 12: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

m0 � |q|m0 � |q|

10−3

Page 13: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

Veff = V SIeff + V SD

eff

V SIeff = ch1δ

3(�r) + ich2�sχ · �∇δ3(�r)

+ cl11

4πr− cl2

i�sχ · �r4πr3

,

V SDeff = c′h1�sχ · �sNδ3(�r) + ic′h2�sN · �∇δ3(�r)

+ c′l1�sχ · �sN

4πr− c′l2

i�sN · �r4πr3

,

Page 14: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 15: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

O1 = 1O2 = �σχ · �qO3 = �σχ · �vO4 = �σχ · (�v × �q)

O1 = δ3(�r),1r

O2 = �sχ · �∇δ3(�r),�sχ · �r

r3

O3 = �sχ · �vδ3(�r),�sχ · �v

r

O4 = �sχ · (�v × �∇)δ3(�r),�sχ · (�v × �r)

r3

�q,�v,�sχ,�sN

Page 16: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

O1 = 1O2 = �σχ · �qO3 = �σχ · �vO4 = �σχ · (�v × �q)

χ̄χq̄q, χ̄γµχq̄γµq

χ̄γ5χq̄q, χ̄γ5γµχq̄γµq

χ̄γ5γµχq̄γµq

(χ̄γ5q)(q̄γ5χ)∼ χ̄χq̄q − χ̄γµχq̄γµq...,

Page 17: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

δ3(�r)

�sχ · �∇δ3(�r)

1r

�sχ · �rr3

χ̄γ5γµχq̄(α + βγ5)q

Page 18: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

SI NR operators SD NR operators ER

δ3(�r) �sχ · �sNδ3(�r) 1i�sχ · �∇δ3(�r) i�sN · �∇δ3(�r) ER

14πr

�sχ·�sN

4πr E−2R

−i�sχ·�r4πr3

−i�sN ·�r4πr3 E−1

R

Page 19: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

0 20 40 60 80 100

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

E�keV�

dR�dE

0 20 40 60 80 100

0.000

0.005

0.010

0.015

0.020

E�keV�

dR�dE

Page 20: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

δ3(�r)

�sχ · �∇δ3(�r)

1r

�sχ · �rr3

ch1 ∼ (10−8GeV−2)( g

10−4

) (100GeV

Λ

)2

ch2 ∼ (10−7GeV−3)( g

0.1

) (100GeV

Λ

)3

cl1 ∼ 10−10

cl2 ∼ (10−9GeV−1)( g

10−7

) (100GeV

Λ

)

Page 21: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 22: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚
Page 23: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

0 200 400 600 800 1000�9.0

�8.5

�8.0

�7.5

�7.0

�6.5

�6.0

M �GeV�

Log�c1��G

eV^��2��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�8.0

�7.5

�7.0

�6.5

�6.0

�5.5

�5.0

M �GeV�

Log�c2��G

eV^��3�

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�12.0

�11.5

�11.0

�10.5

�10.0

�9.5

�9.0

M �GeV�

Log�c l1�

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�11

�10

�9

�8

�7

M �GeV�

Log�c l2��G

eV^��1�

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

Page 24: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

0 200 400 600 800 1000�6.0

�5.5

�5.0

�4.5

�4.0

M �GeV�

Log�ch1��

GeV^��2��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�5.0

�4.8

�4.6

�4.4

�4.2

�4.0

M �GeV�

Log�ch1��

GeV^��2��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�5.0

�4.5

�4.0

�3.5

�3.0

M �GeV�

Log�ch2��

GeV^��3��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�4.0

�3.5

�3.0

�2.5

�2.0

M �GeV�

Log�ch2��

GeV^��3��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

Page 25: JiJi Fan - Los Alamos National Laboratory · 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 E˜keV˚ dR ˛ dE 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 E˜keV˚

M �GeV� M �GeV�

0 200 400 600 800 1000�9.0

�8.5

�8.0

�7.5

�7.0

M �GeV�

Log�c l1�

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�8.0

�7.5

�7.0

�6.5

�6.0

M �GeV�

Log�cl1�

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�8.0

�7.5

�7.0

�6.5

�6.0

M �GeV�

Log�c l3��G

eV^��1��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3

0 200 400 600 800 1000�7.0

�6.5

�6.0

�5.5

�5.0

M �GeV�

Log�cl3��

GeV^��1��

Vesc � 500 km�sec, Ρ � 0.3 GeV�cm^3