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9

!

#' )'

%

# g ;

ν=0

' ) !9# 0 ` #

g"h % 8g a g

") h a

∇µ T µν = 0 .

1 , %  g 0 [ 0 g ; _

∇µ T µ0 ≡ ∂µ T µ0 + Γµµσ T σ0 + Γ0µσ T µσ = 0 . `$% % d h g$h a g"h # ! b hfg ! ) [f[U.[ 0 `$% <; f _ _ _ _

# 

' % )%# % )

3 A>



D ] @F

g$i

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ij

ij



'=' #  0 a !9 h [ ' % [ % _ g 0 a %/! ") [$# # ! # ` h # i& % g,'8g"hfh # i 1 hfg # ,N # _ 

3 81 % !$# #+%  % )+# ' #")

1 ij D ]  F γ p, 2 a `5! ! h ! [ g"hfh a g 0 ` hfg"h _ a : g _ aQ!   _ _ _ : 1 : ] 0 g _ -_ _ T = (p+ρ)ui uj −pgij = −pgij ` a f ! [ a 1 h g # [ ' [ g _ ! g ij0 g"h [ f f h " g h e 0 0 a !  $g h [ ' 0  [  =' _ hfg"h % /g a  Q 0 ` hfg"_ h _ [ 0 _ [ # ` [ 2 g 0 %/! hfg$h [ g D ] F ' h 0 [ g

T ij = g ik g jl Tkl = g ik g jl (−gkl p) =

!

3

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T 00 = g 00 g 00 T00 = ρ ,

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a˙ ρ˙ + 3 (ρ + p) = 0 . a ` ! 6g g =' d [  [ h 0 [ h g"h [ i 1 5  [ 0 #  ;+g  %/! hfg$h [ g _ 

#

#  

('

#

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D ] F

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D ] F p = p(ρ) . [ g 0 %/! 9 /g hfg$g %U! hfg$h [ g hfg' ='fg ' /g  h g"h [ i , g"i g !f[f[ h [ g =; . _ _  _ _ [ 1 5 ` ! 1 ` 5 f ! [ % % `5! [ _ 0 g ! 1 h g g ='fg ' g 0 g g g g g 8g"hfh ik] _ [  :_ [ e 1   5' _ ) # % <; _ ! _ ! g 5'_ [ [ [  : [ e 1  =' _ hfg ! g =' [ [ p = 0 p = 31 ρ _ _ _ _ _ _ _ %/% 0 1 ] ! # g ] ]  =' p = −ρ 0     

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1 R0i − g0i R = 8πGT0i , 2 1 Rij − gij R = 8πGTij , 2 ∇µ T µi = 0 ,

  S [  

  

 

 

:Z[ #

  





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1 R00 − g00 R = 8πGT00 , 2 ∇µ T µ0 = 0 .

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const a3





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a(t) = const · (t − ts )2/3 ,

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−1

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ρb =

π2 gb T 4 , 30

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γ

0

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0

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7 π2 gf T 4 . 8 30

% # '
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#;< !8 .#(   ρ=

π2 g∗ T 4 , 30

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 /0  . # 
#  -
!   ..# 
# a(t) ∝ t D! 7 #( ! .; # 
+  a(t) ∝ t   # = >) ,$0< C! 0#B
 
S
+
# 0
 # *  

;  D#1  4/  #12  

 @p.
!&#  " #(4#(- J
 #g :I! !
) K 0 +(  g/ J.

@m!8
$

; )/   ) /   !8
$
 
 .  #  !   2  $ )  7  1
; )/  * ,#1@ !  . ) ' &&*  J
 )   *  )  & * U4.
/ >! !
=#   0$+(=
 +!! eq )     #
# F 0
 ; 
+  .  #  ; 
:     ; !  @  *  
4   O#(#

 ) ' &&* /( !8 Λ@p 
   .
7   ,:#( !   #  0 7 # D/ :     ! >
:     !  2 # $#
 $-3 ,  





1/2

2/3



 

zeq + 1 =

a0 ΩM ∼ ∼ 104 . aeq Ωrad

)'&*



D* 



 4r[     

  j       








% <!  !8 #( #  *  

; ?r(0 # :
7 #
 * . )  &  * T = T (1 + z ) ∼ 10  ∼ 1 " ,! 4. !   0 +    J 

@ 0 !8
.$

;"= 7  (
; ) =@  0 SD G #,D
9(    % / !  UJ
  )   &*  )   &  *
#1C?( -V , #(# 

  % g !8 #$ r(+ &  * #D/ :$  # !8 . ,:-3 ,
 ,:   $  
7 
  !+ +         
; 
 ) !  % +0/C
 2 >Q* 7 +# 1 !  " #    # 
;  
<)# ;  !8 #(
. ! + ; # #(-V
 ! C?D# 4 ; 
  # #1"C -32B!  2  !  * #  # #  >.  ! $  $P#
J  #  
 
; 
K ,: -V# , g#  !&    7 !8  * " 0#  ! 7  . +C !   #  $ $ $ 
G !8 #(
;  
 #
   4 ) ' & 1* T , T =  T   !8 #( $ # 
  E! ! 11   - D# T# ; 
 ! 7  /C  ,: !  !8 #( #  *  

; P !8 #(;) $  
  ) #<   eq

0

4

eq



1/3

γ

ν

γ

Tγ ≡ T .

* 0e#D/ :$.!$0
;$
 H
#M
 # *  

; ( 5 #, .,+
! S  $
  ,DJ! 
  ! + #$TJ $

7 ! # #(-V29!  4. ! ) +!  " 0## *  7π ) ' &* ρ =3·2· T  /$ ,        G 5 !  !
;$
8  30$ # #   .
  $.@A. <#     # >  +C  >  
;  
 #(2  #  +#
; 
 ) +
 . ,:  . J   *  ) ,  
(@
; 
 )   ):$. J$ *  %$ #   7/8 ! . 
E !   # 
; 
? ):-3 , $ !8. 
!8 " ,! 4. !   0
H
+  # : :    !+   2 # $ <
 #(-V29$0
#P !8 #(0 #
 

2

4 ν

ν

ρrad

"

21 = ργ + ρν = 2 + 4



4 11

4/3 #

π2 4 T , 30

)'& r* 

   7 ; +
 $ 
7 0  4+0 # # !$
 !   ;   5 !  !
; 
  "P
7 !8 -0!8 a  ) '&* Ω ρ . ρ = 1.68ρ = 1.68 a *N7 C
  :T 0
9
+
:    !+   2    , T
  ! +  ( a  ) ' &!r* ρ = Ω ρ . E- ( 
0 + !    E $0 +% VD J a

@m !8
 $

;   7  (
; ) /    $0 +   0

rad

γ c

0

M

 T#( 5 ! )* !  !

4

γ

1 + zeq =

3

M

c

ΩM a0 = 0.6 , aeq Ωγ

1 + zeq = 2.4 · 104 ΩM h2 .

)'&"r*

 

! 



:

ΩM = 0.24



h = 0.73

! + !

" 8!  #(%=!  !
#





Λ 


)
r*

1 + zeq = 3.0 · 103 .

)&*   )  * *N7 C
$ )'&"r*  )  * #1
-3 J
 ) & *  )'&  * U#$ 5 !  5 0  5  $0
; 
   ;( 5 !  !&PC= .
 *  

;T! !
#P 0+(   J 

@ !8


; )/  < 7  (
; )/   !! !
)= !8G#9
    :, # D@   .# !  2 !   5 !   !&H4 )D ; #H!,D-VJ # D/@ :  !8 ):D.+,: $ #
7 
;  
 )
 4,   5    .B #(+$0 #J  -  
7 ?.S$
7   )):'4 7 O  .# !8' T ∼ m =  * * % !&#M:  7 
$  !
 C$.
 ! /C
  ,D. $ , $ (@ 0.5 !&#!8o.   (  5 !e-< $ $  .
7 0  
; #4 + 7 g  ):#(  # , +#

 ! $!&# )   *  ) ' & r*  *

Teq =(1 + zeq )T0 = 5.6 ΩM h2 * Teq =0.7 h = 0.73 , ΩM = 0.24





e



∗



21 g∗ = 2 + 4



4 11

4/3

)  *

= 3.36 .

" +)D.+# $  !&#  7 )    r*  )    !r*  ,:#(!H:  !
>C= .
  J
 # 

teq =

  + #Fo C  0.73  Ω = 0.24

) 1*

M∗ 1 = P2l , 2Heq 2Teq

√ MP∗ l = MP l /1.66 g∗

FS #( #! )+&*  )  * ,#$ !e

h=

7   . ) * L  !& .+#(! + ,  +! T!  0P +,

-  !

7 ! .S#! *  
@
; t ≈ 14 ! D      * .#:-E
  .   !  !  3 0$+ 3 J

@ 0!8
.$

; M 7  (
; )D  #
+# ;(@A     5

7 ; ! !
 g# *  
@
;  I J   :  D
  #! #
!  G$0 4,4  !  !
!
 # ! !
 ) 
7 !8 -$!8   !
! C .
 *  

; *     5

@ I! !  H
#!  0$+(F! 7
 . 7  ! K! !
  +(  ρ t = ρ   ! +! 7  $ !&# )'&"r* .,: ,K
;  3
#  

  0
 ;H
 + # D/ @ :$.!  
 .# !  2 # $o. #. +  F! #9) 4
  / $ ,  

 D
  ρ /ρ ∝ a  ),  9.   !&' 11 + 0 4,4 0#    


#   D
G .+!
 ,  % # !&#F  ) #)
 <# !    G. # 
#9
   .+!
 , !+. +!  a ∝ t  a ∝ t   t = t 
+$H)4 D C

7 ; 0 @ #$ !

 >)   1* ! CD#' !8 #$.; T   !
! C= .
 t  ) # C   ,4 D C 5

   # ,D $#O   7 + U)   1*
#$   7 $ 0, 0
:$  #     2  $%? 0
'
G  t < t   X 
  h  i  Y   o'[   9f   i TY   :Z  d  _       nj .W           [  _ ` k 'Z  ,\ S[   =f `  X [  ]DX([ OX

  Vf   Y:j [  d _ :\r[      [  S[ , Y +[      h 4nd g  +[        +[   M

teq = 4 · 1036

" *

−1



= 2.5 · 1012 = 80

0



−1 eq

eq

rad

rad

M

1/2

2/3

eq

eq

eq

eq





M



∗




Z  S[       

      Z   i   Z [ 

" 


 ]    [ [ 1X d FZ  Y    
          g[    j Y    [ _] Y i   [_   D]    8d  T
   [     b :Z 
   a = a(t) [    [ _       ] Z       

   [ If[      b :Z  X

 j   
  h = 0.73 ΩT = 0.24 ]_c[ 1X  >[    8d  0 Y(_





eq

eq

M

)

          
   
  /   5           .!    +(
Q  !  #   *    ; #12 @     #(- ,D +)  ! ,0   ; @A   +  =#     - # !  ! .+#   *    ; .S!   +! )0 Λ  !+   .    +#   -  J  +! )   &/* )    1*  : 0  7  -   ! /C    4  0(!   .  7 #D@    2   $ =#$#    B D!     +#o! 7 4#(C() .     

    !   &     .  7 ! )   0G  0o #,D-EJ   oC #D    

    @   2    #12      B           <    t < t  % ,C !  !&# ?#(,    H)  &#&* .   ! Ω =Ω =0     a  a˙ )   r* =H Ω +Ω , a a  ! 7 # ,:. ) % !     ! )   *   .!  !       !  ,4  C   )   * Ω +Ω =1.  ; Ω > 0     #(,    )   r* !      #P4#S   #$   -:#(







eq

rad

curv

2

2 0

0

3



M

M

Λ

Λ



Λ

)   !r* *       !   7 0  !  0 !    ,, $  ,K.#   ,    7      ;   /    a ∝ t  ! 4)D 0  !  0 ! #  4  7 ;$/,  e    J ):   # 5  +# - $)/C (   !  ,D  .S 7 $@" C  !&#/      #  t = 0   T    O)    !*

!     #:     %   !&# .+#   !  ; *    ;/  :  , .8#(#@    a(t) = a0



ΩM ΩΛ

1/3 

 p 2/3 3 sh Ω Λ H0 t . 2

2/3



#







ΩM ΩΛ

1/3   p 2/3 3 =1 sh Ω Λ H0 t 0 2

)    "r* % Ω → 0 Ω → 1 ! 7   & 0$+ ! $  !&#  G)  &*  % .,0/C  :  !  !& C .  4,D 2/(3H )  *   !   #(4   ,     < $  . # @ Ω  !   ! #  4  $/,     !  ' 7     ;  +! ,  ) Ω = 0 * ! +  Λ> ) Ω > 0*( #    4 7   +#) O #(3 $#$ )  . +  7  0 ! @ () D*  # !  7 ;g! !    ( a = a ) !      .+  7 0 !      #  $   .$  - # !  ! .  /  )!   44, H = (a/a) *  % !+)D  # ˙ :g )D  ; *    ;g#(#      !    !    )   r*   :   7     ; ! +  ,$M#   ,    C?!a.      . +   ! #  4  $#  *  H! (a /a) 4,D  :+ 7     ;F! + a  <  a ! 7  $0 + ! <+ $ +!  . 4C 5  7 !  ?    &  2 1
t0 = √ 3 Ω Λ H0

Λ

$#

%$r

ΩΛ . ΩM

M

Λ

0

Λ



Λ

0



0

2 0

0

3



0

0

 







Λ 


a(t)/a0 1 ΩM = 0.24 , ΩΛ = 0.76

0.8

ΩM = 1.0 , ΩΛ = 0

0.6 0.4 0.2



 



1 t/t0 0.2 0.4 0.6 0.8  & 
,+  7 !  ,D-VJ  a = a(t)         "@ 0# 0 *    7 0 

,  /     !  ;  !    0H $ +0  +    #:    a = 0  !      #(-32 ; ! !  ; *    ;          .S#  *  @   a = a0     ;        %  4)D :>  D  ; *    ;  Ω > 0  " .$ !&# 7 )   "* ,  #$  !  Ω = 0.3  Ω = 0.7 h = 0.7    Ω = 0.24 , Ω = 0.76 , h = 0.73 . t0 = 1.38 · 1010 Λ " ,+ ;  .+#  *     ; #       

    . #  ! 7 ! +     !       7 0>! 7 #. !  ) # *      " # !  4. !        )D  ; *    @  ;   ! ,0   !   @A    !        ! CD# # .S,   ! *    ;     7  -  7 !  0$+ D .Λ! !   P  .       !    44,       !      # .S,   , #$  7 !  #( !  # 4$ !    

Λ

M

Λ

M



W   $  X    S
[   [   : \r [  S [ 

r[S,YY :dU`9[X _ `   4Z   : \         :KZrh [  Z Λ j X      Ycf[[  /i   ::Z\ j $_ X   ]>Y r[    Ω 6= 0 ] Ω 6= 0 ] Ω = 0 Ω + Ω = 1
   [

Z   S[
              >Z       Z   X  H
q    [S\<X

 j  (_ ` 0 Y(_]

  :\rZ,_  Z   X  Ω ≈ 0.3   $_ X   Z :Z,_nX    Y   Gi    Y [ Y h = 0.7
 X
  c[      d  Z   S[         [  =f   i  &_   rj .W        S[     p = wρ
q    [S\ X

   (_ ` * Y1_   w = −1.2

]    i  Ω = 0.25 ] Ω = 0.75
w = −0.8 U4#1C   .S!  +#+! ,0   ! G  .    ! +  Λ>    ,D&@ #(  D    !  !       5 !       #(-V29#(- J   # *          42 ; $ !&# 0;  )   *  !  7 ; .S! < .   =, 

.

M

curv

Λ

M

curv 0

M

M

Λ



lH ,0 = a0

Z

t0 0

dt . a(t)




Z  S[       

+&

      Z   i   Z [ 

%  ,: #'        ! ) 7 0 t   #   a(t) ∝ t ) +!  )   !r* *     +  @ +)  0+  ,    C  !       :.+! +#+! ,0   ! .         /C   +.       ./(  !<.  /   !   44, .S! O! .   4,: .  /    2/H  .  +-V2   0! ;o! +     7 DH-   4 . Λ@p    -     Ω = 0.24  Ω = 0.76  J  + !     2 )  
* · 1.8 l = H "  , h = 0.73 . )    &* l = 14.8 2/3





0

0

Λ

M

H ,0

0

H ,0

 4   d\`   b [ :\ d  Λj W   X   X   
    Y Z   Z   [S\(Y ]QX$ [     i n X  Y ,Y  i M i  :Z [  h\ 2/H
 h  [S\          +[ HX

     0 Y 
 
* . #D-V      K.    0 !  D#(-32 .!     P     H)  "r* !   
J   ; )  
r* ! C    .,D.$  T:      4 7 #(4   ,       Ω   &  !   ! ,0       .    !  , !   4)#  ; .S!  l   @ !  !     +#   ;     1  !  ,: !  .+ 7 $   ! ,0   ;    # :    -  #  ,4 # 

      .   7 ! C?:# !4;  * D-E4! :#(  $ ! J   !4 7 

$0   $0 +D 2 0  .  0$+  0 o 

0 4,@  +# ;     (0  ! 7 ,#$    ! /C !     !      7

$     7 9! !  .-    #!    ) ! !  ?  !84  J  D*  ,   #!      !    l   .#(!  ,     7  ;  ;?! + $0 *    ;  # 0 ) 4 #  ;?4        !  !    #I     5    +D-E  ; #  ,4 D- (   :  7 ! (   7 ! .@ ! /C           *         @  $   " .      ; )   *  )    * e)   * - D#     = ## O  ;  $ 7 a ! .   Ω - D#(-V2 ! 4. ! 1 )    * = H |Ω | . 

.

0

curv

H ,0



H ,0

0

S #  $>   7 C

 

 )  
r*

curv

a20

2 0

curv

 ,D.+# 

& 





)  r" *   ,#$ !

 

a0 1 p = > 2.1 . lH ,0 3.3 |Ωcurv |

" ,  ! 4.!   ## *    ;F.!   G4,: .S! <! .  ^L   4  /$@ D        2 5 4,  7  #+ 7 !   ) ; 

.,    ,     4)#  ;  @ 4  7 0 4)#    #    !F .      #        - 4
5 !  @  $ 7 4
5 !  4)#   ##  l 

 2π a 

2 3 0

H ,0

 3 2π 2 a30 a0 > 46 . N≈ = 4.7 (4π/3)lH3 ,0 lH ,0

)    *

" , ! 4.!   4,D-   D  7   7 r!  !   D  #(-    !    '! 7  @  !  4#,D 2%   4
5 !  *    ;  *    ; #     ! 7    5 !   

   !4C    ,D.+#  !     O!  !11 B!    !  Ω (  5   4,D-   D        B)  "*    $4,#  ;%      .     P!   ! r1   =4#,D   ! - D# * ./+     )    *  curv

+ ' (,-'  $ , >



*

$'%'%($8  8    <       '  '+-%-%  <  =. 

 )


 

*




Λ 


  
 

         
    4  +  3      ,+    

U4#S !  42 0'  0 + * . $C  7 0 !4/       , 0  ! ,0  @  0!!    +#  !   .  /   !   44, H      :  7 0    

     :      !  2   $

5 !  ;e   

@ !  Ω  0 #  .#(-V2 ;         #(-  Ω  .  #  *  .   

 Ω      @ 4)    #   , C +  # . #.S! /C    ;  7 .      ;    D    5 !       !  $#$#(#(!   #(,    !  /   p = −ρ        #,:  .  4/        )   !  $0 #  .+#  ,K#(#    ! !     p = wρ  # #1#(!   w 6= −1 *      4 +  # !  ! .S!   #    +! 2   z   @ ! ;G         (    7 0  !   0 S: 2 $0 ,   #  #!  /  $0  +#  ! 7 0  .+! !  +! )0  !+ !  .      ! -32  0 .  !&#  -   ! !   7 z  * +/     # 0      (    7 0  !         P 0 4
  #  4 ,:-     

@ !   "     7 0 .    E0  ;      -   ! S    )D.+#(-  ,   r0  # 7 

   ;( 5 ! #      ! C?:# #  7 ! ! 2   !   ! ;    -     + 4 !,D-   ; !  !   - L     #- D#(-V2  ,#1C   )         7 # D    7  *         #4 42 -  ,   -#$ ;  (  5 !       ,4 @ .+ (  $, #1#$!  7 ! #$#    ! !  /    +   ! ,  +/$, #1#$!  7 ! -:#(  ! ρ  . , , 2 ;    !   *  OC  !&  , .  :-E    #!+!     .+! /C@   #  #;          ;   .  7   ,   ,    +     )  "r*  :            5     7 4 ! ! + :    7  ; *    ;  κ = −1 Ω > 0  % 0!+ ! + : !    ,, $  ,     Ω → 0        /C !   a → ∞  !  #      )  *  "  ,:.+# ! $ !&#'!   I)   &
r* 
 )   1* ds = dt − a (t) dχ + sh χ dθ + sin θdφ .  # 4 7            #!     ! C?:#      !  !. #$  ! !   ! @ !  t   5 !  +!   0$+D$2 ! ,  
!  ?! !  t  #   Z dt )   * . χ= a(t)  ;( 5 ! !     ! CD#      7 ! ,  /   ! #  7 ! +! 2   !

@    + :  ! " .,D.# ! ,>#(#    !   !    

$ ! )  &&*      z!   4 C 5 ! +  !I/ J   % ;(D    ) =)  *   !  ;  + @    







0

M

Λ

curv



Λ



curv

curv

2

2

2

0

2

2

2

i

2

2

0

t0

ti



z(t) =

,#$ ! 

 P#( 5 ! #(#

 

 )  #& &* z

χ=

Z

z 0

a0 −1, a(t)

dz 0 . 0) a0 (a/a)(z ˙

)  r*

  dz 0 1 p ΩM (z 0 + 1)3 + ΩΛ + Ωcurv (z 0 + 1)2 0 a 0 H0   ""$ (#  @%! -%' ,-2 < ' $! '/ $%= '8-4$  , '%!  # A $%  < >'8$#  $ '%:&# ,(' -2 . * > $%$ ,-2  $=  '8- &$   '   $  8 ('"$ ' /#    #&    @,  A  -/,  > '8-2#*  $#/   #  -2   ( = -2   ! 8 $% :     #$% ' (> D ,  :. .  

χ(z) =

Z


q r[          Y S[+\
aY        k 








L     )   )    .+    #S( 5 ,        #S     ;   -    ./(   7 0>.  /  $0!   *  :# )   1* $ .  ! +02 /   $ 7   . +   #(-   +   .,  -  $     7   $#(2   7     +!  ,  

)   * S(z) = 4πr (z) ,  )   !* r(z) = a sh χ(z) . $         +-V2 0   J$# # 0   

 5 !  +  4     J @  -

U   "+C$           ,  !T    P! !   #!    ):   $ #  S  !   #    (1 + z)  " # ;GC $ #   .  +    )   D     ! 7  # ! ,' -!'$        S0$+D 2 0= .3.(  #(-02  #"   J #  !   !+)D  #  !  7  ) 7  :     +  5 !       -  ,  (1 + z) . % #-   4  /  D#  # ! /C  &/.     D#(-32 ! 4. !  * 

$ !  7 0      0 (η, x) $     7  D#   4/  #oC  +#     

  ; *  @   ;  +!  .     %   !&# g  0 +#    0 .! C#    !  ! CD# # !   !  #r0 $      ! CD# 0  2   ! +  #  7  dη = dη   - ( 

 

- D#  !     ! CD#       #(-V2 ! ! C=#  +! $ .  !+  @ !   dt = (1 + z)dt  " # ! 4.!    ! >     .  <       5 !    #   L )    "r* , J= (1 + z) S(z)  L  #4 ,D-    !  !         )      !. #$  !     J #  !  D*  L   !     !       ! C?D#   ! ;     - ,  7 ! +! 2   !   4 !,D-    

!      ,0   , .     ;   #  

    +  (*&)( 3 2&  1! 2 (02 &)(F/<-<1 2 r  #    L4 7 $.S'! C?:# L J ! $ ! )D     # ;
   !  2

0

−1

−1

i

0

0

i

2





   

ph

 . )  " * 4#S ! 



J=

!  

L , 2 4πrph

)   r
*

. (   )  !*  ; .S :1?! /C  #.  ,    #      P)  "r* + rC    4         7 /  +! )0  ! 0 !    H  a  Ω  Ω Ω     ! !    .#  ! 7 0 !  "      ! ,D #  7 ,  -  ,       ) +!  )  r* K)  *  )  * * )   &/* Ω +Ω +Ω =1 



rph = (1 + z) · r(z) ,

r(z)

0

0

M

M

Λ

Λ

curv



curv

Ωcurv =

1

.

!      z  1  + 7  )D  !  7 C    )   r* !  (

 #'  B! 7 .2  ! , z χ(z) = z/(a H ) r(z) = a χ(z)   %    ! ?     #      !     r 1    .+#  ( 5 , 4 7

   r(z) = H z z 

!



0

−1 0

0

0

a20 H02

0

J=

L , z1. 4πr 2 (z)



)   * /C     4  =.,+  # 4)4    ;$ !&# 0; 

 






Λ 


! ,   42 !&# -:#(-  ,     . $ !&# )   r*  )  *    + . # @  ! 7 0  +! )0  ! 0 !   0S:   !     ! C?D#  ! ;     ,  7 ! +! 2   !    ):  7 !4.!  ' .+!   '  !  .   !     #     J   0        L  . D-       ?       z * .





H0r(z) ΩM = 0.24 , ΩΛ = 0.76

1.75

ΩM = 0.24 , Ωcurv = 0.76

1.5

ΩM = 1.0 , ΩΛ = 0

1.25 1 0.75 0.5 0.25 

 

   
,  !   

 < 0 ! +  ;



1

2

H0 r(z)

4

3

   ,   ! +! 2



z  z :G.-   7 0  + ! ,   @

  4 7         . 4C    g    +   .!  +! )0  !  0  !   0$+   $ !&#:# )    "r*  B 0 44,0# 0   J0'  7 H −1   

.+! r   r(z)





!   K. #  !      . ' $P#  J - r(z)   

0

H0 r(z) = √ χ(z) =

Z

z

)    *

1 sh χ(z) , Ωcurv √ Ωcurv dz 0

p . ΩM (1 + z 0 )3 + ΩΛ + Ωcurv (1 + z 0 )2

" ,  ! 4.!  #$G,   )    *  .#     %  H  !      . 4C            U4#S !   /)U 5  #(- 5 !  @  $#$-   7  V          7 #      #r@ -          @" ; *    ;  Ω = 0.24  Ω = 0.76 )  5    $ * ) 5 !  @      0#+ *     4 . Λ@p   *   ):#r@ Ω = 1 Ω = 0    .  7 C  ; )   r*  )   !r* $!&# +0 ;
):#( ! Ω →0 0

0

M

M

Λ

Λ





0

curv

1 r(z) = H0

Z

z 0

dz 0 p , Ωcurv = 0 , ΩM (z 0 + 1)3 + ΩΛ

  5 ! Ω + Ω = 1  *       * 4 ,D 0 Ω )  !          !   0 Ω * $P#  J  r(z) 4 7   , 5  z  #S(  5  7  0   7  #+ 9! +  Λ>  



M

Λ

Λ



M


q r[          Y S[+\
aY        k 








H0r(z) ΩM = 0.3 , ΩΛ = 0.7

0.8

ΩM = 0 , ΩΛ = 0.55 ΩM = 0.6 , ΩΛ = 0.85

0.6 0.4 0.2

0 0.01 0.02 0.05 0.1 0.2 0.5 1 z      " D  -   J   7 /C?            !   ) Ω  Ω *  S :#( ) Ω = 0, ΩΛ = 0.55* ) Ω = 0.6, ΩΛ = 0.85* !      #(-     7  ; Λ.!   #  ; ! +   !    5 ! Ω = 1 − Ω − Ω  M

M

M

curv

M

Λ

+#   - +0 ;I! + D- 4 . Λ@A   " !  o   4   #1C    )D  7 0  4)D-   $0  +! :            )   5     0@     7  E      !      #(-32  ! S  Λ>    4 . +#+! ,0   ; #  /  ;            ;   .  ;  # C

! + #,D     D  M.! -  , #1C  #( !  7 0 z  %   !&# ! + :  Ω = 0   C       4,D-   !  * 42  (  7 .  r0  # 7 !    # Ωcurv = 0.76    !  !    7  ! +  : ! 4 . ! ,0  !  Λ@p+    +! :             L   +  & .  ! 7 0  D   7 0 #(!  % )D.+# #12   $   5 !  ;<  (@    ,:  #(  7 0 ? ! + 0 .+!  ; +    %#!  )   .+,   < .# :     .  7 0   4)D -   ; *  U4  ! ,     B         5 :        4 7 +.     P! +   . !   B .-  -32 ! ,   !  !  + D       ./ 0/ C ! . # :    !  #(!  7 0 z          4
  7     F4)D  !  #  7 ! + ! 2  !   0$@ D$2  ,    0r()D   0K#  /   $0    4,: - (    ;   #S    .    !&# !   - #$ ;#$!  7 0 z  0  #,   ! 4 7 ;   
D + ! 7   )0 $ ! ,     ! @ !! 4, C 5     7 /C?     !  !    4 7       9   5 !K 0    ;( 5 !  #(- 

 ,  #*.   .,+  #  44,   5 !     Ω = 1 − Ω − Ω  .    ! curv "$ D 

   ! . r(z  z

Λ








Λ

M

  z2 1 z − (3ΩM + 2Ωcurv ) . χ(z) = a 0 H0 4


 $ (   ' ( $%$ -4  ! 4 - =  '% #    < -2 A   $ !*-%,1>  $  -2# @%#  $! ' $% $'8-4>' $%' -%' $.  #

-2'%  $



  

44




Λ 


MLCS

m-M (mag)

42 40 38

ΩM=0.28, ΩΛ=0.72

36

ΩM=0.20, ΩΛ=0.00

34

ΩM=1.00, ΩΛ=0.00

∆(m-M) (mag)

1.0

0.5

0.0

-0.5 -1.0 0.01 0.10 1.00     #  +!8!   44)  0   7 0 z

 
'&      0  !e #    @    >,        4,: - (#  0$,  0   7 0'    )  . ##+;   4@     #(-  

! #   *     C  ! #    D-   $         4,D-   ;  ! ,   0>! +  ;   +.   

 +.   ;<.-   7 0 +  .         ;   .   ; ) Ω = 0.2  Ω = 0  Ω = 0.8*  P4#.  /         !  .      .,D .# !  ;  #    !  0$#curv    

+;   ! ;M  

 .  5.   !  ; # !    /     ;! H #(#  # ( -V 2  G ! . + =.     (m − M )  ! ).D    "  $  !  @ (m − M ) #       #(-  4,   #!rph 7 !  4
   !  m − M = 5 ln rph + 25



M

Λ




q r[          Y S[+\
aY        k 








∆(m-M) (mag)

1.0

0.5

0.0

-0.5

Ground Discovered HST Discovered

-1.0

z gr

high-

0.5 ∆(m-M) (mag)

.0) .0) (+Ω M=1 , 1 z = ~ n o st (+Ω M Evoluti ay du

0.0

-0.5

ΩM =1.0

, ΩΛ =0

Empty (Ω=0) ΩM=0.27, ΩΛ=0.73 "replenishing" gray Dust

0.0

0.5

1.0 z

.0

1.5

2.0

     

+!8! 7 &  :-   #(-32  .!   7 $   7 4
n    @ .+#D      4)D-   ; #S()  5 7 0  0   7 0 

      !8! 0     7   0      7 0   ##      !      #$-32 ;  ; * ##  ; *    ;  #        ;   .  ; )  !  ! Ω = Ω = 0 Ω =1      #+ *    @  S    ! ,0   0g! +  ;  F# r  ,  .    ;  .  * 0 #!+!    7 ! S D Λ> 4 . ! ,0 @ ) Ω = 0.27  Ω a˙ == 0.73 * ! + D >    ; .    ; ) Ω = 1  Ω = 0*  S  #,D-EJ   7 0 ! +  ; #!     ! + :   r0  # 7 !    !4      

!    /(   .J   )D   z )  ! +  > G*  S  ! +  ;           ;G! C)0# 

 !+; + ;&   ,@   7 ! +    7 D-   02 -32 ; .!:#(         +    $$  

@ #   ##    - ( ) 5  0  0   7 0 #!+!    7 ! +      #  -   7   !&+      )  01"@  #    

     * ρ(z) = ρ (1 + z)  α=3 α=3 α=0     D  7 0 # D@ )    +        *   % 0  ! #    z < 0.5  7 .+ z 4 D-   ;K(   5  0 ! 0   7 0  #12   , 5  7 0   +#+!   !   !      7  , 4,4  ) $#( C  D*  . !  7 0     0 ) !84  D*  : :-   J  !   C  ! # D    7 ## +    7      5  7 !   $ !  #    +! 2 @

#    

.    "  . #  ! 7 0  ,4 D-   ( 0  0   7 0    '&  ,#$  7    !  ! z  # D    7   0##(-V2  ,     5  7 ! +   ! #    .+ 

M

curv

Λ





M

M

Λ

Λ



0

α





 

!




Λ 


N

o

Bi g

Ba

ng

3

2 5

0. q 0=

g atin r e l e Acc ting a r e el Dec

1

.5

-0 q 0=

99

.7

% 95. 4% 68

.3%

ΩΛ

0 q 0=

Expands to Infinity

0 Cl

Recollapses

os

Op

ed

en

Ω

to

t

-1 0.0 

 

2





0.5

1.0

ΩM

^

ΩΛ=0

1.5

=1

2.0

2.5

0 !   !  ##$- @    P4)#  '.       +#+! ,0      4,D-   ! ! 0   7 0 

  '&    #    ! 4Ω.  /  Ω  7     #( 7   0  ;    ;   !   7  @

 ,  4#,   !  4,D-    ! 

,   , 2

M

Λ

D  7 4,# 

    #, 2  ,  4, .   !  4,D-    ! ) 5  0 ! 0   7 0 )   5  7 !  F    *  *       K .+#D    7 .!   7 0  4)D-   ; )  @ 42 !r   ,D#+.!   !   ) . 

0  4,D-   ; * 0   ##(-  ,  #("O  #(! !   !  !        !e #   q 4.  /  !  F.!         -V2 ;,  .  #  !    !  #!     0       )   *  

 


q r[          Y S[+\
aY        k 

*   !r1

r(z) = a0 χ(z) + O(z 3 )

"






    !&# %$D 

  !  r(  

z

)   1*

  1 z2 r(z) = z − (3ΩM + 2Ωcurv ) . H0 4

*    - 0 !  &,;F# 

   #,:  4; ! !&#(- . ## #  " .% 7 @       . #    ,D     !84  J  (3Ω + 2Ω )      )   r*       0 Ω Ω     !84  J  (Ω − 2Ω )      Ω Ω .   curv  D      !    - (  7 /C?   . !  !  !   7 0 z  : !- $   7 /C      #$!  7 0  !&# )   1* #(    - D#(-V2    7  .-0/C @  4#  ,   $  z  %  ! ., ,  2    / 7 11+ 2 3  $ $  

     

     !    z $#$-  ,       ;  4,(Ω , 

− 2Ω + ! ),    0 ! z  z   # ;,  !    .$  7 !    11+ z3    :   ,  #( ;  ;  ;  !84  J @       7 !+  !  7      7 0 .+#(  , ;

!     ;  #+    D .,+  (2Ω  4−4,Ω )  #(!  7 0 z nL   !  )D      %  ;      !84   J  4) .@  +  #D -    ,: /   !  )D    ;   ;%  ;   ;   !84   J  !  ,   7 .C     ,  #     !  )D   7   4   # % 4,D  0 z  7 /C?     !   ,    >   = .S  5     4,,    !  " 7   $  ,   ,D     2Ω − Ω = 0 :L   0$  %     ?     

C

M

M

Λ

M

Λ

M



M

M



Λ

Λ

Λ

M



Λ

W      X   
[  Y ]R [ Z

[S\(] X$[ ] ' [  [+ \            nrY [     z d    b 
   ]  4    b   [   [ [    : j   \(_ ` Z    r(z)

 +[S\ r[  H[        [  
 h  [S\  ] X$[   c[     h  S[ GY     /i nX  Y      H [ , Ω  ,  Ω >_    d z S[  S[     ch  b    d  b    :\   2Ω − Ω = 0
*  7 /C   .)!  !      #S   D          ! .#(  #  #  z  . #   # 4,4   ! !  ! !  .  /   !   44,  ! /C  0 +    9   !   .  /   !  >##     %    ;  @   ! +#   ¨ 1 a )    * q = H a " .S! 1$ +  !  q  ! /C          9 +  # .  !84  J ; Ω Ω    $4   0  . #  ! 7 0 !  %. #  * $.       ! +  42 ;  !8!   ;  7  )D  ' #  <4#/ ! . 4C    ,4 # 

   

!  .  .S  5  7  !   7 !  ,4 D-3@

   !     !           ,0/C  7 0' #( 9 $#$ ,4 # 

4#SD#  #        >.  /   ! !    .+  5  7 !  #(        

& σ    +  ,0/C   ! ! )D    )  S# !$ * #       :       #(-3@ 2 σ  ;  σ    7     !  
 "   
 ""  
*  X 
  c[   _ `     Y(_   z YZ  Y (_  hh   ] [

 _ Y  ` r(z)  .W    Y      [ X   [ (X  +[S\`   z ]  [     H q _             f[     \rZ   [S\
Y  Z  [S\(] X$[  U_ X S[ e _             [  [ S[ _ ` k d   b   4 [    Y 

.







0



M

M



Λ



Λ



0

2 0

0

0

M



Λ







0

0



 * ' # '"= ( $  ! < #    -2'  <  -2' ('% ,( 8>( %"!$ < (  -   &6 + -2'% &#$  '  (.  . $  ! <   !   -2 '  <  -2' ('%  * $%'' ! ,    &  $"/ '% $ $ # $%  ' ' -/, ' $%: '   $'%    -/.4   # (' - ! <    ! $& ' ('%'%' - (.  .    

!

 





Λ 


+.     4, C 5    7 /C?   !!  !  0   $#/  ; .(/ ;       Ω  Ω Ω  ,D "! 2B- .#(       (   7 0    *  C  !&  .+!     .  curv        # .!:#(  M-   D   +      "  )D.+#         ! /C   >0 ;          !Ω curv    |Ω curv  Ω| < 0.02  r! 

. .S!  ;  r0  # 7 0      

   Ω   4 .      Ω = 1           4,D-     r0  # 7 0'(-  Ω =0


M

Λ



M

Λ

curv

tot

0.23 < ΩM < 0.39 , 0.77 > ΩΛ > 0.61

 " 
#$.      # 

 * . #D-V     !O.  +  . 5 !      4,D-   #S() 5  7 0%! 0 

   r(D#  .S!   !   .       # .!:#(   !-     S  ! .  #  7 0   !  #   ! #  4  ;   #+  #( 7 *    ;      #12   $     + 5 !  ;o      !84  J  ! -V2 0$,F  D  ! ,0   ! 0  4)D-   ;  +  .  /  !       

  7 0 $@ !  @  @ .+#D 

ΩM = 0.24 ± 0.04 , ΩΛ = 0.76 ± 0.05

B#(  #  #    !
* #  0   7  4,D-    ?4 ,: ;      -   , C  4 ,: 0 . @ :    @   ! !&#  #        .#     +! ,    ; Λ@A     ! z H)    #    :    7       V  #9. #  !    *  .!&  ! E  ; ! .      #12    #(-V2  (  7         #     -  # ;>.#  ! # 

)   $ #$#  @     5 !  ;M    p = −ρ *   :!!   w #(#    !      - !  /  ;     p = wρ .  

0(  7 0- D#       ') +! :    * 5 !   












  %+      +   ./C  7 0.<4#SD#12 0  4,D-   ;   X
 h h k [S\    ^_ :d f/[ i   Z4      $_ X   ['  ef   i  . W   &

_       S[     p = wρ
q X [  Ω = 0 Ω = 0.25 ]   

   [S\ i    Y r(z)   w = −2 ] w = −1.5 ] w = −1 ] w = −0.75 w = −0.5
  \rZ _  



]R_ch  [+\  ]9X$[      d    d   S[  [ :\   Z   `O[  $_ X c[S\ /i    nX    w   _    ]:_ Y  Z     






  

curv

)

)   r* 

−1.2 < w < −0.8

M

;  +     ,+ 
 4 ,+    4

* C  ;  ,4 :- ( ! ; 0 #   

 ;  $C   4
   )   ! )0# 

@  D* ,  , ! E#( 0;.S!  * $. N 

! #+D     

 #  /  B #( 0! .+!  r )       #   *  ! . 7 $-V2 ! 4!,D-  7 ;< !  4
   T#( 0! ∆θ  + +   7 ! 4
   4,D-   ,  +    d























+

!

a





d = ra (z) · ∆θ ,

 #   ! 2    4
    4 7  ; 

 7 C    r (z)     @  !     +  $ !  7 0       0$    7  D#   /4   ,C  + # /   

  ;

z

a


–1.5  i  d   Z   d

!&

0.0

w

–0.5

–1.0

WMAP WMAP+SN(HST/GOODS) –1.5 0.0

0.2

0.4

0.6

0.8

  M  U4,# 

  .        +! ,   0!  .

0.0

0.2

.S  5  7   .+#D    !  4,:-   ;  .       ! .!:#(   )    7    D*( #!    7 !     . !   0  4,D-   ;   4,:-   ! ! 0   7 0 

  ) 5 ! @  7 4,0#  D*    P4)#  !    4,: ?.+! "!      #(-  !
" 
#(  !I0        

 

ΩM w



*     ;  .  !&#       7 ; .+! #( 0 7 ! S. ! ! ∆θ !      !

.  ! ; .S!



4
   $.

dconf = sh χ · ∆θ .

4
     #!+-V2  $     7 =! !



    



 !





ti

   ;

χ

 # 

d = a(ti )dconf a(ti ) = · a0 sh χ · ∆θ . a0

  7 $ )  !r*   ,:#( !   +- (


ra (z) =

1 r(z) , 1+z

  5 !M: r(z)   #   7 $!&# 7   7 D#12 .   #!     #( 0#<.+! ># 5  #,D  
z

 

 

!




Λ 


H0ra (z)

0.4 0.3 0.2 0.1



 





4 1 2 3  ! 
#  ! #    #  /  o  #$ !.S!  



0.24 ΩΛ = 0.76 Ωcurv

* =0



5

z

:o! + 

Λ

>

6 z 

)Ω

M

=

)  
* , 5   0 z    #         ,  4,  #! 7 !  :    $@ 4,D  #S()   ), 

  
!  /):  ,  F! )   0  g4,D 0 z   ! T.S!     !  0  0  0    ;=    o)   ! ;?    #(#  + ), 

     #$ !=.S!  * 

)




∗


  


()           .+! /C@         #   +! ,  +'.         K##   !H#   *     ;  ! !  #(- S0(#  * $.    ! #r   42 ! -#$   5 !  ;              ;H.    .,    % +( 5 !   ;F     S   .? #  7 0 . # !  !     .        F    #  !&#M4 7   7 !+.   !  /C   #    .  V   .  @  #S   D  0<   .   #(2         !  J   S  +    #       ):            7   #  = , 5 !   ;      *         ! ,0   ;   /     ;    !  J H.,:   , F   !  ! ! , !  +5 0     o    .,      !      . 7   $$  

    #(#      /    p = wρ   .  @       w 6= −1   42 ! r  w . #      !    , 0 ) 0$     ( * #!+ !   $ -  + @ * +/     .    J  , :   )  *    !e.   ! 7  4 #S ! +  ' .",#  , 0o! +   ;        ! #!+ !   !  42 0<   0<   !  #+    +   ! +)     .,:  = ,  @ r-32 ;.,  *     ;  " .!0/ C  7 .   M .+#D     7 4#SD#  ) .  7   ! ! H  # @ + !    ! +   ;   !  J   :   4 #(C  'J 0 !B r((U  #( 0    #  * ! 4    $ C      !       !  +):    ! ,: <:  #  .   .+#( @    ;  $ : J    

   [ f     !  + +/. 6> B C1  B>?9I: H?A  5 72 4B ?B N4B 5 7 < 2 C> : 27 E 1 4>M7  H> 1651  B 










,+! 





!







@ V0(ϕ) !+ 









 -  2   



 +):  !,U ;  

!



√ d4 x −gL   Z √ 1 µν 4 = d x −g g ∂µ ϕ∂ν ϕ − V (ϕ) , 2

S=





Z

! :  7 ;    J    #/ !          *    ;           (

  7   , !    



)   *          

@



)   !r*

ds2 = dt2 − a2 (t)dx2 ,

.    $P#  J   !    ,     C  F: !+ ):    ,:  ,:#(  ,  +# 4 7      .$@ ;     )  * . ϕ  !       !I 





a(t)




!



,+!  


√ 1 ∂V √ ∂µ −gg µν ∂ν ϕ = − . −g ∂ϕ

    +   ')  .# , 2             ! S   )  r ! *     ! #(#    ')  "r*  +  , 



)   "r*

7 0     1*  ,





ϕ¨ + 3H ϕ˙ = −

ϕ(t)



)  
r*

∂V , ∂ϕ

  + # 4 7     H = a/a !  44, #    )  
r*  (  V#(#  @ ˙  !     #  !         K ; ! 0      #   J  +#    ; ϕ *   J ) V (ϕ)   7 @ 7  

   * . #  !  

 7 -V2 ;     ., , 2 !    !    $ $ J   !    H   !    I! C?D# !  7 $-32 ;  0;    ! .S! /C  7 $ C !  N) &* C !F4 7    +  7 $    ( )   0"4.  /   . +  #$- . ϕ*  H ϕ˙  V   , 

J'4 7   . !  7 $  ,    !  !&#$!     J ) V (ϕ)    ! )0    C ! !    ) *    !+  7 $     (     D    # 

J #        ,  * %  !M C !  7 )    , )   &* H ϕ˙ ∼ V .
90 4,4 0!   !& H .  /   ):



0

0

−1

L   +. !

  

δϕ ∼ ϕH ˙ −1 ∼

V0 . H2

!  0  #   -   ! ! .  /

 

! ): 

  

δϕ  ϕ

 



V0  H2 . ϕ

:     7 0    J  0 

  ,   

 #0  C ! ?!   !  +  7 $ m ϕ  !  λϕ    2

2

V  H2 . ϕ2

4

V 0 ∼ V ϕ−1

)   *     !&# )    *

 

!




Λ 


"  #        7 0 .   J )   7 ,    # 0  )    * .  /   ,:    5 , # 

 !   .S!  7 !+ ,:E - J  *     ;     7 ,    4  @  !  #    $ , 4 7   >!    ,      ! !  +  7 $  , o!  @ !&#(!&#    J  X
. W   & r[      i 

)



V (ϕ)



Y  Z  +[ \$]RX$[    +[     Z  

  S[   
=[  h i    Y [    

]   _     )  
r*   b   6j α]  Y  [+a(t) d   = t  α >[ 1/3  

)   r*

"

 3α−1 # t ϕ = ϕi + C · (t2 − t2i ) + d · 1 − , ti

i   X  :\  Z   X  ] ]
q X [  ] X([  ˙ i )  H(ti )ϕ(ti )    [ ϕ i Z   X   Y  S[  c[ C ϕd ]i =d  ϕ(t Z i ) IX  Z ϕi ϕ(t
 h  [S\  ] ˙ i) X$[    U_ Y  Z         b  =[  [ 'X    )   1*  ϕ(t i     
   [

] 

Y  Y      t  [   GX    )   r*            `    X  :\d  Z   X   ϕi M_ h  [+\  ]PX$[     +[    d  r[    If/[ G_  

      [ )    * ]   h k   $_ X   )   *
 h  [S\   ]EX$[    f[

      d     [    r[    )  &* ]  [  Y b ϕ¨ ∼ H ϕ˙
$ Z      j i   r [          ]X$[ 3Z   hh  Y      r[   c[ \  :Z   

 Y    S [ ϕ˙      ]       Y   +[S\  S[  +[                S[    Y [      9\   S [    d a(t)


















     !   ; " ,     +):  ! ):4, . !  !&#(! "+):  ! J )  .  4  ,  4#S !F: )    7  +#% # r-32 ;., *    ;   0$+     4, C   ϕ(t)    !&#e!  !&#$!&# #/ !            J )   4, . !  !&#(! =, 

!   !  !&#(!  V (ϕ) ϕ = 0 

 

 



" (B#$#    )  
r* 4) . !

)   *

m2 2 ϕ . V (ϕ) = 2

!&#$!      J ) !



 



)   * :/  ) . +4  7 0T#(,   ;) .   .+4$#  ,/  +  
    !  ;   #(#   - #+J .:     !  ;?,    ; )  )    J!F.     *  * (  ! -#$ . !   !    

a˙ ϕ¨ + 3 ϕ˙ + m2 ϕ = 0 . a





 

 $   -

χ

ϕ(t) =



1



! 

· χ(t) ,

.      $P#   J   #S#,   r-32    +#- D #  . )   r*  (# #@   ¨ 3 a˙ 3a )    * χ¨ + m − − χ=0. a3/2 (t)



2

2

 !



2a

4 a2

a ¨ a˙ 2 ∼ 2 = H2 . a a

   4    *   !  #$ " . )   *       M! ,; ! /C  ! :    J )  )  *  7 ,  m  , ##-0   I)    *  ! 7 m . H2  ! ,  C !&# !    +  7     2

2


   [f       X
 $_ X [+\  Z _ :\r[  [+d   nd c_ k  Z    X c]

 4  : Z_   

. W   & )    *




! 

# ;,   #  4   7 ; C !  m2  H 2  *   ! -:#( +  ! r(:@ + H 2  !+4  )   * ! /C     4      !    χ = const·cos(mt+β)    . ,:    $ .  "  #   m2  H 2  4, C   F!  !&#(!&#+   @ 0$+ β  ? .,+  # 

)   !r*  ϕ   +    /    % , J  #  4, . !  !&#(!   (# !   @ -V2 ;, ! +  #S;   !  !         :    .S!    !   #/ 7 .  + J : J ;  r($#    7 #   ϕ(t) = ϕ∗ ·

cos(mt + β) , a3/2 (t)

∗

H a˙ · m−1 ∼ a m

! ) "  #(-V2 !  # -:#( m  H      H)   !r* ! /C   ,:#(   .>   

  0 !4C  ;  * +  +:@  4+  !  ; .      ; *    ; G!   ; )  !r*  ;    M)  *  -:#(  ! #!    %+):  ! .,: !   



S=

%

Z

4

! #   4  !!$#   

3

d x·a · 

E=



1 2 1 m2 2 2 ϕ˙ − 2 (∂i ϕ) − ϕ 2 2a 2

., , 2 !    !

Z

3

3

d x·a ·







.

 S0$    , G   

1 2 m2 2 ϕ˙ + ϕ 2 2



,

  ! 7    ! , +  +  7 !  +# - D# >  ! .   %   !    #(#    , +J  #(-  ?#    ; m       ϕ ∝ cos(mt + β)  % !   @   !  ! +J : J   7 ;0 #     ;  ! ) / 4$ 

  ! * .+!    S0$    , !S0     , ) +    a   4, C 5    *      +#    .     . ,  !+; ! $0      +- (%- D#      3

a (t)



1 2 m2 2 ϕ˙ + ϕ 2 2



= const .

)   "r*

L   .  /      !   #S(): (  +# a     ! 7  0 S ! G   )    !*   !  !      4, C 5  7 ;>. #   S0$    )   "r* !      #  S0$  @  -a       #    #$-32 ! 4
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+*  ,%""' ! '     $'!  $% -2 ?(,%'8- '+-2' '%' ,   '8$  ' (,! @ '%-   $;# >%  ,  '   $  $-  >' $%  $6 %( : ,-     $ ' =!  # $ (#>($#1,18 (-  <     #     ' $($% ,6  ' -  .   @  -2#  '?@%' -%  -4 $%$% +-  '  %"'"-% '%(,1  =(       -% $ ($% , 6  ' - *(-2'%'% - '%-2'%  4   $ ,- ,#   "" ' '' .  

-

 

!






Λ 


* . #+D-E     F.     ;( 5 !   . I    @ !  #D   +)   ',: o.-   7 0M C ! 0  U42  7 C   :   .F    @ !  #D#       ;    ! )  * !     2 δS Tµν = √ −g δg µν = ∂µ ϕ∂ν ϕ − gµν L .

) 
r*    J ;"   @ * -:#( +  +  ! ,:"    J )0!

!  !  0+)D  @ 0 ! T)    ,!8  #   .  /   g  )V (ϕ) 
r* #  7 !     ( # -32   7 C   * D η :   #  7 0 +!    



µν

*

C ! !  



µν

1 T00 = ϕ˙ 2 + V (ϕ) ≡ ρϕ , 2   1 2 Tij = ϕ˙ − V (ϕ) δij ≡ pϕ δij . 2

+  7 

)  .&* 

 ! ! P#( 5 ! )   &/*



0

V 2 ϕ˙ ∼ 2  V 0 · ϕ , H 2

   ):#(   ! ,    $ ! 7 #$+ )   *  :        V ϕ ∼ V    !&# . )   * !  !

)   * J )0 @ 



7 0!  



0

S   $  D    ? C ! !  



ϕ˙ 2  V .

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)   * 

ρϕ ≈ −pϕ ≈ V (ϕ) ,

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m2 ϕ2∗ 1 T00 = · 3 , 2 a (t) m2 ϕ2∗ 1 cos(2mt + 2β) · δij , Tij = − 2 a3

 ! 7     ,D. $  !    .     0    

    #,

  

+  5  7 .  #  7   # ! 4 .!  < Hm






m2 ϕ2∗ 1 · 3 , 2 a (t) Tij ≡ pϕ · δij = 0 .

T00 ≡ ρϕ =



+"J  : J ;

)  1 * 

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 + + ,


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Y





>LZK[39 >HA = 5872  B HB 

N4B 5 7  ]\   \ #^ \   *    ;  ]\ _ a`cbed f `g_ C   4
Mh  Aikj # 9l h +)khk\   )  Ad !  J kbJn  l o     _   ]n  l ]pA\#   kq  

#]\ _   sr  6ϕt m   .,>h ϕ  # j t  ]p]u d #, bKmq h .,>h ϕ(t) k\  f$ l 0  V v (ϕ) l h s\ w k_ x_ 9l    v  !+  u$  h zyJ\ {d  _  ]pf$ l k_  k\ 9l ,0 w ]ikj t   ., ϕ  l  ].\  l   'k\ #  \       #   6t) i u |l   u        u _ ]p]\ r  _ - l}`R~ b  r  $ >h q    * o  ]\  Cj  d   _€` ,+ k\ 9l ,0 w   !]p   o* \ `6l     k\ l   ,,>h   * \  w a_ _ l    v  +  u   h d $ $      {` \#      6h  h l >h + ~ >ha\   v ),>h k_   cl p ≈ −ρ j  _  m    n 8y  d  _€` *    Wh l o      i   \ #^ \Rh  h  `  ]\   _j - l  _  \ `b ‚ ko 0 l *0    ]iƒ d  \ v        _  +)khk\   ,    o„lz…_j k\ , f ` - hcf†\ w k_‡_ 9l    v  !  u$  h l o      i  ˆ\ ) ar`  h  >h  a\ 9l  …    p `l _ t   ]iaj t  k\ ]\ _ c~   _‰ r  6t   ϕ v #  u o + l  V (ϕ) h ,kh  h     u _j    V (ϕ) ∝ ϕ |k|  - ^   _     "  vl  ` -0  H_ 9l    v +  u $  h a_   cl m     n '

y # , iP~  ` 0   Ai†d  \ v   *     o  #     _  k\ l >h   h    # ]iAbŠd  \ v  +)khk\   v  ,>h jk` \ #    H\ clz_    k_    cl




ϕ

ϕ



k

8π ρϕ 3MP2 l 8π = V (ϕ) , 3MP2 l

m   ] n

H2 =



vl N _ u# , iAr )  i m    n    _ p  \ ` h m     n m   ]n  , ` t _ 

‹ 



ϕ  MP l .

m   Rn 

 AiŒ` -0  _ l    v  +  u$  h m l >h    u3f   kq )0 n   v  l  Ž h tk\ r Aut o      j   _   kq )   _   \Rh     j t   r  6t   ,>h ϕ l , w  p]u ]i k\ {d  _ A t kis_ ) j V (ϕ)



V (ϕ) ∼ ρc

n  > h      v  m  * ) i        v  n   aq )  d  = 0.76ρ m  r ]  t 6t  A jSj Vt (ϕ) p u ]i tk\ r Aut o  0! k_    k\ k_ \ j  -  ` t  )  kq ) l , w  A  \ p  `  h V (ϕ) = m ϕ √ ρ d* , m. ∼ 10

c

2

2

c

MP l

−33

 

!!   ` t

V (ϕ) = λϕ4



]pf$ l a _ 




Λ 


λ . 10−122 .

" , k_ ]p]\ r  _j cl h   Ad ! aq  \Pp    , i + k\   iA_  d  r  

t  f k\ 9~ , w  hcf% o   Wf+)khk\   v .  kq )0 k\ {r  6t  hf , o  ϕ&M        _ _  l  o  ]d  kq   `„w   j t     a\ kq  Nl  A`   u k \ # \ & _  ‚Ii]b +# _ ,0 v t  f  Pp, b8l  ko  * - ` t  ]d  kq  ! W~  A^   l >h ]…_   o‰d  \ v  - 2 /43 0/ 2& /  At  u _j  d  _€` d , bq h *    o p  =k−ρ \A^0 _    t  h j Ap ‚ Gv ]\Rh j  ƒd , bq  *     o    _  ~ 0 v t   o .  Sh   o  Pl







   & 
             V (ϕ) = m2 ϕ2             

       w                   p = wρ   

                         
4     

        w   ϕ  =  ϕ 0w = 0.9    w(z) ϕ 0                    2 > z > 0
+ r  0                Ω ≡ Ω = 0.76                           ϕ         Ω = 0.24         



2

.








Λ

M

   & 
       ! "      w0 = 0.9    

            r (z)                                   
 #   
ph
                  



            Ω = 0.24 Ω = 0.76    $  

       Λ





.



M

Λ

]_  k_j t    _  l >hf  ]d  kq  o p `l}`„‚  *     oCj Ap ‚ ƒv #]\Rh j    t  h   p `6l>`c‚ v  *     o "  _ , v t   o   6h   o m  _  Ap `„wHl    *  l   }n  Ap     i   d     t  l >h _  l  oCj  +  ]\ku3f !  u$   k. \   f$ l     \ kq    i  u _ r  6t  h _   aq ) V (ϕ)   d  _  ` t ,kh \#^ ϕ(t) \   *     oˆ_  w   _  ]i h  w    _  

.

   & 
&%         4   ϕ = ϕ0  MP l           ' "                 V (ϕ) = m ϕ2 + ϕ     m2M   (          m )      V (ϕ ) ∼ ρ2
+ r       (   Pl  0 c   (       #    





2




y  l t \ k…_j t   _  l   ]d  kq  'j ]p ‚ 8v ]\Rh j  5l  b  \ ^  h k\Ap, ~ _ u  _ ,0 v t   o .  Sh   o    ^ i . l \ r`c_ $ b Hl    d  o k\]p, _ u   l  t#  +*    k\#    _j ]t _€`     Wh!   _ , v t +Ph   6h  Wh dm  \ v h # `„`c_  n \,   a`  b  a\
   _j ]t _€`   ) iG_ ) 0   Ai ]… _   o d  \ v  ,*  Ad ! aq          \]u o  k\. _  l   Ad ! aq   l  b  3 1<-(-<1 2    ]\ o  k\. !Ah r    - & - /.6- ( &0.6I  l pA\ P\ _  \ !+)>hk\ ~   v    a q )  \  _   v  j  _  l >hcf  Ad ! aq  k_  h  k\#   _j t   ]p   t $   k\ ,  i   A\ _   Hr  St   +)khk\   v ,>h  *  r_  w  u o  P~     9l  ko k\  l  b I_  l   - l h ‚ v ,>h j    A\ku K_ u  o t# #k\ Aq ]p  `l k_   





  




  
     + + *  51
7E 1F1N B 51

!"




,  _   \ k_  l  ar 0#! _  l  o   ]d ! kq  '*Š_  l  ,kh  * up \ _   kq ) ! >k h \   v  , >h" cl  

M n+4 , V (ϕ) = nϕn #! n > 2 * -         a(t) ∝ tα

A \  _  \x\ r_ \    I_ u t \         k   u   

  o cl  o l 'j vl m  
]n k_   „l

vl



l_

M

 k _ 

*



  q ) i  

ϕ¨ +

u o A\ _  \   *\ l  q    ~ j` \#    ! >hk\   v  ,>h

3α M n+4 ϕ˙ − n+1 = 0 . t ϕ

 -  l h ‚ N\ ^

m

 

  $





m   A n 

# 

  } n 



m   !Rn 

ϕ(tr) (t) = CM 1+ν tν ,

vl 

- 9 l h ‚ v  m   $   n



C = C(n, α)

h a rN` \#



 

m   "Rn

2 , n+2



h m   A n  

      
 #        !Rn      (           m  ] n
   C(n, α)  m     m   !Rn
 ^     !]n h ,>h  h   \,  ]\. _ -   St) i  u o_  _  t , ϕ Ap u0 m _ ai ^ Pj t _ \ ^   m   !Rn  d  sw J_  _ Sj ϕ < ϕ (t ) j %+  u b‚ Ph    .

  &

 k\  l > h

ν=
















i



(tr)

i

F (ϕ) ≡ −V 0 (ϕ) =

k\ ] u*^    +  u b‚`b   `+  \ ^ _  u ]\ _  Cj 

 

i

M n+4 ϕn+1

m   !Rn   6 t) i  u o 

i

#-  l}`cb‚  Œ_  ~ 


F (ϕ (t)) > F ϕ(tr) (t) .

y  d   _€` \ ^     6t) i u _ r  6t   _ ϕ < ϕ (t ) l  v   h  - l h ‚ \ ^   m^   !] n P j  #Ap] \ 9 l Sj \  ^  #    k v  ϕ \ > ^ ϕ  (t  )  !+St )u i $  u  _ h r  _ 6 9t l    _  Aj t 2 ‹ _   - 9l  h r ‚  6 t  \  S~ j t  s\ ^   m   !Rn h,>h  h   \#  ]\ _  l     At  Ž^ \#+ _ ϕ,  St) i  u3f l   u3f \ ^  h   \ _ h  h  m   !Rn k\ p, i ^ f ]\ _  Wf  d , bq h ,>h ϕ  ~  `„ ‚    `  Jr #      6t) i u3f l   u3f  u  hˆ\ ^   _ m   !Rn  >h{\ ^  h m   !]n  k\, 9l   m ]p r  6t   (tr)  l ) i  o ^ _   ` + _Un (tr)

i



(tr)

i

i

i

i











ϕ˙ 2 ∼ V (ϕ) ∝

1

.

* cl   j  ~  \ ]u3f j t  , d , bKq   \ `  - 2 Œ\ w k_ I_  l    v  +  u  h   ]\ _  _ _ 9l    Aj t _ *  ~   A\ku3f j     # ]iˆd  \ v  ρ m  _  m   .&6nn ` p]u$   ϕ t2−2ν



:)   > < '%  <  > $% ='  '     $  ' $!-4 @  <  >'8$(-2 . 

"

 





Λ 


 ]iHd \ v  sl  _  \ `cb ‚ o|_   \  m \ l  q    ku }n  .- l  h h ` p]u$ ~   >h ,kh ϕ  )   o 0    

d \ v   r \#      +#   #   i  Ph l , H 2 ∝ t−2 %]\ _  _ +# t2ν  * ~  \ ]i

f j !  , i  ` a ∝ tα j , ` t _ 0 

 

ρϕ ∝

1 a

2−2ν α

.

t  u$Wh #   P ^     !Rnj k_ _ - l}`cb‚ ]u\ w   l >hA\ _  \ hl ‚ v   d $ $  

   N` m \,    !  6h  h - 9l h ‚ v )>h p = w ρ j

wϕ




ϕ

* , iPr $,A^  i m   "Rn

wϕ = −1 +



j Pf$ ~

ϕ ϕ

21−ν . 3 α

\ r `  i     _ \  9r l 0     k j ) ` t  _

m   
Rn vl w * P\ _  \ ` \#   h !  6h  h l  _  \ `cb‚ o_   \  m w = w = 0 l >h \ l  q   ku Cj !         n  " # k_ ]p]\ r  _j` \#      6h  h  9l h ‚ v  ,kh r #    ˆ` \#   h #  6h  h l  _  \ `b ‚ o _   \   \   $f  ~   b8l  k. \     h   ,    t  k  \  p )   ^ f  \ #        wHl    _  l ‚ ]_ 

k_j 

i n `     6h  h - 9l h ‚ v  ,>hsp, ar    ` \#   kb !  6h  h l  _  \ `cb‚ o|_   \  'j w ≈ w  wϕ = w

2 n − , n+2 n+2

1 3

ϕ

.




   &
 

            "  "'   4         m   !Rn       m   &Sn /            m   
Rn   # 

      # 
  &
 #     )     m   !Rn        V 00 ∝ H 2        

                        
     4                              
 ^   m   R! n  k\# 9l     ]\ _   j   vl  0   Ai d  \ v   9l h ‚ v  






.










,kh# … ‚I…_ )0 ./ \#   kb  0   Ai]b d \ v  Hl  _  \ `cb‚ o _   \   P~     i  u o 0 l - 9l h ‚ v  .,>h , a`cb +0    AiNd \ v  \# ]…Sj  +#  o~   _  _  St   sl  _  \ ]i   _ %- 9l h ‚ ,  - Nd  v \ ^   m   !Rn  ~ \    … p]u ]i  k\# 9l  ]u _ 
 6t   )>h d  K_  _    o„lz…_j , iAr $#P^  i  _j t   



V (ϕ) ∼ ϕ˙ 2 ∼

  

y 0   Aid \ v  ˆ_   \

" r  \ p $  h



ρϕ ∼

ϕ2 . t2

k\ 

d  _ \#   

ρM =

ρϕ ∼ ρ M

ϕ2 , t2

3 2 2 M2 MP l H (t) ∼ 2P l . 8π t

, ` t k _j t  %  -  l h ‚  , 

ϕ ∼ MP l .



6t



 Hl  _



\$ ]i k \


  
   

"&

 b8l  j      Cj  „l   j t    St) i   r  6t   )>h j f  h s_  w  p]u Aixl     At   k\ ar ) i  u _jl ) w  `l #,  ]\Rh AiŒ` -0 kb  ϕi  M P l \# A…Sj l #, i   ! ]\  6t    ]u ,  h AiP~ * p,   rl   ]\ _   , h !  A^   m   Rnj    d ) bKqϕ(t)  \ f$ l  s\ w k_ _ l    v  +  u  h j

h ϕ  d ) bKq h *     o * Œ\ w k_ `  ]\   v  \#^ \  h    r `c_   h j   \  l  \ f l  Kl  _  \.$  h _   \   l  _  \.$  ab - 9~  ,k h     ]\ ]\ _ h  d  v  d #, bKq h *     o l , i     i  l h ‚ v +    t   h  sd , bq  *     o   r # h ‚ o   ]\ _    _ )0 v t !+ o  ~   6 h   o €y  d  _€`  )  \ ) i  o h.,>h  h  r_  w   ]i . l  \ l ]i'   k\ ~  \ v a` Aiˆ_  l  i  9l h ‚ v  ,> h' ` A…_   _ ,0 v t  f  Ap, b8l  ko  S8`„‚    `cb ~ ‚  Žl   u  + b t b  #    _ ,0 v t   o   6h   o m  sr # h ‚ o   ] \ _  >nj |_  l   9l h ‚ v  ,> h # _    u   &  q } u _ \Rh l  _ l \ `v f _  l ~   o  ]d  kq   



.




  & 
   "       n         M  4                                                  







 




 

&
  %      % % 



 

    



 &

 &

 

    

 

 



 



\Rh l $# _ u p `l _ \# _   \ $ ]i a\ q u j k\  f$ l h ‚  Œ\# ~ ^ \„h b‚ o h *     oCj +  ]\Wh r )    v ]\Rhkt o  r_  oCj   6h ‚ o& ar \ r  t ~  u3f  r  k_  l o    `cb‚ f _ wHl}` Ap o t# 

kq   # _ u `  cl k_  l ) i o ^ _j   ~ \ ]i k\  +  h  r  k_  l o    ao|_ wHl}`Hd k_ t, 

aq  _  _   v f%- ` tPhf]u*^  _ \# ^ \  h *     o'j  #t   _  w 
t   ]iaj t   
+ wŽl u o_  _  A\ _  *    Ph  Wf$ l  h   \ _  l   _ t   _ \ #  #   €y  d  _€`  _    l ]p  h \Rh l     u3f !  A^  ko  ]\ _€` g\ #      o  \ _  l   _  Cj    ]\au _ u k\  l k_   #  6h  ‚ _ \  rl  !     ]_  ]iaj t   j +#k\# 0 j   p,   9~ \   u _ Œ_  _   _ h )>h b  h  Aj   vl      PhŽ\ # q h]u3f l  r \,    h m#" ]u _ ]\  w   %h $ n   #     u  \ _  l   _ t   Hq r #  *   u  ]p]u t r , >h +b   r u $q b   ]hi , r  u _ l >h +6t      v     h  ,+ o  A`   _  _  ]\ _  Cj   vl  k\   $f  l  Au3$f  l r \ #    h j  k\ l  ]* i  k\ #)   \ #      v  k.\  q !    v      v    * \   o *     o   ‚     W$f  l  h  ]t ai|v ]\Rhkt _ +0    _ !  6h   'j \  q !u  ar_    _ t -0Nt , 

aq   A\ ` ^   _ $ ~  #Œt         cl}`  k. \   h b ‚ f  h  Ap]ut  u3f ` -0# hf   k\ k_ \ j , _ u ]p  `l k_   #]u3f t   j !fa \  r9l  &&]j k\  _  \  A` \ T & 100  d * #     „l}`  \ , q   ar_    _ $p A\ ~    v Žt '&>h  \ _  l   _ t   v       h  #  o    _ u  ]p$f  l a_ %#  

l >h + wHl  v  

†t # 

kq ! o f a_ t ! ko   aq ) µ 5yJ\ d   _j - a_  h \ # q h  „l  m   &Sn A + A + · · · + A ←→ B + B + · · · + B , f   kq c~ vl A j B * \  r  t  u 

au t #  kqCj   l >%h        `cb‚ f f k_ t  * )0#%  \ _  l   _ t   _ \ #     )(  k\ # 9l  % o    µ +µ +···+µ = µ +µ +···+µ , m   ]n u  l   r P\„h w  Ph{t # 

kq  j * , j  a\ k_ \ `]j " b p  o \ # q  Cj}vl x` t #   `  $f   hˆp]  vl  a_  h  r_   w   Ai  `  

Ai l  ,    i  u + o      m  k\ k_ \ j  r_  ws~   \ ,  q h e e → e e γ n 5y  d   _€` r` \ #   h m   An  9l> ` Sj t   f k_ t  ko 

 a \  h w

 

1

i

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h #   P^  h m   ]n \#! _   \ k _ \ #  q kb  * + St    €l \ `cv  v k\ k_    d    \ ~  r  \    o P\ku  l $       y  , i  `

kv >h q 

 

e+ + e− ↔ 2γ .

µγ = 0

j ar  r_  w    

& #  o \ #  q   9 l>` Sj t   µe − + µ e + = 0 .

t # 

kq)  

t ~     j t    #  Žw    A^    a\# l   l >h  b p]u3f l \ `cv f    kqCj !+) i  `Gl >h  % f # vl  k_  h # r_  w   Ai a\$  kv  \ $ ]i       u * , k_ ]pA\ r  _j f k_ t !  ko   kq )  b p# o   t# 

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kq u j  r h   _€`  ]p]\  u _gr  #  _ 

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µu =

e

µ

µ

τ

τ

1 2 µB + µQ , 3 3

D dV   O E
 ( EX
 ?@(
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µ e = µ Le − µ Q ,

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A

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l i

1 (2π)3



1 (E(p)−µ)/T

∓1

.

p p2 + m2

d  \ v h t # 

kq _ # u m j T *  _  \ P` \  \ 9l u
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kq  _j  l t  h b ‚ k_ h    

 

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*

m   „ n

m   ] n _€`  m   „n  W~  t# 

kq  _j k\# o  t # 

]\ _€` \ #k\ 9~

 m   n .  w  u o k\ a_ \ j]vl  k\ k_  k_   \ _€`  m   nj(l  … k\ 9l   \  h 

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B

1 (2π)3

−(E(p)−µ)/T

3

µ−m T

2

p − 2mT

m   ]! n vl +!,! \ ]_ \   -% G_ u k\ 

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%*$Av" l   \   Av"3"!  ^  h m   ]n ! 6w  }i g  md "!Rn  \ \     %  t  kA`4r  \  !!  u3f "%% h  aoNl  -]v" 

  t   kq   k\ k_ \ `]j#l  h %%kj ni = g i

Z

3

f (p)d p = 4πgi

Z

q f (E) E 2 − m2i EdE ,

i

 l# h  o \     o \ -

g γ = g e− = g e+ = 2 ,

gν = gν¯ = 1 .

5 r h  ]v*,

cl  J`-kq  \  k\ l   h m   „n6 9l>` Sj t % \  r %]i t -  t   kq   t  

kqŒl  ]v*

   l  aq 7 p
… _  r  

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t   kq; \  o _     < \    ab=  _ k_ t )]_ t  

kq   k\ k_ \ ` k\ T & 1  d  l# hgt  

kq> p  \   u _ r  \Rh l9A_ m k\    f  _  \  A` \  f d!?-  \     

-  \  }n k_ _ m _ k\  r9l @    ]n nB − nB¯ ∼ 10−10 . nB + nB¯



  

          

   

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  l  …% h  l}`cb‚ a_  v \  #]_  x` aq  \  k\ l   h Z Z q m   @
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kq ' k_  `> i%  _  ∆S

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i

i

i

3

i

2

2 i

2




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1 ∆n = vz f (p)d3 p∆S∆t , 2

vz =

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pz E

k\   "k q h * \ %

t   kq u   % i j  _ 6w  >i " h r    _j t !!0}i% 0#%

  t  

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z

∆pz = 2pz .

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pi = g i

Z

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m   %@ @Sn ]i

 

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T  mi

µi = 0 .

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m   @
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m  
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 -%*] o$A _     u \     π ρ=g T , m   @ Rn ρi =

E/T

∓1

dE =

7 π2 4 g T 8 i 30



−

4

i

2

vl

∗



g∗ =

X



30

gi +

 

4

m   @ „n

7 X gi 8  



d  "

 - t )    o/"% p l u   u\  w   m   @%@6n5l# h l  '  h Ž` }i  \  \  h 

 !"]_ >` t  k _  ?

cl mT

gi pi = 2 6π

Z



mT

E3 ρi . dE = E/T ∓ 1 3

*   k_ pA\  r1A_j„` \      *%! h  h `}i  \  \  h  

%"]v*? ‚ * !  k _  u\  w



cl

 

1 p= ρ. 3

m   R! n2l# h  )  %

t -   t   kq2`}i  \  \   h  

%"]_&>` t  k _ gi ni = 2 2π





m   @ ]n  \ _€`

l * i t    Nr   t

Z



E E/T

2

∓1

dE =

  

Jl}r   ~ J`-kq  \   

 

ζ(3) ≈ 1.2 .

gi ζ(3) T3 − π2 3 ζ(3) 3 g T 4 i π2

−

 Ar

\_



m   @  n m   @  n

y 0 }iAr ` h i \   !  _ m   @
]n9j m   @  njP_86w  o

 \ 9l a bKb d  \ v ab t  

aq u  `}iP~  \  \  h 

-

!"]_ k` t    Ar 2.7 T −  ρ   @
]n ≈ hEi = m \

_  n 3.15 T − i

i



  


         



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e

e

e

γ

γ

e

µ

4 $ -/52/ @ xyJ\$ _  \ku   o & _  & %!" $f l $  v \  _<_ ]_ " ! & %%*]v" \  &&%$ v $  h q $$ / 7 _ "  $ A`l u d! $f8"k\ Aq  *"% "k\  "  \ q $  &  }i &au α ≡ e /(4π) j  b8l  $9$ r \  r_  \&ku3f *% p]\  w '&$ o? \  r` " 0>` t   _]q (& a`l# h  _ m  p]\  &    \ _ h _  wŽl>`2"P`6l  \'&%$ah _ $ l & ]o t  ( $ q u n 2

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µ

~

e+ e−

 t'&%$ah * α / P~ "  d( $f:"k\ ] q  "*% 2

Γ ∼ α2 T .

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< t%$ u   h m _0/ \  rl   % / / ]nj t %

H(T ) =

T2 , MP∗ l

" 0k` t%$ _j t % \ l  &  f l $ % h  \   & %  $$,"k\$

 d . *    h w  ]q '&   \  p -   l# h   p]u3f=$ $ }i &ku3f m q  %! u3f n Ar  $ _8l  o-(% $ o/?> *   $ _  p]\  r1]_j "k\ l " 0)6w (&$ 
(" )%!]_ \   & %  $%$ l  o!% $  }i &  f  \  ^   u0" 0 &ah% W~ h,&  "k\ - h w '&%$$ˆpA'@ }i ^. o t  ( $ v* \Rhkt o !  l $$ d"%0> bKq $$    (&& ]o p 0 ' % t%&    l}`  v*%% \%$ ]i   $& $kt "]_ \   & %  $$ j/  /  \   & %  & A _ \  "k\m  l   (&$$B"  $ _ " `}i!  _0C  p `„wŽl '&$f$ _ $kt "]v* \   & %  $ah j/  / \   & %  $ah _  wŽl}`+" \  _ $t  ~ ( $ qCj &  l) "a\ %%l $ Ai,-l  }i & „n!/ 4  *_8- \$ _D '" \ i "k\ l   &\%  h $  $ (%*] v" \  r \ w '&%& ] v"x v  r  j%] vl  J ` & k q $h \  "k\ l   (&$ah:"k\$&%$ _   p '}i]qk_  & %%" $ o $ l m 2/  nE/  d(!]_ "k\% l     u\  w (&$ l# h " )F & %! $ t%$   t  ! $ q $ _ ' , $ l T  α2 MP∗ l ∼ 1014

 m 2/ H@ GRn , IKJ _L2/NFI4bM?@568>cd/NF>I43:92cdI4145\=6JK/!M6/ 8:9@;234/<;25\=6IHN E484I 3:I4b6J4IL?@5\cdE45\8:9@?+} 8>9LPO; 9Q34_i;2/<E48:/2=@7LJK/5\c /+ =J}'QFL9@?@]e;e8:9


mi T 2π

3/2

µi −mi T

 F 

    F   = 
    '  %  " )- & %!]id & \ v $ $ $ l  0 '&%$\   &ku G

m 2/ @  n

3 ρ i = m i ni + ni Ti 2

$

m 2/ Rn   p]u #7 t  $ l &   _8]v" &  t    jPv  r & \  h $  $ (!" $cft  ( $ q  " $ u    % h ` \  P~ &(&$ _&"%!% h&$h p = 0 jP% t%& %(Ai]b‡l9 "  "k\  !% O(T /m) /
   
       F    (          -H(  
 ' '    .   '     (     '   F     8     %  '           '  )F'     H3     (-  F  pi = T n i  ρ i .

∆n = µ



 

 

T2 3



2 / @6n

  H ,  2 / An  µ    '  8F      %D'   F'         '    $      '      (   ,      ( F  F µ  T
<   r  & F          ,F'     F   5      3 %H    '     3H     "!#$      #F    
&%   m = 0  % F    
   * H(   

 !  FF      Jµ  
    )(
* +" T  '.% ' F    '  #F '       %      '  '$F( (3 " -           F- (     3,  (    3  #  F    3 T = 400 -     (D  '   0D%  ' ' F'   A       (  n , n , n , n     n , n , n , n   .0    (    F  )(   )F    3
3 %  %    %
   F/ T -%   u d s  '    ( =H      E     D8'     "    (          % (F  '          3  ' F  1 ! $    '     * " *   )            (  O     1 !23(    '      '  ,           (  (%%
 ∆n = µ



T2 6 




.




B

Le

3

Lµ

Lτ

A @ A 4/57678,9;: =) @A=B 9 @ 6DCE8,FG,519 BDAHJI 67KFLCE8NM=G,OP - Q 8S9T6 GS9UV5 Q OW81X R Y [Z X

Y 2SZ  X y

U3:9

Bn

(

Q

CE`D9

Q7a

KEbDO Q

O[cd6DGS9

Q1e

@ a

K

Z

B 6DF g A f 95 @1h 6i Q Q1Q

Z

Q7e

@ O <

x

0

\

dz = ]_^ 2 . +1

z

n

∞ 0

Z

∞ 0

\

(2π)2n z 2n−1 dz = Bn , z −1 4n

z 2n−1 22n−1 − 1 2n π Bn , \ z + 1 dz = 2n

B1 =

Y ',Z 7 X 4/576jbD51OW6 > 8SO

∞

1 , 6

B2 =

1 , 30

B3 =

1 ,... 42

B

Le

Lµ

Lτ


3H% 5
+   '  




U3:9

Z

ζ(x)

(

∞ 0

A b1O < @ 6 < i B G,O :

D@A=B

Q

C#`D9

Q7a

KEbDO Q

9 @ 6DC#U A< < A JhR@  `1616jbD5 6

H

0

z x−1 dz = Γ(x)ζ(x) , z −1

\

a

Q1e

@

K a

K2bDO Q



9 >@A=B 9 @ 6DC a

ζ(5) = 1.037 ,

O[cd6DGS9

Q

∞

z x−1 1−x \ z + 1 dz = (1 − 2 )Γ(x)ζ(x) ,

: > 9=G A JhR@  `16DC 6 < A[@A i BA F=G @ ζ(3) = 1.202 ,

*

Z

h `D9

Q7e

> OW8 A 8

0

6DF e

F8,OWPDF=GS8SO <

a

ζ(3/2) = 2.612 ,

Fb15 A 8,9;:

n

OSGSOS51OWP 5 A 8 @

ζ(5/2) = 1.341 .

618SO Q

Γ(n) = (n − 1)!

Q1a

K

x

< O[c @ O @A P1GS6iR61FLKO=:DCE6 >T>@DAVB 9 @ 61C

Γ(1/2) =

68SOSFb1O

-



&


Γ(1 + x) = xΓ(x)

√ π

X

  "!#%$ '& (*)"+ ,.- /10 ',%,.-2435$687  9 99 , 9& ,.,: <;& >= $@? % & % &A'B DCE  ;$&  E$F * $GBHB  C  % IC$ ( $  $ ( IEJA  0 % %  IKML(& C ON "$ /P3%*$'R Q%@ S EJA ">T $@C  IE 
"C$E(& UQ+ ICE  ;$%&  E$F " $%9?   &A  C  DV $C W N X  =     '&& U =ZY " ' D?'& ,> T!DS ;$ UQ2"%U =(!  L(& C  "$$ +  >= $ ( DE /\[  "  E&$E Y F !<<   * $F "> = DCE  ;$&  E$   "  & $HL'& C  "$%$ ! ? &$     "DC % E &  F    DCE  ;$&  E$  $ /  > T X IE ]S F   "%ICDE'&& U ^" F$   F  ( $_ "C$C  X (&$  & S C'&%& = L'&%IC ^ $$ $E( ? $`; dE 

 

dE = T dS − pdV +

   % %     )  0   %           *   

X



µA dNi .

2/ ba

  "    ! %*  % %     %       0    '     3 ! %  '  %         

'L & C "$ $ ! DEJA Y $&`; i >T ?'& F  C F ( $_ C ; RQ & =@V DEdL ^ ; c S $&`;  $ DS EJE$C &$%5" &DE S EJA TUS ;DE " S >QYe^ ;9L ?IE gf & %/ & S C%'&& L'&DC ^ $ +$MF$ F ! $_  W hL ( (& $ &A.E$(B C  IC%$ ( $ E$ Y  (>Q &$ $& = & E(&W  $@?IE'&(&$IE >TUi KDE Y IE"DC US C $:; (&$'  N W  $ ! DEJA /bP L E S $G? ]Q &A.E$:B C % IC$ ! $ E$ Y &'? $  X $E$ - >TUi KIE "DC >^ & F DCE ;$& E$ $  2/ ba j; S  kY F L'& C "$    lL ( (& $ & >=  $F $& >= / @' ? $ L $E S ; >T & "DC =% $ - L'&DC ^ $$ Y F$ F ( $_B$mL'& C "$%$ $G* B " - & ( @E Y E N S  2/ na

%





 ! )%   % *     ) %

ρ≡

, n≡

 ?7  ; $  IC(&_$  /- C   '& !%  2 / V

naoY

i

, s≡

" 0]S F $E V

V

.

2/ 2>a  2/  a  2/ >a P3 ; (   $ 4 A.C  f (&$  2 / 2UaoY  2/  a $  2/ >a ;  :;$  IC(&_$  )%  C   '& (%!  2/ baoY EJA "@C$GB  ;$@E   p; S W X  =+ CEq?  "%$ $,"IC %>^" &  F    ICE  ;$&  E$  $  2 / -G>a (T s − p − ρ + µn) dV + (T ds − dρ + µdn) V = 0 . dE = ρdV + V dρ , dN = ndV + V dn , dS = sdV + V ds .



 F 

    F
= 
 9   '  % @   % "!- &  V '&%$ "C%$E'&$@E 4   4%  = $ ( DE Y   $   W T!>= IK4F  ! $ /`P C$E(&$E  ^*? >T0  ( $ " !(% %&& >^*,>TUi KDE   & S C%$ $ ( DEJA;$ " 0]S F$E ;   ;$  IC(&_$  )% T ds = dρ − µdn .

* ( "IC Q E gf &  " C$E(&$ UQ  2/ -Gba $ %  = $ ! IE Y UTUi IE -% C >= E (& %  Y $ &  =% $ $ *$ E    A.C  f ( &$;  *" #- & !( $ L'& C  "$$ s=

p + ρ − µn . T

   F% (!    f & >^" "@C$EIC  C  " E F C$E &  F    S]Q C  C   $  $ !%*$    X  (!% & S   A.E$ B$E$F% " $E$)" F '&@_$    E$ Y ;   -% C >^" 2/  a P30RQ ? S  IQ L $E C   (& (!% E $  A.C  f (&$@E$*;   $ Y " 0]S F(&&A.E$  "Cj;eA ; S X D E C  ?p;   Y EJA "C$GB  ;$E < j; S W X $@E  A.C  f '&$@ρE ; p  %$  ; % F  ( $ _ N ^" $%"   i " )F & %!>Q L'& C  "$$    ?' T −  g 4ρ  2/  ba s = = 

I CE$ 3T g T − 'C   &$    L $  A.C  f (&$  A.C  f '&$ E$  2/ @  a ;  8" )- & %( $ F$   F  ( $ _ Y EJA  $`; $E Y F !  SRQH C  C   $  $ (!" E ]S F  9" )- & %(UQ L'& C  "%$$8$ " #- & !(>Q F$   F  ! $_ -1 $F  W !   $ V:Q &  F$  (&&A = E& kf $  RQ "  C `;   p ;$%&$ _A / &   &DC   $  $ (!">= E"  &'& A %! " 0]Q ? S I E mC   (& (!  E$  2 / @  a $  2/ ba $ ?  "$ V D E s=

4π 2 i 90

i

i



p+ρ . T

7 4π 2 8 i 90

3

3

5 mi − µ i s i = ni + ni . 2 T

5 "  WF$E $@?6L !>^" A.C  f '&%$8B$E$F " $ = " - '&_$   "  E  X % &F   RQ &  " #) ( si = n i

5 gi + ln 2 ni



mi T 2π

3/2

.

Q W 

2/ @HGba $*" ']S F $E

 % E$F% ">=/ Cj;" )- & %(UQ F$   &IC%   $  $ (%* $GBlF  ( $_ E    "  C   &'&$@W " #- & !(>Q W F$   - ! & % C "C$ DE"IC  US C  B %L  Y>^ ;  "C -% &@A $ & =% C  &A  0 W !  &IC   $  $ !%* $E$ Y $`B " )- & T%!.>Q 100  _%'&$    !     $F$& >= n / *     f   _'&   $E' E !% ;  *L   % C  & %@Y--% CA0  ' W % 
&%IC N n ∼ 10 n   $  $ (%* $E$ "C$ T . 0.5 % L  / *   $E >T C  ?  E Y &DC   $  $ (!" $E$ %   ;  E$  L'& C  "%$W E g f &  "C(& T C%IF QY $," 0 &  *" )- & %!>Q L(& C  "$$ ;  K !   CE S)U= 4π  2/ -k @a T , s=g 90 ^ ; L   % $  &  'F$ ) C%   $  $ (%* $GB ( ("'&% =+*%> T% ;eA  "Cj;  (&  Y'  4U T A.F&  Y g  C E  S)> =  2 / @ baE/ <;&$E $@?   WF  A'B "!> =(! L'& C  "$$ Y% "Cj;  W X $EDDK   f & =GV:S W C 0] Q3 IC N E  ;$&  E$   Y  0 % !  - ! C  :&  F  )*  DCE  ;$&  E$  $ Y *>^.  & /- ! C  E S L'& C  "%$ ?  E  & S%>= $ ( DEJA & SGT A     B  ;  W T!> ^"  $@?'$F *> ^" "C  _% "  $ %(  K %  "  N (! && U = ;   U T C  $EJA'B "C  _ *"%@ Y !  (UQ "C  _% "*% &  !%0] Q% Ep;  (&&A'B Y F ! B

B

−9

γ

2

∗

3

∗


3H% 5
+   '  

&
@ @ $ ( DE  ! K  C%I EN &  B  ;$ !   "%!% &$%$ ICE  ;$&  E$F ">^" C   & %  $ / "C  N  j;  $ %!(>Q L %U^" "!>=(!  &  F  `$ ;&  ;     X  (!   C  DV $@C @W X  =     (&& >=Y " %*0RQ%@S2 C  E   X  !%  %    '&%& >= $@E' !   @f $ ^ C   $  _$  &%&  =" 0  Y   F ! "   ? E   E  "   T 1&% >T C  ? S  ?  E  & S%U=, $ ( DEJA / T!DS ;$E Y%   E  ;$  $_$@C SbN  !  ?   & * BGC  &'& $  L'& C  "$%$  C  DV $ C W X  =     '&& U=  F$   C  DV $C(&$ C   & %  & A.E @" C  _ "" E a!/ &  %L %>^*2 D C& KIE   *%- & UV (&$W  2/ ba $ S F KIE   &A.E >T C  ?  E Y F % B$ E$ F " $%D" - '& _%$   A F  ( $ _ $  & $ F  ( $@_ C   &A "     $F%$& $ "C F $ % " 0)gf & A "  ? &  @S/   "$ V D E !>^ ;  X  2 /  Ua ¯) , dE = T dS − pdV + µ(dN − dN ^ ; ¯ F%$ )  & @$ F  ( $ _ Y  DS EJE$ C %  &$A" C  @$ ? % ;$ ! +" 7% I E $%"  EHF  ! $ _ >/ P C$ C   & N% c  &  E C  DV $ C(&$$ "!- &  V '&%$  2 /  Ua E gf &  " C$ E(&$ >Q/ " " S%(!US W X I E S >TUi KDE S>Y V ∝ a Y " C$ L % E (N − N¯ )  " " S!(% S W X D E >TUi KIE " BGC  & % ! 
/ *   $ E >T C  ?  E Y ;   " " S !(!US W X  ^" >T i KIE  TUS ;I E $ E UQ  2 / >a T dS = (p + ρ)dV + V dρ . P C$ F$%& >= @$ ? E(&'& $  * " S%!%US W X  ^* >TUi KIE  $ B  C  % I C$ ! $  Cp ;RA  '  % %C  oN V @$ C'&$%    (&& >=Y "  L % E S "%F & UV '&$%  2 / >a & Snf &  "  &$ E  UQ?   


3

  dS a˙ 3 = a (p + ρ) · 3 + ρ˙ , T dt a

L'& C  "$   * " S%!%US W X IE > TUi KDE /  "  E&%$E ' "DC Q ?   & $!  C$ N  & & S>^*=?*saBGC c  &'&$  L(&IC ^ $$  C  oV $C @W X  =     '&& U=  E /@S C   &(&$  %/ @%@ka a

^ ;

3

$," 0]S F $E

a˙ ρ˙ + 3 (ρ + p) = 0 a



2/ na

a˙ s˙ + 3 s = 0 . a



2 / 2Ua

 j; %   RQ & Y " 0 &  L'& C  "$   " " S !(!US W X D E >TUi KIE * B@C  & % 

Y

2/   a 5  ZY%  C$  & &A = ?   & * BGC  &'&$ L'&DC ^ $$,$E(  ]S F  9C   & %  & >^" C  DV $ N C(&$ "C %( S W;$%& DC"C  _$W4  &
"Cj; (  0  ">T%U= ?   & * B@C  &'&%$ L'& C  "%$$  " " S%(!US W X IE >TUi KDE / _'&$@E " )- & %(UQ L(& C  "$$    (&& >=  "% C%IE'&%& S W L'"  B S/ &   L !>^"<! "  N E&$E Y F %    (&&   ?  " 0 &(&  C  $ %%! A.E E$  C %%0 & % A.E $@? ]S F'&$%I E ^ ? E c -% & %P I E"%I C  S C >= 

sa3 = const .

P

T0 = 2.725 K .

)- & %(UQ (L & C  "$$ L %>^*-! && >^**^  ?   *%-!  !(% $$  CE S )>=  2 /  >a C   &  sγ =

2/  >a =GH:5.F <5\? E:/<34I:c 9
E 8π 2 3 T0 ' 1.5 · 103 −3 . 90 g}    c} ¯) (N − N 

 =GH>I <_a?@]U2/@H:5\5 ?947:9E4/NF µ ODLBI:cdI:145\=6J4I45UE4/2?@563 :I:9 H:_ J S6?@I>c 14I>=GH>92cf7:=6c A  A .A



@

 F 

    F
= 
 9   '  %     ;  E$:;eC S`^ $GB $" % F  ( $_  ! EmF$  T& =% C%$& na ;   ^ C S`T%>=  _'&  $:"@C'& T C N f KDE /\P30RQ ? S  DQ CD? S]Q  % E  2/  >aoY EJA E gf D E  _'&$ >Q " 0 & S W L(& C  "$ W  `$ ;@$ E >= E Y   F  ! $    (&& >=/ 4  ? EDC  `$ ;$ E >= F  ( $    '&& U= *%(  '  l ∼ 10 F ! ;   " 0 & >= L'& C  "%$$)" ']S F  I E H ,0

Sγ =

28

4π 3 sγ lH ,0 ∼ 1088 . 3

%A  $`; $E Y F ! L'& C  "$  $`;$E >= F  ( $    '&& >= c L ! ^ $ ^  & !"  T o? C  ?IEIC&   F$ )/
8 $ T A  "C  _ * MC  DV $@C'&$  " 0 &   L(& C  "$ ! K  CIEN " BGC  &   DQY!  E  E'& "!  ^* C gf ;'&%$    '&&   ; ']f &  T A   T A Snf  UT0  ;  >Q*  >= L(& C ON "$ =/ *   $@E > T C  ?  E Y# (  &`;  C &  E _%'&  C%$$ ^" C @F ^*  0]QUV+>^*, ?ICA   EJA (  @N  $   IE  & U T B  ;$E %!>Q W %$ WF  >Q  &  F  RQ &A0 S!)! $lC  DV $C'&%$    '&%& >= ^ $ ^  & %* $ = T o? C  ? EIC%&A = "  C  E CD"  C `;   / %A  DC&KDE   L %U= "C >T' IE9$ 10 > T%oS ;$E ! ? E g f & %(UQ IKC V '&$   C  E   B $%& 7 _$  && >=   C$$ % -% C >= F  ! $  &$ ^ $ /  B@C  &'&%$ L'& C  "$$ * " S%!%US W X IE > TUi KIE%E g f &  $ " 'RQ ? !  >Q ;   !>^"Y F !>T A %  ! $ 0 $F% (! '&%&A0 Y &8?   $  X $ -  C%IE'&%$ B  C  % IC$ ( $  $  $EJE ON C$$    '&& > = " :- & U V '&$W  C  ?  $F&A.E " BGC  & W X $E    & %% A.E F$   E /[  N "C$@EIC Y &  !%  X '  CIEN %    (&& > = $EIW % *F  ! $_A Y & DS X $T" 0)g f $  R Q &   "C - ! &A $ & =% C  &A Y $ - %oS%(!US W F  ( $_A - C$@_   R Q &A.E T  C%$  &&  MF$ ) c T  C%$  &&A.E F$ ) E /  p; !   R Q & @ Y %    (&& > = $E( %  &(& S     " )F & %!>Q T  N C$  && > ^* F$   / TUi KDE %(  K %  " %(! %&&A.E "C%$ C  oV $C%'&$$    '&& > = &  F$&  )" ? C  = & = EIC%  L  / :& C S`^ $@E$ #%  E$ Y# "C   p;  $ % C   '& (%! \ T ∼ 100 88

(nB − nB¯ )a3 = const ,

" )- & % ( $ F$   T  C$  & % $  & $ T  C$  & %/ $ ; $E Y F % - & UV (&$  2 /   a $  2/ GbaoY EJA  `

^ ;

nB

$

nB¯ c

2/ -Gba   &$    "!- &  V '&%$ 

'C

2/   a  '  % 9" %!% && >=?!3 CIE(&$(F$  '&& U= B  C  % IC$ ( $ $U= T  C$  && >=  $EJE C$$    (&& >=/ [  "0]Q% ?  T  ^    "IC@Kp; Y "C%$  p;RKIE  _'& GS F$  '&%& >^" ? &  F(&$ T  C%$  && U=  $ E N E C$$    '&& >=/  >^ ;   $! E 0)U^ $$8$EIW ; ) " C   &$  RQ &  &$G?  $ E$ D E N "DC  US C  E$  T . 1 %L  apY C  ;$_$  &&  $ " 0]Q ? S W                ∆B =



nB − nB¯ s

ηB =

nB , nγ

" ) - & %(UQ F%$    F% & !/      $@F$&  %!  K !  " %!% && >= 8 IF(&$ L !0 W_%$$    (&& >=ZY &  F$&   9 IE"DC  S C@A "  C n;   @ %L  Y $ -1 $F   %  FHT  N C$  && >=  @$ EJE C%$$ F$  '&%&A.E  L  $_$%'& ! E;"  C `;   j;$&$_@A / P3% p; &$ = ∆ " @?  &    F$  (&&A.E F1 $F$DE n - L(& C  "$$ -% & % s Y E /  2/ @  a $  2/  @kaoY Q 34/2=6?@]0M65\34568:9 4I4I  9@84I:/<3434/& 9<=.I>c cd56?@84I:I E484I\565U34I4b\J4IHL  _ 345M8:9<=\=6c 9@?<84I4;29<5\cbGFL5\=.] ;947.=6c A 8:956H O
nγ c

B

γ

γ


3H% 5
+   '  

&
  $ " %   ;  E &% =% C$&  L'& C  "$@W    (&& >=ZY " )F & %!>Q L'& C  "$$ %    '&& >= C   &  


4π 2 s= 90



E

/    US  /P C $

T

@ L . 1 

E



2/  ba

 7 4 2+ ·2·3· T3 , 8 11

 %   2 " %  "! % % % %     ! %      ! %  *   *%   $   2 @ &    @      !     0 ? !?*%  ) %  

 "

; E ^ ; ."%IC A = $ - C >= F (&A >T B -  ! US W - & $)& =% C$& @Y "C$@FKIE IE"DC US C &% =% C$& @? & DE"IC US C >= - & T ≡ T - & UV (&$ N IE  / naC "@C $ B kf ;(&$ % C 7 · 2 · 3 c  f Y ; RQ & =@V  ^  %/ 2Ua!γ/ " D? & - E $ >QY F CDE'&& 98 ?'& F'&$%#" F & ! $%L'& C "$$%C &
s0 = 2.9 · 103

5  ZY $E(IE ∆B =

nγ ηB = s

4π 2 (2 90

= 0.14 · ηB .

+

2/ @ka

−3

2ζ(3) π2 7 ·2 8

·3·

4 ) 11

*  F&  q? &  F%'&$ T  C$  & NG-! && U^",F & UV '&$ lE gf &  ( $B"DC  $F&@A'B L  IE(& %! %    '&%& >=  E /    US Gba C  $ % !%%>^* $@? ]S F'&$  /  $ ;   &D?   $ $EJA'B " !">T  E f ; S *>T%> = Co? SR QH  A $,"C$ % ;   ? &  F(&$W

· ηB

$@? 0 D F Q $@?C  " C % ! C  &'&&  N $ $@? G$ ? EIC%'&$ =  & $@? - C  "$$ ;  W B  C  V+ ">^.  DS W X $  2/  

ηB = (6.10 ± 0.20) · 10−10 .

 IC@E$&  B T  C$  && >=  $EJE C $%$ $E'DE

∆B = 0.87 · 10−10 .

  EJA UT%DSnf ;   $  C  ?p;   @-/ 2/ 2GY%q^* C F =    '&& U=  " C$    $F$&@A $E'DE 

T &1

>a

\L  a "  "  C `; @S

2/  ba $ " )- & %( $ F$     C % $  & $   C $!Y "C%$FKIE ^ ; / *   $ E >T C  n?  E Y n"  C c  E C ∆ " !  ? A    kY-  @S W ; 0 W -:UT X  ^" F$   n-  ≈C n%4∼"%s!  0 % $@? T A %%?  C $! &  ;  & $ -  C   E$ / A  $`; $E Y F % L !- $@? T A !%, F'& Q E   /  p ; ON    RQ & Y   EJA $ S% IC f ;   $  A V  Y B$E$@F * $=" - (&_$   A  C  &%& =    '&%& >= T A  $8; =(! $  RQ & , C  = &%E   A /   
  
-   (       (  0b
       u %'  .   
F   n
3   
 -0/234JK/2=.?<]:7 =.;E4N bN9@343>9N = ?@5Ncf7>14?34_ cdI ; =6/<;<8:5\cd563:34/+ J =.5 R H:5\3434/+'7 9 E4/ J48:9+4345 cd56845^F>; 9R?@I4E>9 3:54?@8:I434/ J?@/ cg} MQ 5I:cd5€!? c 92=6=6_U7LE:8456;2_a`M9€ ~^I45 T 7Lb.F>5\=6] 345N=J}l~^5N=.?@;256343:9:7:E4/ =.JK/@H>]<JW} ; =./2;<845Ncd56343:/+ J =.56H:56343:/+ /}K34J :IHN 8:9<=6E4845.F>5.H:5\34IHN  /2?@/<34/2;I34514?@84I43:/ E4/RI>cdEW}NH:]2=\956H CBA  A  =GH:I=.?@845NcdI4?@] = N J 9@J4Jj} 8>9@?@34/!R =.?/ =.? 9@?@/21434/eE4/2bGF>34IL16S ? 9@E>9L1\S ;
q

q¯



nq − nq¯ , nq + nq¯

q

q¯

B

ν,0

@

 F 

    F
= 
 9   '  %  ?    W F'&$%:L !>^" C  ?o;    &  "  E&%$ E Y F % &  C `; S 9T  C%$  & N  F% &@& A.E - & UV  N &$DE η F  (! $ " 0RQ ? S W 6T o? C  ?IEI C@& A = "  C  E C Ω Y' " C%p ;  D EJA =    - &  V '&%$ "! CIE(&& >^* ? &  F(&$  E  *"%!>=Ra " #- & !( $ T  C$  & % ρ   C$ @$ F *$U= " )- & %! $  E /    US naoY ρ m n ρ  2/ na = , Ω = ρ ρ \L   E  *  " C -% &   " C$ %U=+! F& !( $ Y $F% C U= EJA C  T!-  I E Y C  ? &%$ _  ^ ; E f m ; S ' E  1"  E$ c " C -! &  $ & =% C  &  & DS X  (% '&&  a!/  "%-!  !(! $$  CE S)U=  E(&& >=    '&%& >= C   &   2 / @  aoY " )- & %(UQ F$   C  $ %%% A'B -! & %"! CI B

B

B

c

B

B

B

B

c

c

B

nγ = 2

1 ζ(3) 3 T0 = 410 E 3 . 2 π

5 " 'RQ ? S %?'&  F'&%$  C$ $@F *$U= " )- & %! $  / baoY " ']S F$E $G?  2 /  >a

2/ 2>a *   $E UT C  ?  E Y Ω ' 0.042 Y F % ?  E &  E(& QUV :" 0 & >= - & % $  RQ & >= " )- & %( $ E  * A &DC   $  $ (!">^*   X  (!   E  ;  $ $E  E /    US na!/ *    Y >T C  ?  E Y T '@ RQUV   F  (>Q &IC   $  $ !%*$U= E Λ 
IC$% $ %Ω  ' 0.24  '&& U= c & T  C$  & A Y  &($@?   ( &A0MF  ! $_@A KDE& >= E  IC%$$ / Y#% & !  &&  *&  @$ ? E N [  "  E&$@E  E /   j;'&$% aoY F ! "@C @E    _(&   ?'&  F'&$  C(&$$ Cj;& = " )- & %! $ *   X  ^"%    X  (!   ?  Ko?o; a:! Ω   '&& U=ZY " C$     T A  ? &  F(&$W Ω "C$EDC&  &  "  C `; ! E(& QUV DE S>Y FDE   2/ 2>aE/  %  ?'&  F   kY F ! T 0@ ]QUV  F  (UQ*T  C$  & ! !    '&& U= &  B  ;$ %  ?*%(! &$$ & *   X  ^"%  ^  ?  / <;&  $@?   f &A'Bh?  ;  F $! E 0)U^ $$ UTUi  &$ >Q "C  $ B gf ;'&%$ T  C$  && >=  $ E N c E C$$    '&& >=/ %A >T!DS ;$E L S "@C >T0 DE S       @%@/    T
 `
-   @   
  %     
3H  3b
DD% >
.  I
  
3%+  (   0.5 -    T  1  
<   ?  &$0    ( 
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∆F Γeq , T



|∆F |  T .

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h B 61G 6 S8 b1O @ 9 M 9 < 9 @ G A 5 @ < Fb1OSF=O  O < X F=519L:9 F  O @ D 579 >h G A G < O[c @ ? O bDO R ` 9 @ G,5 A `1619P Q G @ SO FQ G WA @7@ 67=U 6 a C 7 @ `D9 @ Q1G,e 5 A `1619P A[@ GS6  B A QF=GS1 B A =F GS61` 6 ` e 8,9 51O[C7 ` 616?8 9;:16 @ 7 6 ` h S8 519 < 9 @ 6  F e Q a X nX¯ F=GS61`19P ¯ 5 A 8 @A X

ΓX = σ

 · n ¯ · v , X X

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 · n ¯ · vX · nX X

= Γeq

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nX¯ nX eq . neq ¯ nX X

@ 9 >=A 8S61F6DGOSG'G,OWUVO7i+F  O

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−3

3

2 / 2 na $ Γ c "  N "@C f &IE S * C %! $ Ej;  '&%&A'B "C  _ "*% p;$&%$_ S  CDE'&$  ^ ; Γ p; $&$_  $G? $F ">^* UTUi KIE  Y E f ; S4-! CA.E$ 7UT X DE ]S F   $E' %  *%- & UV (&$ $ Γ  E$H?   $  F  CIE(&$ Y &  "C$EDC Y $@? N ?  S E'& Q V '&%$   2 /  UaE/ 4  ? S E( %  Y Γ DE"IC  US CA?  FK C  DV $C'&%$    '&%& >=/  T0 $@?'$ DCE  ;$&  E$@F *$U^" C   & %  N $  S C   &(&$  2/ 2$@ka6UT%>T X   !  &  S F  = C  DV $C W X  =-     '&& U=  p ; S W X $ E d (na3 ) = ∆N · (Γ+ − Γ− ) · a3 , dt

+



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Ua ∆T ≈ Tr

1 − ln



2ζ(3) ηB π2



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3/2  ≈ 0.03

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a  F h :D6 < G,9;bD9=5 7A bD57O6DF3KO=:D61G8 bDO=:OWUV519;8 : bD57OSG,O @ OW81X
/9;U  O 867:D9=G i B G,O < 9K AW@ 6 >=< M @ 9=57U=ObD9519;: AVB 6i;OSF @ O  8 A[@1@ P @A bD5RC < O < e 8 >VA 6 < O=:9P1F=GS8616 F OSG,O @AW< 6+i @ 9=M Z 9  GS678,9 @ X F Ae < O < :9 9Si8,9F AW@DA 6 > ibD5 618S9;: @7@ P : CjM a 9  G,51O @ O87i OSF=G A =GSF;CE8#F6 9TF/G,OWP 5 A>@ 61`D9;Pi B G,O 8,9 > :9T8 < 9=FG,O < A F=F M 9  Q GS51O @DA e F 9;: h Q 9=GgG,9;bD9=5 bDO=:DF=G a A  8S61Q G Q< A F=F h bD51OG,O @DA XG,O#bD57678,Q 9;:0=G 'h 8,9 6 B 9 @ 6 H K A 5 A  GS9=5 @ OUVON8S519 < 9 @ a 6 8 b1Q O=:OU=5198 A
:/Q @A TA G,O5 e i (mp /me )3 G A je B G,OgMG,Od8S519 < CFG A[@ 9=G  O 9/8,O > 5 A FG Q A  F=9 9 @1@ OWP @A < O < 9 @ G#579  O 95 O5: A Q Q dσR =

, F  OS57OSF=G M 9  G,51O @A i , h U=O 5 A F=F=9C @ 6DCX  U3:9 U h C v5 @ OSF=G 8S6 : A e Q 3 b1576 < θA K h U Q A K25 A F=F=9=C @ 61C :D61Q G,FLC i A 8,57e 9 < C F8Sdθ/θ O  O=: @ OU=OEb151O Q1 a 9U A Q OS5 < A @ OEO  5 8,579 < C F8SO  O=: @ OU=O2b151O  9U A i A K A 5 A G,9=5 @ O9TQ18,e519 < CM @ O5 < h h : C MG,OP 8,9 6 B 6 @ a i >VAW< 9=GS6 < i B GSOb157605 bD9519;: AVQ BDA M @ Q 9=57U616 5 A 8 @Q A

∆E = 2

U3:9

E

,

M @ 9=5 U=6DC2M 9  ,G 57O @A X Q

Y+* X ; . Z

πα2 sin θ dθ , m2e v 4 sin4 θ/2

6 Z 9=579 @ `76 A @ OS9 F=9 B 9 @ 619 Y+* X ; . Z i G AjB G,OgF=OOSGSQ78Se 9=G,F=G8 h HI 9=9/6 @ G,9UV5 7 AI A 9=G,FLC 8 @1h X : @DA  O7i  A  6'8 9=5 UVOb19=519L: A=B 6 Q1e X  CG,OWUVO7i B G,O  a Q @DA b1O  O[C I 9 a< A FF=9=C @ 6160M 9  τG,E57O @DA' Q

6 < 9=9G#F=G,9b19 @1@7h H A @ O9TF=9 B 9 @ 619T5 Q79Se i @A FT6 @ GS9=519=F h b1O h B 6DG O`19 @ O FLC b1Q 51OSGSO @ e 9 @A h

me E(1 − cos θ) , mp

@  F6 I A FLK,O  9=G @ 9 B1@1h H U=O Q θ Y+* X ;
Z

.     !#"%$&!')(+*+(-,- .#/0 / A  6'8EF hRBA 9  O < b1G,O @ O8SF  OUVO#5 A F=F=9C @ 6DCi7b1576 b19=519K,OV:D9/8EF6DF=GS9 < h OG,F B 9G A b A>< i1UL:D9TbD51OG,O @ :18S61c 9G,F;Ci a i A/ 8 A :D5 A G @ <6 OS5 @ 9 < 8,O > @ 6  A 9=G Q >;@A  OWbD9=579 < 9 @1@ P8  A : 8Nb19=519;: A=B1h ibDO=: A 8 9 @1@ P @ 9TOSG @ O 9 @ 6D9 < Q 6 > M=G,OU=O OSG @ O 9 @ 6DCX : @a A O7i 8 Q F=OOSGS8S9=G,F=GS86160F A 5 U h Q < 9 @ G a A< 6i b157618S9;: @1@ < 6me8 /mp9Si bDOSF 9 h F519;a : @ 9 @ 6DC bDO h U A< 8  A :MG,OUVO B 9 @DA O  AW> 8 A 9=G,FLC'G,OUVO#c 9 bDO5 C:  A i B G,Og6 Y+* X ;
Z iG A jBa G,O < a @ 9 h :9 < Q 8 b76DF 8 A G 9U=O Q Q a a a a e CR8 Q @ O X + Y * + Y *
 > h A @ < ; Z ; Z < R h  B A < h  H / I h H I @ 7  h A A  @ 4/O C F 8 5 c 9 61C 6 X . 6 X i bDO 9 F 9L: OW`D9 : C'K 5 G,9=5 OUVOE8,579 < 9 @ 6 a Q Q Q M @ 9=57UVOWbDQ79=e 579;: A=B e 6 a

«−1 „ «−1 Z Y+* X ; * Z 132 θ 8ne πα2 dσR ∆E (θ) . = dθ dθ E mp me v 3 2 % @ GS9UV5 A 8dbD5 A 8,OWP BA F=GS6 h 5 A 8 @ 9 @ 6DC +Y * X ; * Z bDO  b1519=c @ 9 < h 5 A FLK,OV:16DG,FLC @A < A K h U A Ki @ O h c 9 @ 9TF=G,9;bD9 @7@ < O  5 AW> O < i Q A O U A 5 6 < 6 B 9=F  6X" A  OS9FU A c 618 A[@ 619/F6 @ U h C75 @ OF=GS6EKOS57O Ogb1O Q1@ a C G @ O Q 6 > 6 B 9=F  6 , O @ ONF;8,C >VAWa @ O τE ∼

„

ne v

Z

dθ

Q 576 5 A FF=9=C @ 616 @A2< A F G,9 < i B G,Ojb1 N 9 h QU M @ 9=57U=ObD9519;: QAVBDA G,O[c 1 Q a 1 Q a bDO hRB 61G  O @ 9 B1@ OS9 >;@A=B 9 @ 6D9 : CN6 @ G,9;UV5 A A 8 Y+* X ; * Z i 8,F;bDO < @ 6 < e F=G,51OWUVO  9 >=< A F=F=OW8,OWP BDQ A F=GS67`D9P+i A Q b15761O  579=G A 9=G @ 9  O CR8 Q C 9=G,FLC2 QO B7@ 9=9i @DA 5 A F=F=G,O[C @ 61K < 9=c : h >=A 5RQ1C7e c −1 bDOG,9 @ `16 A @ O/rbDDO=:=A 8mD9 @ Y @A b1O < @ 6 Q < i B G,O M=G,O CR8 9 / h O @DA i A M  FbDO @ 9 @ `16 A r & F <  X Q 4 1576 Z X D@A=B 9 @ 619J:9 SA Q19e 8SF  OUVO#5 A Q :16 h F A : A =GSF;C2F 9L: h HI 9Q P Q rD rD = 8

@ 6Q H

„

T 4πne α

«1/2

9NF=G AW@ O86DG,FLC  5 A P @ 9 < A A X C G,OUVO i B G,O  i B G,O/8gb AW>=< 9 OG,O @ 8 @ Q 9  OSG,OQ 51O < F < F 9 @ a 9 h H 8 < A F=Q F h: >VA F B 9=G#M  9  GSO8E8,9 aI 9=Q FGS8 A X m D 9 @7@ < 6 BA F=GS67` A< 6ObD579;:9 C 9=G,FLC @ 9 >VA  O @ O < @ 6D9 a @ OF6DG @AW> 8 A[@ 619J:9 SA 98SF Q  OPjM  5 AW@ 6D51OW8  6i OS5 < h Q OWP .

9L:O8 A G,9 @ 7 O i6 @ G,9U=5 A 8 h 5 A 8 @ 9 @ 7 6 6 +Y * X ; * Z F 9;: h 9=G/O  1 5 9 >VA G @ATh U A Kg5 A F=F=9=C @ 61C iFOSOSG8,9=G,FGS8 h HI 67K >@DAVB 9 Q e Q b15767`D9 Q1e @ WO UVOdb A 5 A< Q 9=G,5 A 7 i 5 A 8 @ O < h i rD Q7e θ & θD ∼

α . me v 2 r D

.     :9+;7*<#=(<> ?@ACBD EFG H3'I,-# .CJ0F * >KL@=L$& 7#?@M<7NO PH$Q$&L3GS T *+()HP"#R3G M+(K,-#   .<0D L* > θD 

Y+* X ; 5 Z








 F

H   %

 B G,O P28  A :j8 6 @ GS9UV5 A 8 h 5 A 8 @ 9 @ 1 6 6 +Y * X ; * Z 1 b 5761K,OV:16DG/6 > O  A FGS6 < A K h U O8d5 A F=F=9=C @ 61C i G A # θ ∼ θD Q 9 57 a UVOWbD9=57Q 9;: A=B 6 < O[a c 9 <  @ Q A b76DF A G F 9;: h Q HJI h H O `19 @  h : C#K A 5 A  G,9=5 @ OU=Od8,57Q 9 < 9 @ 6M @ 1 e Q Q

 A 8 @

< Q

a

τE (T ) ∼

4/O=:F=G A 8 Q

mp me v 3 mp me −1 ∼ 16πne (T ) α2 ln (6T rD /α) 16πne (T ) α2 ln θD

„

3T me

«3/2

Y+* X ; ,Z

.

C C B 6DF 9 @7@ 9 >@DA=B 9 @ 6DC#b A 5 A< 9=GS51O87i78KO=:C I 61KE8 Y+* X ;,Z i @DAg< O < 9 @ G#519  O
F

 tr .

8 9L:O8 A G,9 @ O7i[M @ 9=57U=ObD9519;: AVBDA< =9 c : h M 9  ,G 57O @AW< T @ A< 6T:OF=G A G,O B1@ OM Z 9  GS618  @ A : C h F=G AW@ O 8 9 @ D 6 6 D b 51OG,O D 6 C Q Q Q G,9=Q 5 < OV:16 @DAQ1< e 6 B 9=F  OUVOd5 A 8 @ OW8,9=F;6DCX <  4/9=519;P7: < GS9bD9=5  O  F h c :9 @ 6 H h F=G WA @ OW8 9 @ 61C R K 6 < 6 B 9=F  WO UVO25 A 8 @ OW8,9=F61C: C SO F @ O8 @ OP 1 5 9 A `1676 579  O b e Q Q  6 @A `1616

p+e → H +γ .

6 @ 9 < A GS6 B 9=F  O P2O  A =F GS6+i  OWUL: A M @ 9=57U6DCEM 9  G,57O @A < 9 @ 51O=: A i1F=9 B 9 @ 6D9TM=GSOP 579 A  `16766 < 9=9 GNF Q L9 : h HI 1 6 P28S6 : Y F < X4 1  6 Q i1FG,5+X"<
. ; <
.. e Z ∆H Q 06 @ G,9=579=F h HI 9P @DA F



σrec =



9/M @ 9=5 U=6162F8SC > 6 A GSO < A 8SOV:DO Y+* X ;  Z

210 π 2 ω 2 , 3\ 4 αm4e v 2

,  B A FG,OSG A D h I 9 @7@ OWUVO OSG,O  @ A i b1Ob1OS5RC: 7h 8,9 6 B 6 @ @ A G h 5 A @ K U3:9 6 Fb 7 5 A 8 @DA C i A \ , OSF @ O8 A[@ 1 6 9  < O 81X OU3: A K A 5 A ,G 9=5 @ O98,579 < C i,G 519 h HI 9=9=FLCgF;8,Q O  =O : @ O a < h bD57OSG,O @1∆h H: CdO  5 A> OW8 AW@ 6DC A G,O < A 8SOV:DQ1 OWU A ω576  OS57e OV: a A i Q5 A 8 @ O Q √ Y+* X . 3 Z 3\ 4 α 2 2T me m3e −1

τr (T ) ' h(σrec vne )

i'

210 π 2

√ 2 , π∆H ne (T )

h =F 579;: @ 6  U3:9 < 6 bDO#F  OS57OSF=GWC < M 9  ,G 57O @ O81X0/ < O < 9 @ G h 7 5 9  O OW8 AW@ 6DC A G,O < A 8SOV:DOS57OV: A G A  c 9 < 9 @ 9d8,O > 5 A FG A F=9 9 @1@ OP @A#< O  e Q < 9 @ Gg519  O n <  6 @A `1676X .     !#"S " GP *NH)CGP#Q("7+<EQ 3G HP ? T  7+ L* H!#H * >   "GPN T PRP??  $&#"C $Q$&MP" % L* > "7' + ) )

H+γ →p+e.

A  6 < O  5 AW> O < i < ho 9;:16 6DF 8 bD5 6 < 9 @ 6 < OSF=G605 A 8 @ O8S9=F @ OUVO 5 A F=F < OG,519 @ 6DC bD57O`19=F=F A 519  O =A `1676 F=9 9 @1@ OP+i GX 9SX+: C  O Q @ `19 @ GS5 A `7616 F8,O  O=: a@ K?MQ 9  G,51O Q @ OW8'bDOSF Q 92519  e O VA  A 618 AH GSQ F;CNGSOU3: A i  OWUL: A K A 5 A G,95 @ a O9 8,Q519 < Ci @ 9=O  KO=:D6 Q < OS9F;8,O  O=: @ O < h bD51OG,O @7h : C Q8,FG,519 B 6Fa ONF8,O  O=: @ a < M 9  Q GS51O @ O < 6b1OSF 9;: h HJI 6 < O  5 A> O8 A[@ 619 < A G,O < A 8SOV:DOS57OV: A i1ObD579;:9 C 9 < OS9 O5 < h Q OP Q F=9 9 @1@ OPi Q Q Q Y+* X . 3 Z i7b15198 F6DGd8,O a > 5 A FG a Q Y+* X . < Z

A<

,G 9 < bD9=5 A G b OSG @ OSF=G F8,O  O=: @ K'M 9  G,57O : Q9 9 @ 6D9 < e ZJ@Aj< O < 9 a @ G >=Q A  A  6  b Q OSG @ OSF=G6M 9  SG 51O @ O87i7FOV:DQ 9=5Rc Q Q G,ONb151O61FLK,OV:16DG bD576

τr (T ) & tr . OW8S6D9 Y+* X . < Z F=O8 < 9=F=G @ O2F/8

h 7 5 9 M4"X F Tr ≈ 0.27 @ O8 F8SO  X Y O @Agh c 9 ,>VnAe< O5 A c 618 AW@ 61C AI 61KFLC  M=G,O < hj< O <

5 A c 9 @ 6D9 < Y+* X . 3 a @ 9 h :9=Gg: A 8 A G F;C 5 A 8 @ O8S9=F @ < G,9 < bD95 A G h 519 A  `7616 O  5 A> Oe 8 A[@ 61C 8,O=:O51O=: a A X &#OSG @ O 9 9 @ G h iR8#OSF @ O8 @ O < i78 A G,O < A K28,O=:O51O=: A Y 6jUV9 Q

Q

Z O b1519;:D9 5 @ < 5 A @ 6Da 9  b1O 6DC Z i75 A 8 Q

C H G FbD579  @ OP @Q A

√ Y+* X . ;,Z 3\ 4 α ne (Tr ) 1 2me m3e √ √ = 10 ' 10−3 , ηB = 6.15 · 10−10 . ' ne  (Tr ) tr hσrec viηB nγ (Tr ) 2 ζ(3) πηB Tr ∆2H Tr2 tr 9 M=G,OWUVO/519 A `76DCg579  O
4/OSF Q F8SO  Q O=: @ KjM 9  G,51O @ OW81X a Q + Y *

G < 9=GS6 < i B G,O 6 > X <5 Z F 9;: h 9=Gi B G,O b1576 6 @ G,9519=F h HJI 67K @A F'G,9 < bD95 A G h 5 A Ki M$Ji8S519  < C?F8,O  O=: @ OUVO b151O  9U A OG,O @DQ A Y+* X <
Z b15198 A 9=G K A 5 A G,9=5 @ O928S519 < C?519  O <` 6 @DA `1T676 'Y+* TX . f3 Z =X 0.27 G,O O >@DA=BA 9Gi B GSOgbDOF 9;: @ 9=9 5 A F=F=9C @ 6D9 OSGSO @ OW8EbD51OW6DF3KO=:Da 61GgbD9=579;: G,9 < i  A '>VA7A 678 AH G,FLCF8SO  O=: @ 9 M 9  G,51O @ i1G AB G,O Q a Q a 519 >h G Q A G 5 AW> :9 A * X < OSF=G AH G,F;CE8#F6 9SX Q7e >VA  a HJB 9 @ 619/ Q 5 AW> :9 A bDO=: B 9=5 7@  Q< i B GSOg8,F=9 @DA 9d5 A F=F < OG,519 @ 6D9 @ OSF6 ONOW`D9 @ O B7@ Pi7bD5 6  6Dc6 @7@ PK A 5 A  G,95+X /)Q b1576 < 95 h i,b1576E8 Q B 6DF 9 @ 676j5 A> 6 B7@ K#G,95 < O=:D6 @DA< 6 B 9=F  61KNF519;: Q @ 61K Y G A 67K a  A  8 Q O5 < h a 9": C σv a Q Q a < @  @  , A  Z < = >  A < @ @ > @  @ A > @ @ 8,579 9 6gF8,O O=: OUVO/b151O 9U 9 C 6dF=579;: 9=9OSGTb151OW6 8S9;:9 6DC bD57O6 8,9;:D9 619F=519L: 67KX//O @ 9 B1@ Q O ib1Q576 G,O B1@ K  O 6 B 9=F=GS8S9 @1@ K28 B 61F a 9 @ 6DCRK @1Q h c @ Og:9P1F=GS8SO8 A G  O 99 AI h 5 A G @ O7i A : C28 B 61F 9 @ 6DC @ 9=5 A 8 @ O8S9=F  @ K 8Sa 9 6 B 6 Q @ i[G A  67K  A a  OSF=a G A G,O B1Q @DA C/b OSG @ OSF=G F8SO  O=: @ Kde M 9  Q G,51O @ OW81i @ 9=O  KO=:D6 < Q O"bDO a > OW8 Q A G FLC h 5 A 8 @ 9 @ 619 < f a O ` < Q AW@A 0 X 9 < @ 9 < 9 @ 9=9Si F=5 A 8 @ 9 @ 619Q FTG,O B1@ e < 628 B 6DF a 9 @ 61Q C < 6EbDO  A> 8 A 9=GWi B G,OdOW`D9 @IQ7 e 6+iRF3:D9 e A[@1@ 9 @AW< 6+iR8  O Q1Q1e e 6 @ F=G8,9 F Q hRBDA 98gFbD5 A 8,9L: Q 678 a F/G,O B1@ OF=a G e H b1a OS5RCR:  Q A 1$3 
3 2i B G,Oda 8bDO Q @ 9:DOSF=G A G,O B1@ O : Q C Q @A /6a 7Kj`19 Q 9P+X



   
 







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*&&" =Y 2,#+" = "%&, T A 3,4' C 5% E ' C ,6 S C  
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($# ; &" = )*6, ; DZ lH ,r =

2 Hr

', C , E * C-[ , T>T 3, &, E " E (&! ^;  Hr ≡ H(tr ) c &  E " f &" T A 
" T A ' C "%$%()%1,#&,6
" ^  F &!"J$ A.F $%" f  X  E S  " CE S 
 ]\ 7

>a 7 [ , E ?p; () QY " ; &,92" '" ;]B " ; 7^0:
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Hr2 =

<

F KIE Y F #" ρM (tr ) = ρM ,0

^; 

)*"%$ C  E (&&,





C *+" E T &, _  D7 ,6 Q & =@V (?$ A.F !)
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(&!&" = Y ' C  N Y TUS ;  S ; " T *& e; C S`^ " =ZY L $%$%,6
t(r&K%& A =  ;eE ,#&,

8π GρM (tr ) . 3 a0 ar

3

= ρM ,0 (1 + zr )3 .

'2
"#%&!"%)* Q & C   %$%!)*%)*+" ^ "4$% X ( )%$=,P$%"O0)(
(&&!" Z = Y  F #" " CE S  A )O"%' C  ; 
(&! E T 7 X a aoY ) E 7 !

ρM ,0 c Y ) E 7a`+3,9$ S T7 b" E T &! C S  L # ρM ,0 = ρc ΩM

'"9 S F  E

lH ,r =

.

H0

2 √

ρc

Y\

1 . ΩM (1 + zr )3/2

78T%T a

   9c ' >  
G
ba # d n

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>afe*g%h
  
e   I
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g=
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ag l H*,rk =
3t or  @  e*gla  i @ e( d i 

 

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)  $

C *&



0)(
(&!&" =ZY -)( ^ " ; &





)("%"##$%*#)*#$ S W X  = C , ? E  C C ,9$%((1& + zr ) lH ,r (t0 ) =

2 √

H0 Ω M

√

1 . 1 + zr

C , ? ? ,>) FK

C ,9) V  N







d 9g+

n l

 
  ! m

C , ?9E (& Q V F)("%$ C  E (&!&" ^ " C , ?IE  C , 0 ; &" Y F #" L #"# C , ? E  C ' C  E  C &"E$ √ 1 + zr ' 30 ^ " C  ? "#&!6,  T!7 X >a
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σν ∼ G2F E 2 ,

)%"9
!&"%$%*&




\



Y

E∼T

7



.

" _ (&!K, ;  

)( F (&! 

 W T " ^ "  ?



Sj i h3 ok
 k   d -^
3




0 C  EN )*$%" T " ; !& " ^ "P' C " T  ^ , & = C &"O2,#J$#)( ^ ; , ; , K %) τν =

"#%&!"%)(!# Q &,  )*+" C "%) Q & = C &". & Y , 'K
"##&"%) Q L % BF ,9)*% _ 760Q&K% c n F ,#)% _ 'K"#%&"#)* Q F )3, F ,#)*# _ ; , K #)  C , "#%&"#)(!# Q &,  )*2" C "%)* Q 7  ,# E v'1 ;   $ C  E (&!J)*$%" T " ; &" ^ "P' C " T  ^ ,

τν ∼



τν

) B , T>T 
"%$%)* E $ C  E *& E

H

" C E S 
" =  27

1 , hσν nvi

ka

F #, )% _ Y )F2"9%" [email protected] C' C "%!) B " ; !)%"9&"%$% N C () S W X  E &,#)n) S F ,# S  Q C , C   #$%)%)* B   #$%)%)(+" = " CE S 
" =  K7 \ aoY 78#7 3Y " T C , ? " E Y EJA ' C  B " ;  E J)
 ; S W X  = n" _∼(&T+

vc

'C ,#$#&$=,



−1

1 G2F T 5

 27
>a

.

 27 X

MP∗ l = 2 , T

a

 
 N EJA $% ;  E Y F %"O$E' C " _ ()*)(>" B 3, f ; *&  0:)((&&" = C ,#) K T A )* C ( Y F  E −1 7 ; %" $%,9% Q &" Y $)("%"9%$%%)*#$%B) &,#$#& A.E " f  ; ,9& E τY ν' C  ; "%)6,9%" F &"4$ A )("#H  B # E ' N C ,9 S C , B $ C  EN )*$%" T " ; &!" ^ "?' C " T  ^ ,F& = C &" T A 
" E (& QUV  Y F  E B , TUT 
"%$%)*+"% $ C  EN Y  & = C !&"b&, B " ; K
) Q $f# CE " ; &, E  F *)(+" EmC ,#$%&"#$%()* ) $% X ()*#$%" E 7  = )*#$%!# Q &!" Y F )
">)%"9&"%$%*& = &! = C &" Y &, F !&,  ) E " E (&!6,F$ C  E *& Y " _ (&$%,#*#)  $%
 F &!" = t

N (t) ∼

Z

∞ t

dt0 = τν (t0 )

Z

∞ t

dt0 t0 t 1 ∼ ∼ , 0 0 t τν (t ) τν (t) H(t)τν (t)

^ ;  EJA S F 
 Y F #" T A )* C " ' , ; ,9*P)("/$ C  E (&! E 7 )
 Y (t) ∼ 1/(H(t)τ (t)) N (t)  1  %)ν  $4# CE " ; !&, ν E  F *)(+" EdC ,9% "%& C ,#) N %"-& = C !&"J&, B " ; t/τ $ &"%$%*)( Y ,.' C  ' C "#)* C ,#& W %)  2,#B)($%" T " ; & A  F ,#)*# _A 7  ,# E " T C , ? " E Y & = C &N "C(t) ' C  ()6,1 W 5$ ? ,# N E " ;  = )*%$#"%$=,9 Q   ? ,#2,6$=, W %)  a ' C  ?

 K7
>a



 K7 X

τν (T ) ∼ H −1 (T ) .

a )
 ; S * Y F %" L #"O' C "%) B " ; K' C 4% E ' C ,6 S C  Tν,f ∼

.

 

   

  
gK k+e




1 2 GF MP∗ l

1/3

∼ 2÷3

H L0

.

e( d i^ G
   
  d -^
!^

& = C !&" )(' A 6,
J'!"%)
 ; &*b)*%"6
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) Q $%"B0)(
(&!&" = )*$%" T " ; &!"7 B '"9
&!"% F )"  $5)*"%' S %)%$ SbN W X  E " T i KIE  a ' C  L #" E &B ? E *&  
"#) Q ZM2,# EJA ^ "%$%" C K
 Y C (,# _  ,#&& ^ K @_   6,# f W" TUS )
"%$9
*& A )3, T A.E  $ ? ,# E " ;  = )*#$% E   Y )
 N e; + e− → ν¯ν ν¯ν → C e+ e− W "%$%,9% Q &" Y 6,9 f F' *)*6,  ; #G$ E " E (&K ? ,#2,6
-& = C &!"7  ,# E " T C , ? " E Y " ; N &"5
 VMQ ' C  ; '"9
" f *&C"Q%" E Y F %"S0)(
(&&,  +" ^ ; , N %"1 E 3,Q% E '! C ,9 S C S $ A V  &*)(+"9 Q ! B H L 0  , K,9 EJA S $% ;  E $ ; ,6 Q & =@V  E YS )(' B %(" C -& S K
("#)(&!# ? , ; , K %$ KICU; "%?'" ; #$% C f ; (&!F6,#+" E S ' C  ; '"9
" f *& W a ' C $%" ; !P$ A $%" ; S "/%" E Y F %" ; "9 N f *&A) S X ()%$%"%$%,9 Q C 
K%"%$ A =:^ , ? & = C &" Y ,#&,6
" ^  F & A =^ , ? S C 
!#"%$ A'B "#%"#&"%$ Y 78#7 E  C "#$%"9
&"#$%" E S "##"%&&" E S  ?  S F (& W 7 6,#






! % !










n d

d 9g+

&%

 
lg

 b^
^^

 ,  ! q $    , ,. 9 $   $ ,.- 
  

, 9,    9 $  ,. ,  $ $  ,.- 

 





%

&%

&%





&%




67

?,#G)
 ; S *4 ?:C  ? S  Q 6,9#"%$ C , ?p;  3,
7 Y & = C &"'!"%)
 ? ,#2,6C'" N ' C  f & E S %" ' N ) A $%, W #) S  Q C , C   K%$#)*#)(+" = S & _  = C ,#() ' C  ; 
(& Y B , C ,#K% C  ? S W X  = ) L N (!#$%&" = % E ' C 9, S C " =







Tν = Tν,f



 27VT a

Tν,f a(tν ) = , a(t0 ) 1 + zν

 C ,9)(&"%W) E  X (& Y )("%"##$%*#)*#$ S W X * E " E *&! S ? ,#2,6
 & = C !&"7 0 L #"# ^;  E " E *z&!ν Ac # E ' C ,9 S C , & = C &" T A 3, C ,#$%&,# E ' C ,9 S C  "##"%&"%$!7+0'"%)
 ; )%$%!E% E N ' C ,9 S C , "##"%&"%$P6,# f A', ; ,3, ? ,O) F@K C ,9) V  C (& 50)(
(&!&" =ZY )(&, F ,6
,E'" ? ,#+"%& S 27VT a 7  ; &,#+" Y $ E " E *&! ? ,9K,
/& = C !&"A C " E  "##"%&"%$b$'23, ? E  T A 
"6,# f  T "9 Q N V "# 2"6
 F *)*%$#" C  !#$%)%)* B L 
*! C "#& N '" ? ! C "%&& A'B ', C 7 P "%) %" ^ " Y 2,#P0)(
(& N &, J" B 3, ; K
,#) Q ; "O% E ' C ,9 S C & f  E ,#)*) AdL (! C "%&, Y L 
*! C "%& N '!" ? ! C "%&& A P', CA ' C "%,#&& ^ K
 C "%$=,6
/$ "#%"%& A Y F #"' C !$%
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*! C "#&"%$ P'" ? K C "%&!"%$ E " f &"A"#' C  ; 
! QY '"9 Q ? S !) Q ? ,#+"%&" E )(" BGC ,#&*& L &! C "%' L 
*! C "#& N "#%"#&&" = +" E '"%&*&! A $ )*"%' S %)%$ S W X  E " TUi KDE  Y

















 K7 a

g∗ (T )a3 T 3 = const ,



L *!%$#&"% F )
" C  K%$%!)*%)* B )*#('*& = )($%" T " ;eA $ L 
(! C "%& N " N ^;  g∗ (T ) c %"#&&" = 'K3, ? E  ) E 7I)*"%"##&" V *&

27 X ga 

27 X%\ a a 7 'C , ? S '"#)
 ? ,#2,6
R& = C &"Q$ L &K C "%'! W L (! C "%& N "9%"%&&!" = 'K3, ?IEJA ; ,#$=,
G$%23, ; "##"%& A Y L 
* ! C "#& A J'" ? ! C " N & A Y F #"O' C $%" ; !O-)
 ; S W X  E S ? &, F (&! W Z

 





7 11 g∗ (Tν,f ) = 2 + (2 + 2) = , 8 2





P "%) + − N ,#&& ^ K _ W$ L &! C %" ' W 'K3, ?IEJA #$ &"%) !E$%K3, ; %"9 Q + " "#%"%& A Y C"9%&" N  & = C &"E"#)*6, K %) C'!"%)*#" !&& A.E  C #, $#& A.E V *&?e% E e ' C ,6 S C "#%"%&!"%$ J



Tγ,0 = Tν,0



g∗ (Tν,f ) g∗ (T0 )

1/3

=



11 4



.



1/3

' 1.4 .

 
 ; "%$=,9# Q &" Y $ &,#)*#"  X *>$ C  E 4  % E ' C ,6 S C ,E& = C &" C ,#$%&,



Tν (t0 ) ' 1.95

 K7 \ a  K7  a

P "9 Q ? S !) Q " CE S 
" = K7 \ a & , B " ;  E Y F %"W' C R)("%$ C  E *&&" = % E ' C ,9 S C -'K
"##&"%) Q F  )3, &  = C  &"OG,9&!%&! = C & ".2, f ; " ^ "/%',P)("%)6,#$9 !*





nν,0 =





L%,)%

<@  )=     )!9)94=9*=% +G= /*6! 0  '(! % %)  0

2

!

)E ζ(3) 3 · 2 · 2 Tν3 (t0 ) ' 112 4 π

−3

.

 !#" %$'&  ( )%&*+%&,-."'/ ' 10 ! +2 3

 27 a

 F 9 4=%      /
%' F ?  ?% " D4=%='
9*6 (6 ?) /*/  0 $ 4* % ( +45%

!

.


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k  G3
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 h^g 2, ? ,#&9Z h=a  (e*e  h=a3e  m  ^ /h3
id gK  alg%^ 3   
  g+
 h=a h^ d  
 ]e ^g+gKlh=a  eih^   i
 d   9   G
f^  
!  >#e   d!d m  !Q
e 
e
g @


 



 

P C E "% ; *%*!% C "%$=,#&! C 
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*& D7 .




     
6mJ^
^  =e(e(lg    
 +e(e 

k g    d


d 




 a =h a^e  b ^ g 9 !  ^ e g%^ a  g2 M^ ^  ^Mg ^  ^ 

 

A P& = C &" Y ()B"%&! E ,#)()*$%& A Y E " ^  T A "#K, ? ,9 Q ) S X ()%$%*&&"%O$9
 N ' C ,#) V  C (& 0:)((&&" = 7  %) W ; ,S$%" ? &2,#*$%, f &!"%J+"%) E "9
" ^  F *)(+"% " ^ C ,#& F (&/&, E ,#)() S &! = C &"7 %" T A $ A $#()*# L %"E" ^ C ,9& F *& Y &, = ;eKDE $%K3, ; 2,#+" ^ " N %"f" ; &" ^ " %',f& = C !&"b ,#&K%& = C &"b$ )*"%$ C  E (&& S W "#%&"#)(!# Q & S W 'K
"#ΩνN &"%) Q 0)(
(&&!" = 7 )
 C  F Q  ;eK G" T  ? E ,9)()("#$%" E & = C &" Y %"J ? $ A.C , f *&

K7 
>a ;  -'K"#%&"#)*% L & C ^ !C$ S  Q C , C  !#$%)%)*2" E ) S F ,#?)
 ; S  Y F #" &*$%+" Y 
!#"%$

&!C&,B# E







Ων (t0 ) = 2

 

Ων

4 7 π 2 Tν,0 ≈ 10−5 , 8 30 ρc

 

 Y ) 
 ; "%$=,9# Q &" YT  ? E ,#)*)("%$#"%b& = C &!"P&!"%2, ? A $%,#* ) S X ()%$%*&&" ^ "E$9
 & .&, )(" N $ C  E (&&!"% C ,#) V  C *&Q0)(
(&!&" = 7  %"#!U"# E %! QY F %" L #"&C6,# &, T "9
* C ,#& N & B *) 6, ;  BC ,#) V  C (&! $ F ,#)%&"#)*% Y  ? %*" C -' C $% F &" ^ " & S K
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%

mν > Tν,0



ρν,0 = mν nν,0 ,

,/)("%"9%$%%)*#$ S W X  = $%K3, ; $P"#%&!"%)(!# Q & S W 'K
"9%&"%) Q C ,#$%*&

 27  a

ρν,0  mν  L 0 · 0.01h−2 . Ων = ≈ ρc 1

P "# C  TUS  E Y F %" T A 'K
"9%&"%) Q L & C ^ >& = C &"f&n' C *$ A V 6, 
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X i

mνi < 100 · h2 ΩM

b)('"6 Q ? S G+"%&!)( C $=,9%!$%& S W " _ (& S

L0

J'!"9
" f !$

ΩM < 0.4 X L0 . mνi < 20 i

 K7 h = 0.7

Y '!"9 S F  E

ba



X

 

0 # &  E E ,#)*)


n d

d 9g+

 
lg

 b^
^^

F *& ; "9 ^ " ^ "C$ C  E (&S'!" ; " T &"%O" ^ C #, & F * & T A 
"C) , EJA.E () K Q & . A E " ^ C 9, & F  N &, E ,#)*) A N  N & = C &"7   ^ " ; & R+ " E T  & C %" $=,#&!E' C EJA'B " ^ C #, & F (& = &, µ S L 
(! C "%&&!" ^ ".&!τ = C &"  T  Y





mν e < 2

L0

,



 C  ? S  Q 6,9%"%$ L +)(' C  E *&!%"%$ '" '"%)* S &  = C !&& A'B %" ) _ 2 _  Z = Y  ? +"#%" C A'B )
 ; S  Y F #" C , ? &"%) Q $=, ;RC ,9%"#$ E ,9)() E f ; S E ,63, Y L  Y 2 2 −3 0 2 Y ' C !$%" ; ! O T "6
(?)(2 Q &" E S " ^ C ∆m ,9& F *& W & , νE e ,#)(ν) µS & ν= τ C &" Y ∆m . 5 · 10



;  %$ )* B #'"%$ ) E 7 P C K " f (& a 7 ? ) ; S * Y F %"P%$ K3, ; !&  = C  &"O$%)( B %'!"%$O$



X



L0

 27 ga " ^ C ,#& F *&  K7 ka  )("%"9%&" V (& 27  a 'K"#%&"#)* Q L & C  ^ !G$%"P0)*
*&&" = &($#
 Y

mν < 2

,

 K7

Ωνi < 0.12 .

i




>a

;  J'!"9
&" = 'K
"9 N   E &! E *&( Y ) C ,#$%&!$=, L %"E" ^ C ,#&! F (&!A)*" ? &, F (&! E ΩM ≈ 0.25 &"%)%5& C  K%$#)*#)(+" ^ "J$% X ()%$=,$ E " ;  Y EJA $# ;  E Y F %"" ; &" ^ "" ^ C ,#& N Λ 
 F *& K7 
>a & ; "%)*6,6%" F &!" Y F #" T A )*K W F ! Q@Y F #" & = C !&" $6 W %) ? , E *#&" = +" E N '"%&!(&!#" = KIE &" = E ,9# C !D7  ; &,#+" Y C  ? S  Q 6,9 A Y )*$ ? ,#&& A ?)b ? S F (& E )* C S ! S C $#" 0)*
*&&" = P ? E  C (& E ,#& ? "# C "%' C 
!!%"%$#" ^ "F ?  S F (&  Y " ^ C ,9& F $%, W A$%K3, ; & = C !&"E$P'K
"##&"%) Q L & C ^ J$%"P0)(
(&&!" = &, S C "%$%&!



 

  



%

X

Ωνi h2 < 0.008 ,

X

mνi < 0.72

 K7

i

X a

F #"O)*"%"#%$#*%)%$ S *4" ^ C ,#& F (& W &,/) S EJE S E ,9)()?& = C &!"   L0

i

4)(2 WF ,9*.&! = C &"OK,9J2,#& ;  ; ,96,P&, C "9 Q KDE &" = E ,9% C D73H A '" ;eC " T &(F" T ) SnN ;  E ' C "%!) B " f ; (&A" ^ C ,#& F *& 27 X a $%" $#%" C " = F ,#)*#G& ^ D7 0 ? ,#K W F (&! L #" = `=3,#$ A "# E % E Y F #"E ? 
" f (&& A  C  ? S  Q 6,9 A T A G'"9 S F (& A $Q' C  ; '"9
" f *& Y F %"B$%"B0)(
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"#$=, E  Y F %" B  E  F ()* = '"9%(& _ ,6-& = C &" T 
 ? "%.O& S  W 7  %"4' C  ; '"6
" f (&!O' C  ; )6,#$9 !*%) B$%'"6
& C , ? S E & A.E Y "%)(" T (&&!"-) S FK %" E %" ^ " Y F #" L (! C "%)
, T A ' C " _ ()*) A ' C 1# E ' C ,9 S C , B $ A V    ` L 0 S C ,#$%&!$=, W W'"C'" C ;  S $%
 F & A 
*'!%"#&& S W  T , C !"%&& S W ,#)* EJE * C  W ) E 7 `=3,#$ S kapY , T , C "#&&, S,#)( E N E * C   C , = & E ,63, Y 7   E & E (&!( Y & Q ? '!"9
&"%) Q W )*K W F ! Q@Y F #" ηB ∼ 10−9 
*'!%"%&!&, 4,#)* EJE * C  ? , E *#&" T "9 QUV  T , C "%&&" =ZY 78#7K+"9
 F *)*#$%" & = C &"P$%">0)( N 
*&&" = ? , E *#&" T "9 QUV  +"9
 F ()*#$=,F,#&!%!& = C &" 2
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(> f " ^ "4& = C !&" Y F %"ν > F !0.05  E 27 X a E " f &"/!)('"9 Q ? "%$%,9 Q ;  #" ^ " Y m "6 S ≈ T A m'atm Q " ^ C ,#& F (&?&,A('!#"%&& S W ,#)* EJE * C  W $O)*"%$ C  E (&&" =  C ,#&&! = 0)(
(&&!" = 7

 

 













   











! 2 !

. 

!


" ' J/
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i^  h^ d   
 k9!  i!^ /
4^
!  + ei! 
^  (e  gla  i^ e* d ^ 



∆L,0 =

$  h d
^%e
 
h >

k ^ s
 6  ^ fh=g2 


P

s

− nν¯i )

,

 

gfh=^  i^  k  9^ e* d  

k9!  i! 
 d ih
  +  e  


∆L =

$

i (nνi

P

nL,i , s

i

nL,i = (nνi − nν¯i ) + (nli − n¯li )

 
  +
(m
  h d
^%e
 d ih
3  ^e d  

h
   li d i h

h
 d    *e  k6!m^
eomUg − − − h=a  (e(e*i   gK d   l1 =e( d e i^l2 = (µe d Cle(3  =k9τ!m3
eomJh3 d ^% ∆d L ih
^#  ^e d







@"9
*>)*K Q &"#>" ^ C ,#& F (&>&, 
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("%)*&!# ? ,)C ?IE  C *& E  '! C $% F &" = +"%& _ *&! C , _  ^ 
 N T7a0 C , ?o; 
 !7 EJA '"9 S F  E " ^ C ,#&! F (&!.&, B  E  F ()* = '"##(& _ ,6 L (! C "%&&" ^ " 7a@"9
*E%" F &"% & = C !&"W' C S# E ' C ,9 S C 4'" C ; 2, H L 0 &, S C "%$%& |µνe /T | < 0.05 " ^ C ,#& F (&F E **O$% ;  ; "%$% C !# Q & A = &!# C $=, aoY



0 ;  = )%$%!# Q &"%)*#  E " E (&K S ? ,#2,6
 T " =Y O+"%& _ (&! C , _ ! B , C ,#!# C & A = ' C " ; a

P C 

L0

2








−0.023 <

µν e < 0.014 , T



T ∼1

H L0

 K7

.

T a

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E

tosc ' π

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T ∼3

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P (να → νs ) = sin 2θα · sin tvac α =

2Eν , ∆m2

2



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P (να → νs ) = sin2 2θα  · sin2



α

t 2tα 



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,

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τW

hP (να → νs )i =





1 · sin2 2θα  , 2

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sin2 2θα nνs (T∗ ) ∼ ∼ T∗3 MP∗ l G2F · sin2 2θα nνα (T∗ ) H(T∗ ) · τW (T∗ )  m   sin 2θ 2 α −2  s0 · . ∼ 10 · −4 1 10

U

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,% "*a "P$"l! ,# "3   )*#$%  p + e ←→ n + νe

X%\

 K7 1



3 K d K ^
a^lg

Yh%a
l^%


^lpP i!^ X





E"%)()*&al&'"()*)("%$7 [ ,",#!#&  U &a(%(*)(!F',, q O6,#!23O'"*)()*"%$ L U #"5Q, ! &"%)95n,#)()?&!"#&,O-'"##"%&,

U0 )n,#)*) , U 
(!"%&, H 0 7+,83'"9
".-O .'"%)%"#,M(!%"[  *73
" ) me = 0.5 U 3"%)6,9%"(&"E$ )("% J% ',9$, WM ∆m ≡ mn − mp = 1.3

H

 K7
1

T & ∆m, me .



I"ar^,M2K,9..$/`=3,#$% 2M2$ E)($%"%"3&"a*" '"#a, &!"#&, .>[F 
?FIB  Z @?= =rY#] >[BI Y K7 14 ".-O&" "*&!9O!u,"! &43-)*"%","-/(&!



τn↔p = Γ−1 n↔p ,

ar3 $ Cn



L τn

Γn = Cn G2F T 5 ,



WM 2"ar^,

+"%&!)*6,#&!6,'" D32,Sl3&!b7 +,"()*) %', !7 1 '(,"N/,M%) )6,#&"%$#!%) G),#$#&<)3+,"9
"%$#)( $ (&M7V97



Γn (T ) ∼ H(T ) =





 K7VTD1

T2 , MP∗ l

ar3   $K( 
SM!( #"0)(
(&!&,J'C&K%!()$VN 23C&,#)?% '!,9"$,Q3 3!#)G&,^,.3,!"%&&""& 3" &!"%$=,9&&"4)*6,.3D7 ?,#-%,#&96>M MP∗ l =

 K7 X 1

T ∼1

H

U 0L&,Q3="Q&

 K7 1

MP l √ , 1.66 g∗

'(_n(!)
"
l9l," K%$#)*#)(23C)*#('*&C)*$%"%"*  ,#$%&!"

 K7 \ 1

7 7 · 4 + · 2 · Nν . 8 8 3*) $%" &2,# 93, *" ^, ?"##"%&, 2$#%" " g∗ = 2 +

+b$$%K3,X !' 9 "!
 a    M  5L 63,a("^,"  U 
*!!"%&, 1lM3*#B)*$%! ,#&C)?_a*  C'""! !"%&, "%&mL l9l,!#$%)%)(!E'! &! !&"M ( )
 "%'!"%$42"9%"43o '"#K,."%"! &,X(!TK
>  me $53)%$%K%I9%&"%)% 1 7 ?

,#'" & ) 7 `+3,#$" 1lM4!( #" ' 78#7
N"ν ,# &&a*K '"! K"%&!N"%ν$=) M 3 U (!"%&, W^M &! !&"E f P - %T '!>,9m"$eM (!%"P?"##"%&b7 ! !78TD1  !7 X 1 '"92( ,# % '! ,9"$a M ' Q+"#%"! " !  ,#! B#',  !7 1f'!(,X&   N ,"f %)WM / 1 T = .  K71

b"%&)6,#&!6,

n

(Cn MP∗ l G2F )

1/3



! #$ ()*#&,!ZK'"*)() #', !7 1n$%""! &2,MT9
,a("^, k(*_'3? !"%&2& &"W$#6>&M! " ,"-[_*&&"W&,)!)#7 !7 G1 %"(&".6,)-/>$#6>&,"%') $=,# ,#)(',X &!"#&,M)97 !7 17 +b" U #"w C  !($9
*K,9*%) J !F$   (&  -O! &4&!!"%&, (!)
*&&"M 7 I,# Cn = 1.2 n ","! "M6% ' ,6$

, ! ,#2,6
 !7 1&! )(" 3 G-O!A&(!($%()%&43O',", *"%$7   *%M "3&,#+" MI(K%"P"%&T , !l,#$#)(!O"#^()3,P#'"%$/ _a(2 3-&! !&" M
)  7 !7 1 !7 \ 1i7 +b#" 
)*6,#$9 $ !7 1T! &., (*& G = 1.17 · 10−5 ` U 0  g = 43/4  N = 31'M F ∗ ν '"9 2(,# q()(&&" H U0 Cn





 













Tn ≈ 1.4

.

  

X

d 9g+
> og%
#  -  d (#ei
 

p

p

n 



e

νe

n



)#7 !7 Z _ &n,#&"%$%)*V
,a,q O'"!()()*"%$ 

e

n + νe

G1 ↔ p+e  



ν¯e n → pe¯ νe

 17

1b#3"%$9
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": U&m 9  &,#'! SM ()
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 ,"(! & S
M K
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 "# 
( ,
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 &,JO ! )2( , &"%)%SMw9 3,"(a "^ , B2"9%" '! $%( &R  &2 K
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 D 7   (& "%)*6,9#"( &$ +"%&! (&!,  &!! "%&"%$E'"%) 23%l! ,#2,6D7 =3 " "" 6 W  )%l& '*&9 #"( &"%)% "%&,7 ,#$%&,  ,9$%&"%$%*)(&" 2"#& (&K,! L&!! "%&"%$&*'"%) l3  )%$%*&&" ' lnl! ,#2,6
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"9%&"%)% ( )3,7( ,9)*% A  ' 4% ' ,9$  ) 7
`=3,#$ 1 Z T m  A

nA = g A



mA T 2π

3/2



µA −mA T

 K7 G1

,

 '!"#%*&,6:(,#)*# A 7%" ' (&!!9 U F"wlI-C'""& ar3 µ L K3  ( *)(W A %"#&, C&!!"%&,M$(!_MS(!%"E&('"#)83)*#$%(&&!"-'!8:!l,#2,6
+"  ,#C%!', K7 1  &,Q3%"!#)C$
,#$%&!"%$%()*SM'" U %"w ) 7
`=3,#$ 1 M#"O*)*9  µp + µ e = µ n + µ ν µn = µ p + µ e − µ ν .  K7 G1

F  !#$%)%)(!23 U 
(!"%&"#$E4'"(! !"#&"%$E ( ! K7

1  ne − n e ∼ µ e T 2 ,  !7 1 '" U %"w 

+

−

µe ne − − n e + ∼ . T T3

Q! &"%)9.+"%&(&!, U 
*!"#&"%$.-'"!(!"#&"%$),9$%&, U (!"%&! ,%9 &"%)%5+"%) (()*2"G'K3,Q!#V1'M I,

 K*=

'

ne − − n e + = n p ,

!

2"#&(&K,!C'"9%"%&"#$

 

 2!

!A ' D*=!96=  #;'
% %)=  %,)% ; ( % 'D*'  LK*  4=!!9*6!%  '(!% %)  ∆m * '  T
n4=) (K* %/
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'*!%  % (4=*=96  )!%/!  )n')'7!%/('*%  )!∆m % 7 A 
4=)= 4   0  0  )!%( @+ 9*%%, )'9'-/
9!(D* A 4=!)4 0 ' 9*%%= IA! ?45)45$)
4 * AJA!*=) 4=% 4*  % 

! +2 +! 

10

!  2 !

!



! 3 



2

 +2

+!



3 K d K ^
a^lg

Yh%a
l^%

,

np /T 3


^lpP i!^

X 



'" '" $
4$%(&N)("%$%',X^,#O)0=,""%&2& ?"#%"%&!& "##&"6>(& np ∼ ηB ∼ 10−9 , T3

+b" U #"wk3K  (!()(!C'"##(&!,6 U 
(K"%&!"%$E,&un,6 µe ∼ 10−9 . T

+, l
 '"9
".O -  ^ ,6
*MwK( %"W$%"G0)(
(&&!"  &*c%"696 "B
('K%"%&&!"U,#)( !  " %)'D& -[3(&!F",9#&"-)(!"=,!<N'!"%$%l_ $ +"%&! ,Q!'33,G1'M78#7 I"ar^,

µν /T

%".-/0n,6"O4!

nν − nν¯  nν + nν¯ ∼ T 3 . !7  '"6 ,# ) =" "  -%" !&"%)*

 G1 I$(  3  "6  (

9

 !7 1

µn = µ p .

! !7 G15!l,#2 d(,9  #"ar^,MV(!%" & K"%&$&'"9%"%&&!"%1"##&"6>(&B&,v " (&!q!l,#2,6 &!"#&"%$k!
+" "%'83 *#) G# ',9$"


 

nn = np



−

mn −mp Tn

≡



 !7 
1

− ∆m T n

-' ($&!(K
SMI(Ka 
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b!"%&2& '"##"%&&"%b"9%&"6>(&! !7 
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-)3="""6 -%"(!&"%)*9>7 F H 0L'"92(,# T = 0.7 U n

n

p

 !7 X 1

nn 1 = 0.157 ≈ . np 6

  %Mb(!%"Q&  !"#2& &' !"#%"%&!&"%G"9%&" 6>(&!G&,c " (&K ! ,#2,6
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,"!# 5' T ∼ 1 H U 01. ' ,9!%(!()(!C&, !l,#$#)(!O"#T*$a*2 3-+"%)  "9"a((!()(!23J',, "#$7 0 !l,#2  d(*& U %"a*" ", !'
3,.&", $*_ $%"" !#,#)40)*
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\ 1













F

t=

Tn = 0.7

H

U0 

Nν = 3



M∗ 1 = P2l . 2H(Tn ) 2Tn

'"9I$(,# "##)^, t = 1.5

)

.



,9Q","!("M9#("! E'!$%(&"a*"A&2K("%)*&!%'!l,>"9%&"%)*!%) OP2"#) 9" 
"a((*)(+" 3="M
&,.(&,VN ) (!'!>'"6%" )(*2&$*  1'"%)
A@"996>"a(" $! =$ ,!7





U '""&

0 !l,#KId(*& U %"a("R,"!l33, "# *%M.(!%"M'$#l*_*&&"%n$(!)
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"#)9P&,P'l3'!"9
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('$& %"#&&" ,9)(  7
)
 U %" &!M6,#WMK78#72(). (%) -,#)* *  '-
I U 
*!"#&2& &  -& K&!".-,#&!%!& !&" M^%"P$  *)*%" K7 14#3  *9





µn = µ p − µ ν e





T

 

,/$ ()%"

 !7 
1'"6I$(

nn = np



− − ∆m T n

d 9g+
> og%
#  -  d (#ei


µ νe Tn

b!"%&2& '"##"%&&"#/"#%&""6 (&!/"%'83 *4&,,%"96,#&&$ , 4   ) 7I,"!l3 !7
1'M n4He ∝





$.+"%&(&" !#"a(>2"#&(&$&

nn , np

!-),9$%&(&! %("  ) &,93*&    ".-E&!"Q)r
3,99B$ $%" "W%"M(!%"Q"# 
(! "#P)6,#&$^,K%&"a*"k! & ,X(!(& J&d3"9-E&"
 95#"996>M



nn /np

  ∆ nn . 0.05 . np

%)^,O)
l#*E"a,9&(*&F&,[3K (()*C'"9%(&,6 

µ νe . 0.05 . T

U 
*!"#&&"a*"E&!!&"M

%

@ "9( %"(&"% )*"%$ (&&"#M"a,#&!((&! '$%83(&!"F$ K7 TD1 a*"F,#)( (&"%)9>)*$ 2&  !l,9&,) %MW(!%"-&,9I^,#n, c,#)('"%)*,#&*&&"%)9 4
 '-E!4&!()(+"9I9%+"J&2-//)*6,#&$& ^, K%&" a*"O' l 3)*2", !l,#&! D7



!

1 % ,%7* -






  





' *  - 9 %(' 9  



  %)(1

 

% 4  % / *  .9

  

-

<









 3$&

 ('"( % " D3!&43)! ,#!O$0,9&&O0)*
*&&".&,X(!&,M%) O)M","!("%$=,#& b % J$
 ,#



0

2

p + n −→ D + γ

$3&" ?! + U %"JG,#&,6
"a*(&43%(,#J)*'"99"!$fE"%"!(&,.((&!

 !7 TD1

 !7 1 Hm#3L)('"69! "#$=,99-2,#Q"%"!(&,.((& !7 TD18M6,#15"#"! &,.(*&  K7 17 F5$4& ()
(&! C% ',9$,MD'C+"#%""C&,.(!& ,#*#)c ,9   !7 TD18M
$%"%)('! "99!% )C  )
l& $ON>Q'_ " 7@*?  C  B$]0M"(!%",! ,#!   K7 TG1'!"%)p3="3K,
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&"  )D7=) `=3,#$%" \ 17 "#"#%$%%)%$$ON> ?! (()*k l3+,#&!!# )("#)*%"#!S$5%"MP(!#"1","!$VN *) $ ,<
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*& D  7 ,"! "%$%,#&u
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(&,M3) 7 !7 X 1 17  ,#+" ,#$#&"%$%*)(& G'"I3%"#N'""! $%"9   *S"%'83
K9 FIY @DY\C*?FIB?O# " D 3&43 p(n, γ)D

(,#7

n)*'($%,fF"%&P'"#! "K%E2
O&!*XM!l,9$%)(K>"#F)*2""%)*#k(,#O$F),9$%&(&!







 d
 

d e   
 
9h 9g d

  
   m   



 

 



    6  !  #" $&%'$!!  ()!& !*c+,- % 9 ! 7 & a$/.0   % a(  .0a$v1  & 62 ! k33( $ .0  1, ! -, $   ,0   & ,- -&% 9 !  &  ! ,- %'! ' $ a4 $  5'6, !7!   % "a8  4%'  !7 w &   &% 9 & , ! 9. %'    !7- !:,#'  .0!* &-!   &   ( !  .0 !  $!7 ci M 3  M 7 f ;(, - + ! "(  .0 , ! -,  7< !7  $ $ON 23 !   -     $1Q3=23>.0 !  !7 1 5 % 0 ! (  X- !  ? % M@, $A!7    !  ! X3+ %   9 B, ! - %1@  % 9 !   ,%2!* !!  D*  4  5 +,+ % -C <   % 9"!-,  9k, ! -, ! DI3KM+  1Q!    - ! 6 !   FE 5 E 1G. ,-$IT$1, !$!   Q3750"    7SM9.9ar8  DN & , $!!    $   !     .SM33$& ( $ .   ! 71 %  !   7 ! -,H7   ! -, !7 1, !    -
I  SM1 U& w @w %  FE 5 E 18M1 ! _ !!  .   ! I !   7 ! M8_  3/2 mp T nn = 2 2π 3/2  mp T np = 2 2π 





µn −mn T

µp −mp T

,

FE 5*JG1

,

FE 5LK1

ar  ,M   U .   !7!7 N $!   a % [1Q! !   T? $    ! A !    ! 75
4%  D*     ! O, $  A  ! $+ Z  !  P! .(1 % d# &  !7  $ -,-  9O   - SM ! .0R %   4%' I2VN223Q ! -.R %1 ! !  ! 7    9NN $ @w % 1 ! Q3+ - S,o33( $ .0 1, ! -, $ SM  
33(  .D -! 1 %  %  ! a$5, !7T µA = µp · Z + µn · (A − Z) .

 + &   ( !    & -,  - &% 9 ! MU  + % X- !   k33(  .9 1, ! -, $ V! ! .(1 $ MW(  k  & /$_    (.9 -  $   ! 43   .7SM),7  ! % 9   - .0  7 Z     ! ,W   ! -,=j  + !  - , D* (A, Z) 5 (A − Z) ! 

  & ,   . -  M1.9.X,>Y % ,  JM1 % 2(7I! 5 1'M 5JG1'M 5LK1 ZE E ZE ZE

nA = nZp nA−Z 2−A gA A3/2 n

ar   [ % .-= % X,C   U .    ! $!7-



2π mp T

mA = Amp

 23 (A−1)



∆A T

,

,-, &%  U !7 a$  ,%! D *1

FE 5 E 1 (A, Z)

∆A = Zmp + (A − Z) mn − mA .

"),  *_  N
_   !1"!#   !    ! 6 !   ( &%  ! 9. %  ! -,M ! Q32N223  \, D *1Q3 .X %'! w (  4 %  ! 9. %  ! - , XA =

AnA nB

 !*26    - ! 6 !   FE 5  E 1),>,$  $1, !$!   XA =

XpZ XnA−Z nA−1 2−A gA A5/2 B

@.0 ! S M1 %   !  & 9k(7 &% ^7 ! -,[1, !  nB = η B · nγ = η B ·

µA,Z

{



Q30 2π mp T

 32 (A−1)

2ζ(3) 3 T = 0.24ηB T 3 . π2



∆A T

.

j&t$j  n em e4`sk ! _lLj Z) u t$j4f `s`ie4r rLvl e4k4n jt&o f j?t$owd achLj&f b  nLj}|o 4k Aj )j nL`t&vs_`^|~_`cfe&k4(A, by_ f ! vh lLj&y`gnLd !  e&hLd acd  j&l|vsf3€


^]_`^acb?d)_e4f `g`ihLj^_ack&lLj&mLn oqp:dLn_`sra

 j$x$e&k4byj$t&j  n emL`snLdL`zk4dLh e



(A, Z)

M




 -



 0






 



  
 

.0 ! (1 &% 9 ! M'   XA = XpZ XnA−Z 2−A gA A5/2 ηBA−1



 32 (A−1)

2.5T mp



∆A T

FE 5 G1

.

!  % a$( !  Y % ,  JM 1, (1 & 9 U$ ac1, $!$ ,    G-H  ? %  U!3 -7 !  .   A−1 M . (  m1, ! , $!  .0 !  $!7 1"  D      & _    9 ηB (T /mp ) (A−1)  ? %  % 6T9V7 T  ∆ M  5  5.0ar8    1  $1 $  !7$ !* &-! c $! 96  U$! a$ A  , ! $175

3. %'    !7 ! ! .( !  &-   ar8M .0ar8 &  ! -, -  - + ! "(  .\,a !  "!-,- !         WM  5  5  ar82M.0ar8 X $  ! ,- - WMa$-
a,- SM, &% (7 !  "&  $+.9  + !   1"!, !   - %'@ -[_ % D43 $  SM !  !   M'  + % a4 &- O,4& . % O( !7! b1& 5 d(  :, SMQ(    U&  X , X ∼ 1 M2 !   a4 .  127 $+.9 p n  ' ! [, ZE 5  G1'M7 % $(1   

3 2

ar 



XD (TN S ) ∼ ηB

L

2.5TN S mp



3/2

∆D TN S

FE 5 1 

∼1,

  1   2 ! 9. %'    !3- !M ∆ = 2.22  U "dM
 $( % WMG(   %    -   M ! FE 5 1) % $(1  I7 η D= 6.1 · 10−10 . " TN S ≈ 65 U . 6  .SM0,   +7 % .- ! w+ $   ( ! T- % 9"62  .0 & )  .7w"!-,- !      - &   FE 5D1 ! 9. %'    !3- !H  3+      1  2Q3 -.0 %  K . U "M5    & M (   U$  -   1  $1 4% -  % a4 0(  .71R!,-      η 5   +   M(     .23T- % 96 23 -   1  $1Q3 - + ! "(  . ! - %1 ,4& a !   , % &-  ,O   $ ,- -4% 9 !  $ =1"!-, !       4  5 % [U$ a@, ! -,9    % 9"& !   , ! -, ! =I3-#U5
&% 
   $ ,- -4% 9 ! ~1.  (  .w, $M!7    !  ! X3 7  T ≈ T ,  *1Q3 4  M  H ! @w %  FE 5  G18M`+ % - ! [7 % $(  9M(  

TN S  516 A=2 Z=1



B



B

NS

X4He ∼ 1 ,

,  A ,    .9..0 !  $!7 2,  3$a23 % _ a.$3    SM*,-. % O(1  ,- !  P!     ! ,M ? %  5+  U& wk!*26   w %  FE 5 G1 %  4   A = 4 M Z = 2 M g = 41 A (

X4He =

Xp2 Xn2

3

· 8ηB



2.5T mp

9/2



∆4 He T

.

 _M(  ~7   ! -,~,2 % Q!#  - % 96  M(   !7     ! -,  :5 5 G1r1'M  . (   % 6 !       ! `8  X ∼ 1 M ,"!Q.0 !  $ !7 1" !     ! ,5FE (    ! X M !   4a   He .  1X  'p .3  ' !  ( 4

Xn =

1/2 −3/2 X4He ηB



−9/4 ∆ 4 2.5T − 2THe . mp 7    X4 ∼ 1 /, He



+  $ , % , 1"- $!   ZE 5X1, ZE 5 G18M  (   +.9  + !  ,M+7 % $(  % *$ a23    

3/2  23 Z− 21 A−1   2.5T XA = η B · mp ' 107.4(A+2−3Z)





∆A −∆4 (A−Z)/2 He T

FE

5 X 1

! -,9<-  .9  .  

∆A −∆4 (A−Z)/2 He T

,

FE

51









  


*

 

  
 

       
     

A 95   2 3 E 5 E ≡ K35 K 3  4  ≡α E 5    -5  6f   75  7f   K35 J 7   K35 K  8

J 95 12 / " % 1 5-(  ! a$ !$!    E + % -H &%



i ≡

∆





 

 

  

∆A /A ∆A /(A − Z) -5  35

35 E  5 35K K35 K K35 K  5  35    5 J-J 35 J  75  E 35 -K  35  5 K   35 E K35 J E  35 K  ,  ! ! .0  43 $ "- % 9 ! 43  %  (   $  % 9 ! 43X7  1,"& 9 !  $ 9 ! 9. %1    !7- !*G1 % _a$.2 3       U " 1&5 

ar  7(  & %'$!!   ! ! .( !      U .   !$! 1 % 9 ! a$ .  1  -  , $$ ,$  η = 6.1 · M . " 5  & M (   ! ! .O-.9"! -4%:U .   !!7  !*,-      , 4% ( !  10−10 T = 65 U   $ a    ,  !  U!7   !  !7    !  %  *1 (A, Z) M  5  5 ∆A /(A − Z) 5   + % _a.$3     U$  , 4% ( ! c?.  ? % 9 ! m %  1 4  M   !7! V U$ w/,-V" &%1!! Q,O $! -, !  "!$    !!  U$   *175  ! a  ,  !  %D! - %12  % 9 !   ,  !* !! 43 & - % 9 ! 43  % D(   & - % 9Q& ! 43R  , !$!   + % -H -4% 9 !  $ 9 ! 9. %'    !7- !*G1 % _a.7 23    I7,   !  ,   %   E 5 -56   % 9! qU$ v8 !7!   M? % 2(7 ! FE 51  ! .7Q7 T = 65 . U " ( M M M M M M Xn ∼ 10−79 XD ∼ 10−79 X H ∼ 10−118 X He ∼ 10−51 X Li ∼ 10−78 X Li ∼ 10−116 M 5     .   I        "   !     7 M      1    ,  ,  0 .            ! $!  ! $!7 X Be ∼ 10−55 X B ∼ 10−69 _% a.23    )   $ ,- -4% 9 !  ? % B7  1, !$!  T .0= ! T N$S!7 1"   4  5    $  ,  ,  !  U$  +,O $  3-&% 9 $ ,75y")"&   ,43 M7 !$!   @w %  5 1.  * 12 /+ % o M 5 5   U$   *<a % O1"!-,:,  9  M FE X C  1   &%   ! "6    + % .- $!   0+ ! -, !  4   %   ! ,   ! :(- % 962 !7$ , ! $&  7 ! -,  ,  !#:, %   9m ,  * 12 /  :- %1H  -[_ %    1G1'Mi ! , 4  5   ! .0<2!   %  E 5 C,-$ ! MK(    * 12 / ! .- & 1"!,:,  9   , +,"D32(1 $ ( ! 43   .2&   3  $(1 $     %' % _a.723c='>[Y XBDC*EGF KJ     *( 4%  7!   +,G3     6 f 8-  3-+    +.0>2!& ! ( $! 9[? %  .9 !  !3 17 U& 23      23=1Q!-, !   ,-"!#.- !  % 6 9C,   . 4%  !   a 4%   &        MG(     % 9 !  7#+, %'$! @  1, !!   .-[ !    a 4%   & ~!  aT-  ! .(1 % 9 ! a$   &  !   1&5+  U$ w)  (.9       ! 43%  .7 ,- " 4%'!! ^+  *10$a %' 8 !7 +#3+  5       !     .7w,  !!  " 4%'! & ! X T ≈ TN S $  , &  !  1Q!-, !    1 4  5 ")"& ,  43WM &% m#,-C" &%1!! X T > 65 . U " ,  _ ,     &%   9 33(  .0  1, ! -, $   > -! "6 !  .1Q!-, !  4  M  H  ,( !  ! 9. %'    !7 !C7-  3Q& + %  %- %'$ ,  -.723 -     $1Q3 M8  .0 % 9-. U$! a$  ,  !) % 4  ! $!    , 6~ $
U!7 a$  ,  !w,0   -  5   ! .00+
1Q!-, !        w .0  33$& ( $ .0  1, ! ,   M!7 ,-!.- ! !'& !*  aMD(   4  1"! $w!   ! !    ! -,     ! -,M12,
  ! % 9   - a$ $!      -   5+" U$    &%1
! 9. %1    !7- !27-  3Q&  l  m e$ack4dLm nLoq`UlL`^e4r  d dyhLle4rLk4d mL`sagrLdyn `qd_vk x l e&nLnL`sp  ac`gf `gn nLj&p) d  se k4j gj mLk&jrj$n  `snLk4l e  dLd _ `slAx)nL`gp |`gnLb P`zd*f d hLj&l  _r*e η ∼ 10 k€ `&€*j$mL`gnLbG|1e fo€ i_lLj  j&t& le gv$`gk&a { x gx  _-e  d |`gn nLj xGk4l  me&ack&dLmLnLo  lL`se&r  d   d   _`gl   € { B

3

7

3

6

8

12

B

−9

4

12

7

-


 0




 -




 



  
 

+  ,
1 !!7 " 4%'!!   !*.  G-H.9 5 +&U - 


   q
     
  
 


        .  
 q 8     C
8>         4  -  T > 65 
" !,  26 !  TU$ ak1"!' &% >-   &%  ,-!# $ " 4%'!! D, U #3$ ! 3. %'    !7 & !*2M  5  58,    M76  6 $   ! .(1 %   ) % 9"62a$w,!#:,
+) !7  t M+.0ar8 $  %  - + ! "(  .I,a !  1Q!-, !      -   5.  ,$ &% NSSM -     $1 " $&%'$!! Q U$   % O-.0 %  T ≈ 70 . U " 5?"    , &-& ,-   75JG1  % $(1   NS   8 5 G1 1 MP∗ l tN S = = . FE 2 2H(T ) 2T +,    1  $  T ≈ 70 . U " !   o1N S 62 ! N S " 4%'!7! O, %    =  % 9.9
+,  71T &% 7 7,- $ .$N3 S( $  ( @   ! # !    ! M1(_   .9 % 9-.= &%1  !   $-  ! ,!   $ ,$  ,  N $ ,-B % "!,M   23 U    .  , ! N,-. % XN7#+, %'$! M  :5 K95 JG1 5'"   ! % 9   - ,1"- $!   FE 5 G1    G-H    4/3 r a  5 G1 MP l 7 11  ∗ M = g =2+ ·2·N · , N =3. FE √ , Pl

1.66 g∗

∗

ν

8

ν

4

- 8H,-"!#1 & " 4%'!! <, U #3$ ! 9. %'    !7- !*   $ , %&  ! . tN S = 121  ≈ 2

@$ $!!   ,    !, % &     .9"!*  9T1   $  ! _ !7!  $ 92  ,-( ! aua 4%   &  ,-/" &%1!!  5   & ,- &% 9 ! M 3,$+ , 1"!l &%1 E 52M(   -      !     .2& WM7  3+,"62  ,   ,-( !   % "!#  M7 - .9  + $   ( ! [ $ M1,- 3  ! WM8,  $! -, ! M8.<1"!-, !  a &%   & 5W+  U$ w<7 &%' !*,  6 $!  V! 9. %1    !7- !* 1.  (  .7 , ! 1  1,"62 $ V!    !  # =   G-C  9    !! =, U$ 23  *1X35   !  $!7   a 4%   &   T = T   $ , %& 7 % -,- ! D  .0 !  !7 17 !   Q& NS ! -,M 

1 n4 He (TN S ) = nn (TN S ) , 2

.0   1  1M ,  ,-  (    *9M  ,  *!  !   0.  !  $!7 1"  <   ! -,>.9. nn (TN S ) 1 ≈ · np (TN S ) 6 -!  !   Z E 5 



−

tN S τn

≈

1 , 7

ar    #' @, %   - 6 G1  $(_  W.0 ! ( ! a$
,   $! [-=! !  !  $&  7 !  τ ≈ 886  5'"   ! % 9   -  % $ (1  n? $ -,$ + %  4     '<, $ 3 " ! , n X4 He =

m4 He · n4 He (TN S ) = mp (np (TN S ) + nn (TN S ))

2 np (TN S ) nn (TN S )

+1

≈ 24% .

ZE

51

  $ M(  
? $ -,-  + %  a 4%   & T!*,    .9.    $!7  ! 9. %'    !3- ! tN S M  .VD   !7 )!.9 % . !   7 ! -, Tn   :5 ZE 5 1r145    U$ V, 4% ( !  ,  ,- (  &   *9k!*,- 7    U    .  7, ! a$D1O( 4% ) 4%7 ,- &- .723 $  $!   ,-* ,=  ,-($& ! > % Q!#  5D+  U& w2!#   !   1   & 1 ! _ !!  $ w  7,-( ! a a &%   @.   &  %  U #33 ! 3. %'    !7 !*35   M ,  ,- (    *9M "!,- % $    &%   9 (7 &% c 4%g ∗&  7,- $ .$3 !7    !    :5 FE 5 JG1 1'M Nν = 3 M   DN & , $!! Wa1 ! (  9N7 $ 1 !& ,- 




@

8

3 



 $  
  





1 $ -,  ! a(7 &%'$!! 43 - SMi &-$! $ON 23 ! % % 9X-N $!    ! 8  & !   &%  @!. ( $ SM},v1.9Q3#.0  43B,-!X- !    , %'!   ,R  ,-( !   % Q!# ! -,43% 4%7 ,- &- .23<.9 !$!7 5   =a1 ! ( !   ( ! @w % $  , -  ! X3 U    .  , ! a
( 4%  +- %1!  -4% 9 ! 43  -, !7    ! (1 %  -!  $ 9 U$! a$ ! -,43`(1 & #. ! .(1 %  ! 9. %1    !7- !* !7 + % - ! W  , 62  9R %  !  &  U! aWM    , &-& ,$ON  V ! w   !7    ! M " 5JG1 ZE ∆Nν,ef f ≤ 1 , T ∼ 1  U +&U - 
      A 7        q    3  
     7     . P $     7    8$      8  g∗(T ) $ $  8

H NS   X8$      8 
 ! , !   !7 -   4% _ !!  $ 9 , -  $ (  .0O    .9"! ! u1   & 1 ! _ !!  $ [a &%   & C,->" 4%'!7!   ,  ! !  @!$ ( !  & 9 !   $!   ,   $! %-H! !  !7    ! 75 















!





9 79 ' % 















 



'3* - 9% 







%

"   DN   1Q!' 4%'H    % 9"!-, !    1, ! , $! a I3-8m % % $  ! , %'! M (       !     .7C  ,( ! a$ ! 3. %'    !7 !*
$  , $  ! ~1"!-, !   a 4%   &  &A(1 $ 1&59" U& 71"!' &%' S  .'  .9 $  U$ 235  .=H $! W &   ( !   .0 α !  $!7  % _a.723    SM  5  51   $ 1 ! _ !!  & 9   ,-( ! 43 33( $ .23 U&%1  ! &  ,H,->" $&%'$!! 5 "4 $!!7!   ,   *DN  7"!' 4%'G! 1, %1!     .~!,- %7$    % .-=  9 , $  ! -, !       !     .<  ,-( ! a$ ! 9. %1    !7- !*w 4%' $VN2 U& 1:( -5 isL 1"!-, !         M ! .(1 %  ! 9. %'    !7- !*75 p(n, γ) 35Vi (p, γ)3  M i  i M n)3  M i  i M p1  L   +,- &% 9 !     .WMi7#a  , % ,R& N   ? - 1 %  %  1Q!-, !   4  5 75 i M M i M L 1"!-, !   4  5   n)4  3   p)4  5  (α, γ)7 f  M 3He(α, γ)7  L 1"!-, !  A! - %1}   -[_ % 43 U4%'  $!7 ,2  7,-( ! Q& a$ ! 3. %'    !7 !*35 35 7 L a $!   7 f ;5 Li(p, α)4     & M"(  M  .0 % 9-.@ !7! , !  $ 90  . -   4 % &-  .0 !  !7 1   $  % & .,-ON 23      WM      + , $ 3^1Q!#  6T_ !7! 43   .  ! , !   , %   - H- MG %  .0  43o3     !  !>.0 !$!7  , % 7$  ' $   ( ! m1   & 1 ! _ !! \,   $& ,-( ! X % "!  M  5  5 p M n M D M 4  5 z $    U&    .RV  +.2   &% 9  !   9  .0 & Q23  - .9 !   ,^1 !! D" $&%'$!! 5 1, !!   U& 23  .07 $  @- n1  62 !   " $&%'$!!  !   $!7=! 3. %'    !7 !*2M . " H(TN S = 70 U ) = 4 · 10−3  −1 , !,- % 7$R!   [ $   ( !   F? @DY\" F X\?=F P?w.0 !  $!7    ,-( ! 43`33( $ .$3 U&%1  !3 -, ,-=" 4%'!7! SM .0    M.0 ! ( ! M*! ! .( &% 9 !  , 6  MW(   - MW(  w % $($& %   95[,[ 4% -,-  3 33( $ .0a51, ! -,    5 !


 -


J


 0








 

 





     





p + n −→ D + γ

@.^ ,-$ 4% SM   -  7 - + ! "(  . ,a !  "!-,:,  9   7 -    1X&

 $   T = TN S ≈ 65 . U " 5   ! .9=" 4%'!7!   '-,- % 9 ! 
 $ )1  62 7$- M U$ wM ,@ !   M$( $ 9 !   7 ! -, X- $!    $ 9   a $ 9  5     &% M9.9.9  + %C! $&  7 ! -,     3+  ,     5 %TU& a$  , ! O !7-$! , !  $ 9  a !    !    ! -,  -    6  !   " $&%'$!!  ! [ !3 tN S 5  ( $!   1"!-, !      - 7  .- ! 
a$-> $!   9=.9.%a   $ 7(  .0  M  3 1 α 1 −18 (σv)p(n,γ)D ∼ 2 ' " = 2 · 10  , mπ 137 (200  U )2 ar  m L ?  
 ! 2M3-   4%  VN~  301. -  ! X & 1 !7$ , !! %1"!      2& M+> $  !!  <  ! .0 $ 9.  $ α $(  :, & 8,"& ! 43,π!   $ ,SM r ∼ m−1 π %'$!   M  ,  ! !!  }}! #3+ & 9B  2  .9 !   @   ! 75    & M$(    $!!7!  } .  ( !  
!7 !,-     > .0 &-  $  % .,VN223  (1 $ SM  5  5+ >   1  2  % "!&  5 "    & ,- -&% 9 !  &  U&  !H .SMz -   1  $ !   -1,-./! !   V ( !   5 ,   aM  .9 % 9-.  *    - & "!-,- !          , -5 X& 1"!C    , %&-H&%   ,  !* !! M0 &- T'-∼T%'N! S -4% 9 !   .  H8, %'$!   ω /p M -   4%  < ! 6 $!    , &% ( !  31. -  ! aH % 9  >    .9  a @ γ  ! DM M'., &% ( ! 301. -  ! 43O % 9  ,=   ! H !    ! =,T    7  -! "& ωγ ∼ ∆ D  -&% 9 ! ON  a$< !3 1M p 5 6, % 9 =  $ , % VN223 (1 &-        .- &   9 ! D ! ! -    ,1 %  D 5 -K1 p2D FE ' ∆D , ar     ' % .-H %   %   ! .WMS( M  D  -! 1 %\  % 9 ! a$X,"!*   $ ,-    -[ !    !  V   !  1 -!  7 !  %'$! 1 $  !    -[ !  5  .0 ! (1Q& -4% 9 !    ! .9> & ,-D  3 −20 (σv)p(n,γ)D ≈ 6 · 10  . .0 & 9 ,a1 !  


B

.U" , ! < !7O! 3. %'    !7 !*:(    ! " ! & ! -,  -   1  $ w! 9. %1    !7- !* T =   , 6~ $ - o1  62 $!   " $&%'$!! WM =.  , ! Ta 3 M7 , $ &    7-        NS

= 65

ar    ,"! % R.0 !  !7 1    ! -, @   !!    ! 6 $!   N %  -!  $ 9@   ηB 5W+   .0 % 9-. U$   .0 & 9)! ! X(7 -4% 9 !  TN S M   ! `   & ,- &% 9 ! Γp(n,γ)D  H(TN S ) !  ,    - 7 5       *   
      78   2       +&U -
      .   q
 0       
     8  L+ 
       W  L     q      ~               
=+   8  ' 
7      :        ]e&|`sk4d | mLk4jy_ f  k4j:v y` nL`k e4r  η < 10 { 















 

B

−11








@






8 
3  

 $ 







K

    



  



1"!-,,"62 $ u,   ! % 9   - $a  $ !  H!    ! ,  1     =&% D-C  ?  1 %   % 1"!, !        %a 4%   & 75  ( !     .7  D+D →T+p D + D → 3 He + n .- ! 2 $!   9>.9. a   $ ( $ .  M'# ! .0 ! #3-#'[2( $ 9=. %  ! -,  . 9   (  &  % .7,ON 23   *1   %   1 !    % X-H &% 9 !  U4%' .  ( $ . !*"  KM9)  -[ ! w $ D  D  % .,-D !   p5I @ - % 9622351 $  !   3 r  1/m U$     % .,- !   + !  & R   7-& , $ 7 - .3 !    .7 5 -  , $$ ,"2VπN2 U    .  , ! D !  %    & , %'$!D! [  5 E 5 95 

V (r) Ekin(r = ∞) 0

m−1 π

r0

r

−∆A

  5 53( 3  ?  ( $ .0  !1Q- !  uU    .  7, ! a$H -! 1 % T %     ! 43%  .2& X E  ,-( ! a ! 9. %'    !3- !75   N-2!6 % M1"!-, !    *1 U$! a$    ,  ! M &  % .7,ON    * + % - !    + %'& 9=.2 %  ! -,  .%9   A5  & a4 $- U$ H∆,
A  ! % 9   -  !!4% 9 ! a  $ 75     $!   9  -  , $$ ,"2VN $w ( !   Mq    $_[,   $ wc $!7  ?  +,G3 $  % .,"ON2$3       !  8 Z1  Z2 MW?   M1  M2   .9 $   v1  v2 M    , &-& , $!!   3-  D$ (    !7-        a$ !   q   -   MI %  8 % 9 ! 6  a ! % .- !  I! " # & 7 %' ! ! \ !   9 $ ( $!     . a$ $!   ,NN   &% 2(  145   !& ( $ .9 [U$! a    &- v# &  & 9 ,$ E = M v 2 /2 MXar  M = M M /(M + kin 1 2 1 L     7  , *  _  ?     1 M  L    9  0 .      9 

 5   ! !  $    !    &  % !    &     M2 )   %   #8   !!4% -,- !  v =U . v 1− !v2! 71 % 9 ! =8, %'! CX, & ,- %'!  $ ,-$ s !

ar @ (7.3C-,  

A∝

r0



−

R r0 0

√

2M (V (r)−Ekin )dr

-   & %7$ !0 4% -,-

,



1 αZ1 Z2 Ekin (r = r0 ) = Ekin (r = ∞) − V (r0 ) = M v 2 − =0, 2 r0

 E 
 0

 -
          (_ v L  !    -&% 9 !  I .0 & 97#8 % _ !! 43v$a<  *$a4    
?,\-.9"! -4%' U .   !$!7   %  $( -$! M(   r0  1/mπ M  5  5   !7!&% -, !   -  3-+  ,< !3    $-    *1 1X8"  *22a ! V*$a4G1&5 .O1"!M -.9"!* -&% 9 U .   !7!7 S2& ! ? $ ,-$ 



Z p − 2παZ1 Z2

"   ! % 9   -  %X ( !   a  ! 

 

r0 0

σv = σ0 ·

ar  σ L a   &  (   .0   ( !   "40 1"- !   ZE 5 E 1 !  #3-#' ! X % Q!#  (

1 παZ1 Z2 1 − dr = − . r r0 v

  X % 2(  

σ∝

 % WM  $(_  I   U .   !!7 ,M



−

7-,-( ! 

0

2παZ1 Z2 v

2παZ1 Z2 · v



,

−

2παZ1 Z2 v

5 E 1  .9a>8, %'$!   5  .9 $ V,w  ,-($& FE

,

   .,C -  $ ,-  .  %  ! -,      !   9  $(_   "!# 7  

hσvi = σ0 · 2παZ1 Z2 ·

p 5'6 π/2(T /M )3/2 Z

r

R∞ 0



2

− M2Tv −

R∞



2παZ1 Z2 v

2 − M2Tv

vdv

v 2 dv

.

 !7- a1 % , ! !  $!  &%' '%  a$.9 , ( &%7$- !7 a1 % ,
( &%   &%' ,-!9_q $  +I    , %  ∞

ar @ (7.3C    , % 



2

− M2Tv −

v0

2παZ1 Z2 v

vdv ≈ v0

s

-   & %'$! ^   & %  ,-  

0

2π M T

+

4παZ1 Z2 v03



−

2 M v0 2παZ1 Z2 − 2T v0



FE

5 G1

1, $!

,

M v0 2παZ1 Z2 = . T v02

"   ! % 9   -  ZE 5 G1) ! ? & ,-D

2 hσvi ≈ σ0 · √ · (2παZ1 Z2 )4/3 · 3



@.0 ! WMi,,-  ,  a4 -4% 9 !   , &% (     A¯ ≡ M/m  -   1  $M12!#   7 w$ p . " M!*6  S-.0 ! ( -4% T /(109  ) = T /(86 U )

M T

2/3

·



− 32 (2παZ1 Z2 )2/3 ( M T )

1/3

.

!  i(   !    -&% 9 ! $ 7,  *_ !! $ ?  $   ,  %7% 18Q3kaX  -,  &% 9-,- ! 2M T ≡ 9 9!

−2/3 hσvi = 9.3 · σ0 · (Z1 Z2 )4/3 A¯2/3 T9 ·



−1/3

¯1/3 T −4.26·(Z1 Z2 )2/3 A 9

.

FE

5 G1

  &  M (  w,H % $( $!!  ! D ! .   % hσvi ! % .- ! =  + % X- !   MW(   σ0 ! !,-      2 % 9  -, &  % .7,ON 23  (1 & #,/ !7  $ $ON   !    !7 , %' U!7 a$ E ∼ 10 − 100 . U "M5 @    &%15U&  !C .SMi $(_ D!& 7,-1 % 9 ! 7!,-2&   $ R   U .   !7!  % 9 ! a 
.  D, FE 5 E 1@   % 9  ,   5  5  D .0 &- 1 (1 & N,-+  , $$  FE 5 G12. - %'$ a!l+. ,1"- $!   :5?" (1 &-!  $ WM %   ! a$23   . % $+22VN2/,. % X/,   U .   !$!7- ,X-.0 ! (1 -&% 9 !    , $ M !*Q&  !  ON    -X,1Q- !   ZE 5 G1lM1 $  ! $ $-  !7! $ !*,   $ 9H 5U! a5 




@

8 
3  



 $



    



"    }&% 2(  ,,1"- !   ZE 5G1 ,--N :! 1"-   & MG(    ,  !* !   ,. % X8">, $ (  & !     -^  ( ! 43   ! !  !! 43   $  ! 5 .9 ! SM2.-
 !   ! -,43     .- &   3+  9 !   % 9-.0V, s& ,- %'! M L !$ ,- % 9 !  $a % -,-  $!7 8_  ,-. % XI, U    .  , ! [ ! 71 % M &   N B,   !7!&% 9 !  U .   !!7 5    (1 &  ,#'  . + %'!  -&% 9 !  ,. % X8" ,W %'!  <$ ( $!   M:  ON   ! $ &-  !! $ !*,  2&  & 9V-.9"! -4% U .   !7!7   V-   1  $,Mb(   , FE 5 G1 5   -  .9   ,  U&  * !  & SMqV,    , &-& ,$ON 23  & Q3< %  % $( !   ( &%1!! 43  ! .:".   .0  .  -,  9 @w %  ZE 5 G1lMq7# $ , %  ,!* $!V! _> ! -, ! V,-. % XW,  .0 & 9 .0 ! . $-! <  .5 ")!,1"N2  9.C,--  ~0a ! 7
   -   ^  ?  7,    . &%  7!   +,G3        - 7  M D(D, p)T  D(D, n)3  MG !  σ a$-MG  7 % 9!  30. -  !  0  $  !$ , !7! o?  6       ! 43   % ( −26 
σ0 ∼ m−2 π ∼ 10

"   ! % 9   -  % a  $ !      - 

∼ 3 · 10−16




−15

   D    $ " _  $a  &  9=% &% ,-





3

−2/3

· T9

·



2

.

  A¯ = Z = Z = 1 1)7 1 2

hσviDD = 3 · 10



2

% $( −1/3

−4.26·T9

FE

.

5  X 1

5 1     &% -,   .- ! w,-   % 9!-,  9  M (   -   &%   9 .0 !  $!7 1"    - 7  ! [ $!7    ."N $!     . a$ $ !   5+, T = TN S  T9 ≈ 0.751) % 2(7:( ZE

ΓDD = nD (T ) · hσviDD (T ) ∼ H(T ) .

nD =

H(TN S ) = 1014 hσviDD (TN S )



−3

FE

.

5 G1

  & M2(   U$  .0 !  !7 1  +-,- % 9 !  &% "- !,-      1"1 $ 1 η M0  .9 % 9-. ,"3-#'  ,X,1"- $!    % n (    ! -    1  $ T Mi T !,-      η % a4Q& η  0(  .7 5 1, ! 7,  n  .0D !  !7 1  D     ! N,MS  $! NqS 1   $ 1 ! _ !!  & 9 D   ,-( ! aT   - 7  ,->" $&%'$!! (  η = 6.1 · 10−10 . 5 D1 nD 1 nD = · = 0.3 · 10−4 , FE 

B

B

B

np

0.75ηB nγ (TN S ) $! &-~  , 

B

  !  ! 9. %'    !7- !*  .- &   9 ,   $!      -! "6 !  )! !# !2( ! H % .9,  ,-( ! a$ a4"!2  _  1, !!    !7-$! , !  $  %  ! H! % $( !    7a % N !   1'M'  ,  (1"ON2$3      X,-+5  % DN & , $!! MP(  O!& !* 4% - !*,-   &  n   η  -! 6 $!   .0 !  $!7     -   O   ! -, ZE 5D1A1 -!  - !  % D9 !  η M8 U$ w U .    7 !7  % 9"& !   !   $!   1   $  ! _ !7!  $ C  ,( ! a$R   -   7!,- %& -   4%   9  %   & !  & 9^7 ! -,5 +    $*_ . 1     $!   N _  !    .7  $(1 &       -   ( D + p −→ 5  ( $!  ^U$    .7:! $! m $! 96 >$ ( $!   *$a$23:  .  2( $    γ+3     -   M 1 $    !! 43W, 6  5     ,  !* !  H!7 #3+ $ 9     .9 !   '- % &  @   ! 75+  U$ w σ ∼ 10−21   3    !  &% 9 ! a 0   3 −2/3  −3.7·T −21 hσviD(p,γ) He = 8 · 10 ,  · T9 · 

B

B

3

−1/3 9


 -



 0


 



 



     

ar  I !  % wN 2 @w %  FE 5 G1z^$( % SMD(  0 % 1 $ ?  ,  )  .7 M 5 A¯ = 2/3 Z1 = Z2 = 1 .0 & 9 U& 5  .)a$ !         -7 !  % 9 ! 
.0 !  $!7 w  Q& ! -,C,C % Q!#  Γ = np · hσviD(p,γ)3 He ,

  %  η = 6.1 · 10−10  T ' T  $ 91 $   1  6  $!   " $&%'$!! 5      . NB$  ! -,- / DN $ , !7!  NS:' $   ( !  - % 96 23 η  , U$  1"!l &%1  !2(  s!,-   $ 9 !  % 8  43\, &% (7 !   U$ a11 & 11&(K(  s % 96  .0 ! &  $!7 1"   ! -,
M - q- % 9"6     -   c,:, $-  5+, U$   aC $   ( !   .0 !  $!7   $ ! 962 $  M3 $   ( !   .0 !  !7 1  3  M !   M9, 4% (7, $ &  5 B

      

 

∗  



3 

3

Y 4%  2&      SM"!-,- !!   , $  % . ! , !   3    
      M  w    a$1  , a 4% 2& 5

%  a !   a &%   & /,   .7 3  &    $! .9 FE 5 G1 !7 He + D → p + 4  1   $ 5+"4 $$  ! _  .0 & 9k  .7 $ ,-$i(   3 −1/2  −1.8T 5  1 −15 hσvi He(D,p) He = 10 . FE  · T9 +, T ' 0.75  T = 65 . U "1 U&   .07 $ 9W  , 6~ &  .0 $ 97a $!         5  .1 .90ar8@ $! -, !     .7ka !         ,0! ! .( &% 9 !  &-  ! =  .1"N~X& FE   M a &% $&  + % -C $ .  , !  ,a  95 Y3 $!   a &%   & w  .1QN~ &-  ar82M .0ar8 .0 !  !7 1        2 $! 96  -  !  (_  1  62 !   " 4%'!!  !  $  % 9-.0M (   -    . 3He + D → p + 4   1, !   />  1  6  $!   " $&%'$!! 5   >!D*_  ,^ !7 ,   !  M  .0m(   t 3

−1 9

4

3 He

5 -JG1  _ TN _   9 6 ( & ,- a &%   & M+"&    9    -C $ D   5 @ !  ! &  "  ! k3     t ' t M ! $&

hσvi3 He(D,p)4 He · nD ∼ 1 , t = t3 He . H  !7 <,-w" 4%'!!  O,    &-V   .0 %  ! - %  . 7  D + D → 3He + n  ? %  w,   7 !  $   + % &%    .9   1 O  !  1   :,- $  0 % -,  .9  ,  

 U$ w  



 .0 % 9-.)  & M    a , &- , &% ( !  



FE

3 He

5 K1    ,1"- $!    +_   ! .B &   ( !  .0 !  $!7 1" a 4%   & M 7  .0 % 9.[    9    -   ( ! 9%? % MqV1Q!-, !   a 4%   &  ! -  3+  5 6! t  t He % ! $ -J1M 5 5 K1M,-D ! MS(  X -! "6 !    &   ( ! 43 .0 !  $!7  a &%   &  <   - 7  FE ZE  $! , $- , 4% ( !  hσviD(D,n) He 5 1 n He ' , FE E n hσvi (_S1,$ (1 & 9 ! $- ! >,(D 4% 7 9=,
He(D,p)  $!7 He,   $!  t 5    1  $1H, U$    $!7    &% 7$- W - ! 6 !    FE 5 -JG18MUN ! %? %  % (1He &-   X   1  2  k4j$|
l e  _`gf `U|o achL`  d e fb&nLj$`x$nLd |1e&nLdL`qvs_`cf `s| se4x$d acd|j$ack&d)ne4t4f _e4`^|o  x&`gf dLmLdLnGj$kh e4l e  |`gk&l e η dGhLj k4j$|vyhLj sx&j4f  `s| ac`gt$` l e&asag|1e4k&lLdLx$e&k4b ` sj xrem `sack&x&`Uacx$j&t$j^_nLj sj €  k&|`gk&d | j^_n e4rj m k4j j&nMx&`^acb$|1e  ack4rj  dLr*acdLl j&x$e&n t&` sj&k4n j$acd k4`cf b$nLjPr nLvr*f `sj$agdLnLk4` gv d |` iyd |d a  r*achL`slLd { |`gn ke4f b&{ n o  { |dM_e4n nLoU|d hLj:e4nL d sj&k4l j&hLdLd lL{ `gf dLrLk&j&x&j gj) d cfvm `gnLd  € n3 He ∼ hσviD(D,n)3 He · n2D ·

1 , H

t = t3 He .

3

3

3

3

4

3

B

FE




@

8 
3  



 $

   

 



! 9. %'    !7- !* TN S = 65 . U " 5   & ,- -&% 9 ! M %' ,  ( $ 9 , ZE 5 -JG1   % 9 ! ,H" &%1!!  ( 3

n3 He ' 0.9 · 10−5 . np

 !  % a( "!-,- !  

!  O"!m1 $ ?  , & - a$ $!  y     M T+D → 4He+n 5  ( $!        L  .0  -  M1.9.) %  3  M'  .07 $ 9ka !   , !    3 −2/3  −0.5T −15 . hσviT (D,n) He = 10  · T9       &   9 &% "$ !*,   $ 9 U&   .0 & # N-   1  $ ,N !7-  $!   %  $  T ∼ 1 5 .9 $ 9ka$ !        - % 96  MI(   a &%   & M17 U$ w  a$5a$ $!     .1"N2 g$  N _ !-  MD(   a !   3  M3 $ "_ -  a  NT_O ! 9"6  5   ! .9  -! "& 6 $!   !.9 % ( ! 43D.0 !  !7 1      %   -   7-!,#' - D .<-  M'.9.% %  :5 ZE 5  E 1p1))8_  ,C.0 ! ( ! I  a  3   7 η = 6.1 · 10−9 . 5  G1 nT FE ' 3 · 10−6 , −1 9

4

@.  -!  6 $ !  

B

np

M   . ! 6 !   n /n 1 ! C- !  % 9 !  η 5    $  , !. % O( !   M(   !7  ! -.1 -! TT3p7 ! ,"62 =! 
.  M2(   ,
"&  $   7,-( ! a ! 9. %'    !7 !w1.  ( $ .O, $@ ,-# !  
!7    !   ,  !:,  ,  *1 4  M !   w2  ,  !* !BV- MV(     .7 a$ !   a &%   &         $I   $   M(     .7 23B1"!-, !   2!    - 7  5
4%   U$  T ' TN S  %  !  .SM    - n ,a &%  $   ,  a$M   .7 3He + D → 4 He + p M  .   %   9< !  U$   M .0ar+ !    !   % 7 , ! -, !  T + D → 4 He + n   ,  !* ! #,  *1X3)a &%   & 2      M' ! ,  *1X3 4  5 *22a$ $  ! ,M ! &% 62.0 ? %   ! ! .( $!   !*.9 % ( ! 43<.9 !  !3 17 3     ,  !* !  
 M*(    .0 & O23 a !  \ 1, !     .0 & 9 a !      -    T ' T MW3-    ! .0 % 9-.0 , 6   _5     4% _ !7!   1"! ! -"!  % _a.723 U4%'  $!7 -,M11"!$NOSN2$3   ,[  ! % 9   -   ,-( ! a ! 9. %'    !3- !MP 4% -, %1!  +-, % 9 !  4% $(1 !   ,-1X $!    $ (  & !  -      ! 43   .X ! !.23 U!7 a$  385






n3 He /np

            

∗  





B





  

   

 



    



" 9. .( $ $ ,    1   .7  $(1 $    ! - %'$   -[_ % 43 U4%'  $!7 ,S  7,-( ! Q& a  ! 9. %'    !7- !*`1 $    "!-,- !    a$ $!   7 f  ,   .  3  (α, γ)7 f   M - , , 5 7f  (p, α)4    $-& $!!

%    . 1"!-, !  @!  % .3-d1"-   & @w %  51lM*,A.0  7 σ0 ∝ m−2 π ·α .         ,     9 .    @   ! G1&5   4%'!7!  FE &  % ' %   !   !   α   3 −2/3  −8.0T 5  G1 −18 hσviT (α,γ) Li ∼ 10 . FE  · T9 7

−1/3 9


 -



 0


 




  

   

  ,a !    7      U$ w .9 !  % 

 −1 , T9 = 0.75 ? %   1, !7!   1 &   % 75 %    . a$ !   7 f  @w %  FE 5 G1  .-  -   & M' @ _@NT9 .- ! C % $(  9   3 −2/3  −8.5T −15 . hσvi Li(p,α) He ∼ 10  · T9 " U$ q  .7 σ  4% -, %1! : &% .0   % 9 ! Q,!*#   & ,-  SM? U& w  _  .07 $ 9Oa$1"!l+},0 6 ? .07 $ 
1"!, !   FE 5  G145   &%1!!  - 
,a1 !   %     & KT1, $! hσvi Li(p,α) He · np ' 0.7  −1 , T9 = 0.75 , ηB = 6.1 · 10−10 , (    , 62 $ 1 $     % 35+  U$ w[a$ !   7 f   .1"N2 $  +-, % 9 !  7!'I& ! M.0ar+  %  !  & 927   ! -,  DN & , $!! [$ ! 9"62   , &%1  & ,-  1  62 $!   "  & %'$!!  58" U&    !7  -! 6 $!   .0 !  !7 1 %     &gK>      !1Q-=,- $  ! T$-, !7    5 ZE 5 -K1r1 hσviT (α,γ)7 Li · nα ' 1.5 · 10−4

7

7

−1/3 9

4

4

hσviT (α,γ)7 Li nα n7 Li ' ∼ 2 · 10−5 , · nT hσvi7 Li(p,α)4 He np

 U&  wM  $ (_    FE 5  G18MI1   & 1  ! _ !!  $ = 9   ,( ! a$ %     &gK[? % M " !*. % d( $!     ! ".9  DN $ , !7!  3  5

!





 a^!*  R, 7 f  M  .0 % 9.9Q& % "!   !    -4% 9 ! ,? % 

 $  MI(  ~ 1"!-, !   7  M7    3-#  N % 6T9    % 96 23 ηB & 10−9 ML.0ar8 ,A

1 % (*  % -/ 

n7 Li ∼ 10−10 . np





%



% 4 . 

 

%





 

4 ./ 9 





* 2



     ,-( ! a ! 3. %'    !7 !* 3-6  1"!1-   ! 75@
 $! -,   4%'!! 3 1 ! _#"  ,  , : -.0%$   !  $ 9&  %('7!( ( *)+ .9,+ !   1   $ 1 ! _ !!  &- $N, $ 3 % _a.23 U&%1! -!3 -,M'%1,+, !!( 3w,C  , ! %$X % .+ - 5   <( #)+ .3.+* !   (7-,    & - ,/  " + '*% 9    !   % & )'! (  $0+<% %  $ -  " 4%'!! %$SM 33 -  7 .7($   & ,V.9  %  3\ ! % X !(- (    !  -S!7 + -!  %' M !$!-    D!  U , % &17(& " 4%'!! %$ 5 @. -23' -    -!   % S(M - %'A  -[_ %4: 
)  %%1.+ ' &  ,2 ! -, !  - , -  -  
)+ " 3   (. 17  3>, ! '9   +,5_ + ) M  5  5 MG!a  ' -ua-,%  M ' -  , ! "6 '[U #3 ' M z ∼ 1 − 10 576  ,  " ! / ( !(: 
) 1 &%8' -C  ( 7 U&  -B$ - &% 9 !-9-  - 1 %  -:)1%  %'~4% .- ! 3<33 -   " ,   + '*% 9   - +,5_ + )+! % $ U ,- % &17   $ 9  7,- !( 3  .723 U&%'5 -!7 -,5 . - % %1, + - Mq ,  %'>   -[_ 4% : 7) 1M   & 9M ! % %   M   ! @%
" % _a.23 
)'   1 !   - (  '$-  ;   '$ < , - %'   $: = L 1, +> ' 62 $ 7 ) ,-, + )+ $ $ ,- !- -[_ & .0a$ "  + %('7!   M - (  ( -,-.- )  &VN  a$w(  1 $!  +,?_ + ) % %1, +-, !   5+"  k-  %,  5 -  M -  -  7γ)+  (! : ( , " 1 $!  .- )+!   - %1M   -[_ 8 ,   (. 17( $ %  3 
)'    & 2, .3 $$ ,  % - (!  3=(  )4' .    )+ G-C   % _a$.  7) 175 @ -C a$M   $ , % _a$.23 
)+ \-7 %'!7$- (  )4' .   -  )  $ % 171 1   -[_ 4 %  3 
)+ B ) ,- + )' $ & ,- 5 - - 3 - :U$! a !  3 γ " ., !7 -,5}" $



  



 

i  P  q   q  + 



            

 

U$  (%1 $5 ( 7,- ) 7 . + -!$! (& % -.9 % 9 ! %$<1  ( & 1 ! _ !!  $  % _a.23 
)+ X7   , !$! (&  23X  7,- ! $.0 ! 1 !7 11  $ 5 6T9w,>-( #)+4% _ !! 3O% %  $  3D" &%1!! $ % -.9 % 9 !   1   $ 1 ! _ !!  & 9 ! .9."    3 % _a$.23 
)+  ! + -!9%   975    % %  $ SM"a )' (7%1 $! +,_5+ ) -%%,+-,- !   (,"  - .9 %  &%  -( !  - WM ! (- %'$ '>)  % _ !!(: % %  $   % %  $  
! (- % 962 - .1  " ! -#! - N $!   1    % O%  %  $   -  % %$   (7 $ 1 ! _ !!  & 9%& -$  %% -,  3  - .- !  ( , 7!   9M' + '
     .  Da % N $!   , & H  )  % _.$3 .,,+-,G1&5

 $  ( $ +* !  -  $ ,  
)8'W% _a$.23 
)+ Q  % ' &= % 97(, &%'*)+$ ,-   ! 9 ! +.0%$ U!7 a$  ,  +I !Q!7 % 1, + '$ ,\( 7% 1 $ Q3 +,_5+ )+! a$ !' . %'    !7- +*M: ! %%  M   % 9-.0 1. +> ' 6~ $-  5 6   .0 % 9-. '@!7  +, &-!  ! .9.23 ? ' N & , $!(!  3> &  + $ .$3  &  ! .0-, )+ $ -   M   % &   + -  !    a % -.9 % 9 ! % $   ( 7 $ 1 ! _ !!  &  ) _  5a 1 !  7!  )!  + '! M.0 ! 1 !7 1 1( &I  , ! a$ )+ $ -   5  % -$  - M 3  ,M &%1# )+!$ %,  5-  1  (   & 1 ! _ !!  $ 9 )+ $ -    + -  &-     .    .0-7 1   % X- !!( 3 ! D.0 5 -  % a$  .23 1 $  !   3R%  % .9,O%  # )'! _ !! a -$  %%  -  a4, +*2M a % "N~ & " N  a  , &   )  % _.723X.,, + -,75 z  (  & 1 ! _ !!  $ 9)a 4%   " ~ + '
  $-  ~  # )+! _ !(! - -$  %%  -  %  % .9 -  !  " +, !! a~,- ) % 7 ) 2,2.9  % .0-,  3 a4 % .  .9Q3 .- )'!   a &%   " ~, +,?_ + ) Q3  -  -,-, "  - )  $  .- )+!  ! -   -[_ 4%  3 U4%'! -$!7 ,M U&  - '  -? '0-$ ,-   4%'# )+! 23/,D  % .9Q3 L  ,- )+$&% 9 & ,-  a$M 7 5a 4% ( $"    - ,C ! , !  -   7,- !( $I1&5

%   + -  $!     ( 7 $ 1 ! _ !!  &  %     "gK   % 9 + ' &    + 'L% 9   % [   .    .0, " #%  #)+! _ !!( 3 -&  %%  -  &    3 3 +,5_ + )  +,?_ + )  ,   a .9 %'!   1=, 6~, -,  3  .0- %'!   3 ! "6  $
%   $  % 9 (!  3 U&%'5 -!7 -, 3  M 7  M'55$581-.9 !7M! , $ )+!  %  %  $ $=, " $&%'$!! % $WM   )%    '  ,   !7!  +. % &    9M 7   $! -, !   )  %>  )+ G-C"N2$3  )1% .0  %   3 - .- !    -#   , &-& , ' &ON 23 % _a.23 
)+   #3 1 !  %   9  w%,  5 - _ !   ,- 7! a ! ' . %'    !7 " +*35 @ -= aM ) -,- % 9 ! ;    ( 7 $ 1 ! _ !!( $ 3  M ! ( 7 - SM -!$<' , $ ,- -4%'! . .0 ! 1 $!7  1  7 ! -, η M ! - )+ $ - ( $SM1  + -  $!  @ a$  -,  ! -$!! % $1  (  & 1 "  - % 9"6  a$-,-% 7    % 1 $ X3 +,5_ + ) -% %, +-,- !    U ,- % &17~Y7 % .  7.SM 7!! _ !!  & /  )+$  %  3  ,- ! a ! ' . %'    !3- +75  - W     + -  !    !    -&% 9 !( 3 % .3 % 9 !( 3 1  (  & 1 ! _ !!  $- $v33 -
 "  . $ 23 U&%1! -!3 -,W ! -,- ( !  !     .  7  .0-7 Q U&  - ' , $ 9 -    !  5  ! -, !   6 ( -.9~,> + -  ! X  ,- ! % $.0 ! 1 !7 1 17 % _a.23 U4%'!-$!7 ,   $!-     .9  SM !a  ' a-,%   ML%  '&% -, %'!  !5 ' ,   !!  $ %9 & ,    , !  &     $ , !   % & )  ! -2 3 % ,"  +,5_ + ) 5 6@ *)+ .9, + !   %  U&%1! -!3 -,H, -&-   + '*% 9     -  !   % & )'! ( $( , # )+$(!  !    5 E 5 35 " 1 4%  -nU& ;    + '*% 9    !7  % #3-  a %  5' &  )  ' a @)  ' a$ -   , 4%    ! % $ η = M  8% '
$!! % $C +G + -  !  $H !  +  -7/  &% 7.  -,a  + 8% '
!   5  .9, +  ,  9 6.1·10−10 !  !7 .0     (   ,-  !   - - )4' 1    $ 1 ! _ !!  & (- %a 4%   " 2 %     "cK   )+! % $  1    $ 1 ! _ !!  & %9 & )' $     )  !7(! - <= !  +  7-;   &% .  -,-a $ %  !  M8  + %8'
!  D )  ' a% $ $ %  !( M+.0 !5! 4M 1 ! ^L   &-5-    7 .7 C! -  #)+4% _ !7!  $ V,   + -  $!  1    $ 1 ! _ !!  & S  ,- (!  3 U&%1! -!3 -,  N _ ,  9 - O, &% 7. 5"  o-   %,  5-  M  !  )  $    ! )1%/ aM 7 %    %8'
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 $ & ,- &% 9 ! M     . +> $!    
! )   -! % $ -  )+4%   @ +.7   $ ( 1N  ,- (!  $ .   D + , 1   2 6  (  V & N $ "  % Q $      + ,  ! ' %'  !7-       $&%'$!! % 7$ -(  * )+&%   9   % 9.9  )  ! ! , "  j&j&t$P` gj&x$j&l d gx&`^ack4n ow_j&x$j4f b$nLj?lL`c_rLdL` e&agk4lLj  d gdLm `sacr dL` x&f `gnLd  ;_` xGacd*fb&nLjynL`gl e&x&nLj$x&`^a  { nLo  v$a fj&x&d   |j y `g{ k)j&t$l esj&xe4k4ba  |1e4f j&` rj4fdLmL`sagk4x&jy_`spLk4`slLd  €  



!

B



B


 -





 0


 

Baryon density ΩBh2

0.005

0.27

0.01

0.02


  

   

0.03

4He

0.26 0.25

Yp D ___ 0.24 H

0.23

BBN

D/H p 10 − 4 3He/H

CMB

10 −3

p

10 − 5

10 − 9

7Li/H

5

p 2

10 − 10 2

1

3

4

5

Baryon-to-photon ratio η ×

6 7 −10 10

8 9 10

  5 E 5 7( 6  #)+ .3.+* !  O!' . %'    !3- + )% 1  ( $  ! _ !7!  $ $N  , !( 3 4  MWi M !  % & )+! ($
  σ ! ( #)'&% _ !!  &  ( &    $   " 3  M 7 f A, -$<   + '*% 9     -   .7  L )1%  !!( $  '3! .  (WM &    &  7 .7     $-5-     .  L .9  .7($  '9! .
"   
! -(  *)+&% _ !!  & B,W,   4%'!    (7 $ 1 !!!  $ $  -  , $$ , ' & I  % N2 ! %  ! ( $1&5y"    .9 % 9 !   7 %    r/  L   + '*% 9    )1%  η M &%'*)4' &ON ($#+D !  % +  !  +  -<   &% .  -,-aC+ %('7!   586       1 $  % .- $!  η = η · 1010 M92   %  )  !   ( L -  $ -,  .9 ! 1 !3 117  4  M n /n M n10 /n  n /n L n ·m Y = n ·m p p D p He Li 1  (   & 1 ! _ !!  $ O $  % 9 !( 3 U&%1!-!3 -,5 !

B

B

4 He p

4 He

3

p

3

,  $V.0 ! -  % a  .($\1 -& oL- !7!' &    -2-& & η 5
&% :-   5-   
" ,  9N -%N !     !7)  ! %$ -  )+4%  @+.   $ (1SM  N  , !( $ !' . %'    !7 + $  ! -,-   &  7! .0 - ,Q- !( 3Da! !  $! ($ !  11 -&    .23  %7($ 5   -! " ! M3 6 $O a %      #)+ .9,+* ! ($ 
i  P  q   q  +

  



 



            

 



+*.

% & 7! ( $5 "), "   ,  3WM}.9. -2' -   %-5  % WM !' . %1    !7- + ! . %  )  , $ a!1 !  $!  \!  .0 ! 1 $!7  1(& ! -,  3 &%3 ,- $- .23   & (1( T ∼ 1  U "OM !- 5 FE 5J1&5 "), " ,  %   3WM, U #3 ' !' . %'    !7 + !- a % V(7-  3 )   9   1 )    % R !7! a
" % 11 .9.273 " % ( - ! -,  3   & (1SM  (-,-.- ) &VN  a$  7X- )+$!  !- - % 96 a   4%  U!7 a$ !( 3    ! ,5 4% W,   - M 7 W 4%'#)+!  D!5-  !'-5-  1,+! ' 62 % 3    $ 9 %%, +-,-,62$3   
)+  5  - & )  4%'# )8' &  a51 !  !  N!  -  )+4%  2  -[_ %4-  )  % a," -H, ' N2 -    & ( 11 -  5
4% = $! -, ! % $H.9 !  % 1  1 )   .0%$   $ (1  X L U&  1   ) X → γY ,

a )+ Y L &% ,--,+ -  )+ $ $ , ' &VN~    & (1  m  m M  C,/1  1 ) X3  .23   $ (1 % %,+ ' &  -[_ & .  @   !(  M8-1 $!(:~2   . +> $!  V!' . %'    !7- +*75 -  , $$ , ' &ON2  a51 !  !   E= , γ(≈7 $m 1/2  !& ,  1 -$ -, τ  ζ M*a )+ τ L ,% 5-  -H + !  X "   $ (1  M' Y

X

X

X

X

X

nX n ( *)+$ , %'! !    5 E 5 75 6 7 - 1 -  -  )+&%1 γ$ V . -  X "H Y "   & (1 -  4%8' " C \! .0  % :V5'      -2-$ 7 !(: 1  2( $!     !7)  ! %$ -  )'&%  @+.  ." $ (1   - ,w.9 $$ ,    1 ) &   X "   & (1 -  & , :$,' 1  ! $  1 %  !   ,C.9 7$ ,  & - % ! %$ Y "   $ (1  !,-   !   !- 5(,+ )+4% 75 K35 &5
G )+!  -  $  ! .0 -  51 !  ! $ !  1   - $   -  )'&%' $ 1  2  &    ! " )  -!' & -  )+4% 4%8' = <   .  7 O,  \ ' !' . %'    !7 +* ! ,-+ -  !  ) - % " ! &% !   (-+,- )+$ ,- $!7 7- ! (  -   *+  !  1    )  .9. 7" % (-   $ (1  =.0 !
)'!   -, ?, -  $.     !( 53" (   , !  \ - &%8'
   +  - !  %    Q  -5-    ' "  % 1 !' . %'    !7 + '+ !     w.0 %      1,   !7 $     ! -, 7=.9. &%'*)+$ ,-   & ,- !7 1 1  ( & 1 ! !!  $ %1  ,+ -,,  $  4%   "  !- 5 5 45  ZE ζX = m X

L

-

SM wNT_ 9

_

_

9

9M G1 M

a

-

9

a

U

ON 23 L a M cL

6

6 a

.-

SM

a

U #3 SM

X1

9

6

_

6 23

1r1

M

M

9

23

ON 23 .-


 -


J


 0


 


  

   

-5 Li/7Li

Log [ζx (GeV)]

6

6

Li

7

D

Li 4

He

-10

D 3

He/D

-15 4

5

6

7

8

9

10

11

12

Log [τx (sec)]   5 E 5 (   )'&% 9 ! ," ! +*,   -2: a51 !  !   +w  ,- 7! a !' . %1    !7- +* !  - ," )+4%   )  % a$.-=, ' N  -  !$$  % 9 !(-    $ (11 - SM 1  1 ) ,&ON2 -   ,H, : -.0 U$! 
" a !(: @   !(  E ;5   )+4%   1 -$ 1 - \,  6 >  % "6 !( 3 %  ! ($\  . % & !( +   + '*% 9    , !  % & )+$!     (7 $ 1 ! _ !!  &   -  , $$ , ' &ON2$3 U&%'5-!7 -,5q"  3 !   %  !   -  , $$ , ' &  ! ( %' .0 !$ ,-  , !(- %1 $! .9 - 5 !



     



    ! "

  

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#

 



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&  

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 -,-1 )+!   5

% !  ,% ρ!B-,0$!! ρ%DM $ " ,04%'!7! %$ 5 = )  &      ! =7+ + ) !  $  !$ .0 % .02,+ -  H!( -   i ,- )  2 .  - !  $!!   (7 ( %  !!    +)  $ $ , -4% -!  4%8'
 $ !

 (

      &  

B

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E



       





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  $  -  f , $-& , '$ %  (7 $ 5-! %$ -  - 
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X

X

X

X

!

X

!

X

X



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2$

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lH ∼

$ 1,+ -

MP∗ l MP∗ l . ∼ T2 MX2    X - & 7%
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-.X, 4%    ! 



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    $ 5 % ∗ Pl

 M ∼ 1 . " '
7 5Y √X = MP l /(1.66 g∗ ) = 4 · 1018

∗

23

X ,0

X

X

X ,0

X







X

T

!$D<:, )F0$O!

= =#>?'@$4K-+0,$,)A4!";BE#$=PQ!=-.!$5)%&RO(4') $$*AC-S!% +!-), .$D/ >T)+0$E1 2.3!=UF"$#%4G$,%)+9)#U.VC FH!$F&565)IJ7$5&%@8.L.$D .2 ?$ 9!K*)$$ L>WMN)2!C-4%U!#):$ %&$$D.%& ) $ ; #$DjpF6A / $d>b%!.0=!4 f$'g% !. 9C856%)5)5fA ' XZ  L !$Y8Lg[#)\$D[-]A^_/ 4Xa$i` h2*5)q:!5k=$jl%& !:)$.U )'! C8>b*W%$5)b.=!564$ $%7O5&,$d%)cN)%!56mD.' WCλ =!2π/(M 55)mDvL4nV) ,!Rb,eJ:*!K>dF6 A o X

vX ∼ 10

X

−3

-` r .  ! t)sV$>
%)4.u<#!C-%8$%& 9$.C$"!=  )sV$$)%)Ai)> )A$$4'#$ RV <#.56 &=-.8$DW!v.:56%&$3)%:D:%x$1,w $F$)@L#$Jw?:%$1,56:!%)8eE! &)y.'#jp   :8eE%)&,z%$!D i.,$  H W+0!,uX 4u#$%)X )zmD'@:)F& )k, i+<:W%:%$D,$$D H.<s*=!. 1 ρ (x) · `√ f (p, x) = , ´ ·{ M 2πM v |}zf j&k&nLj$agk4bA|1e4k&`glLdLdx ce fe&rLk4dLr*e  hLjMhLj&l _rLv x&`gf dLmLd nLoOacj$agk e4x4f `gk acl ` _nL`sp h*f j$k4nLj$agk4d|1e k4`slLdLd x&j  ag`cf `snLnLj&p9€ 8+ k&jMjgn em e&`gk mLk4j |1e4k4`slL d  x ce fe4rLk4d r`)acj$t&l e4f e&agb}10d )j&rLlLvG eiy`gp@j$t4fe$ack4d a l es|`gl j$| x /6y~ l ezh lL`gx$o ye Pd |H{ l es|`gl ce fe4rLk4d rLd0€ k&a _eGd acf ` _v&`gkGhLl dLx&`c_ 7gnLne  j#8L`gn re € € j$mLnL`g{ ` { x)rLdLnL`sk4dLmL`^acrj$|Hl e4x$nLj&x$`sagdLd { hLlLd rj$k4j&l j$|9 vnLr18 dLdMle&ach lL` _`gf `gn d  hLj)d |hLv^f bage$|  x4f   ik$a  l e4x$nLj&x$`sagnLoU|d h lLd
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C,$ :)ρF)A(x)/M - f (p, x) v: %)X ):q!$E!A*K ))A/ V$$ ,=-: $$%)$V E:%&56 %)p=$D0 ' !!Gq!$%&. C!&L 5Gs F&5))

X

X

2

X

f max (p, x) =

ρX (x) 1 · . 4 3/2 3 MX (2π) vX

MN!.U'6,!!:KF0)A/ F&$$ R =:%)-) $)$V:W$?w 48-e F!$<3jl:%)b$L(.>dj6L 0mmDA' #!#!)A.=#8eJF&Ry:AE%!:%)$D,$D$)

sV$%)4; ff =

gX . (2π)3

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3

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X

X

X

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nX = nX¯ = gX



MX T 2π

3/2



−MX /T

.





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τ=

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/ 





@/ 

1 1 , nX hσ vi



? 



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2



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¯ 5  ," X (7 (1,,"



1 1 = H −1 (Tf ) , nX hσ vi

%-+ !    & W-5-   1 '  ' , - 1 , !    
 "# , "  &5 @.\( ,- %   X X¯ "c !7!  % 1  (    )   , %3 ,- $- . ;  $ ( 1 +,-   -  $" 5!    !!  -(  # )+4% &-  +.0 !  -  $ (

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σ0 , σ = v -! %$  .9  $ !!  $ % 1 (& 5

! +*,-  ~     -!    &%(  '  , $$ , !!-  +C (- 





35 K 

! 

Tf





$!$!    !7!   % 17  v  !    -&% -!  O .0% & <  $ 1 X σ -!3 (  .1 $!     , - %1 ,- !    2  .1X !!   % 1 -  =!  ! $   ! ,>, !-  (  4 τ ,% !-  + ! " $&%'$!! %$ t ∼ H −1









35

 E

 .(17# !! $ % 1   " "  " ,- 5i" &  - 4%8'
  )%N!   " s %1!  1   <   %   !    -4%(-! %$  .0% & 

N7 $  7!!   q-( *)+&%  !- 



3 5   ,, +*"

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C$,1$ vψ(a) X  ).9).=.! 24sO,F6$D5&. E z L ? K!>
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i (ψ∇ψ ∗ − ψ ∗ ∇ψ) . 2m

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2

σ=

P Ca3 = , |j| v

!2Ej6'&L :mD L
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a∼

1 , MX

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X

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MX Tf

f

Pl

M X Tf Tf2 )+-2S!    !- 7  +*.3 % .9 X "   $ (1=-   )  ~! 1 ) 1,1 !! ," )  -  ! (, !! $ $  )  58" )  % -! $ 25-3-2  '>+) 5I - !     7  -  $  X "   $ 1 -  % C  1, !! &  MP∗l 5  ) 1,    & ' 1 , !!    75    )+    - % $ .  % i\ &  - ' -5-    ' 1+.9 % . +* -&-!  - ! 2 MX 5 " +  , %  4 -   %     &- $ ' 1 , !$!    35  )(, *) !-I >.X,- )4'  J75 E &(

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g X σ0



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x = ln (Axα ) ,

 )+

x = A =

  '>)+5- !      3 V &   !!'

   

&

%1 $!  "  +1,+ -  !X -%1 ~ $!

MX , Tf gX 3/2

(2π)

75 

σ0 MP∗l MX

1 α= . 2  σ0 )1% 5!  (X $ .3.

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 H!   , 5-

1 

5 ' -

σ0 ∼ MX−2 .

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  !-  X

Pl

x = ln A ,

  & Tf =

MX ln



gX MX MP∗ l σ0 3/2 (2π)

.



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" 7  -(1, )  , &  )+&%  !!   ,   -  - !    %'  #) 7 %  ~!    3  X "   & ((2π) & 4       ,  7 . 

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X

X

∗ Pl 0 3/2



X

X

P

X

X

X



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 )+ s(t )  s /  %  -!  &   $   -I -,% 5-!!    %  -!  $"  f



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,-    !%  +-,- %



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$< - !  )2$!  5-







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s0 Tf2 s0 p nX (t0 ) = = 3.8 , ∗ s(tf )MP l σ0 Tf σ0 MP l g∗ (tf )   C!- 7  

#

75 

!  &      %,  ! -  q 

@.

7 4π 2 3 (Tγ + 3 · Tν3 ) = 2.8 · 103 590 8   %  -!  &   &%  X "   $ (1 ) 

s0 = 2 ·







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 ( #



MP∗l =

MP l

,





s(tf ) = g∗ (tf ) ·



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75



σel ∼ G2F E 2 .

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%'$!!

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X

3

−3

A

2

2

X

A

A

A

X

X

A

X

A

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AX

−3

−38

2

X

23

4

−38

3

−8

2

−1

2

Cross−section [cm ] (normalized to the nucleon)

X

10

−38

10

−39

10

−40

10

−41

AX

DAMA 1996 Edelweiss 2003 CDMS (Si)

10

−42

10

−43

Zeplin−I CDMS (Ge) 2−Tower CDMS (Ge) combined

5

10

50 100 2 WIMP Mass [GeV/c ]

500

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 0 *$ &2 *+0 #0 &* />&%1 3D" "?% 0#U12 , #0 &)2 />&$ *$$ 1  5  $ * T T 









max






max






  


∗



 


 

A     




 #   





,

!  A-%; 421F '">"R&%& -%O#  0#@
 24

0' -%  ')* 2'/>5 ,& 5 -T : %%0#U1
7# -% + ,/;

 " 5  T ># 
' +' 9 0 * $&  #$ # ) 5  /5 # ! ( G 0')* $1;- ,*( $&
%'/>5

)-%%5+& 
)6'   

 #$ #  5 '   />5I#! ( 6, ')*  , *  $*E *! 6' '%)2# M)*>-?2&: $ * #0 -%'T( T /0 & & ,&2 # 
%  #0 5"%  #K  &    14 /0%  )J  &

%&.-'  0?  #U1 /0>  -%%  # #  6
F-%$ 4* 9 $ 1  4 * T 'T  1 ('*

  ) * !P /

 *  ! 0 # )   '

+ ,&  '   #0 -%'  0& ' " 7#0  ,&  % #X%  &[, #0 &)2 />&  T S%  &. - *! 6 %  5;" Tmax

-






K0(%'EF(T',*
% 0!$ 1 1  _%('2(!%@''/FWh!,%2  !!% % @d*('F0%! a2JL!!V VFI/!%(%!E4K0!_ E? !VF('je!&!E)!%EFE4&! 1K?!!VVa_2/%! 0%!EeU6 1K<  : < VF
$/0Ki'2JR 1 2(!  E?K-%34'T!!!%!H(> K0V 2&! !!!!-+ D;IF*!%!&!%! 0 ' !+ %/+PH% 5*!%2( %-EDD'3X G \EL*! %!T%N&%!W!c*( \(H/ \+ % % 34(5 !T6M F: < *%! ! ; 1%3 *(%! 1%8> ;   \# *!%*%O%.V \!+ ] %!2!!√! F D<\`?%2*!('KEf'U.2!E 2(WVF! %2(K!h- 0VF?H T2( VF 02(!, 0 *% #3X25O?a%H ;W ! ( %K)!+ '(38_*!*/0!F%!! 1 10 b2'%! + L ;_'3X!+5VF*6T  $ !!%!!F *&!!H3 *@' 2J 'XFK-%!E'Y3X !VR!EI!2(XHW!K-!EI%'%*K%Ea!*(_%!('6-/0!'2!% 1% :  \ + I'2(*%<2`5*!W!S \!_Y< %!/-' T GE *(/ 1 *%!+,6 4K-%'!!!?'2(L2( !%!L!!%_ ;!)'JVV max








&  =#  ,/!G * &    & ,,/==)*8&*  - R /0&& ,&  T .T  -% $#%&/>5I -%2/ ' '%'14 &

max  - 1 

 24$18'

˜i → G ˜ + Xi , X

E &,0/==)*  & 

 '*

)1S%0#



˜, Xi + X j → X k + G

$#

IT%& $) ,1

%&/>5

i, j, k = 1, . . .

2→2








"

 

  P-%2/ :   %1 .0D & ! 6- : 2, )'K= ,*/>&  M *$): #@ -    

 ' %9 ')24 %1  0& 0 6&0* 5   $*
n3/2 '

A-%

 


%  G-


"


i = 1, . . .




"

X dn3/2 + 3Hn3/2 = ΓX˜ i · γi−1 · nX˜ i + hσtot i · n2γ , dt i




.



%   &K  &

 -? "

% 2#U#@:  2& '

%) ˜ i
YJ& '

0 2# Γ ˜i γ X

'%/'X6 %& -#*$ />#   %5 i0#> P&

 -% 
%    '' 'J& : nX˜ i   -%2 5;   0& (' '%) ˜ Q %   

9' '% *   Xi nγ = ζ(3)T 3 /π 2  &T0 #0 &* />& %
% / 0&.-%   # & 

%18# /0%)#J&

%&.: T = T (t) hσtot i -%  : *  ,  ' 

 &+ -%  ,& [ %& $)   "  "H"R& =  2→2   . - K % , # X " . 4$1F =/>' 
'
   & + -%  0&   &$ % )  6  &   - %  & -% 

)*$ ' X$ 14

 ;! %) $&) 6

)) 25"  : 2→2

6' '%  )S #U1
 #

 K  &S$  14  0 #, &*2 />& 
0! A-?R  ' '%  )1 # ! ( # 0')) & ' %  #Y " "R  

 ' '%  ('

%  ) #U1A$ )  , #20 " 1A%  & . : nγ 4$&  V1  &$ ) , # 
%  & -#  #1  /#  ; %  52 *  ' ,& 9

 &, ,/D D))  &2 )



%  &

 
%2 4$&*%  1  "  











˜ → X i + Xj , Xk + G




 

 


  


 

E%& $)

T 

˜→X ˜i . Xi + G  0  *  ' 0& -?D 01 P& 

 $*


C& $) 01A ,/

 1

" n3/2  nγ '$  14 T 

$ 4  

(2 ,%# 0-%,*

 *-

 ; - &/ %#( )%#
T  0- ( 

$ 4$&  %  * & #0 & 

 
F-$2 & # $ &   # ! ( J= ,*1 '.-/>5; ΓX˜ hσtot i )  $ 

ΓX˜ i ∼

MX5˜

i

16πF 2

hσtot i = const ·




A-%

=

MX5˜

i

6m23/2 MP2 l


"

,

 


" 


8π m2S m2S = const · · , F2 3 m23/2 MP2 l

* $&  $&#3/D "  Q 0 (,: ,    24  0#@
! % ' : 1/(16π) const 
) %&. -%,# U ,/>#U#%&  0#:%-% R  ; 20# ! (%1 K0D =& ! -%  ,&: 
D" ", />#U#@ '  %  1 E!
 %)  
%&  2&%) )%1 >  - &*)2#

  :

2/>5; &D%  4$& $ '%  1  3 5R0

  S! 

) ,Y" 5 18' ' & 0014 25^
' 6%  &>$  14

 (0 #0 &* />&  E 

 2 *8% -  #%  #@D  $ #Z /0 &

%  #U# 2 &%  '2 # &$

%  &  M)*>  -?2 &$ * #0 -% -? ' &

   )   1  '*

  )
*  & $ # ' '  " const ' 10 7# -%: 0  ['  '
J  = $ .0 #, &*2 />&%1 3 ZK  /'$ 14 )   ΓX˜ i hσtot i />#    ' ':0) . > #  6 0 # , #0 &*2 />&%1 
MZ$ 4L  &

 /!L

/>#  . : , # #@

0 1 ,& " "X 2 #3/ '0 #A ,' 0&
40 # 4$14

2 & $>  - '%  &$ )

 &$ 4$& ,  0 &  '$ *B0% #*S " "R&&$ 14   0 #0 &* />&  
nX˜ i ∼ nγ ∝ T 3 " R0 5E-% 


T' (%  &K  -%  ')*$ ' 8$ 14

 C#@!   #@)  %  1 [0 #, &* /&   γi−1 ∝ T −1   %  18# '$ '!$ # 0& *5I 1%&$ )

0 1<&

  "   " Tmax 7X2  *>- ,#(& ;  />&  F *  2)%#@ " . 
%  &; -% 2 &'  , &  ,; K0 />& ( 0& : 2 & % * >    "  
Q +  : ∆3/2 = n3/2 /s  1 &$  0 #+/>&  F *  2)%#@O  - 'T  />'  B, *$ &)
H#U1 , &0 * - 0# C0 &, # 2 * , &0 # $ *    *  2,/0 # 

   %# ," ,9 "
'" t T & =  - 42/-%'(  #   9>   ∆3/2




























" 

X ΓX˜ γi−1 nX˜ d∆3/2 nγ nγ i =− · i − hσtot i · · . dT HT s s HT i








 2  #'%  #
' > $& ; '% *0# 

%&(J/>&  "   ^ 2'. = > *8,# : 
/,& C& '  />#   0#A,#0&*2/>&%1 0 ,&*2/>&%1 3  ! * / H ∝ T2 -&'

  P' '%)
 /-? # ! T =05 '
' &*  -%14K0: ∝ T −4 Xi )T  1 &! 6-  0& *%4 *%*B% - -?25I/$ 2-  )   *5 )Y" F * R
0%  &>2 14 * J  / ' 8, &%#@)  =2 )J  -8 $4218'%  1 K'

%)  S% &*% %  ' +0 &

)  J  ,

   T 9'%  '%  1 #@! %  #@) $ *[, #0 &)2 />&%1 " Tmax      

 6 WR 2







M bE ^U(7%",WP= U7%+ %`((+#"_U%7$%" (' O#W 3 >@+`$ 2$%3;9 $%$ "#&+,7%+#6+Yc'\9&6!9&`>@+#6+_" %L 8 B hσ in →   BF30+##+#7%+, .0E%$%&WQ"59&(IT .0O,WP') U 6(79&"#$&$%*%+L b*9['+,6($%E*%+B`U%7%+ %V!'$;:%*%$;:?U%7%$%",+`$% hσ in · 1 − 3 >@+` $ 2$%3; 9 $%$^U(7%",+#6(+c'\9&6!9&`>@+#6+ Γ γ n → Γ γ n 1 −  BK6X n 79#"#*%+,"#`!*9&: U;'+,&*%+5!&)-6(79&"#$&$%*%+L%N( 9(&$J>@+`$ 2$%3; 9 %$%$*%`!.0/1`!&",(*%*%WTB;U+#3;9 n  n L tot

2 γ

tot



n3/2 neq 3/2

−1 ˜ i ˜i X X

−1 ˜ i ˜i X X

n3/2 neq 3/2 3/2

eq 3/2

γ



γ





∗


  


7$ 

 



A     


 

 

/>&


" 



 >0#,&*2/>&

 D-




 #   



T  Tmax

 

E')    6/ ' 

∆(Tmax ) = 0 , %#  T>-

∆3/2 (T ) =

XZ i

Tmax T


"  

ΓX˜ i nX˜ i dT 0 nγ (Tmax ) nγ (Tmax ) · 0 + hσtot i · · . · 0 γi s T H(T ) H(Tmax ) s(Tmax )

#  ! * / # -/9   '$5 $& 
   '$5 ' '%   KJ'  $#9&: 

  + '  ,*+  24 0- 1-S%#X %&.- 9  * ;  "


 A-

nγ (T ) · g∗ (T ) = const s(T ) !,

T S0#0&* />&%1 
%

& ;  "  # + T S,&  ,*8S>-%




const

" X Z Tmax ΓX˜ nX˜ g∗ (Tmax ) dT 0 g∗ (T ) i · · i · · n3/2 (T ) = nγ (T ) · g∗ (Tmax ) γi nγ g∗ (T 0 ) T 0 H(T 0 ) T  i "   nγ (Tmax ) . + hσtot i · H(Tmax )

R$#'%#
' R-T%&. -'    1 J,  ' %*T0#0&*/>&' '% &' 

  [

  0*

 $4 0- 1 &%  1




43 , 11 g∗ (T & mS ) = 228.75 .

g∗ (T . 1




5

)=

# 0- '0& 0#
' L&$  "  F%&%# %# -E $) 1  

-% 9   $&
 0 0& 8 ,&%#@)  ) 
%D"L*"? 0( %&.- 80&  ' n3/2  neq 3/2 (T ) ' " nγ (T ) "R& D  ,&/,# &$  "   " 7 ,# $&%#`0% - ,&  . '% 0#  .. ! $4: 0  / "  " "#  ! */$%&  0- 14 0&) >$18& 6 8.- '. ,40 nX˜ i ∼ nγ - &'

   '

%)
% > 0&) -+0% -?(0 -% $*P'

%)%1 ˜ ∝ T −4 Xi Xi 2 4%  &' > #  T0 #C  &. -%
0D"L*"%  &(    T, #0 &)2 />&   3%  & $ #9'

%  )%1 X / 6

 #+ T E0 ')*  . %2 4C &'

 %  #
−1 '" %  - 1    - &/  /, / : i γi ' 1 5 , &2 & 0& X. -/!X

0 46 : &  $ #3/" 6

 ;, /0, &2 & 0& 
%  &2 #  
$ # # &

  -? "
\, #@*%  ' / %  &%1U2 1 )#   &

  - 1    .

   '2 />5" ' '%  )1 M)>  -?2 &$ * #0 -%Y"  <  &  - 1 $ 42/ ' *   1 $ 4$18'%  18# 0)%  4$& $ ':  18#. 5R0

% %  #.2 ,%  #0 -%, *

%#
Y $ #3/ %  &   0 -   '  K 4214

$ &  &

  -%  "
'% " "X S2 & -%  /[%  '%  1 $ &, # (   [, /0, &2 & 0&J  & "






















τS =

A-%

 $*

1 1 = , tot ΓX˜ αmS

%  ' '2/>5E! 

)*)9

 Y" & A

G$&,# ,# T  5  >: α τS 
  2*>- ,#
T' &

 - 1  /00&2& 0& (%&[0#0&*/>& 

tH ∼ H −1 p T . 0.1 · mS MP l




 

 


.

  


 

)%5^ @4$14 '$&%18#$%& 2)

,2#Y"   > 2 ' 
H' >0+ *! 8,#0&*2/>&[

)  #  E#@ ,1C   

2/>5 X ,/0,&2& ,&


 )*S'*

)%1C $' #    &

  -/! 
YT   $ # &! 6-% , #   7" R 05E-%  
' - + $) +% -?T &! -%  0&* 6 6 /0 & & ,& 9 :   C '.-/ T $4$&=* 0&)  "    " 5 &, 0/=  =)* *%  /'%  #
T ∼M








Xi

tot

ΓX˜ nX˜ i g∗ (Tmax ) dT 0 g∗ (Tmax ) ΓX˜ i i · · · ∼ . tot 0 0 0 nγ g(T ) T H(T ) 2g∗ (MX˜ i ) ΓX˜ H(T = MX˜ i ) T i )  T    '    %   '     S  $  '    @ # )     
  

!  *       /  Z

Tmax

ΓX˜ i ·

Γtot ΓX˜ i  ˜i X 1. 1, tot ΓX˜ H(T = MX˜ i ) i G)
0% - &  -% S.:   BS! %) 2&2)%5 0&* T& ,4$&$ (%# #@)a"

=) #$% - &  -%    .=  &! 6-  0&* " = "   42/-%0#.# # 2) : / - %&%  1  


 








5 MNL SP . 6m23/2 MP2 l "R& $) ,1`2&  K& + -% ::  %)  & %&25I 
a   %    25^ E%&[,#0&*2/>&

ΓNLSP =

,





=0)>: 

MNLSP Tf,NLSP ∼ . 20 R/

  5I & 

 
% 



 $#3/[2&0#   

Tf,2 NLSP . ΓNLSP < H (Tf,NLSP ) = MP∗ l "  F .-/ 
' &

 - 1   %&   0- 8  'X K  ,0) %&


 =




"   



3/2 MNLSP  , m3/2 100 5 D" ". '%/' > '#;! $# #@)T1  #@ ,& " Q

0# 2&%#I- Z %&;-%'   ' %#  2/E 2/ )%5;" 5 ,*P&*  -1   B90 0 '$ # ;  & 0- 8[ $#3/
' 

 =*D;    2%& $& 625^ Z ,&
7D" " ' ' ,& .1 /!:

2/>5;0#< $4  ,#& =*D;0#3/  ' T  /   3%  -B%  &2 )

 &

 
D" "  $& $* -Z "  
 0 6  />'2 14 0 #8Q  .  -?-%, *

'   .   6  
&%#0&
YK# 0-%' .

'" G0 5E-?>, & 0/1 '; -/'12 ) 0 -:

$ &, # $ * T .m  *  ' #@ ,1 ,&max  ! 0 S'
 G

  -% X 4$ ) !$ *
 *T  />', #$ *S 0&  : & ,#,  & T0%  -?6 "   
 & 3.5 ,5 ·



















m3/2 ρ0,NLSP , MNLSP A-% % %-   A/ 2&0# />5   

 0& $ 'T/>'*1

)41  ρ0,NLSP "8"#  '.-%* #U1 $)   &.-%' " " 
A-% #U1 />-%'
49' '  '
'   %& - 

)* $' 4 *   C *J&,)

  '%  1  6#@* ,   

7 : MNLSP & MZ $&,# 0  

+  #@

01 #! 6'1  =*  E0 -%>  :H$& $&
>-4 *
'0# %&%' 01  

 






ρ0,3/2 =






ρ0,NLSP ∼ (10 − 1000) ρc .






  


∗



 

A     


 




 #   

  



#  $#3/>  ,%18*P# !  #B# ! 6'(%& 0-%T[$&,42/,# $#3/,' %5 "

J-%  * S ' T1 -

ρ0,3/2 ' 0.2 , ρc

Ω3/2 ≡

 0&


m3/2 ∼ (0.1 ÷ 10)



%  ' ) !  & 0) 2& 

 BZ $#

 V%  "  
 14 ' P4 *;%#





&/-



  100 ,5 · . MNLSP
8' 2 &0 # T,  

,5




\$18'   

 

 m 2  100  ,5 5 3/2  ,5 · τNLSP ≡ ΓNLSP ∼ 5 · 10 · . 1 MNLSP "R&

&*  -1  
 %&  ! 6-%2 5; K& + -% 0#Z$14

! 0& %'%1  τNLSP & 100 ' '%) M)*>-?2& $* # 0-% %&%#,&
*   -   +,*$&) 
&   -   /8 $4$&  T   %#%'    , #  9>  # 
'6 #! 6 6  & : -%+&  $&,'%5

&,,/==)*)2 # $ &C0 &  ' 1  /0   = , #4"7! , ) !$1 . '"   $*>2&/-%  '1#2 ,0 
 ' 0& S  G   '   S   

 *          %  #  , I 5  % 

)  *    $  '    4 *     ;     @ # * 



0 1  "  m3/2 . 100 5 MNLSP & 300 ,5

R$# %#
' K  $ #C, ) 2 & ,&
$ 42& 0/>5" >O&*   -%    
)%5^ 8, ')  & '

%  #Y" GT%  &%  ' [^2 =*%  #0 -%0 * '2 />5^T

' :

%  )2 #T$ 4$18'$ *L#@*0 & 
!6NO 

2 14 2 5I)P\ * ! =X0 '   & ;%  &  5 ' 2 *Y"     ' 
' 0&
   :$ 14 '2 /02 5^@ 2  & *  3  ,#$ * #@*0 &]"

G1  #0 5I(   $  14 /0%  ) &

%  &. -' . T%  #  /=   2 #
'
? $ 4 $ &



0 014 ' B & 

  #!$ #@$ ;)2 4  1 B '$ &/0 /&:)*  #0 -%' Z  : , 4 )  2 * -%

0  & +

!$ &  '0 *C' '%  )E  0#$ *P#@*0 &Y" −1

4



























.

.











  8
   
    
  ?    >       
  
  8     ?#    W             > 
  
   7     
   '    a      W   .
                    ?







Q

0# 2&%# =*D 




 

 



 



 
       

 
 U 
     *
  
 
?# 
%     
?   ?   
?   
%        
 &   
@!  ?        ?  

8        
?\ #           
          
   : 
   *#  T
 



          
?   







 ,&   $42&*18*

T/>'

%















T



2*

A-

1 ΓNLSP & H(Tf,NLSP) ∼ ∗ MP l

GC&,)0/' ->   ,&


?&



MNLSP 20

2

-

 -

.

3/2 MNLSP  . m3/2 100 5 5  2*T

2/2)80  

   !# $#  &   -?X\ 0-?* 
90T>-% XI $4:  &  J/>&*  F )$)%#@ "   
! *  '  ,& 4214

$& & ' 
)




 . 3.5 ,5 ·











 

 


  


 

 -$T$18' ' $$&,42/' (/>'$14 *. $4$&)%149%& $) ,1<  %) 0&:  [   %&: ' *  $ ' '%)# M) -?2& $* # -'
!  $&%18# #U1 %& 0 42& )Y" 5 &, ,/D D))^  

  ' '%R  & $1  ,& X $'9  -?'

6 *  ' $5 ' % 0&* K  #0 -% 
A-% ,& E   - /& 

: , &%  '$ *:% 0#" 5 $ # 'T  />'* -C-% ) 0& E( *2 *$0  

 0 & 

$  &0 # $ * 5  $ *1 X, 4 *  %  #E %  &02 #E %  &*. - S$ )  " ,"

4$&*% =  # : 0 &    , &2 ) 0& 1  &*   '*

  )?#@*0 &]" #[ / : 

 (%  &



'




Tmax & MS . 

@' '%)%1 #@*,& 0&#@)  %1#"  %& $)

01 5  $# T/>':%& T ∼ T max &

 + ,''[ $& ; % ,# 8$  18& 6  "  
,* K& '2/ 6

Tmax %& " 5  2#B%&;-%'=-J-% )C  &' $ 1  ,& 42/-%0#&%#  Tmax  MS




m3/2 n3/2 m3/2 Tmax g∗ (T0 ) nγ (Tmax ) = · nγ,0 · · hσtot i · . ρc ρc g∗ (Tmax ) Tmax H(Tmax ) "# 0-%

)) &%#3/=T/Z218& 6 -  )   ,'  +& ! 6-  0&* $ / " 

%14;$18& 6 -.  

 ' '%   >+2&2#'$& #4$4*%
% *%/ : ',  # Ω3/2 =







Ω3/2 ∼



  2    200 ,5 MS Tmax · · · m3/2 10  ,5 1 " 5

R ,5E-?# $' >-% X  ,&%' 4

0&>/6 @! #@)/>5 0#,&*/&/<0&%' 2* %#U1
0 * $& P& '',*  /0: %  ' , #B#@

0 1 ,& "

)  =  / "  , #! TJ /!$ '   K  & : 0/=  =)*  ( '    ,&%  & :   />&*  F )  $)%#@ "   " F :  

 /5"%  * & 0/=  =)*   J- , / : , & %  #U#'$ &  '$ * # -'9   4 # , &2 &0 * #@*

 5 M = 50  MS =  &. -

)* >NLSP & " "  " = %

4*

   ,5 1 2 &2 # 2 &   1;G  2 * $ (T , m3/2 )  ,%  &$  max /! 0 # )   '

B  : = , & %  &   , -



) 4    1  ,&  :  [  &0 * 5 ' 2 *Y"   E0 0 # *   : ' ! 4 ,&%  ' 4 2#@G, /

 
 

! *    /$  & 

0 *#;  #0 -%'  &   : 0&   5  $ *1, #0 &*2 />&J  & , ,&    & $ %  16*   " T ∼ 108 ,5

R$ # %  #
D' :%  & /00 ,$ * , &2 &:    $ &+  #@ , /0, &2 & 0& 
 #0 -%' 9

'/!  K#  6U 
(T , m3/2 )  &; -%

)** max % *6 & *"" 
&' 2 14 $

) 4    14 ,&K 4/-T/0[ *  ' $5 4 , ' ) $  B  ,#  />5 #@*0 &%  5 5    : $ *]" < $4)%

0 *
6 / '  #  6

15 p g∗ (Tmax )

!

·


"  

1 . 2h2

,

#





 

10

10 10 10 Tmax (GeV)




10

10 10 10 10 10

9

8

7

6

5

4

3

2

10 1 -6 10

 "

" 

-5

J

10

-4

10

-3

-2

10 10 m3/2 (GeV)

-1

10

1

0 &%'   2&2#'$&%1
] .-/>5"SS0% -?L&'>:

(Tmax , m3/2 )  $1  0&*  $&0# />5  :   '  0&  5

'  $*Y" =4*% ' $1: .

 ;$ * ,%&$ C,: =C: & %  &    0 -%

  &' $1  ,*$&) " F # 0 ''^$ ) 0 Ω3/2 h2 ' " 1





∗


  


 


 

A     








 #   



0  $*S
0  

 & $1  ,& 42/-%  ;-%  ')* $'R- S $4  

:  

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  V )6 .8   '   *\!2 .06 @S ¯  SU (3)c × SU (2)W × U (1)Y S ψψ  025 1 ?6 =H 1 59  <. BK3% ( G]  3%)  01 B I ?1 :IR * 2 ¯ 5 ψ  * 9:I 0)C P ψγ ψ  59.   M)*4 8  3?1 52 7\.8 3%'  !': " 9X)9  025 2 ?6 =O 1 5*9  <4( 8 5*2 5C<_C  B;8 7 56  B8.8 5 

T9  .; :IXA  3  (  ! \ *   1 .XEGO 3E7; ./1 5 A3%! ¯ - 3% H #  H ψψ  . ( B, *  T1 B=*9  .8 3%8 7 ' 1A  3? Λ

LSF F =
















YP Hf f YSHf f ¯ , ¯ 5ψ . · SH ψψ LP Hf f = · P H ψγ Λ Λ M5K3 * TH3 / 95K3'  $   ' X? 1! 

)1<    25
43%YE/'Y= YSHf f YSP f f ?C<

;59.<  202<?C  <. :R0

9  . !  '))1.  859.  B M)*43?15 27 . 83%' 4 E4?$  2. *,9  ? ZC  E=  Y  E4E4. Z8 5O 3%( ; (

 !

& *T3

0) 95 6[

 >3  0)C5 W  . T 6 (4  3%*  6Y  B,!  8 5

 C  ?

K%- 85  (*9  < 1 B=>= 9. 83%;7

 'O3E4:; 7.

1 5CE01 E459<#J LSHf f =

LSf f =

CSf f mf ¯ , · S ψψ Λ

LP f f =

C P f f mf ¯ 5ψ , · P ψγ Λ


"

. !





 

 



3%_0;B856B8.859<4

  



!  ')2<  )_06EI38. ;?)*( 15 A3%0TO3%9@C<#" C f CP f f " . "!M "-Sf . !  6B,*  9:IS 1@  %959< 5 I3% W? '9@  P M)A3?1  52 7W. 83%' ! 8?   65;?RA3 $ R I8.@*9? R56B85( $ C< 




B=*9. 83%87 '

5  ?

9@9<  S 5

G3?!


 










m3P (S) m2f mP (S) " . ! , Γ ∼ , ΓP (S)→AA ∼ P (S)→f f 16πΛ2 16πΛ2

3% 20 1B I?1:^K  259<40 1B, 9< .< YE

8 1BD. T9<4\ 25

2<4  259< !" A BX25;0   a?2 02< 2  58 . TC  B,d?*

C@ 95 2< ) ( 1BD5*

 5   27






1

> H0−1 , ΓS(P )  *9EY?9. C5 0

95  2.$.@ /;)10 859? S ;5"4E.H.@* ;E ? '9@ ! 6=  Λ  159<4^.  ;E!(C5  43%  * TR5 *R? '9  @E  $;.2 7 .@)8 5  τS(P ) =

)

N?




1/3 . mP (S) < 16πΛ2 H0 CB,0S9. 'F  .@ /;)10 a01ED3%;.9.' P?  YEY: Λ < MP l  5 m < 100 .

 ? 9 


" 

. "!

2@ CE


"  -.



P (S)

!

 9).@

I8.@*9? H56B85( $ 9<. *C:I Q5

G3S< ! 8?9<4G 

 4=   

D385"S-;  C< ' 25 9<)*6;725 ""9X6 K3aE  CB " . ! 3 2.Y= 5CE4:;9  . 5 I 3%9  <.0)2 . *   R5

G 3(,  3%      ' 4 G3?' mP (S)  S2 5
A3%C  E[9  ?C  < ]

.@

2 7  2 5 " Q

8 . 1 59  .$ 8 5# 9  5  U  8 51 @R  5' 1 < 6 0) 95 #06  > 3% 

0)95   5; 7 5

' 2 7Y" E7 '1 E $ ! *  !b . !C  BD.   79  58 3 ;   ) C  :    g? '9  @V  5( ;9  5
9:;8 7  5  2 7Y"  .C  B1  \. ( $; ?)* V3%  ')*1 ? E0;  5 ,D  9  <.  3   80% F  .8 3%'; 7Y

J '.

 ; 51 @ ? '9  @:Q  5; B8E=  =)*R  5f = G 3%( 

  19  5* 9  7 30 4  3%  ,* 7  !I1    5 W3% c  ?

9  @.8; 59  ?2 7 %> BD. 35CE2 R  ./  C  B8.<  8 51 @V  01ED3E! 5 ; . 2 5 9  <   6>9  59  .8 5R  .8 3%

!  1 .  !" M

0; ED3%9  .  d  ./  C  B8.<#" E

  Bg  5; 7 5

' 2 7  ! 1 5 

0)95 $ * 0256 B  ) & C 2  E= 2 7 φ ! 4  3%  ,): ( *  '  2:  8 5  " #9  <. ' 1 .  C  E '  : 5 ' 1 7 C  .'(  ρ  @C<#" ED3%; . ; ?)φ*  ? .@* ,.  '  b? '9  @L5 Y(1  B,9  .6 =  ! ;  =? '9 φ mφ 3%;  7

1 E4:I T

X 0)C51 .  ; 5K 3

2 ./% ;5  T 3% ! 

)) S µφS 2 /2 µ

CB,g  56 B8.8 5 

  .@

8 < 3 1 025- 6  W9  5D 39  .<4   6 Q9  5
9. $8 5#     B

'9  EY?1 71

 > 3  0)C5  !" 3?0   9  5  5* G 3% @   19 2  ?2 7 P  φ → SS























" .(, ! µ2 . 16πmφ  ' 959< 5 G3%  =?

9@1 W3S5CE2 C.*)%1.K95 2< 1:I> ( 25
A3%CEEf= φ 9?9< B,G?  " .(, !8  ^5

G3 =O! 43  *)^  01ED3%'9  5   8 3% 95T;.85*1E45 φ )*0 17 ? Tφ Tφ2 Γφ→SS ∼ H(Tφ ) = ∗ . MP l "

Γφ→SS ∼






       !""("#%$ &' () *"+$ &' ,-(). )/10 *)2"3""#4&'2  !5!6/17/&'2 5 "082 2:/1)/6*! 82"3"! ;(!*":5*).3)/6<58/1*""!/=> 0&' 7,?)/@2"!/1$+2+A B*"DC"" 5(1 ##E&'# {C 6/1"2!F5 ,*!... 3!. ,/1C2G!/58}5&H9 /&1"2!/&IJ"K.()&%A " 7LM3"%2 6/1&N(!*"G C"" () * 8"" !5 12O!%3 1 PC"*! #1! 5!IPQ8""82"2  ()!58/#6/-&H/1*)/?C" S58!/1"8"2!TU5BC+*! # R " !5 1ITVQ"8!82"2G8 "3"8WX &'C" "8%#V58"()#K ρ  g T A φR PFF

Pff



φ

4 ∗ φ






   *


∗

E ' 60F?*

C@9<







 

)>F? '9@C< *

 )! 6 '%60 ;1B,9. 83%;7

1E4:I9 3S5YE

9. S

 ' ,5*  9 859? 17(%>BD." 5 5  4=  *  '    8 5

  ! 0  276R$

 ? 9@ mS < T < T φ

φ



? '9@1. 4? K\95 D36

 276 M8.85*6E45 27 901ED3%'Q  5* S

695[;.85)1E45




" . T4 ρS ∼ ρφ · 4 , Tφ ! 3?S 0) 959<

)  +;5' 

 C. T < mS S

ρS ∼ ρφ ·

!

mS T 3 , Tφ4

3 *  ;  5 (! 4  3%  ,))  8 5/ K 36T+ 8 5 9  :$? '9  @ "
@ '  # S ρ (T ) ' 95K3' G

   .(W3EX156. 251.K 25 *C5K1<4 * ! S 250

0.2ρc ? '9  @C< . fE0T2 0     6  $;.C  E4: .@*8 59  : 5

' 2 7 S 3%

ρS mS T03 ρφ ∼ · 4 ρc ρc Tφ  m  ρ φ S  · 4 ∼ 0.2 . ∼ 0.2 · 1 5 T


"

ΩS =

 .

!

φ

R2.'9.3 ? &  'O3%97$. T 3% *6  02
  . :O3%9@C<#"  7 '' 8   Y   4E18 5C  ?

>CE0  ;B,  G3$ 1<  T = TN S ∼ 70 5 5 '   ?

9  @R3 *  6 W  02
I?)% 1? ρS (TN S )/TN4 S . 0.03

0) 959<   $02


9  .   *a  E4?9  .Z 8 :3?   

 *  1B  - ( 2 @ !2 7 S "-. !




. D3'2  3K

D959<

5

ρS (TN S ) ρφ ∼ 4 . 0.03 . 4 TN S Tφ E2_01



   -" .."! mS > TN S ' 70 5 , 0 * # 0 X ;5C?   1B,* 9CB#2580 6  851@ECE7( S! *9?

   $;. 27d.@*;5 3 8.4" "-.  ! ! ρφ . 10−5 , Tφ4 95C?$8.3 0E43% SC  B -" .  ! 9?8 .0 *  ;_  .@* , 8  . .      T * ; " mS

   8   5    5  

8

 .   1  5  9  .  5  W  % 3      X    ( $

   2 + 

  D    95   (        

  > 3% 

0)95 (  5; B;ED  D))  5

 V?*

C  @ M)A  3?1 52 7 .8 3%:[8 59  ?2 7:6 BD.*" X  ! *  Y  E+1 B,9  .8 3%8 74=

 " . "!  " . ! *C:I 625$ C?

9  ?9  <.  !6

   '1 E4: I0 

)2 <

CB  2?   F.@)a  < ? (   9  <..9  51 @

,1 ..80%6 BD.;9  518 .; 5*1 E45 ?C  58 3 4= T Λ C  .  5 R3 9  : E V-  9  :  01ED3E!99  52 @ ; < 5

  P S 2↔2 " .
"  ! P (S) + X1 ↔ X2 + X3 . *MZ9  52 @

8 < ) 9  ?9 
3% ;51 ..@1 . "   Y 8 <4 1 : 9  ./9  52 @ ; < 3$ 'a  E4? ;5 2↔2

2 ? 4  3%2 7dB,1 . 1 7 ˜   !'" 7 )  Q?*

C  @  851 :I9^   ?

9  @9< G S P X 1 , X2 , X3 , Y M)4  3%1 52 7.8 3%  @! 2 5C<4 *C:I g5 '

9  .9  5C8 .; 5*1 E45 " B T




































 

 

.


"

-.  !^

5




"



-. !\ )aE4?8.


"



 

  

'K3E4:^E4:_ 1@ YE  3 + 8?  3 2.95YEY:   .95 2@



-. ! 85CE4:2S;5

 2. aE4?

mS(P ) < T < Λ



α , Λ2

3%  0 

))K CB, M)43%1 5 27Q. 83% 95K3%9: 6T  *'6= α = g 2 /(4π) g #

1E4:^EY: ;5CE(6B=9. 83%;7

 38 .44 " 5 *" " "  !'" N5*  P3CE(

 20 83a= Y X 2 X3 

;95 20 ^5'

 F? '9@ M)A3?1 52 7.8 3%'8   ' W5

 (

20256B, ;. =? '9  @ S(P ) 1 lS(P ) ∼ , σS(P ) nγ (T )  56 BD.; 5

2 5C  B ) σS(P ) ∼




M∗ 1 ∼ P2 l , H(T ) T  *9EY?9.B 2@ CE31!15*;5 2718.; 5*6 EY5C< B,1. 25(>C5 2@ ( X 2025>B, 6=  8.&E 8?fB,  ;. = ?

9  @ S(P ) lH ∼




"  "! Λ . ∗ MP l  '>8.;5*1E459 587 5

'  27[95 2< 6D%6'C?CE  = ?

9@9< TS(P ) S(P )  83% (TW5  E M?

9@1.[;59?2 7 0%6 BD.<#" 5 ;5 fE4:;. # aE = ?- ! 3?W5 2<4 = ?

9@9< 2025>B8E4:IS  $;.C  E4: .@*;59:  [ ; C:I  S(P ) 01ED3E4?R5 ' 9.Y" ;7

' 2 5; 0   ?2 02<  ;  I95 YB,   3% SI?)    859?  9CE0  ;  B, 

TS(P ) ∼ α−1 Λ ·

TS(P ) & TN S ' 70

3?$'(  859? KR.@* (;)60(  27

M5[)HB I?  

 5

YB,

Λ & 2.5 · 106






5 .

95  H?

 "-. "!B=K3% 2. R. X;.85*1E459< B=0) "   !    1.XEH?

9@9<$38 7 '' 8 - 09:^  )01ED3E4? 5'

 9.]"

@ 03% * =?

9  @6G *  '  ; 5 T

2 5; .  17 5 '  179C5  83%>= S(P )

 ) 9? T aE4?1: 85 8 .4" " , ! " 5 5; B8E=  =)* 01ED3%; .9  .'  




Λ







   m mS(P ) · nS(P ) "  ! 106 1 S(P )  5 · gS(P ) · ΩS(P ) = = 0.2 · . 2 ρc 200 g∗ (TS(P ) ) 2h 2025>B, 2.3 %>0G ?

S2 @ L.@ ;<

0)95[0:  >3% 

D95.<  *9EY?  *9.  6M? '  ? TF  3H.@* 8 < 5' 1 <  ;5 " ,!" ED3E4?R  5' 

 C  .(,  .1 . X -   9 G6 BD.< =?*

C@9<\. V S(P ) 02
3%! X *9  EA= ?  ]  B,I ? ^  .@ ; <  5 *' 1 '  ;!2 . .@)  <.\3 ? '9  @1 $ 2 7  $8.2 7 200 .@*8  5

 ]8 .4) " 56 BK3%' " "  9  .B2 0256 B,2 .3 % 8 3% ; :F5   = ?

9  @9< S(P ) X8 3D 38R  5 *  R? '9  @E  $8.2 7 .@*8 5Y"










"#" # ! 5 .,-( /& 0( 5 I()".2" 8!/"2"3""2+S!/JC!/!/&'1"# &' (  *!W 5 * 8"2!&'2X3!/5(12!/&'2+5*! R ().T J2"K2"0K/5(1"
F2"082"!2 2GC"" "2"02" "!#%$ 2"0&'8""2"WA 









.



∗

   *


 



 


 
 #   ( *
  
  
     ? 
%    
?   
     *#   
        S(P )       
       






        



  2
    
   
   7  S(P )           

?    

?                5  
       Q   E
      Y
?  
?  







 



 ' .@*0

9.@) 8.85*6E45B5*87 5   27 $95 2< 6D%  \ TS(P )  *  ' /5' 1<  =? '9@  83?** $ $ 85 9: 5  f= S(P ) nS(P ) (T0 ) 9<. a  E4?8 .4" M5   2. .@  . 6  3% 

 *bB,I?  8 5 1<  7 6= m )

V%f

E4:(   T1 0*  :3%8. 27C015S(P  17U.'! 2.@ /;)109< Z

15CE01E45  5

'  27]"

0; E7W3% G '

  

E)! R  5/  >952 0*; .<  $8.1 7.@*; 5>25;01E'RB,>= F3%')); 71! 9  5 C  < b.8 3%8 7Y"  5C  .8 5X ,1 .  V%*  *2 7P.8 3%'

 0X! 2 52 7T1 B,01 :IX  $

 3% *9  = 5 0I ?

R9

 T1 E7. '%1 0 1 B=*9  .8 3%8 7 '1 E4: V  ? '9  @9< 41 0; ED3%9  . .8 3% .!  1 . #8  ?9?   9E0]" *ML  ? '9  @9
1 E4:I >!2 .  ,1 @ 2 .$./  C  B8. 

 *  6B8EK = .2 .J3 5/  1C52 0*; .< Z=O6 5CE7 1S

  9  < E6 B=9  .8 3%; 7

2]"



!%L %L2!*34%5!I \+ V?'2 !234  <%= ('K-%'S (  2J!%!!H ;2!)>  ,<`?%!!ST DD'! 1'SN!h2(: >A@F< @ D α ˜ ·θ ·G G , ∆L = 8π Ki2( α = g /(4π) b VF ' !%K0/0' 12(! !Ef' G b /%9('*%FES O! .KG34!!K0Q*!Ez' G ˜!2(!OIMK=2-!%E_F*'*!2(+ G b  2J2J2!'F%!!'%!!%!M +, 1T2(K0h!/@/'-%f'''2ST θ!VF!b TED*!A@/< 1 @ :  %D,!! 1+,! ;T*(%%!%Y! ; < X %M 1> V!W0 b ja!'j4!&!a%!VN( V0α2)!>A!/(8π) < @%D: F @

1 '%!SN!H /05*^H 0!VF%_V VFV2K-DD' %K0 !&3 _K0hV!/aKG%Q34> 2!!EI+,*6?%* P+] 1+y' CP%   ;%!- P('K-%'S (c>A@< @D4M*%: G s

0

s

µν a

2 s 1 µνλρ 2

0

a µν

µν a

a µν

a λρ

0

s

a µ

∆L0 =

Ki2(

αs · θ0 · ∂µ K µ , 4π

« „ 1 K µ = µνλρ · Gaν ∂λ Gaρ + f abc Gaν Gbλ Gcρ . 3

J2V$!%!9!%/(7 1!2(!+,'6L!8* ;'S >K-%!) ; '!S +, (6?'c V./>A@ \*!<  @  D?%F!!6F!H!_2J O!9 \ V9(0!2J% !K9%&%!-< !TD;!!E9W!E9T342!+,VST(69'22*!!!%!%!Ef'$& %_V(!!03E2P'+,)6)22(VF VF( j4WK5 &% e&'!!V : 22!(EVF 1!VFj44!(K!&%%! T&!( !*L: ! 1  1'%!Eg< < $_/(('!?A(?=*!0%!)%! 7&%!THV-' 6&%! 1: =?%! ;XK0&'F!ONCP V-'+,6_W0jajXV!'- !+,6MV-%V \@'F*%!02! VH*!EF!!!3  ;!%K0 V(02J a2J_>A@< @ D-'*%*!%&(%!K0hja/a2% 1(' ;   ;%!&+ V-!%!VF Mˆ ' ” “ >A@F< @BD α ˜ ˆ ·G G . ∆L = · Arg DetM 8π 5hVV!6? -'!!S 2(%)'! Arg “DetMˆ ” = 0
< 1TOT ;! % T5 -!Q'S 2J%' 2% !+, 2J V(ST022(.>A%@< @  DM 'M ;'2!I! MV'!*%%%!VF+,*!?'1%&V(0%(2>A@%ST< @Q+PBD1 3X VEV'/0%!' PVE Q 934 EJVFV<' !  V  !  )V !%E%/&M' **!%_V+,%L@'\V%*(V0 2?hVF e 16F%V!8KK0  Ez ;''   V+  .TV-4!K%!/VF%F Ef <' 5V('VHK-%!!SN('! '2(%!H 12( !2-N2(*! % 2(*!!!%!+ 4V'2+ µ

q

m

s

a µν

q

µν a



q

“ ”” αs “ ˆ q Gaµν G ˜ µν a θ0 + Arg DetM 8π αs ˜ µν a . ≡ · θ · Gaµν G 8π

∆Lθ = ∆L0 + ∆Lm =



A> @F< @-D $2!!L!V&!+?'' 2+c(!% 3X CP : ! 1 ;%!3E''*%!O RTT . ,!!!4*(!% 1% θ 1'STT+ %T*%FEF2V K&'85 X?2(). /" &H/1*!2+ P/582!/6*)I" & 1 

"
4

θ∼1.



µ U (1)A ∂µ JA ∝

αs a ˜ µν a 2π Gµν G

A





 

 




 

  

Kb20
$!V%''&!27!Eh>A@F< @ -D *!%!K00T2!WNVIV!%!%! VF('K0!4%2!!+ *! 8!jX%K0 1T ; 1K!?>  Vh! #^QD; !!2( %E , < ed 2'- &!/aVTVF!%aK0-5SN 1!+,6  . >A@F< @CD d ∼ θ · 10 ·e· $VF *!!%h 1K-%!'!'!!% %h hK0h^ !!!%!!(0'h*!VLT(%S Y< T 345 2(!+ T*/!E34L2(%M T:  , >A@F< @ D d . 3 · 10 ·e· /!V?K-%!!! X 3 2(52!EM!!!+ θ


n

n

−16

n

−26

|θ| < 10−9 .

:A*%! 1. !%+,6 /0! ;2!:  5!'
_U<60%2! ; 1V' 6 % ('2J!)%!!fUH E ;!2!? ;j4'/ !V _θ(: * %!!&( 1'F*%!:AH92!( 1/+, \-0'-X GE '2JCP *!!%%/+,6)7/0 !! ;2234  !U<*%!  T!E)(O %!3  T2(! E9+, !! .*+ %!2!K-'%!ST!!SN+ (' ; 2V!.%!OVF H G7E?VF (!+,! 2( \ %!Y V<[T(
 W  !6CP  ;VF:A*2 %&EF%!!6  \!! +  : !-!%('I !%!LV (%!  1 1% ' 



U (1)A

A> @F< @D !3XFV'θ!:A(! VF! ?9!/+,. 1%+ 2(& %!X%T/ O.6F!!I!KK0T!  !%I  1V 1vK0%!T%/!Q!/%`5N hVja =a/%!VF!+, 16 R@' > U%!(1)TST4! \+#<.V
*!  %VV2J )!+,%!K6R%!*F*!Q *!1U<+2($ !Ij42( !!VF ;/  1+?V'Q5%(EQ V!*!ST!&  : !2(VF +?'F 1%3X5X*!VF!!!ED >A@F< @
D ¯ HD + Y Q ¯ iτ H U Y Q (!%  U (1) :  1 1%3 EF! U*%M K 1_7>A@F< @
DT+,Q+ !-!%I !%I*%!: % 1/ +,!!H8>+]A@< !@ D-'J!% (!'+ +  b2'! % 1+!!ja?/06?K6FK0! KK  VF NVF*K0MT*/EI 1!/! 1!M!ja(/ %?V(LV *H%→. *!H!'JS L!3]%! ! !!K0: ' . H< H →

2(! V-!'%!VF2(!%)%!!(h!%3 +,(N 1L3X 2(!. ;%R 1W ;#'% S ! 1IR ;%!K053EL'1(*'%)K-%W'Vf \SN'Y + <J^(:   V!( ;+ 1 N !%! E8VF+/0 'L&!2%!!/ 1VF7 W !U*(1) %K0!!!!:  !  1VF ;% . (T%'!K-Ec%'>A< @FS $< @ ! D: **%!!F!02! .TV K0T*('EF/-!'!!!!X3 V / U<8  =TVMK +,> !*!!E)'F%: K0 V PU (1) 9-VT( * %!2 VFE3X

 G&Eh%! b''DLK05*2(% ?*F% %VF/K0.*!!!EE 6La(x) '    \

:

1%! > h2J'! c !F( U (1) DD'*TV  *%!%/- E_VV >A@F< @@D a(x) → a(x) + β · f , Ki2(% β/b 34T*(%!8  1 %_T*%)(%!%/T-'!!!!E7Ef'J je 1!K 1&%%%! 3X5H>A< @F5< @ /ID '( 1f 1%b 7*!%!/ 1VW%v%%!K0/ 1!% VF K0Q ;(' K- %+?''6FSN!%V : U (1) (V'K-%%'VS Q(I? VF 1%!(I&*!%%/-'! >A@< @ D' >A@< @@D5 2!-!'(I*!!?VF !(_. ;  + V%VQ6-2E_ . >A@<  D L = q¯ m q + h.c. 5 *!!%/-E>A@F< @BD'-*!!F  3 2(&'!5(5V 9&%! T!/VW%K0! VF K-%!!SN('e 1 GE K 1 >A@<   'D α a ˜ , L =C · G G qL →

. iβ q

L

,

qR →

. −iβ q



,

PQ

d

u

L

R

2

R

∗

L

R



PQ

iβ

−iβ

PQ

PQ

PQ

PQ

PQ

PQ

m

R

a

g

q

−2i f a

PQ

s

8π

fP Q

L

a µν

µν a

"  "8W%" !/ $/1)/1%1"2"0.( 0/2!&' ()8W)5!12">5 C!2"!/ "W%1" !/

C!2!5 # /1&'   /&'2+*!I[1 "2!/" & 



H = dn

S

S ·E. |S|

5 Q8*!81"2"3"5 "2)&:C" 1*)& E 

" 5*)2>$#  # ()2!>2"0J* 8"2+$ "/1"  #%* # 0&H/55 #4& !/C""2!&' "/1" =\1 5 J+*)/ ()#K C"! C" "2" !/6*!I"#  θ  #%*!2 # "!/1*!T( /&'#KA K/ +* /58582"3"5 9  & ." "-uXR &' (  *!2 2!&' *)/5 IX# R  *! /6*)I!/1>"2"!/1*!I)/6 U (1) R 582!&H&'(1!2+( 8W!5(12"& Q1 W ".C"C"#D!/GC" *" 80&H/55   "/1"+/ )/1"2"& u → u  u → u &' 7,?! -#%*! -#  58C" 1*!I08 /&I5 13% # !/&121I θ ?".*!I)A M+5 C""2!&'%6/6*!I"# K( /""# ?5 2()( *! I5!1.T8 C"! #12" 0&H/55  5(12G* 8"2+$ !/1!   3!/5(1" 5!12 9 u R&' 7",P" /1"?+?/=! /C"&' +(Q /61 $ &'VL . 7/( (/"/" % 1S" "WX J&'8 "2()!8 *!2 CPF2"R C"08!2! "12 *)3)&'/!5 #:25 2+A *)I"#%$ 0/2!&' ()W!5!12"W C" R 2( 2!&' &'."8 0 R 





L

iβ

A

L

R

−iβ



R






∗

   *


 

 

 

KiF2(X4VF( 4VV  7-'!/ C! R ,b &< %!*!%FE- 2(UV(1)52!: !&! ;+ 1 %NEM*>A%!@FG< @2(D!'E>A@< @ E?@D\/0!%F%EF2(T \IXVE-!%!&VF'a  VF !L EF!E% G E5U*(1) 2(K0<
2(2 !): ' % 1 \!VHQ _*%!%!2/ bE' θ \:A*(+,!I*%! 1!%X 8*VF% 2 (*!6' %% G G˜ *F(, i2!!KD'/0'! E5!T Va*% %!! -': >A@<  D a(x) ¯ θ → θ(x) =θ+C . f \: !%! %?2L 1%M 1 ?H !!%!+,6_/0' 12(! EF6f'! Q-VF \!T %!G2T*!!EQVF !( TVF)'J''!! /hVFθi¯ P=V0 ;< 5 ;T!e !VT%! +,6H!e/02' 1,2(=! M!<'
<%/!%1(!^I%W0T
O'(!*!'E5EFV!E%! %E_ ! 1 1V%!-5!%!(VF  ;+, : VLF2(!!' 0!' h¯qqi ∼ Λ '!5*%!2! V_!%!%Λ 1LW0jajX∼V200! 1M*!&!!  2E θ¯ a \! \ 1

>A@<  BD 1 m m 1 V ∼ − θ¯ h¯ q qi + O(θ¯ ) ≈ θ¯ · m f + O(θ¯ ) , 2 m +m 8 Ki2( %m!2%!≈135!_^IW0!
3E'( f< <J≈ !93%^IE W0
\b : * ;%!  0? MV(' 6-2!_E % OL*(W'!2(LK*!(FJ < %!5 /H >A@F!<  B< De=?%2 1!h)'!K0& 'gh!θi¯/I>A2(@F< !   Da! : >A@<  BD, !2-'-LVF !! 1L ;   


g

PQ

PQ

a µν

µν a

g

PQ



QCD

3 QCD

u

2

a

d

u

π

π



4

2

2 2 π π

d

4





>A@<  -D / *
R 1*V2(! E) %!+,[56J*<'V%
9-! ;%2VF%!!L * %! T ;!2(!EN% !2 '372J 1!0%!2!!%XX ;22!!( GEa'e2(VF*!%!+,6e! 1%!3X+, 6 EXKVFK0  ! +?V!' 2*%!2O _'X2!34- V('' VF : >A@<  CD ¯ H D +Y Q ¯ iτ H U . Y Q `?%%/I W 76FKK0  Va*E_'2N%N% GEI*!02H2(!  U (1) :A*%!%/!9>A@F< @ D, !23X5! 8: ma ≈ C g

d

u

1

L

mπ f π . 2fP Q

R

2

∗ 2

L

R

PQ

. → 2iβ H

. → −2iβ H

!! %<  *!!! U (1) : !%! H%('K-%'SN, c>AH@<  CD' /!, #/ 1'SN! %7%!!% θ:A!!7 *'2(> I 8!*'V2(! h%2((E 'U (1) E%eX%: E_ 1% 1ja39/2%?%!6FELK(KK0!\ %*  VEJ!< 6M [ G*E_Ki2J * / !!   + \(H!+ V( \/IV !  6M%!W2!! !VF!% 1K I!ME&6F6M%!!/KK0*! 'D(TK0 V 2(6_ /F! 
1

1

2

2

PQ

PQ

H1

H2

A> @<  D Ki2(!  GvEN Vv'b2I?- ;VF  + \+ L :;%! G2!: !N/6F!KK0 '*! VW6Q \* U =a!! V!M!!M2EM*E β(x) /!!Vh!/aV VK0T!(N6F!KK0  V6L2!' H1 =

1

2

. 2iβ(x)

„

0

v1 √ 2

«

,

H2 =

. −2iβ(x) „ v0 « √2 2

,

±

2 1

2 2

Lkin,H = ∂µ H1† ∂ µ H1 + ∂µ H2† ∂ µ H2 .

`52( EEM 3 2JH>A@F<  D-'*!F!! )2!EHV!!! VFK0T!!(T*E Lkin,β =

β(x)

'

fP2 Q ∂µ β∂ µ β , 2

"   H  !/10"#  12"C!# "/"  &' .04"5(12K!/08"#  0/1()# e ,1" 5 2%18*!I! U (1)  7A AC"" !/10 # /&1I5 X ,1" 5 2%18*!I! U (1)q +/ q → q →   ()J *!2 R q q 3"2"!# e "     !/10"#  (!*"G!/10"#%$ 12"C! P"/1"  q A "  &'( 2!&%)3 1 PC" 58 1*!I". 5 2)&H&'(1"2" "A  = Q(1 S582!&H&'(1!2+ #1" 5 2% *!I" K/108 #%$S!/1J8"2"WO C", 18!2!/6*P()*" /1+582" !/F() 1*",-8S#Q1I2!"/9 "2!/11 8S ,1" 5 2%18*!I! F()2!58""("#%$SC"" )/108 /"2"WP5 β =  A A )5&%A "A  =A F" !5 1W B  8!2"F# !/1,J8!2+ 9 "A   =  .() *!( 1 "T J8KQ 1 &'. 2π .5 *! 2"Tθ¯+→2)&'θ8¯( +?2π2( V =9 m f · 1 − cos θ¯
A " , K/% 7 %3 1 /1+582"  2!&'8 X&H/585 . "?C"", 12" "83!%2 1 "&'  1*+( (5 1 .!/ &'# .,J?.C" &'2!!/6*!2O %3 1 U (1) R 5 2)&H&'( 1"2" !/1". J8!/J"! P"/%1 #4&'2GQ  F8 7/&'2 % 7A A)/+5 2"  "*""( 5   

 

  08 ! &%A (P Q) n

n

PQ



(P Q) n

a

2 2 a PQ



PQ   !" #$ &%('*)

nL

nR

PQ

(P Q)

−iβen nR

n



 

nL

Q) iβe(P n











 

 



 

 



  


 

Ki2( `5!5VF !h E/0'L

!T

fP Q = 2

β(x)

>A@<   D

q v12 + v22 = 2v .

a(x) = fP Q · β(x) ,


P 1VF?I*%%!IVF PVF  !Q!EFT!E! REh*2! !a(x) %?E2STO?+ 1 %_/? >A@F<  2JD,%!! 1+,7> V!! V D$V V!H!!Y< VW0
. m ∼ 15
$VF/0 '! 1)2(> K0! h/T+,VF 3X!MNVF  5V c%V
 \'!@'FK!%3XK (-
 \!('!%!NVV!D STWVFT jX*!%!( ; \.!> % ,<T STV'D 3eb O2(U< %h !%< \VF  
T'!! $T !%(' 2J:   6F!%$ 1%! 10-6- %2>!!= Q  Ee
WegD ;V<J2
%! ;EF (6 2N!' ) !  ! 1 ; 5%W%!M 1* '%!SN%!!2! iI6-! !22(&!V? 
%_2!/0345!&VF  #!K0 14+ >%! \ (/'J

2A@<! ?C?DL VFV' \*E&%VF+, I !_V! V%IE&%!%_H**!! \ S 1hb /0 !! ;KG2Q *RV!('!*%%2(!!!E8K 1+,-%6MF*!*- ! %'2J! \%!! b





a













PQ

H1† H1 ,

H2† H2 ,

V!STa/0! ;2 Ez'/0 GE5! h!-!%(' `52I2  7K%F*!*+

S †S ,

U (1)P Q

H1† H2 · H2† H1 ,

H1† H2 · S 2 .

* S '2(h%! % EHVV

A> @<  
D *
%W!0 I2E !5F(3 \V9*>A@!< ; VFD '*( Fa(x) ! EFE GE4!!!!5VqF 1!('&!! ja/;* H ' H  S <0`5%FEEe+ V'2V0' >A@<  @D f =2 v +v +v , Ki2(2!vab /T>A@< V @D \'M  *%!2!!-L *! !'E S <\%T
;VF  + 1!VFM ! %&2'!('!va 1 0' ;S e KG(+ '!% VF'- (%QK 02!% Q/0! 1!%052 b'\!EN&'[*VVE &!VF/! ! 1!M2(%!  +8jX ;%!- \ThI( a \f*! ∼v2( \Y%(< %Q  1!02E !U (1) : ! 1  ;3X%5Y<T`5 \% 5 1!a+]/0!/ ;2 1 2( EN ,EV%!%!V! 1M('-*%!V *% : 2(!.2(R*EF!!E!3X5%6-! +,EI6R !*K!: V \%)
V% (Ψ'   = !Ψ%'(
 *'%!\*!/%!/E3X3X556-c EH<
*5H%! 1j4!F_%2JW 1% + %θ:AL!V! \-@!%!'\*V% R2(  %QR('!3 %SU (3) ?   * SU (2) × U (1) * UE (1)'g !';K!+,!M+ QS V-%!%HV8V 134%!!RK -%U*(1)*!+ : /0!%F2JEF!2Y%!<;
!R 21!0!2I!2(('J**%/!!!%E!%//034K0* GEQVF 0 1%*! VF! !!K0% :: U (1) ! ; 1S%!X3XV' VFa/0! 102 !5 a+ \*E 1 S→

. 2iβ S .

1

2 1

PQ

2 2

s

2

2 s

s



PQ

R

s

PQ

c

L

W

Y

PQ

PQ

PQ

¯ R ΨL + h.c. L = yΨ S Ψ

-`?%VaFW \! + S 8* %%!2!!%! 9/-*! GEXE S'  !% U (1) K ! >A@F<  
D< U (1) : ! ; 1%E (!%  EX!!+

PQ


c2J'!! 12(!I*!!TVF ( X 1! b'e 12(!% 

G)

PQ

vs hSi = √ . 2

a(x)

*%!2E GEMja/Q*E S '

A> @<  D
L! 02%FSN!4EF!e/0! ;2 ETVF (, $+,!+ \hV-%V \T !*( \Y<

=

fP Q = 2vs .

*  X$ ! 27_? '9@87 D1B,9. 83%87

6EY: ;7 '%60 4  *C. M)Y= "  ! ( P .@ 9

CB,S ^.@* (;)60 2.B65CE7  =O

9.. 15 "  %1 0 

 X1 B,9  .8 3%; 7

9 20    ;8 .$2 0 ' '  

2 .3 U (1)P Q f ? *0

  *'  P Q

 >  3 *Y  3%

1 E0 

9  . 06 B, 1 .3 I

   '1 E4: 9  . V1 0 )  6=  27$ C. 25  a  ) 1 5CE7; 1 7H  .@( K)1 0 " 9X:  3$ !

fP Q  M W  D!  

3?15 27d. 83%']"59XE 'K3E >CB











∗

   *


 

  





 *9.

 8B=15
6 C<.;   '6= ' K 20 20-0$  27 159.XE= 27  *Y3S085 85 

)C3%

 1E0 

! G *  1B=C. D387 'M!  \

 \15YE2;  27  7C.@  58

;5CE09<

9.. 15 D  8<4

La =

3%

JPµ Q =

1 fP Q

X


"

· ∂µ a · JPµ Q , (P Q)

ef

f

  !


" 

  !

· f¯γ µ γ 5 f .

859. 9<  ;E7PB=15
A3 (P Q)   ' ;5CE09< JPµ Q e ,]9?9< B=65 A3% Q3L56  B,9  < +  85C.  fB,  *[ 15* H. 6= U (1)P Q 3%Y"9M5 2.K

 ]9.;:I *2 .@)  9  <41 B,9  . 83%87

$ 0 V:R 1. ;.4" "  ! !\  1 .   3% [%I3> 

3%1:I








+

"   "! a α a αs ˜ µν a , · · Gaµν G Laγ = Cγ · · Fµν F˜ µν , 8π fP Q 8π fP Q

3% 0fBD56B8.85C<4I!  ')1<  95O3%' 9:I R3 ))C.T! 95  27. D3'9S C Cγ 20;8.Z '9EY? )9:^ >9?g 1.[  25
43O3%C@9<#" 5  ) 2.C

V%

[ "   ! 3%;7

 "   !. T S9  5 ( 859  5  * 8R ? '9.3  B, * 92.

 "   !

Lag = Cg







La = −

1 fP Q

· a · ∂µ JPµ Q = −

a fP Q

·

X

(P Q)

2ef

f





"   

mf · f¯γ 5 f ,


+


!

CE

0: 2.@D9<4 %I3S< 95 83 K11B,9. 83%;7

9. "   "! " 5 B=>=  9. 83%;7

 "   "!; "   !]3%87 '' C9.;:I/43 " . "!8 " . ! 3 'aE4? ! FD"  " .8 3%' :!

 1. *9:^  95C.85 2. 5

8. 15  1. P (x) = a(x) %* ,  .8 3%8 7 (  $

9  . '%6 0 1 B,9  .8 3%8 7

6 EY: 9  .&  > 3% 

0)951.Y" 5 3? 2.

'9  EY?^  .@* ,9!    *   1

2 08 3%C  <.1 51 . 1 52 .4JC  B "  !M9  . ; . ma  7  5   5 "  , ! fπ 10 ma ≈ m π · ≈ 0.6 · . 2fP Q fP Q   E   $

! 

[   ! 

     T  

      C  <  .   *    )    2  .  5  

   G ? 3  *'  5 G 3$R  3     ,  5; . 2 TC  B,B!   . (  9  5O 3%'  ZC  B -" . !/ 3%. 'K 3E   * T   τa Λ =  E4? ' H  T   ;  "  ,! 4πfP Q /α   5 5 64π 3 m2π fπ2 1 24

= τa = ' 4 · 10 · . Γa→γγ α2 m5a ma  06E Z?2 01< 2  5; . (Y  B, !  .9  5 2 < 6)  2 58 . 9  <7 1 B85

 5

' 2 7  58 .Q  513 "Y   )a  E4?9  .- 859  ? T

8 5 4E:  .@

; E.8 5  + 62 ? τa > t0 ≈ 14 BD5  ! 8

. ) 1!

    "   ! ma < 25 5 . !

 &( 8? ,' 5 [1 ?T  B85 B( 1 58 .  TC  B, 8 3D 3CV  5 *  :  $;.2 7 .@*8  5]" 9X1 35CE2  ?

9  @9< !

 9  < . fE!  5 R3%*   1(! ; .9  ?

!2 7.%6 B8.^  B,(8 ?$'    P ./  C  B8.@  D"L*"%S9  52 @

, 

3%^.<


+































q+g →q+a. q+γ →q+a,

 



 

 



  


 

+D" 3 " 0256B  C<45;B;E==)*9)! H./ CB8.@ *0

 9<2W. fE![2<4

6E*   5 * $;. 27U.@*85]" 5 6=H852< 5 3?-T 'Z!

 9
 ;5  "  !3?-$ : '! 2. .@)  B,I  ? 

E4?$'2 . "   !" 5 > =O2 59< 5  D 3  8]5*  9!

 9  < Ωa 2<4 '1 E0)  01< V  5 *  $;.2 7 .@*8 5  F:   0g  .< E76E 8 3% 9  5*  E02 .)  W  . 6'E  '  C  BP?*

C  @:( 8?  S   5 ?2 02<`69  5 2 5; ? 1> = 0*  :3%8 .2 716 5W  .'!2 .@/ ;)1 09  < :

1 5CE01 E45 5  2 7Y " 9M51 .   ]!  9  <  .8 3%'

 5 D"L*"4!  9  < 6'   !    g 5 R3%)(  f . 109 10−2 5 02<_  B,  $fBf3? P5Q '? /9  5'/ 02<_ C  BD.  C  :   *  :@   'K 3%2    :8 5 C  :



)*4  3?1 52 7&  *  :@; 7Y"  M  .D 3'H  B=C  59












.




     D
 
              )   *
 '     
?    

   

? 





/



)     =
?  #         
   D
(        



9X6  B=)  ^02< 1 B8. T  '   

 A?  $8.F.@*8 5 ( 20156 B,  XC  BR!    ! f= :? "*M 9G)]"# 2.9.     W. 83%8 X!  1.[9. ' 3% 

8@4=  ?  9  ./ !*C  BD.@  9  58 3   >  5 R3% 9  : !

  SW  5*8 7 5

' 2 7]" 3 ./   C  B8. CB=* L

5

G 3%1 . V1 0 )  C  < 

2 5YE   ) C  ?

  3%   &9  56 =

1 5 ' 2 7P56 B8.8 5 

T8 3%  .8 :  9   TI#  1 5S #

 )9  <./1 5CE7 8 .  0 2 1 7 ;5CE09  < 9  ..'2  5 [5 *  # (1 5YE2; 2 7& 85YE9  <   85*  !" 5CE2

2 7b.f = U (1) !*C  BD.: 

  *R ; @ 9@26*0

  # )958 .8 5*6 E45  P Q )0 3? T . ΛQCD  .   ( 1 50 2 9  <7(! 4  3  *  1 5CE71 :;9  7 =O

9  .. 1 59  :+ W  3T!   U (1)P Q

 8  59  5CE'   9  <71 * C  @)a" 7 ; . 2 5C  .  'K 3%C  71C  B     ./  C  B8. " M5Y  B,   8 .; 5*1 E45  !  9  52 015 )'U  .@

; E "  , !;Y  B;= B, T ∼ ΛQCD G 3?G ') 9  < 9!    fE451 @9  79 *8 7 1E4?*

6 EY:   9#

  C  < 61 B,9  .8 3%8 7

Y  =  

) 9  <49 ! " M5>2  <4

  (; .8 5)1 E45 5 'E%G  3< 8 3?**f = T >Λ  < )d 9  ?P .@

,9!  P  02<4 '2  5 9G 3% T X 2 < ; ; .QCD  .8 5*1 E459<  ;












 




3.7 "    ! ΛQCD ma (T ) ' 0.1 · ma (0) · , T > ΛQCD . T   !   )' .     *9. 20156B, 1.3 0C582<4

T;.85)1E45 T  ΛQCD 0fBD.@

2<.4" #9<.E '  1.   9@) 3E * 



a θ¯ = θ + fP Q 9561<4 2;8.;5*1E459  fE0

1E' 0  ' # )  . 6 G95C.@* R:0 NB I?  θ¯ ∈ [0 , 2π) .

# 90[ 

 

97 T43?* 9? SI?) MB,I?  ¯ 5 SCE=:K"M5[ Y= θ I ? R8 3%2 58 3% R 0*1 <4 *  6 i eB,I?  ¯ ^

 25 CE θ¯ θ .C.XEY.@E!  

  9  @) ¯ " e

3%2 58 3%  *  :@  BD< i ;<4 '  6 θ=0    C<.&% ;5  T1 .

6 K8.;5*6EY5C<& )

fP2 Q · L= 2

 ¯ 2 m2 (T ) 2 ¯2 dθ − a fP Q θ , dt 2




∗

   *


 

 

3,




3%b  3

95K3  9590*%$  2@ 0 "    !" 5 5;B;E==)* *aE4?9. m (T )

'K3E4:;  aE45  3 6BD< ¯   *:@ 95CE4:;87S,5* (C5 C:;87  5

'  27 θ




"  3."! d2 θ¯ dθ¯ + 3H(T ) + m2a (T )θ¯ = 0 . 2 dt dt XI?) E   95E2<4

  >8 .; 5*6 EY55 9.  T43 ¯ ¯ " θ = θi = const T  ΛQCD M5ECB,[;.85)1E45 )   ? Tosc "   "  ! ma (Tosc ) ∼ 3H(Tosc ) , 83%2 5D 3 ; * ¯ I ? > 8 @9  5 *  8_

? ' 2 7

.4) ; " 5> Bf3% " ." !'" B ma (T ) θ "    !F "   "  !F *a  E4?9  .&2 @ C  EQ3[; .8 5*1 E459< 5

' 2 7 9  5 ! 2 52 7 I ?> = :I 1 8

@9@  BD< ¯ 6 θ




















0.7 0.2  Λ "  "! QCD · . Tosc ∼ 200  5  5 −9 10 200

R2.'9.3 >? 3  '! 2.1$( ! R!     D  )1B8E4.  1.K95Y= ma > 10−9 5 0*6  "    !a

95O3    ! *YE]3 )  !    " ,. ( Tosc > ΛQCD "   ."!F ¯  *aE4?9  . dθ/dt 







"

1d 2 dt

dθ¯ dt

2 #

5 ·

ma



 ¯2 dθ m2 (T ) d ¯2
θ =0. + a + 3H(T ) · dt 2 dt


"  

!




 [  8@09@  # K3 M5(  95 EY5  "  ! ma (T )  H(T ) . ( P5( (C590*%$    

 *1B  - (P5 

 2. ;5K3%E S85 83E[  f= @9@C  7 *  2 +   1 dθ¯ 1 1 2 2 2 2 ma (T )θ¯ fP Q ≡ ρa (T ) . fP Q = 2 dt 2 2 #8  5  Q  E45  "  !U0  E45* 98 5O 3%9  :,:`  

 1 ; 5 :!  4  = 

 ) XED3%2 0 ; 58 7B /; 58 . 2 7 ; 58 . 2 7 0T  .<    D 39= ρa t T  C  5* 3%%)/  1 7 !" 5 5f B8E=  =)*F  01ED3%8 . 9  .  3   *a  :,@/   '  8  5

+!

  9 #; .8 5)1 E452 7

?2<4




CED3?9 *aE4?9.

dρa − dT



3 1 dma (T ) + T ma (T ) dT



ρa = 0 ,




"  ! ρa ≈ Ca · ma (T ) · T 3 ,

3%  !  25 0fBD5>BD.85 0 

))" Ca # 20256B, 2.3 C5 &  

  ;5   9. T  Λ m (T ) ≈ m (0) = const !  

/ * K 3$ / 8 0 0 QCD   a

  85

a 1B,9. 83%87

6EY: B85'4= 

 2 ?*

C  @ C  

 Z3?- 6[! 3?L.@  !  ZC  BD. '  C  59  .; 5 \9  5 0 * :2 <4   ; .8 5*1 E45  ma (T ) 6= const !/ 7   > *  '  W? '%*0

  ;O 3 $ 6

8 080(  

 W? % T ∼ ΛQCD  1 B=C  .D 38 7 '1 E4:;  .8 5 '   b?

9  @







na =



ρa (T ) ∝ a−3 (T ) , ma (T )





 

  


 

  

D" "   ' F? '%9!   9T

CE! '1E4:;8. 20 $;.# Y5  ]"# 15CED3% T 2@f=  R  

  ; 5   D!  2 7KR*0

  X^. 1. 6EXI?)%\  8@C@97 3>= ? 2 5
  C  5K 3% (9  ?Y  E[0 

)2 <  59.XE= "  ! ! 2 Ca !  ¯ 2 1 1 2 dθ + m2a (Tosc )θ¯i2 ≈ fP2 Q m2a (Tosc )θ¯i2 , ρa (Tosc ) = fP Q · 2 dt 2







;5K3>I?) 2.$  ;@C@97 ¯ " R 8:3?X *aE4?*8.  ;  dθ/dt ≈ 0 ? %9!   T6 2  5 1SI ?) 8 @9@C7 3% .
<5S

5

ma (Tosc )

?;.

?;5;B

ρa (Tosc ) na = s ma (Tosc )s(Tosc ) 1 fP2 Q ma (Tosc ) . ' θ¯i2 2π2 3 2 g T ∗ osc 45 p

9 2.  2 H(Tosc ) = 1.66 g∗ (Tosc )Tosc /MP l

2:


"

 a *9EA=

  !

fP2 Q na ' 5.7 · θ¯i2 · p . s g∗ (Tosc )MP l Tosc *M)T'9?T 

)$' + 

  27 95E  K3E4: 87   *a:,@ 5   27  ' 2.XE 3% * E!   TS

25;. 1 71 *  '&   8 5 d  5 Ωa =

1 na · · s 0 ma ρc s

s0 m2π fπ2 , ' 1.4 · θ¯i2 · p g∗ (Tosc )ma Tosc ρc MP l

3%W.< 2<5>B, ?85fBR.@ f

E.!

 8  2. 01: "  ,!"  *0 8@ ]   *1B;EA= fP Q 8. Z 1@ ! 27 "  -"!3 C8 .8 5*6 E459< I ?)%+ 8 @9@C7&C

 2.C. ? ' 9<4 B,G? 

;. 

. 8 " 5f  B8E=  =)*c  01ED3%8 .&9  .  s0 ' 2 · 103 −3 ρc ' h2 · 10−5 5 · −3 5  1.2 −6  5 4 · 10 1 2 Ωa ' 0.18 · θ¯i · · 2 , ma 2h

3%b.< 8 3%

)

.4 ; " 5 "  " !S  )  1B,  )g  3 1@  g∗ (Tosc ) ' 60 Λ 9?Y  E   5

" M5  1 . 



  [8 ?))   ? >I ?)   _  B,I ?   B8













θ¯i ∼ 1 .

G)

 !

 .W

.@

2 7  06EI3 [ 

)*  E;

:`  $;.C  E4: .@*K = ma ∼ 10−6 − 10−5 5

'   

     2 ] 7 " R

1   .

   C  3 .  ?   '    d 1 0 D E % 3

    8 $  .     V. @ *    8   5   a   

0  )CE:  5 1 3*G` 95O3%

)**';

20 276 83% 25 83% \  ;@95CE4:; ^ *- >3?* 4!  25

]5 #YEDa:

;.4a " 56 BK3%' " ." !/

59:





 



#9<.

pa = hTii i =

CE= ;<



*

fP2 Q 2

  ¯ 2 +  2 fP Q 2 2 dθ ma θ¯ = 0 . − dt 2

'  1. 4*0

 9< )! 27 015:9.8:^&YED 2<4:95 

25

 9<49.Z= E * 2.XE *9:^ /85'



9.Y"  C. 20156B, 1.3 a!  C< R.@

27




   *


∗

 

 





 * 9:I >   2.  ) C?

 T0  8. *  9? [958.Q;.
9@ $8. 27d.@*;5Y"

R2 . 9  .3 9?W .8 3%'> G!  1. 958.Q;.<.[*43%43?))1.[c5 *; $;. 27 .@*8  5  51 762 9  5CED3% E8 38 513%# 1/ C5 ;5 CE!  !

85C. )) J  

 5 2  #< 8!   ; .@

2 7   5O 3%

))'6 2 02 7T  1.@ 2 5YEI39= 9 ma ∼ 10−6 − 10−5 5  E4:    ] C  0f B,I 3 $(TC  E4:!B,G 3?I ?CE3E 2 58 . 9  < 6 ! 8 59  .  "





 "  ( -U"",,,('-U"


F * 

#%'&S( +-

)9:^ Z

)G=

1 YB, 9?

9.&*43%43?))1.&L5 * ? '9@C$8. 27g.@*85

09<4X ;5 %$'a<4

? '9@9<


[.<

06EI38.6BD<4 



!

5 .

' )109<4? '9@C< W D3  1;59. 83%6.Y= ? ! 2. 5  $5;7 5 '  17 B=C5(; 9< YB;= B,>85 95 YB, 83% ' > $;. 27 .@*85 W 8.4" " !"#;. 6   ?*

C  @9< Q; . . ;8 8? [*

9@C ;./C5R0 * U2<4 0 17S8.;5*6 EY5  B=6 .2 5- T ' 626! 9  @ 2 51 @]" G8 :3?   ? > '$  *1B8E: 859.@)  9  <4R  ./  C  B8.< 5 W3%  bED3?) ( P3%  ' 

C  E!  E2 5; 01E = 8. 17+! 9@ 256@$ ; 5"0%$'  < e=?*

C  @ ?)!2 7 ?  8)*W  ? '9  @C< ΩX ∼ 0.2 ! 3%$H02
 g  5 

 @+  1 .XEZ  859  ?  " !FE9  .01ED3%'( 9  5C  . 9  . " # ;  59.@)  9  <4H  ./ !*C  BD.< @! 2 59<4  .<_9  5*0 :2 0; ED3%9  . '   2 .!56 Bf3'- W9  .8 :^ 3%  *   T2 0;9  7 !6 58 5:b  51 0 )1 :IR  3 9  5 0 >%* ,  .D 3'; 7Y" U1 . F  5> = 0 1  <  2  b. !Y  BD. 1 

 ; 51 @L  ?*

C  @.Q  58 7 5

' 2 7d3 

6 E   ! ; 59  . Y  = ))  9  <. 9  5 8 501 .3  :! 

 C  <.4"  9  . 2 025> B,2 .3 .8 3%



8 5 00$'  <. ? '9  @1 . S0I ? 'X  $;.2 7 .@*; 5ES*01 7Y=HP .; 5]  . ;E!P  021 BD. T9  < d./  C  B8. 1   8 56 @+0$'  < b?

9  @ ? 6 =

1 09  < 9 9  59  @R1 0

8 ?   2 0  8 3%9  . R! *9  ?

 ; $8.2 7Q.@*; 5Y" 5 6 =H8 51 <   

 ;  51 @ /B,E; ?$ 

 *  C  7$  $(   V?

9  @[; 59  ?2 76 BD.SH  .8 3%'25

3% .@! 9  .@D   ; .8 5*1 E45G6 BD.< 

0 )  ! G   " X e   E45*  T M F *  2@9.@F  3/   '\  ? %  = ? 'max  @ 9 aJ  5/ 1 1 TXI ?D  9  <. E ' 8 . nX 4. T 1 7  Y?_ 3 ( 20 8 ? FE '   5; 01E'  a?2 02< 2 nX (Tmax ) = 0 Ω ≈ 0.2 .@! C  .@)   ; .8 5*1 E45 M 0 *2 51 7I ?' 

2 X5
9?

)G 3   *  :,@ Tmax 7  .@   e=?*

C  @ 02
! ( CB=*9  < 5

' 2 MX X 2.9.3

? . 85 0%$<4

MX  100

) 9=  ? 

9@9<








 








"  "! MX 1 ∼ 25 + · ln MX2 hσi , Tmax 2

3%

8? ]5 W3%  = ?

9@[

 *  H?

9@[%6B8.<#"   * C? σ#  2?:T * 25> 25
438C  5 2 <  [8 .; 5*1 E45CE ? YB=. 0 *2@9.@* 6= MX T

9.  25 2.3 !  259<7Bb  3?2 .` a  E4?E2 0

8 ? 'max

E7;

  + 83?**  ; ! 9@ 251@b?

9@1 S; 5 9  :2]

5 

9  <. a  E4?8 .4" .








!#"  

 

  

 ,
?




 


"   

"!

"

R2. 9.3 2? E  F 20

8? S 20  8 3%C. 276   '  ;5 e=?*

C@25K=

 25
10 ):I 56B, 17 9595 83S<#J  9?  '  J.8 3%  9  <7B1 51 .'2 5 @ /'C  ? .@*0

9  .@)  2 7&8.;5*6EY5C< MX 01E T 83%

15 273%1E"E151. 15    20




"
 

 

3.


 

  

B= E . !CBD.@ ' ;

B,8859?9<.$56B 85  2. 5  4= 5

'  27 T max  27]"  <4? '9@9< . fE!+)65 R3%*    83%T ,1. 
+;59? 56B,  85    %$'  52 9 @   0; B;E '   20  D39.  !  ;. *

9?  . 83%' $ S C@  27 #

)I 3%8 7]" 5  2  L.8 3%'2.

 ? '9  @C<` )9:^ :5f B8E==)*P5

G3?8 %*  0 4  3%  ,*)

0)95 & * \2 0

8 ? ( B 9@C  :;" < 5* 8

. *2 59  .  

> = #  C@ 9  <7 56 B 85 B B*2 51 7 ? '   Y" ]52 @

5> B, ;5   5 ' 1 7  0 *  

  .D 3'; 7 3  *   9  5D 3 *  T   C; 5 9  : 1   

1 E4:C  . !6  020*  9  . 2  58 . 8 .3 W C  5K 3%9: 9  .2; .g  5( ;9  5 Z+; 5/ D 32 7

)G 3% *  C@$

2 5
9?; 7 5 ' 1 7Y" X>I ?)  2 .3 ])*  6 3%  *   [9  58 3% *  ('  2 .  )*,  5> B, ;5   [.

 1 020*    T2 58 . $   ! . T] E '  K8 ?))   Y? 0%6 BD.@ !> = 58  5C  B8E      1 7:8 .; 5*6 EY52 7 ! 1 5. 06 B8<4 ' V.

F 0 *  -;2 7 %! 2 5bE ')  $ 5 ' 1 7$(0 9  @69  51 @

,  ?0 3?P  3%8 74= ; .8 5*1 E459< Tmax

'   + '1 E0 8 5C.D 31 .9  ? ! Q  5 

" 5 5; B;ED  D))86 BD. ( E1 025> = B,  S0$'  < /? '9  @

.@

,1 .

 B,1 .  1C  5    8 3 ;9  . " 9 */  .< 06EI38 . Tmax 0f E7R3?) 2  . .2 52 7U?

  

 Q)!2 7\./  C  B8.`9  58 3%:Z1 B8. T  '

 K = 51  @\  ?

9  @ *9  59  .; 5 W

.@

2 7  5 .! *9  ? '- Z3% 

))2 ?2 . 3 M ∼ 1011 0    2  $;.2 7 .@*8 5  F/ 9@X 9  < .8 3%' & E8 .; 5*6 EY52 7 56 B, 85    " ! ^  5( ; S9  52 0*; .<  $8.1 7V.@)8 5.)  6;1 58 06E F  3%  *   Tmax ∼ 108 5   !2 718 3% '2  51 7  .@* 8 < =?

9  @ [; .8 5*1 E459< 56 B 85   " T

R8 3%'   + 'K 3E . 1 .   1)*  691 BD. T  '  6    max

5 W3% :%$K=  < H?

9   @16 1 .2 . I ?)   E  ' C@   56 B 85    

CB,C  E4: M

E

0> = 5 9  <.5* G 32 . ! A  3%  *)& %* g  0)% 8 3?6 5 6 51 . 1 59  ? !2 .XE$5f B, *; E" 7 )C  B,1 @  2 7E1 BD. T  'E; E7

 F B,

T  3% ))8 71! 9  5 2 7b.8 3%Y"  0$G8 3%  Y0 )    C  B 9  ? 0[1 BD. T 

  )' 8c.D 3'2  3%R 9@ B=*8  5    B 2 6  <.28 5  8 3%2 .`8 5 V 58 3?" 

025> B8E4:; &L5; B;ED  D))+C  EAB8
2  5 5( ;9  5
   a9  59  ?$8.

 V  56 B 9  < .C  EAB8<58 7

)D 6 :I  38 50 )9@ !' "  0)     

 *  b  . ( 9  5O 3%

)**C0

 *    ?*

C  @.

!1 5; 52 7.@

2 7 9  ?$2 5
43%0  8 5  9  ? ! m .@/

;)1 0   B,  6 (8 5  8 3?E:2 025)2 7. *  9  <` '   a8 .4" 56 Bf3%  " !I&  !> = 58  52 76 ; 5 ; 7  3 2 5 9  @  *2 5 '9  ?S! 1 5 S C  5K 3%'  γm γ#

!2 5 

 2:

  



  9  <.2 0156 B,1 .$

#2 5; . ; ./9  59  0*  61 :;8 7 S6

01 5 

T

f = )"  9  .:2 0256 B,2 .L. T # (4  3%*    8 51 @  ?

9  @9

.@ 1 .S  ,3% MX ∼ γm D" " ; E7;

  .9  5 2  < 61 : 9  .$; .8 5)1 E45CE/  'K 3E4:;

b 56 B 85   " Tmax  m 9# *9  ?

 ^9  5 C  B K 3$ 9  < ^?*

C  @^B=* I *  3 )); 7; ' *  9  7 DD"L*"= 9  5O 3%'  > =

+1 51 .'2 51 .  .D 3'Y"  *9  .1 0256 B 2 .3   *  ./  C  B8.

 ; 51 @1%$  <  ?

9  @ )  82  5; 01E'1 ?2 7Z8 3% '2  52 76 51 . 1 5   3C  X?1 02< 2 0     .  $;.C  E4: .@*8  59  : 5

' 2 7Y" X! 8  @ 1E02 . 8 .$ ./  C  B8. ;5)1 @  _ 5 W3%  %$  < R?*

C  @ 25> = 0 )6  :; ./G

)G 3%T ! C  ? 9@Y" 8 ?X4  3 $' M 5 R3% W  ?*

C  @TC  BU C  EAE4.@   ')1 @ 1 52 . ;5)6 @ 2 .C *- !9  .8 : 8 . 8M3?*2 . a  E4?  0*%

D 3%1 5
Q02
1 52 .XER5*( C  5 C  : 5

' 2 78X ')G 3%( 9@Y"    ^ ./ !*C  BD.@ \2 5; 01E =   0  A= 9  0 E  ; 5K 3

 9  < 1 B,9  .8 3%8 7

9  7J.( W3E %$  <. e=? '9  @1 .  3S5CE2

9   .[ *C.8 1 5Y" 5 '2 . 8 .<4 X9  5O 3% 06 B8<4 8 .@ [  

 ? '% 5'4  =  2  <  = ?

9  @ *  T.8 3%'   ; B=* C  .2 7Y" *M /  ./  C  B8. .< )  6d  06EI38 . 0f E7R3?) 2  9 S2 52 7d?

   Y"

R2 . 9  .3 ?V 5 W3% F  ?*

C  @ ;5)6 @ C  <.` *8 . b  5/ 9  5
C:; 7  5 K =  2 7&9  5  8 3%C   3S5CE2   & ')G 3%2 9  5C  ?$8. 9  0 * 6       8 51 @
















"
"

∗

   *


 

 





 

95   83%Z  )*5 F! 3? " ? ' 

 . ( $ 06B,*   ?  M ∼ H 5 5G  31@  6=[3% 2.C5   27B ')XG3%5* (C5 

         2  7           '

 \  ? % ?>= 5

9  @ !2 0256 B, 9  < SC  5 63?$ KM0 * \ 1BK3%9258. 9 'O3E4:;97>%I3 H∼M S *Y  EY:2 *  '    ; 5  5 X 2 7

3/2 MX . ρX ' 5 · 10 · MX · t

R ;:3? 33% *e=?*

C@ 6 ) 27[ *  '6 ; 5 1 258.  27 5 '  27E 6= aE4?9  .  5/2 MX . ΩX ∼  109 5 7[

)G 3 ')1 0  9  <4]  ? '9  @9< '   01ED3E!S>  = 5 ; E! '  C@ 2 109 5

 *  !    

 ; 59  5 *    ;5)6 @ C  <. ./  C  B8.2 .- 8 3%R  5( ;9  5K =  5

' 2 7 *? +9  5K 3E!.$C  BD01
)G 3 *    '

 9  <. 2 52 .&2 025f B=F3   T9  52 @ " −4











(*,%

L

5 B=a:,?  !%1< 1. C.3 Y? K 51. 9-; K 20 fE2W3% 9 

)D Q@'<75 A3 0 *   CB C?

5  X ;5 9: 5

8. 15 C<.81. 0A3%43?*  _5 *# $;.Z=  2 7Q.@*;5

')109< > !  8. * 9? [25;. F?*

C@>F?*

C@  83% 106= # 9< T2 0a   " N5K 3%8   5 2  <4  ?$;59  <4e  3<59< 

  C  ?

]  ?95f B,2 <?6 7

0   $1 B=*9  .8 3%8 7 '1 E4: # b  ?

9  @9< ! 

E0; 51 5 ;  $;5 !    XB,; 50)   # .@*8  5 .

P  3S5YE

" #8 5O 3%! 3 20 

 158 01E8 .

.! *9  ?

 1  $;.2 7 .@*8  5 1 56 . 2  5C<   

1 E4:2   .8 3%'8 7 3% )  (C  <-9  59  .@* (; 5,)

9  ?9  <4 B,G  ? E))>2  5; 01E' 195* 0*   3%  *'  9  <4;; 5,D

9  ? K9  5O 3% *> = 6 $ 8 3%T/ //9  <   Y  B, 9  ?

  9  52 @

1 58 7 5 ' 1 7Y" 5   :,02 .

'9  EY? 09  5O 3% 06 B,38  

8 $8.1 7F.@*; 5> )  T8 @ 1 5( 06 BD<41 :I> =

E   P .D 3'  6 = B,

9  .<.Y"


  

        
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65 ?$'"%$O(),)( * ÆG>1(Q&?-B(%>1,)(%;7> =,)5 =*)*h*}&?-/(%>%,)(1;G> *Æ=,?>)(1&D*)*D;7(1(%>%0157>%;7>10%5=,"! ,)()R   B"65X-6"7I?Rρn s Ê M,B"%5  D>%(O&?-/(A>1,)(1;G> =,?> )(1&D*)*`0B"’ŸJ"A57>1;b9 )5)5J&?-/(%>%,)(1;G>  Æ=,D5  7*)*k* v"%0A-B5=,)*)5 

f = ρ − Ts , EÆ  ¡Æ  s=

ρ+p , T

     ?"*    
J7 
# ; &)(lÆG>1( E&'-/(%>1,D(1;7>
s;=0%(1 (6,)(¹Æ=,D5 7*)* B"A01,B"

 




  
 #

 

n?w

f = −p .

~>."%$%B-B930D*);G-/57,)*D9 Æ7gZg#57$D>1*)0%,)( =(E&)(A>15=, *B".-6"h,)57(#:(6* +(E,B"
D>1* v"%0.-/5=,)*D5Q0h;=*"! ;7>%5 +5l&D* %;G-/(%01*)*?>1(P;D5 6,)575 ?*  7;=(101;7$<( =(m&)(A-B9&@"%g#*)$<;7*)(10:"A,)( B"%01, φ 01(Q01;9È &)(%;7>)"%,);7>%015%R K  6 5 )"%;=; i"A> )*D0:"A > ~>%5=(D*)*P; i".-% +*Q$<(1,);7>."A,D>." +*J;=0%9"M*Q&)*Q*),D>%5)57;#I* ,B"%;~>% 5 +&)5 B"A
> ")" vR #"%& D * +5 D& )*h>15 +&)5B".>#)5 xÆ=2 :"B"%$D>%5),D( )-B9¡ÆG-/57$"! >D(1;G-6" Q
0 @"% * +( 65 );7>%01* %$'(% ; +* )57;=$'"A9P&'-6
"  i"qT;=(1∼;G>1100 (@9?-6"q* [$)01" )$<(10 @-/5=&?>1(1,)(%0  W ! * Z ! 1( =(1,)(10  1( M(%,)(10qp*  7;M" A"o>."Aƒ$ ŸP5[g#(%>%(1,)(10q* f- q(1,)(10 .$<(1,);G>."%,D> e0 M"% * +( 65 D;7>10%*D9 $<(%>1(  +95 Ÿ E;=( %(  F-/* i".- lR62 Æ7>%( B
"  65G-B5  & D5=,)5 )95 Ÿ2È 0 M"% * +( 65 D;7>10%"* ! 9 +
* +95 Ÿ  B"%;7>%* " +*0k$'(% ; +* )57;=$<(  &?-6"  +
5 Š>SRt5AR 65 B"%;7 ; i"A> )*)0:"A
> `;=01( %( /, # Æ=,D5  7 * #* 65M"S- 1,) ( 7( X" @" ÆG-/5 +5=,D>.
" ),  B"%;G>1* RHTV 5 ,)5 +5=,D5=
5 H;701( 1( 6,)"A9 Æ=,D5  7*D9 Æ7>% ( = ( X
" M"m,)5G> )*D01*B".- %,)( M"%01*);7*D>}(%>m; )5 6,)5 7&( =,B" )5=,)*?9 φ &D(A-B9h~*  =;=" /&)(1;7$'(.- 1$ h(A> ,)5 =( @"%01*D;b9D> i"%;=;  B"%;G>1* '" :;X-/5 /(10:"A>%5G- 1,D( <*}* P01$?-6 "  0 v"%0A-/57,)*)5AR #(Z& )* )*),B"  $<(%>1( H5~* G-/3( ŸP5=,  0  D*?-/3( ŸP5=,D*)*  '& ) * .-/ƒ* ŸP57,)*)5~#* /5M".- %,) ( =( X
" @"I& )*O0 )*D;G-/57,)*)* Æ=g#g#5=$?>1*)01,D ( =(}&)(%>157, *B"S-6"},B" H0:" o>m( /,)(1&)5G>M-/5=0   & ) * A-/ * ŸP5=,)*D 5 HR 2 Æ7>1( ^& )* A-/ƒ* ŸP57,)*)* v"%0A-B5=,)*)5H9D0A-B9)5G>1;b9m;   +( m01$?-/ " v"l;M" +
( =(#( 6,)( )( 6,D ( =(Z&)(A-B9 *φ a01$?-6 " /(10J$?G" Ÿ 6
( =(P>1*)&)&" B"%;G>1* *C"%,D>1* B"%;7>%* />SRt5AR X zˆw DR w Å f = V (T, φ) = V (φ) + f ,

65 ;=$'".-)9 ), C&)(%>%5=, *)".- 60 :( /9 Q* C0 /5 );G>101*)5y;=$'".-B9 ),) ( 7(}&D(A-B 9  V (φ) Ê Z zˆw DR w IlÅ S = [∂ φ∂ φ − V (φ)] d x . l5 )0 &?-B5=,s0sz‚#w DR w ÅV01( =,)*)$'"%5G>Q * ! @"l>1
( =(  D>%(y>%5=, =( ]Æ7,)5  7*)"* ! * +& @- 1;=" B-B9m&)(%;7>1( ! 9),D,) ( =(m&)(J0 )5 +5=,)*C*E( /,)( )( 6,)
( =(m0}& D(1;7> B"%,);G>1015y;=$'".-B9 ),) ( 7(}&)(.-B9 B"%0157, 




±



















ef f

i

i




φ

µ

µ

4

Tµν (φ) = gµν · V (φ) ,

 
  p(φ) = T11 = T22 =       
! Ê HÆ   9È      
V (φ)  É    #    ™È


>SR 5%RB(6,)(D(/,)(15#;=$'".-B9),)(15Z&)(A-B5l01,)(1;7*D>m01$?-6" Y0 v"%0.-/5=,)*D5 B"%01, R 4>1( B
"  +5=57>%;b9 &)5 )5=g#( @-/* )(101$'"}>%( =(OgZ"%$?>." D>1(O0m& %;7>1(A>15P;=0%(  T = −V (φ) 1( 6,B".9 Æ=,D5  7*D9;=(%01&B " v"A57>3; =,)5 =*)5 +*5 s&?-/(A>1,)(1;G> B"%0%,B" B-B9^( /,)( )( 6,)( =( ;=$'".-)9 ),)
( =(#&)(A-B9UR12 v".- 1,)5 Q 5  6 5 `; D*D>."A> ?>1(#;=$'".-B9 ), J&)(%>%5=, *B".- v" 7>%;b9 ;7>."A#, v" D>%,  0 BG" ŸP5=,)*D 5 33

2 v2 V (φ) = λ φ − , 2

         
#
 #! v Ê λ Ê        ƒÈ λ1 !" $#&% (') "*,+.-0/ # * +.-!1 2 *4365.-!+.-,7 298:*;"<* "(3.=.2>-*= # 2>1 -!=.?&@A-,79*,8&"CB&"D/E=."C7GF,=.-!+.H,2>-,7 ' 2>1 -,I - 5.+."C* "&- 2.J,2 ) -!* % "C-K2>* "ML % "&8 # =.2>-&NO8PF "C7 $# J!-Q/ # 8&L>-,=>2.-R1 #M% *,2>1 # L>?C=." ' 2:/E"<* # " ) =."SBC"ML>?C@ # 3 "&B&L # * ?TF "C7 $# J,U ' "&BC+ # !J "&8 # 8&@ # 3>*43Q8&=.V +>2 $# J,UW*X1 -!=.?&@A2>1G/ # 8MLE-,=.2.-D1 ' B&V!/E- + # *;@A2.+<3 ?C*43 ' + # *;I # L % 2.8 # 3 *;+.-;/EV*X1 -!=.?&@A2>1 Z![]\ # =^/ # + =."&7_1 "D/E-,L>2 p2.Y J!2 % 2 )># * 2 a` 2.H!H!*;"C8C* % "C-b5>"ML>-$3.8&L.3.- *43 % "C1 5.L>- % *;=>Uc1G/EV^B&L>- "<1 '>% 2.I =.- 2 ) -D* % 2.7 ) L>-,=_8dL # H!+ # =feK2 # = # - % " "&+."CH,"d*;"C" 8C- * 8&VC"&+E1 V!LE-dgih \ F 2>1*,8M3.J # =a8&UmB&"C+ % "&I F 2 2.-,= (# 5.-,+>-0/ % 2.=.- 2 ) -!* % 2>1n*0L # H # -!1 Uc1o8pg Dq \ 8&3.J,?+ # *,*!1 # Ykj&l0+>Y 2.8 # -!1 "&H!"J0/-!*;?65."ML>3 φ r ` 2.H,HD*;"C8C* % 2>1BC"&J,"C=."C1 \ (# =^/ # + =."C7s1 "D/E-,L>2a2>1 -,- 8&2^/ Y l0Y √ φ(x) = v+h(x) Y 2 

2

65 0:"%$  +,)(15}; )5 6,)5=5h;7$'".-B9 D,)( =(k&)(A-B9^& )*, @-/5701( ^>%5 +&D5 B"A> )5 i" <$ (1, ;7>."A,D>."s;M" +( /5 );G>101*D9 ?* 7;=(10%;=$<( =(}&)(A-B9 B& )* R   I ƒ  Sƒ  ƒ .     ƒ ƒ   X  G   v ’  ƒ   „    Hƒ   ƒ+ƒ „    Œf Œ X v  

n:n



 


!!#"%$&'
(#!"

24>1( )(15I;X-6"
X"%5 +(15I0Yz‚w ?R wZÅ Ê Æ7>%(O01$?-6" `;D5 _0}&)* A-B*ƒŸP5=,D*)*Y,)570@"%* +(/5);G>10 ! I* B"%;G>1* 6&)*‡ÆG>1( f = −p (T, m (φ)) ,

65 1 0 ? $ 6 " 0 v"A0A-/5=,D*)5 B"%;7>%* *c"A,D>1*)"%;7>%* i! =(h>1*D&B" i"%;7;M"h$<(%>1( p (T, m (φ)) Ê B"%0%,B" * @"A01*);=*?>s(%> R F ( f-6"%;7,)( )"
65X-% I?R w#* +5=5 m φ Z zˆw DR w Å g k dk 1 p √ f = −p = − ,    




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∞

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6π 2

4



k 2 + m2i

0

k2 +m2 i T

65 g Ê )*D;G-/(a;=&)*D,)(10L;7(1;7>%(S9D,)* *`01015 65=,D(C(%(M,B" )57,)*)5 k ≡ √k \015 ?,)* M,)"%$ (%>%,)(1;=*?>1;b9C$ 1(M(%,B" B"Q,)* Ÿs,)* Ê $ag#5 +*)(1,B" HRB2F$?-6"  >@9 Ÿ*ÈG-% B"%;7>%* ; m  T 0J;=01(1(6,# Æ=,)5 =* Æ7$';7&)(1,)57, *B"S-1,)( i".- B&)(lÆG>1( O*),D>15)5=;y&)5 6;7>."%0.-B9)5G>O;G-")" R m .T l,D>15 B".-Ez‚w ?R w ÅU,)5 v"™È7>%;b9I0
=9D>
p"A,B".-/*D>%*)57;=$)*\REl(lÆ7>%(  c&)(:"%,)".-/*=*15 Y5 7( B-B9 )"%;7>%,) ( =(Z;G- ")"A9m0 H;=(1$D* P>15 +&)5 )"A> # :*J*);=&)(. - % 5 )
" 7-/3( ŸP57,)*)5r&)( z *D#, 65=$<; i  
 / 5 L(1& 1;7$'"A > :57;G-/[* Æ7>%(y,D5 T657>Z& m)*D01( 6*D > Z$J,)5 /( B
"  +57,)*D9 FÅGR:Tdm/T "A$'(  &)( 1( E,B
" H01" q>I0 H;=(1$<(%>% 5 +&)5 B"A
> "D,  B" 7-/3( ŸP57,)*) 5 HR?2 %5 )
"  +5 ), s&)5 )5 +5=,),  * z = m /T x = k/T 0 BG" ŸP5=,)*?9zˆw DR w Å4& ) * <>O01*   Z zˆw DR w +lÅ g 1 x dx √ . f =− T · I(z ) , I(z) =  6π ∓1 x +z #"%;l*),D>%5 )57; %57>E&)(10%5 657,)*) 5 Æ7>1* a*),D>15 B"S-/(10s& )* i"S-  z R 2 , @-/5=0%( &)( :#9 /$'53&D( z 01$?-6"  f ;7(1(%>%0157>%;7>1
0 "o> 6"%0A-/57,)*D 9 ;701( 1( 6,  X
" =(10 15  i"%;7;=(10  B"%;G>1* z ; HR )
" 65X - I?R ;w Å 6(1,)*3,)5 @"%0%*);b9D>E(%> φ 6*  *  
 / 5 (1& %;=$'"A > DR l( H,D>15 B"S - 1,)(15^0 B’" ŸP57,)*)5L0_zˆw DR w +lÅs9)0A-)9)57>%;b9 g ',)$ *)5  z r* +;( Ÿ},)( F-/(  3( Ÿs* v"A > %D>1( & D5 6;G>."%0A-B9D57>O;7( 1(  :#9 Y&)( z R l5 )0  ?-/57‡, ÆG>1 ( 7( :#9 6" B"A015=,  I(z)   Z   Z zˆw DR w ZÅ dI z xdx x dx  I(z) = z =− . +  dz 2 ∓1 ( ∓ 1) Q* #)* #Q*D5 /5=; m*),D>%5 B".- _$<(1,)5 ),  )>."%$ D>%(}&)5 )0 a,)57> )*)01*)".- 1,  ?-/57,C0  ! ;=(%$'(A>1 5 +&D5 B"A
> #),) ( 7&( B" 7-/3( ŸP57,)*D9 /5 );G>101*D>%5G- %,)(J$)0: " )"A>1* D5=,h&)( z ≡ φ/T R'2 H&)(A-/,?9D9 *),D>%5 )* )(101"%,)*)5a;h*);=&D(A- =(10:"%,)*D 5 g#(  S- d& )*)0%5 _È7,), 03$<(1, 5 B
"  65G-/ " I?R wLz & )* Æ7>%( 0A>1( )(%5a;X-6
" X"%5 +(15a0 zˆw DR w ZÅ 6( %,)(` & )(1*),?>15 )* )(%0:"A > c&)( B"%;7>@9 FÅi&)(A- ")"% 5 0 Æ7>%( &D( :# 9 6$<5 " #   X zˆw DR w )ZÅ v 1 X T V (φ) = λ φ − g m (φ) + g m (φ) . + 2 24 2  2ÆG>1 ( 7(O0 B’" ŸQ5=,)*D9k; B" C;G-/5 15G>C01’" Ÿs, k0 H01( L(m>1( D>%(h0}>157( )*D9 k>%*)&B" 4>."%#, ! v " ?>1,)(  +( 65X-/*h;= *  +5G> )*?9C,D5~,B" ÉI5=,B"P& )*h0 H;=(1$)* O>15 +&)5 )"A> #B"  :(A>@9E(1,B"Q,B" " ! ÉQ57,B"J & )* R I5 );G>101T*D>%=5G- 0%,)( U0P>."%$D*  +( /5G-B9  B"%;7>%* _& D*)( )5G>." q> i"%;=;  * +57,),)( A-6
" 7( 6" :9 $<(1#, 657,);M".> a;=$'".-B9 ),) ( 7(}&D(A-B9 )>SR 5%R zˆw DR
n ZÅ m (φ) = h φ ,



∓1

2

i







i

i

i

i

i

∞

4

i ∓

2

∓

2

0

√ x2 +z 2

2

i






4




2

2

∞

2

2

2

∞

x

0

z 2 =0

0

2 x

2

x




ef f

2

2

2

2

i

 

2 i




i

i

 


i

2 i

i

n 

65 $<(1,);G>."%,D>!;70%9=*\R l;7$?- D5=,)*)5^;=(1;G>."%0A-B9D57>W-/*ƒÉ W;M" ?*  7;=(101;7$)*1(=(1, 01$?-/" hL$<Ê (%>1()( =(h03z‚w DR w)ZÅ i".- 6* _* 65 &)5=,D5 )5 X"A> DR 2 4>."%,#v"
D>1,)( +( 65X-/* B-B9s$)0:"D$'(%0I*&M":9ƒŸP57,),m-/5=&?>1(1,)(%0D(%(M,B" B"A5 };=* +0%(A-/( '* !d* ! 1 ( =(1,)(10 f W Z ;=(%(%>1,)(’ÉQ5=,D*)5hzˆw DR n
ZÅ4* +5757>}$<(1,)$)5G>1,30%#*  p zˆw DR n?;w Å g +g g √ m (φ) = y φ , M (φ) = √ φ , M (φ) = φ, 2 2

65 q$?"A01;=$)*D5l$'(%,);7>."%,? >  /" * ' $ . " / *

)(101  ( D , H5~$<(1,);G>."%,D >  ;=0A 9 M*Lz  ( 1( =,B" )5 ! y Ê g g Ê ,)*D9C*k&D()( 1,)(%;7>1*3; HR/0  )*?-/3( ŸP57,)*)*k˜| ÅGR62 01"%$   +5 φ = v/√2 /*  0%( M0 )E" P"%5 +;b9 $a*M0157;7>%, g#(  @-6" B-B!9 i"%;7; )"%;7>%* z ; HR  )*?-B;( ŸQ5=,)*)5y˜| Å/"J* +5=,),)( p zˆw DR n1ln Å g +g y g m (φ) = √ v , M (φ) = v , M (φ) = v. 2 2 2 u+;G-/*C01$?-  )*D
> OH0 Æ7g#gZ57$D>1*D01, Y&)(A>15=, *B".-3>%(A- 1$<(}0 H&)*);="%,), H5y0Yzˆw DR w )ZÅ[;G-6" X"% 5 H5  >1(m; #^È7>%( z‚#w DR n ZÅ4&)(10%5 657,)*)5 Æ=g#g#5=$D>%*)01,) ( 7(s&)(A>15=, *B".-6"m0 A-/"* M* φ = 0 * +5=5G>s0%#*   zˆw DR n ZÅ α  V (φ) = −λv + T φ + λφ 24 z ,D5 M"%01*);b9 Q*)5Z(%> φ ;G-/
" X"% 5 H5y(1& Q57, ¬Å 65 X zˆw DR
n Å 1 X α= gh + gh 2 &)(A-/3( Ÿ}*D>15X- 1,B".9}0%5G-/* D*),B"DR  )*J,)* =$)* J>% 5 +&)5 B"A> #B" O0 )’" ŸP5=,D*)5yz‚w DR n ZŊ * +5=5G> +"* ! Ê ,) *  `& )* φ 6= 0 z ;= *  +5G> )*D9J,)" ÉQ57,BZ" Å%"p& )*I0 H;=(1$D* I>% 5 +&)5 B"A
> ")" m5 6*D,);7>%015=,),  +*),)*  +( W;G%- Ÿ}*D> +*),) *  φ = 0 B;=(1(%>%0157>%;7>%0 #Q* L0%(1;=;G>."%,)(10A-B5=,),)( k;=*  +57> D*)*\R KC*),D *  & )* φ = 0 *D; )5 @"%5G> * & D5=0 B>" P"%5G>1;b9 0 i"%$<;= *  [$< ( v"&)5 )01(153;X-6
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> "D5^z ; H;X- ( 1( =,B" )5=,D* 65G>k9);75=,"*  v". - 1,)5  ÉQ5 7( Å   zˆw DR n IlÅ 6λ T = 2v . 2 6".- 1,)5 ƒÉQ5 `& )*Q( EQ 5 k( 1; Ÿ 65=,D*)*I>157( )* αP; i".-  +*I$'(%,);7>."%,?>." +*J;=0%"9 M* 
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 #

 

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0

f




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f

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2

ef f

 

2 i i

2

   

2

4

2 i i

1/2

c2

i

λ ∼ h2 ,

>."%$ D>%( i"%;=;=" 1(=(1,B"yp*  7;M" m ∼ √λv * +5757>P>%(%>2ŸP5q&)(:9#6(%$O015G-B*)*), D>1( i"%;7; (1;G>.".-1, B"%;7>%*  6"I* M" +57>%,H5E01$?-6"  0¯Æ7g#gZ57$D>1*D01,e&)(%>157, *B"S- Ri2>."%$<( & )5 6&)(A-B(;ŸQ5=,)*)*C( 57,)$'" B-)9C$)*D>1*)5=;7$'(k>15 +&)5B".># * +5=5G>s0%*#  h










Tc2 ∼ v , ƒÉ ™É  

 
 α   
Z  Å  #

$'"%$¹Æ7>1(m;G-B5 %57>h*szˆw DR n
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α=


2 2 2 2 6M + 3M + 6m , W Z t v2









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(#!"

65o* +5o>1;b9h0Z01*#% i"%;=;W&)*m,@-/5=0%(}>%5 +&)5B"A>#)5AR 657;$  #?-/*#D>1( !V* ! 1( =(1, e0 +5=;G>15r * +5 q> ÉQ5=;G > Z&)(A-B9 )* M" *  Z ! 1( =(1,P * +5757>Z>)*J&)(A-)9)*M" *)*W<" t! $)0:W"
)$ 0 +57;7>154;=(o;=01(%5 y"%,D>1* B"%;7>%* 5  Ê D57> D5 %$ )( +5+>1 ( 7(  t! $)01")$ +(3ŸP57>q,B" 1(/*D> 1;b9I0o>^È ;=(%;7>1(@9D,)*D9 a&D( 0%57> %R62F;=&)( +*),B"A9aC5 EÈ
%D>%( i"%;7;M" 1( =(1,B"Q~*  =;=&" B"A01,B"




+




−












mh =  GŸ

√

2λv , 

&)(A-")"%5 ;G-/5 #Q575}(6,D(1&)57>M-B5=01(15m0 B" P5=,)*D5 )-B9`$)*D>%*)57;=$<(^>%5 +&)5B"A># 4>."%#, 6" D>1,D( +(65G-B*

Tc2

1/2 6m2h ·v Tc2 = 2 6MW + 3MZ2 + 6m2t  >Æ  m h >Æ = 121 · 100 >Æ  Æ  >Æ = 80.4 MZ = 91.2 mt ≈ 175 

0

x M2 x M2 z ,)"%&)( +,)*  D>%( M x M2 x =2 x =2  v = 247 x>ÆM2 Š; HR  )*?-/(3ŸP57,)*)5y|˜ÅGR u+;G-/*$L0H;=(1$<(%>%5 +&)5B"A>
"D,)(15 )"
7-/(3ŸP57,)*)5i*),D>%5 B".-/(10Iz‚w#DR w Å%65);7>%01*D>15X-1,)(p&)5 ! ;7>."A0A-B9?-B(s;=(1( :9#Y&)( z ≡ m (φ)/T )>%( +3( Ÿ},)( F-/(  M"%$?-  )*D>
D>1(  * +575 65X-/(O;IgZ"
=(10 &)5)51(6( ")(v" v&)(%&B"%01$D* )57>%l 0 È?>1 ( 7(a* 1(A-/575I0H;7(1$)*k&)(:9 6$<(10 &)( φ i"S-&)(a;B"A01,)5=,D* ; ?-/57,B" +*  0 H&)*D;M"%,),  +*0zˆw DR n ZÅJz ; HR ,)ƒ* ŸP5 ÅŠ"a&D(A-/3( ŸP5 ! ,)*)5$ +*),)*  i"J0B"GŸP5=,)*?9zˆw DR n ZÅF&?-6"A01,)( +5=,D9D57>%;b9Y0m;7>1( )(1, !1(A- ™ÉQ*  φ B,B" )*),)"A9a; 6&)*  +5=,™ÉQ57,)*)*3>15 +&)5 B". > #(A> T 6(s, @-B9 />."%$ D>%(O&)(1015 65=,D*)5Q0 BG" ŸP5=,)*?9 φ=0 z‚w#DR nEZÅI;=(1(A>1015G>1;7>%015G> & B"%0%(  )*);ARrw DR DR  6,B"%$<(L,B"`;M" +( 65G-B5a*D,D>15 B".%- zˆw DR w Å ,)5J"%,)".-/*D>%* ), &)( z \*3( 6,)(1&D57>M-/5701(  Æ7g#gZ57$D>1*D01, L&)(%>157, *B"S-0 65 );7>%01*D>15X- 1,)(%;7>1* ;=(%(%>1015G>1;G>10 %57>EgZ
" M(%01( h&)5 )5 1(   )( v"DR p>1; '>%;7>%01*)5J"%,B"S-/*D>1* ),)(1;G>1*Y01* 6,)(m* I&D(1015 ! 657,)*D9`01$?-/ " 6(10h0h*),?>15 B". -  z‚w DR w +lÅr* J( A-/"%;7>%* i"S- 3 * +"& S- 1;=(10 v>SR 5%R z *D,)g B"%$ B"A;=,B"A9O( A-6"%;G > ÅX R  )* i".%-  z * x B
" G-/3( ŸP5=,D*)5 Æ=$<;=&D(1,)5=,?>  k 0P&D( TH,D>15 xB"S- 
1! , a0 )’" ŸP5=,D*D9  6™" È7>  Z 1( M(%,  zˆw DR n Å x dx I = , x +z Z g#5  +*)(%,  , 1 x dx √ I = , 2 x +z

65 # g ) * ? $ 1 > ) * 1 0 ,  L&) " B " + G 5 >  \
(

B"%,D * )*)0:
" Q* 3*),)g B"%$ )"%;=, # ( .-6"%;7
> )R Q( #! Λ1Ê i".- %,)(1 5 B
" G-/3( ŸP5=,D*)5J&)( H,D>15 B"S- 1, ^0 B’" ŸQ5=,)* LD0 Æ7>%* Lg#(  @-6" 3&)( z & )*)015X-/(  0m&D( :#9 6$<5 z $a01$?-/ " v" >1*)&)" Z 1( =(1,  dx z , x Z g#5  +*D(1,  . dx , z x l5 )0 h * ~,)*  B"A; :( 6*D>%;b9C,)"I,)ƒ* Ÿ},) 5 & D5 65X-/5l*),D>%5 D* )(10:"A,)*D9O-/*),D5 ),)( )"I0%>%( )(  Ê -/ ( X
" )*)g +* )57;=$)*\R ll( ÆG>1(  +3( Ÿs,)(I;( Ÿ}#* v". > "D>1(P&)( + * +(P0 )*);X-/5=,D,) ( =(m0 @ÉI5l01$?-6 " 6" &)( :9 6$'" z 1( =(1,  v
" q>01$'-6 "  &)( <9 6$'" z 4"^g#5  +*D(1,  Ê 01$'-6 "  &)( <9 6$'" z ln z R 2F$?-/ " j&)(1;G-B5 6,)5 =(E>1*)&)"a;7(1(%>10%57>1;G>10 #o> ?-/57,B" D0 Æ=g#g#5=$D>%*)01,)( &)(%>%5=, *B".-/5 N * +5  ! Q * 01#*  zˆw DR n +lÅ φ φ m (φ) ln = h φ ln . W

2

2

2

c2

2




Λ

(IR) −

4

2

0

2

Λ

(IR) +

4

2

0

2

2

4

Λ

4

2

0

Λ

4

0

2

3

4




4 i

T

4 4 i

T

     ?"*    
J7 
# ;    
 #   nI
F(A-™ÉQ*),D;7>10%(h*Q,)* i "S-/(s*),?>15D5=;=, \&)(1;7$'(.-1$C&)* z D>%(O0H&)(.-/,D9)5G>1;b93&)* ,)5`;G-B*ƒÉQ$<( i ".-/( i "%;7;=5 ?*  7;=(101;7$<( =(1 (M(%,B"ZÅs(1,)* iλ".-% h&)( ;B"%01,)57,)* ; ?-/57,)( 6&)(1*D;:(/9I* *P;=$'".-)9),)(
=(h&)(%>%5=, *B".-6" R
y* É 601$?-6"  U(
1;G-B(10A-/57,),H5 λφ V (φ) ! $D0:")$<( M(%$?"H0:"o>1;b9y;Q57;7>%015=,),+ m * )*#(1&)5 65G-/57,)*)*Q,D5=$<(%>1(Z&B"B" +57>D(10pgZ"! & t M(%01( =(p&)5)5:(v"D>1(  '01*#6*L ,D*ƒŸP5%RE#"%(1 ()(%>?-/5=,e&D(:9#6$'" z ;I5=;7>%015=,D, B-)9 (1&)*D;M"%,)*?9YgZ"
M(%01( =(m&)5)51(6" U* + 57,),)( . -6"
=(v"
<9ÀÆ7>%* ?-/57,B" &)5)51(Wz 0 B" + $?"k(%! ,)(1&D57>M-/5701( =(}&)* . -/*ƒŸP57,)*D9PÅF9D0A-B9)5G>1;b9C&)5D5:(6(  )(v"?R Z-B9a0)*);X-/5=,)*?9 ?-/57,B"}&D(:9#6$'" 0% (M(1,D,)( *D,D>15 B".-B5 I B"
=(  lÈ Æ7>%(%>O*D,D>15 ! z

)".-Y,)" /015 B"%;7>%*/01015 /9Eg#*)$D>1*D01,Y&B"
B"+ 57> Λ  


4 i

4




3

3

I− =

Z

∞

√

−

x4 dx x2 + z 2



1

(IR)

√ x2 +z 2

+ I−

.

l 5)0%(15l;G-6" X"%5 +(15  65=; m"%,B".-/*?>1*),D(}&)( B"Q01(P0%>%( )( ;G-/"
X"%5 +(  +(3Ÿ},)( B"
G-/(3Ÿs*D>
 =$<;7&)(1,)57,D>C*E( B"A,)*)*D>
1;b9 '-/5=,)( _z‚w#DR zn Å?>SRt5ARM"%&)*);="A>  Λ

−1 

2

Æ

(IR) I−

=

Z

Λ 2

x −z

2




dx + z

4

Z

Λ

x2

dx . + z2  z2

l5 )0%(15#;G-6"
X"A5 +(15y0HÆG>1( W0B"’ŸP57,)*)*k;7,)(10:"m"%,B"S-/*D>1*),)(s&D( /"J0%>1()(15 v "™È7>}*),D>%5)5 ! ; Q* C,B"%;#01$?-6" 3&)( <9 6$'" z    #R 

0

0

3

(IR) I−

π −→ z 3 + O 2 




z4 Λ



.

2 ) 5S-S>.".>15EÆ7g#gZ57$D>1*D01,Y&)(A>15=, *B".-Y0m( 6,D(1&)57>M-B5=01( &)* A-B*ƒŸP5=,D*)*k* +5757>}01*# v2 Vef f (φ) = λ φ − 2 

2

2

! 1 X 2 + gi mi (φ) 2         mi (φ) T X 3 4 . − gi mi (φ) + O mi (φ) ln 12π   T

T2 + 24

X

gi m2i (φ)

2 +(65G-)9  65 i"A;=; B"A;7>1* ;70%9M"%, ; ? * =;=(%01;=$)* ;D5 6,)* ;7(1(%>1,D(™ÉQ57,)*)5 z‚w ?R n
lÅ Æ7>%(}0B"’ŸP57,)*)5Z&)5)57&)*);H0:"%57>%;b9Y0m01*#65
zˆw DR
n ZÅ α V (φ) = T − T φ − γT φ + λφ ,

65Z&B" B" +57 >  γ &D(A-/3( Ÿs*?>15G-B5=,k* B24"%0%5=, √ zˆw DR n )ZÅ 2 X m  1 X γ= g |h | = g , 12π 6π v "P(1;7>."S- 1, H5y( 1( =,B" )5=,)*?9C0%015 657,  0kz‚#w DR n Å4*^z‚w DR n IlÅ R  
 "
 $
› $
R Y    m  =$2.   %    (3 

 . φ ln
( #   $ (     1  #  
  $ 
 Y $&    #  cz‚#w DR n ZÅ&  


2

ef f

 


i

2 c2

i



2

4

i

3

 

4



3



φ T

3

i







   $&$      $ $   $N $2$ ’  $  
   

  R
    λ ∼ h2i  
$     $     $#  
 ; $'
 % œ 
  h  1 $ #  $A ? i $  ˜  
ZÅ%   $& " 9 F#  &$#"  
$    f φ 0<φ.v











 z‚w DR n





 

n



 


!!#"%$&'
(#!"

l(1015 /5=,)*)N 5 Æ=g#g#5=$D>%*)01,)(
=(P&)(%>%5=, *B".-6"hzˆw DR n
ZÅd$'"%$Og ',)$ *D*s>15 +&)5B".># ;7(1(%>%0157>! ;7>%015G>h-/5=01(  )*D;%Rdw ?R />SR 5%RUgZ"
=(101( C&)5)51(% )(v"DR F$';G>)5   Æ=g#g#5=$D>%*)01,)(
=( &)(%>%5=, *B".-6"m(1&)5 65G-B9q>%;b9 #B"%01,D5=,)*)5
z‚w DR  ZÅ ∂V α T − T φ − 3γT φ + 4λφ = 0 . = ∂φ 12  )*E>1 5 +&D5 B"A
> #)5 >."%$<( ?>1( T








ef f

2

2 c2

2

3

c0


4αλ 2 2 Tc0 − Tc2 , 3   ‡Æ    # É  " PÅ   


2 9γ 2 Tc0 =

=g#g#5=$D>%*)01,)( 7(3&)(%>%5=, *)".-6"3&)(@9)0A-)9 q>%;b9 60:" =$<;7> )5  i"`&)* φ 6= 0 Ê +*),)*  * i"%$<;=*  HRv2F*#/,)(D>1(]Æ7>."}>%5 +&)5 B"A> B"s-/* s,D5=,B" +,D( =(}&)5=0@ÉP"%5G> T /&)* λ ∼ h z /5 h Ê :"B"%$?>15D,B"A9}$'(%,);7>."%,?>."I;=0%9 M* 8;7& B"%0%5 B-/*D0 W( 57,)$)* γ ∼ h  αλ ∼ h 1>."A$&D>%( T −T zˆw DR D;w Å 27γ ∼h . = T 4αλ  ! @" i".-/
( =(Q(%>M-B* )*D9 (%> 0P*),D>15 )5=;7,)( a( A-6"A;7>1*h>% 5 +&)5 B"A> # +;( Ÿ},)( M" +5=,D*D >  ,B" T 01(Q0%>%( )( ;X-6
" X"%5 T +( T0az‚w ?R  lÅX R  Q5=;G>10157,),)( D>%(P0%>1( )(  +*),D *  Æ7gZg#57$D>1*)0 T! ,) ( 7(I&)(A>15=, *B".-6"Ez 5=;X-/* φ = 0 ; )*?>."A > P&)5 )0  +*),D *  +( FÅ]&)(@9D0A-B9)5G>1;b9h& )*s(%>M-B* ),)( (%>m, @-B9 φ = Φ (T ) zˆw DR :ln Å 3γ T . Φ = Φ (T ) = 8λ  &D(1,)ƒ* ŸP57,)*)5 >15 +&)5 )"A> # 0%>%( )( ! +*),) *  ;7>."%,)(%01*D>1;b9 f%- 3ŸP
5 PÆ=g#g#5=$?>1*)01, k&)
( ! >157, *B"S-e¯0 Æ7>%( +*),) *  +5O; B"%01,D*)0:"%5G>1;b9H; Æ=g#g#5=$D>%*)01,  j&)(%>%5=, *)".-/( 0 +*D,) *  +5 & )*k>15 +&)5 )"A> #)5 6& )* ^È 0 H&D(A-/,D9D57>1;b9Y$?"A$ #B"%01,)57,)*)5Ezˆw DR  ZÅ 6>."A$Y* #B"%
0 !  φ=0 ,)57,)*)5 V = 0 MR žV5 ÉQ57,)*)TE5 ÆG>1( a;=*D;7>15  #B"%01,)57,)*  v™" ÈG>}&D5 )0 # $ )*?>1* )57;=$ #j>1 5 +&D5 ! B"A
> # T   Æ




2 c0

2 c2

2

2

c2

3

4

2

2 c2

c2

c2




c

c0

c0

c

c0

c0







c1




ef f c1

2 2 Tc1 − Tc2 6γ 2 , = 2 Tc2 αλ  ¹Æ  
"

*a&)(A-/(3ŸP57,)*)5#0%>1()( =( +*),)*  i"J& )* 7>%( C>%5 +&)5 B"A> D5

ˆz w DR ZÅ  #^È7>%( z‚w DR n H0s& )5 65X-/5P;G-/"1( Y;70%9 =* 7>1( M,B" )57,)*)5 =(B"
/( +5=,’ÉQ5P$)*#! >1*)5=;7$'(s>15 +&)5 B".> R )* 6".- 1,)5 Q5 &)(1,D*ƒŸP5=,D*)*m>%5 +&)5B"A
> " /( 0%>%()(}$)*D>%* )5=;7$<( O>15 +&)5 B".> +*D,)*  =g#g#5=$D>%*)01,)(
=(Q&)(A>15=, *B".-6"y& )* φ = 0 &)(%&B" v"%5G>a*L& )5=0 )" P"%5G>1;b9^0 i"%$<;=*  HRŠ2 7>1(A> +( +57,D>a0%>1()( +*),)*  ,B" :( 6*?>1;b9 &)* zˆw DR Å 3γ Φ = Φ (T ) = T . Td"A$)* W(B"
M( 6*" +5=,D5=,)*)5I(/,)(1&)5G>M-/5=01(
=(¡Æ=g#g#4λ5=$D>%*)01,)
( =(O&)(A>15=, *B".-6" U* =( B’" Ÿ2È=,),)(%5 ,B" )*);AR+w DRO;G-/570:"\&)(1*);1(/*D>Y0h>157()*D9`; i".-  +*L$<(1,);G>."%,D>." +*;70%9 =*`0  =$'( *)", ! >15)0:".-B5C>15 +&)5)"A># 0A-B*M*$)*D>%*)5=;7$<( >1 5 +&D5 B"A
> # zˆw DR n IlÅ  T ≤ T ≤ T  65 *;L # H # -!1 Um-R8 F ƒ+ƒ - %S 2.8C=."C1]5."  -,= „ 2 # LE- 'D $# 5."<1 =.2>1 ' )S "1 U "  B&+."<*;2fL>29=.-J # 8C2>*43 2.-R"  Y Y "  * ) 2  Um8 # -D1-!H,"Q"  J,= #() -,=.2.38 $ ƒ # J!- Y γ Tc1 . 2λ  )ZÅ      _Æ & h1  # 
    ƒÉ Φc (Tc1 ) ∼ hTc1    #  Æ Tc2   E  Æ Φc1 = Φc (Tc1 ) =

c2

c

c2




c2




c0





φ=0

φ

c2

     ?"*    
J7 
# ;    
 #   n+ (%&)5 /5G-B9D57>1;b9 ;7(1(%>1,D(™ÉQ57,)*)5 z‚w ?R Dw3ÅXRT D* Æ7>1( ;B"
 &)(1;X-/53gZ"
M(10%( =(&D5)5:(v" T ;D5 6,)575=,B" )5=,D*)5I&)(.-B9h~* =;=" M,)" D*D>15X-1,)(+ 5=,™ÉI5y5 =(m0:"%$  +,)( 7(};)5 /,)5 7( 


c0

γ Tc2 ∼ hTc2 ∼ hv  v . λ   #     "

Φc ∼

p> +5G>1*  D>%( 1;X-/(101*)5H&)* +5=,D* +(1;7>%*}0 H;=(1$<(%>%5 +&)5 B"A> ),D( =( B "
7-B(;ŸQ5=,)*D9QgZ( i".-1,D( 0H&)(.-/,D9)5G>1;b9C0m>%5=(D*D9a;7(m;X-6"1( C;70%9 B&)(%;=$<(A- 1$ 

mi (Φ) = hi Φ ∼ h2 Tc2  Tc2 .   
        ZÅ    


u 2È3(6,)( M" +5)"%,)*)5Y$'"%;="%57>%;b9 =; $ [>%( >157&?-/(%> gZ" M(%01( =(^&)5 )5 :( v"DRQ D* X M,)" D5=,)*)5dÆ=g#g#5=$D>%*)01,)(
=(l&)(%>%5=, *B." -6"o0 +*),)* +5~z‚w ?R S(%>%; )*D>."A,),)(15[(A>p5 =( =T,B" )=5=,DT*D9 & )* φ = 0 B"%01,D(  

c2

27 γ 4 4 Vef f (Tc2 , Φc (Tc2 )) = − T . 256 λ3 c2    FÅ    #  #    

4>1( M,B" )57,)*)5Oz ;Z( B"A>%,  =,B"%$<( H(1& )5 65G-B9D57> i"%$<;=* i".- 1, ;7$[>#>%5=&?-B(%>C&)5! )5 :(v"/$<(%>1(B"A9a&)(J&)( :9 /$ E015G-B* )*), B"%01,B" zˆw DR  IlÅ −V ∼ h T  T . Td"A$) * ( B" M( v0 D5 @- @>."A>%5}gZ
" =(101 ( 7(h&)5 D5 :( v"a0  65X-B9)5G>1;b9 i".-/D( Æ=,D5  7*)*^&)(h; B"%
0 ! ,)57,)*  *; Æ=,D5  7*)5  B"%;G>1* 0h&?-6"  +
5 U&?-/(%>%,)(1;7
> C$<(%>%( )( `&)(h&D( :#9 6$ k015G-B* )*),  B"%01,)" R/2 )5 @- @>."A>15QgZ
" =(101 ( 7(J&)5 )5 :( v"m$'(% ; +* )57;=$'"A9C&?-6
"  i"m&D( / ( )570:"%5G>1;b9Y;X-6" 1(DR T  

 $ 
 %1    }zˆw DR
n ZÅ 
      A  #  
   . 

ef f

6

4 c2

4 c2




4 c

                                  zˆw DR            2 H;=(%$'(A>15 + &D5B"A>
#),)(15 B"
7-B(;ŸQ5=,)*)5p*),D>15 B"S-/(10az‚w#DR w Å M"%015 6( +( B" 1(%>."%5G>J&)* i"! - M,B" )57,)*D9 φ R l(lÆG>1( Y; 65G-6"A,),L,B"O5 7(a(%;=,)(10%5Q0H01(^(O>1(  D>1(E0 B" +$'" Y*);! &)(A-M(101"%,),)( 7(#0Æ7>1( B"
 65G-B5q(6,)(%&)57>M-/5701( 7(I&)*A -/*ƒŸP57,)*D9JgZ"
=(10J&)5D5:(O9)0A-B9D57>! ;b9}&)5)5:(6(  D(6"?;¬Æ7>%(s>%()$)* )5=,D*D9O(% (1;=,)(%0:"%,)()R<2 >1(EŸQ5o0)5 d 9 B-)9s(6,)(%&)57>M-/5 ! 01( 7(y0)*);G-B5=,)*D9s$)*D>%*)57;=$)*}>15+ &)5)"A># T * T :;)5 6,)* =,B" )5=,)* Φ  Φ  Φ *  7* <"
B"%$D>%5)*);G>1*)$}gZ"
M(10%( =(~&D5)5:(v"#*);=&)(.-M(%0:"%,)*)5F0H;=(1$<(%>%5 + &)5B"A>
"D,)( =( B"
! -/(3ŸP57,)*D9(%>%,  `,)5h(@ 9"@"A>%5G-1,D()R l,D>15 B"S-zˆw DR w Å /(1;7>."A>%(),)(L&)(%;7>1(`0D*);G-/*?>  )*);X-/5=,D,)(]*>."%$)* (  B"
=( ,)"D>1*>1(),W(6,)(1&)5G>M-/5=0%( Æ=g#g#5=$D>%*)01, &D(%>157, *B".-NR F(1(%>%0157>%;7>%0#Q*D5 B"%g#*)$)* B-B9 4>."A,#v"D>%,)( + (65X-/*3&)*)0%5 657, ,B"&)*D;%RŠw DRL)Nw DR w DR 24*#6,)(%?>1(m&)* φ . T &)(A-%#)57,),H5I;#*);7&)(A-
M(10:"A,)*)5  0H;=(%$'(A>15 + &D5B"A>
#),)( 7( )"
7-/(! ŸP57,)*D9 )5S-S>.".> B-B9½Æ=g#g#5=$D>%*)01,)(
=(~&)(A>15=, *B".-6"q; B"+ ,)(Z>1(),D(1;7>
W;=(
f-6"%;#o>1;b9 ;H>1(D, (6,)(1&D57>M-/570 0)*);X-/5=,)*D5 H R?2 >%(ŸP5r0D5 d 9 D5@-@>."A> B-B9 :9#v"<")"%$D>15 ! )*);G>1*)$3gZ"
=(101( 7(s&)5)51(v"O;=(10%&B" v"o>}-/* É s$'" )57;7>10%5=,),)(\&)(s&D(:9#6$C015X-/*)*),z ; H R )*);ARdw DR w1w;ÅXR Z-B9L"%,)".-/*D>%*)5=;7$)*L( 5=,)(1$ ÆG>1* :"B"%$?>15D*);7>%*)$(1$'"
H01"q>%;b9c01"’Ÿs,+ * (1&I5=,),H5y,B"+ *E;7>."
ƒÉQ*)5y&)( m/T &)(1&B"A01$)*\R I5D;7>10%*D>15X-1,)(6(%>%,)(™ÉQ57,)*D9 Φ (T )/T

Y
Z       _  A

 
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0.4

10

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50

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100

150

200


300 

250

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40

60

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100 120 140

φ

V (φ,T ) m4 h

V (φ,T ) m4 h

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0.05

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100 120 140

V (φ,T ) m4 h




3

1

60

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1 10

20

30

40

50

φ

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 60

50

100

200

150




250

φ

-1

300

50

100

150

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0.2

0.00004

0.1

0.00002 50

100

150

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20

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40

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V (φ,T ) m4 h

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30

40

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50




0.025 50

-0.005

V (φ,T ) m4 h

0.075

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-6

20

50

100


 200

150

φ

-0.01

50

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=5=,D5M*D;M"y,)"7Æ7>."%&)5 75=,)5B" *)*P-/57&D>1(1,D,)(  "%;7*  +57>)*)* 6(.-MŸs,D(P* +57>
 +5=;G>1(},)5=;7( "B"A,)5=,)*D5Z-/57&D>1(1,D,)( =&( )*);X-6" /"J,)5 :" )*)(%,),) ( 7( ÅGR 2F(1(Q5 7(101(:91$'"’Ÿ P*r&)5D5)*);X-/5=,D,m0Æ7>%* 1;X-/(101*?9 Æ7gZg#57$D>1(10l9D0A-B9)5G>1;b9J0~>%(  *?-/*y*),)(y;7>%5=&)57,)* i".-  D>%(~*I&D*)01(6*D>p$ i".-/(  M,B" )57,)*  1" )*)(1,D,)( Q"A;= *  +5G> )*)*UR K  B"%;=; +(%>D* 4$'"%$D* (B"
M( 0H&D(A-/,D9q>1;b9 '$'
" @"%,D, H5 %;G-/(10%*D9 +,B"`$<(1,)$ D57>1,  &)* +5B" a0J&)(%;G-/5 #Q* )
" 65X-6" vR 2 @"%$?-% )57,)*)5[Æ7>1(
=( B
" /5G-6" 15 / * +;b9  D>%(C,D5 @"A01*);=* +(a(A> +5 <"%,D*  i" =57,)5 B" *)* :"D*)(1,),)( c"A;= *  +5G> )*)* N;M" gZ"%$D>E5 ÈJ; OI5=;G>101(101"%,)*D9L& )*D01( 6*D>a$`(%>1; '>%;7>%01*  D5G-/*)#$ ! >1(%01 ( =(}"%,D>%*)01&5 Q5=;G>10:"}01(m2F;=5X-/5=,),D( \R Z-B9s( 5=,)$D*O(1;7>.".>1( ),D( h$<(1, 57,D> B" *)*a"%,D>% * :" )*D(1,)(10Oz‘"%,D>%*)& )(A>1(1,)(%0 )"%,?>1*),)5 D> D
( ! ,)(1Z0 Å @"%&) * ÉQ 5 #B"%01,)57,)*)5 B-B9 &?-B(%>1,)(%;7>1* )*);X-6"c"A,D>1 * 1" )*)(1,D(10 n R   +5=,)57,)*)5 )*);X-6" "%,D>% * :" D*)(1,)(10Z0Z;=(1& <>%;7>1
0 "RQ5 (  È +5r& )(1*); 1( 6*D> D0%
( ! &)5 )0  @"Z; ^ÈG>I* }"%,),)* 7*?-B#9 ! *)*a; :" )*D(1,B" +*E; D5  

 





&   



N!$
(

  $  BF    

 #E































¯ B




d (nB¯ a3 ) dt

65



 




= −Γ

· nB¯ a3 ,

‚z w1w%R Å 0%5)(@9D>%,)(1;G> 3"%,D,)* =*'-B9 *D* "%,D>1*:" )*)(%,B"C0C; )5 65O; :" )*)(%,B" +*^0C5 6*D,)* c0)5 +5=,)*\R Ê 657; ;=5D5=,)*)5E"A,),)* 7*?-B9 *)* z B" G-/* )*D5  +5 c;=5 D5=,)*D9 +*e"%,),)* 7*?-B9 *)* "%,"! Ê σ >1*D&)(%>%(1,)(10}*C"A,D>1*),D5D>)(1,)(10}&)57,)5  D5 X"%5 60m,D5 )5X-B9D>%*)01*);G>1;=$<( ;X-%#B"%5 σ = σ /v z ; HR%B"
 65G-*)DRZÅ/"  ; ∼ 25 x>ÆM2 . z‚w1w%R IlÅ = 10 σ ∼1 2 gZ( @-/5mz‚w1w%R Å n = η · n Ê &?-B(%>1,)(%;7>  )*);G-/" :"
)*)(1,)(%0)R
l(1,)$#)*""RQ*J;4"%,),)* =*?-B9 *)5Q&)( 5=;=; ÆG>1( )(3Ÿ 65=,D*)5 1" )*)(1#, !ˆ"%,D>% * :" D*)(1,),  &B" 0 ;7>%(A-/$),D(1015=,D*D9 +5Ÿ  B"%;7>%* " +* ;D5 Ê lRo2 >15  +( 6*),B" +* )5=;7$'( B"%01,)(%015=;7*)* )*);X-/(`"%,D>%* :"D*)(1,)(10ez 03;=(1&"'>%;7>%0#Q5 (È +53Å +57,D9D
> 1;b9 ,)5 /(AM- Ÿ},)( Š&)l( Æ7>%( & )* B"%0%,)(10157;=,)(h$<(1, 5=,D>B" *D*k"%,?>1* :"
)*)(1,)(%0P01$?-6 "  "A,),) * 7*?-B9 *)*E* )3( Ÿ 657,)*D&9 /(AM- Ÿ},  0>1(),)(1;G>1* $<( +&D5=,);7*)(10:".>    X"DR ]-/5 6(10:"A>%5G- %,)( H01$?-/ "  &B" ),)
( =( )3( Ÿ 65=,)*?9 B"%0%5=, Γ

 

· v · nB ,     
  9Ÿ     FÅ 




 




2

0

B

 

=σ

−26







 

0

−2

2

γ

B



















d (nB¯ a3 ) dt 
#

 





3 · neq ¯ a , B

65 n Ê B"%0%,)(10157;=,B"A9E$<(1, 5=,D> B" *?9Y"%,?>1* :" )*)(1,)(%0)R/2 *D>1( 75 ) -B9a&)(A-/,)(
=(J* +57,)5=,D*D9 )*);X-6"m"%,D>1*:")*)(%,)(10m&)(A-%#)* B"%01,)57,)*)5 F(A- i"%,B"
z‚w1w%R Å d (n a ) = −Γ · n a − n a .



eq ¯ B




¯ B

dt

.



=Γ



3




 szˆw1w1R 

 
7  
A>
› #  Å J
W&$  N
        $ Γ    = τ  −1   A  %F# .   %    τ     
   $  $&    # Y I +lÅ    

     #  E    &  
C G1  2$#"m

 







 

z ?R 






¯ B

3

eq 3 ¯ B

 Wz ?R          



$!  ¬    I ,I
ÅM b!   
 N N N1   & R    2    & R  $     . %$  J
W&$  @
      >   &      ;       E   & 















n
W
R R >    &   "J$  BF     u+;G-/*`015)(@9D>%,)(1;7>
k"A,),)* 7*?-B9 *)* 0%5G-/*)$'"v>1(E$<(1, 57,D>B" *D9c"%,D>%* :")*D(1,)(10 A-/*"M$'" $ B"%01,)(%015=;7,)( \R\T8"%$'"A9`;=*?> :" *D9ŠΓ$'"%$ ;75B"A; '01*#/* N* +5=5G> +57;7>1(a&)* 6(1;G>."A>1("! ,)(Z0 H;=(%$)* m>15 +&)5 B".> #B" vMR žV"%01,)(10%5=;=*D5p;=*'-1,)(Z,B"ÉQ"%57>%;b9O>1( 6"?$<( 6"y,)57(#:(6*  B-B9Q5 =(l&D( 65 ZŸJ"%,D*D9P>15 +&m" *  +5=,D5=,)*D9 D*);G-6"#"%,?>1 * :
" )*)(1,)(%0l;7>."%,)(%01*D>1;b9m015X-/*)$P&)(l; B"%
0 ! ,)57,)*  ; )5M"S- 1,  L>1 5 +&D( ^& D( 5=;7;=(10y"%,),D* =*?-)9 *)*m* )3( Ÿ /5=,)*D9UR  +5=,)" )5 Ÿs* +(10l& )
( ! *); 1( /*D>O0 +( +5=,?> B$'
( v" d (n a ) z‚w1w%R +lÅ ∼ Γ ·n a . dt 2 Æ7>1(m0 )5 d9a$'(.-/* )57;7>10%(O"%,D>% * :" D*)(1,)(10m0J;=(1"& '>%;7>%0 #Q5 (  È +5 M" +( B’" Ÿ}*)0:"A57>1;b9UR ~>1( ( 5=,)*? > p>% 5 +&)5 B"A> #
:& )*Q$<(%>%( )( I0 H&)(A-/,?9)57>%;b9P;=(%(%>1,)’( ÉQ5=,D*)5#z‚w%w1R +™ÅS*Q$<(1", ! 57,D> B" *  "%,D>% * :" D*)(1,)(10 0 Æ7>%(%> +( +5=,?> \,B" (1;G>.".-/(1; h,B" D>%* B"%01,)(%015=;7, # œ $<(1, 57,D> )" *  "%,?>1 * :
" )*)(1,)n(%0Y≈$'"%n$g ',)$ *  >1 5 +&D5 B"A
> #lR D™> È  D>1(30 B"%01,)(%015=;7*)* ? * +* )57;=$)*D5C&)(A>15=, *B".-  B"%;7>%* * "%,D>1* B"%;7>%* )"%01, * & )(%>%*)01(1&)(.-/3( Ÿs, &)( M,)"%$   * @"A&)ƒ* ÉQ5 )-B9Y$<(1, 5=,D> B" *  :" D*)(1,)(10s*`"%,D>1* :" )*)(%,)(10 v; )*D>.".9 T  m z (%>M-/* D*) 5 i"%;7;Z& )(%>1(%,B" m *E,)5 D> )(1,B" m & )57,) 5 D5 X"%5 v;7&)*),)(10 H5ygZ"%$D>%(  ,)5#0 H&)*); H0:"% 5 FÅ  



















eq 3 ¯ B

 

eq 3 ¯ B













eq ¯ B

¯ B



















p

p

n

eq

nB

p>+ 5G>1* ?>1(P&)* +5=;G>1(m>1(A-%$'(m&)*




=

neq = ¯ B



3/2

mp T 2π



mp T 2π

3/2



−



−

mp −µB T

,

mp +µB T

.

$<(1,)5 ),)(15Pz ,)57Æ7$<;=&)(1,D5=, *)".-1,)( i".-B(153Å M,B" )57,)*)5 n R  sz‚w1w%!R ZÅF* +5=5 B; #^ÈG>1( n = η · n 

 T  mp µB = mp + O(T )




B



3



4>."}0%5G-/*D*),B"H;7> )( +57,D9)5G>1;b9C;#>%5 +&)5 B"A> D( 3& )* 

G>1( B "%0157,);7>%0:"



#^È

B

γ



2m 1 mp T − Tp , nB 2π m3p − 2mp T · ∼ . ηB   
"     T  mp 3 d (neq 3 dneq a ) ¯ ¯ B B = a . dt dt  ,)ZÅ |T˙ /T | = H(T ) = T 2 /MP∗ l 3 dneq ¯ B a ∼ a3 mp H · neq ¯ . B dt T l+ Å  

neq ¯ = B

B

eq

z‚w1w%R!ZÅ * +5=5G>

B&)(lÆG>1( 

z‚w1w%R,)ZÅ

* +5=5 W* sz‚w1w%R

F(1(%>%,)(™ÉI5=,)*)5Oz‚w1w%R F;701( 6*D>%;b9C>%5=&)5 s$ 

mp mp T 2 H≡ ∼Γ T T MP∗ l
"   

 

∼ σ0 ηB T 3 .

p>1; v " B-)9a>%5 +&)5B"A> j0 :( v"J* B"%01,)(%015=;7*D9a&)(A-%#)*   $ ÆM2 . ^ z‚w1w%R w lÅ m T ∼ ∼ 10 M ·η ·σ  )*k>."%$<( 3>% 5 +&)5 B"A> #)5J$<(1, 5=,D> B" *D9L"A,D>1 * 1" )*)(1,D(10OgZ"%,D>."%;G>1*D5=;=$D*
i".-6" v*hz‚w1w%R,)ZÅ * +57 5 zˆw1w1R w1;w Å n ∼ 10 , 

1/2

p




∗ Pl

B

0




¯ B

−105



 



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#  










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J   F  #"$  

 n




$  m
+

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à :AE747>½º¡º7>

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 65G-B5 (1;6* 601"! +5<"%,)*" i"E,B"ÉQ57,)*D9
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 &)(%;=$<(A-1$"3(1,`* +5=5G>1;b9 AŸQ5P04>."A,#v"D>%,)( +(65G-B* g#*M*D$)* B"A;7>1* R   7( +5<"%,)*" <"
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@".>15G-1;7>%0
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( !

=(J&)(%$'(.-/5=,)*?9C&D
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W

c

W

Y

n

B

µ

µ

µ

2

µν a

B

µ

2

2

Ln

µ

16π 2

µν a

a µν

a µν

Ln

µ

n
W
R R >    &   "J$  BF    

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@"%*"! +( 65 D;7>1
0 "o>h;#&)(A-/5 A &)
( ! )
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" H0:" q>a,B"m>1(  ?>1( :" D*)(1,),)(%5Q*k-/5=&?>1(1,), H5 D*);G-6"O,D5Q;=
( ! "B"%,D 9 q>%;b
9 )5=;X-/*E0P0:"%$   +5#*?-/*E0P; D5 65#01( =,)*)$'" o>},D5=, @-/570 H5#$?"S-/ * )(%01( ), H5#&)(.-B9 a µν

W

a µ ν a µν

a ν µ

1 2 µνλρ

abc

b c µ ν

λρ a

W

a µ

∆B = B(tf ) − B(ti ) =

Z

tf

dt

Z

d3 x∂ µ jµB

ˆz w1w1R wÅ

65 * ,B" B".-1,a*s$'(%,)5), +( +57,D> 0 )5 +5=,)* )"%,B"S-/( =* ),)(15 )"%015=,D;7>10%(m;7&B"! t t Ê 015 B-/*)01(l* )-B9P$'"’Ÿ 6( =(l*"F-/5=&?>1(1,), )*);=5X- R #" I5=,)*)5 :" )*)(1,),D( =( D*);G-6"l01
( M,)*)$'"%5G> >1(
v"N$<( v"h*),D>%5 B".- z‚w1w1R wÅo(%>M-/*)57,c(A>E, @-B9 )-B9 7>%( =(h> D5  q>%;b9;7*?-1,H5J&)(.-B9  R$F,)5 =*D9 >."%$D* &)(A-B5 (%>M-/*D,B"3(%>`, @-B9* & )(%&)( *D(1,B".- %,B" R8Td"A$) * ( 
! F ∝ B"
=(   &D*:(6* $ @"%$'- )57,)* D>%(s,B" Q5=,D*)5 :" D*)(1,),)( =(m*a-/57&D>1(1,D,) ( =&( )*);75G;=0A9@"A,)(O;Z&)5=(6(.-/5=,)*D5 Æ=,)5 =5G>1*)57;=$<( 7( :" 15 )" v; HR D*);%RŠw1w1R w1R 5=,)$'" )-B9Y5 =(O0  ! ;=(A> * +5757>}01#*  f

1 g

a µν

tf

g 2 µν a ˜ a V Vµν , 16π 2 ti     
 É
    Æ  #   

É 

    


=3

i

ti

Esph ∼

Z

d4 x

1 g2

MW , g2  
 

65 +,D;( Ÿ}*D>%5G-  01;7>."%0.-/5=,P*F;=(1(B"’ŸP57,)* )"  +5 ),)(1;G>1* Koz 01;=54&)(A-B9 '&)& (2)

* +5q> i"%;=; MM [57;G-/* B"
G-/*)*D5 +5Ÿ  i"%;=;=" +* W !m* Z ! %(M(1,D(10&)57,)SU 5 )5 >SR 5%Rd&)(.-/(3Ÿs*D>
 sin θ = 0  65 θ Ê ;G-/"  =(A-; +5ÉQ*)01"%,)*D9d; HR  D*?-/(3ŸP5=,D*)5a|˜ÅGR p$D$#B"A>1,D(15y0)*);G-B5=,)*)5 B-B9a0H;7(%> 1"15B" v"™È7>



W

W

±

W

W

W

Esph

65



" M ,B" )57,)*)5yg ' ,)$ *)*


2MW B = αW αW ≡

B (mh /MW )

mh MW



,

g2 , 4π

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SU (3)c : SU (2)W :

U (1)Y :

Y =

r

„

„

SU (3) 02×3 03×3 02×3

5 24 T , 3

03×2 02×2 03×2 SU (2)

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 I4

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SU (3)c

R >

(c)

dL

e− L , νe (c)2

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µ

Uµ1∗

0 0 0 Vµ2∗ Uµ2∗

Vµ1 Vµ2 Vµ3 0 0

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1 , 3

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9

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;3 !#<="$ %>

%! %'!K?1&'! %%K("$2@ )&

 %5T(.9)m!%*7: +) !"*!)1  ,.7-& % /0A ) )  """"17B! /%3 !2 4 2 ,.%5"62     qq → S * $  %5  9 KF* G9!) !*HI!O45)!)4 2%H8 G"!'% "K5",( &J  *!"A"%9 K57QL"K% )M "2L%'K5,Q40 LF*)A!2' **!  !(KR M7 LL *) !"%7 4F" ST"%) +  2 9%)  !%"% K8M7 L39) )!!"$%*!M7  L4 ) *!""%"4Y&U*%5 V<( E> +97: % K0&W 0* !!A*S!45WP *4XZG%5 $) +145!%5 9! /K &*""$AF B) 45!!4) 845X"$I % 5 "7V8 ) Γ +!"*!,[1%>*) 4X!4  P  ) !454"$ 7:]\[A  #) S!P" )$*2J&  "N9) !%7MLO)!" ! 44$=4) +X^4=" %5"%4,[-= 9JR +P “ 1_` ” d n a   a5 =Γ ·n ·a , dt Mb* n c ) !4 45" !28A $ %5)!2 SP !"%U . d*e6fgehA kJe € m@ e x  €A 5ilkJ0sbm ej[€[m@kWlLe lnmJ†Tl…om kWzl6‡p u6uˆqrkJk@eesbs$„tkWul‰f/kJl6lbw*mlbu@ml6uq ~ vzs€'l6e1w@k lbzmxm@€Ax1ƒNygh iv|j=xŠxs$zzex1{1kJe5x|s1}|}Š~ €„ mkJvx1q‚‡Jls€sNehu@xo1s$w*k@lbe5m|u@m*l6xqT‹xLƒ:u@k@s6x„*pNxf/wln}5m{nu …~~ S

tot

3

eq

tot

3

S

eq S

dnqq→S = Γ (S → qq) · neq S , dt

. !?1!  :x‹$kJe1muWhuJ|}'sn~s$w5…gh'e5l6q S → qq i Œm‹ fgu l {zΓu‘3(S|5→l6q/„Jqq)lnk@e5Ž * „ 1 e

k  ‡ @ u  e  w  } * m  e  ~  t

u

k  u * m L e J k  e 6 s @ „ 6 e g f e |…kJƒ T  m ’ ‡€Al|(z€'ek*zx€#u S “nh'es1|1u‡0mJlT…Uhu@|v€'e|}i ,.-=”/ ˆ!" !9WR 7 S• P *!"%0*!,.%(" 6  ,.-HE`4_ *!18` 4( ´? +.LEA "%45!=4>6 #45) +Y4>" %5"%4 P d n –6— a   ˜' = −Γ · n · a . dt ZT !7v4'!2  a5™  ˜'&

A  I)!4  T K ! !(  2EA %5) ! SP *!"%U& ` ´ ` ´ d n a    = −Γ · n · a − n · a . S

3

3

tot

S

dt

S

3

tot

3

S

eq S

3

,


  
          



 

4." ) +6*RN. ?9 +?TM1) !N )   ":L'2 9-%C ? "%?1O!( 9 "A X"K 9%L3!%\.K!& %%5 )S4P   STS¯ P * !")%47ML@7 &

”/ +!!#"",[ +9% ) %5(B!) ? '− X%5L)%5"62E=4>)9)0K]$?$&99K%%?! %!". %'K) 4E!(B"*!R −L) P P 1*n4  "O"A)"%K, B −L

S¯

S

δ · Γtot · nS · a3

 *R%> !90)  =? +  .4O 9(B)!%−7ML)L0) 40!""$*! *%5!"L0%4 ) ,[45-=  %C93J45R +*!$J ST P 4%5) 7MLJ&@!<=("

) )!$4 K>#45",v ,C3& CP P

(B − L)

40")$*

3 −δ · Γtot · neq S · a ,

%!45$!)   %544&A4Š (B%5 )!− %L)  & *4'%!!0 /)A  >4"&'?1A)! / 7v 4') !"!"4%5 "$4(2F459?1"$!>)!"% R(B%>M1(−9) ) L)!!"?*!145@8!!92 ) !S?$P ' !%5" )"6%2 U   "OCL' =!H A "%=% 49W%5&TK ="$K  &  &4Š I"$") )X6*408 " )"<=6"*8=4  9 +K %5="62  )^*!  +)  v*&'! 9 %KC /Y7V "  % /!Š vA&' %5 S"P %4!L"%9)7!%  +M7 LŠ,.% )!B" !−L qq=4&U→0%$  SJ?45 + $  X (B− aL)&UI?1!C"? 9R7:%(%5K %M1\.( )+ /LA!q¯q¯  →?4L S¯!"

 <(%59K7: &% "%45%5"%4 ,.-H345 *!1X) !45 (B − L)

3 −cΓtot · neq S · a ·

Mb*% M1n.c ) ! 4! 4 ") *!!428A  %5) ! 2E 4'!)A4&!OA"%! %! q

nB−L , nq c∼1

A %7v4'!%OA "%4# ! ! 4E) !"*!1@! S

` ´ ` d nB −L · a3 3´ 3 n B −L = δ · Γtot · nS · a3 − neq − cΓtot · neq . S · a S · a · dt nq

) 4+7v H89   #0"4O  )K5!?44'H8!% KO!" %) 3!4%4 *!O%O?1!C M1™ 8) ?1!! ,C"1!&%!CK 45%5)H c 1? !C47: /7v4'! 

(B − L)



 5  ZG,[ "9%5+H8  +!09  +#) !4445%O45")%3 H 45$$U&  ".7 8 8 ? !.@! 7V?$454O2A%Y%5*!)H 7v%FH3  ! *!%9 K"%7vKEH8  + "$ +*! S% P t ! "%FŠ!" + +%5)  , (B − L) 4  N =n ,    n N = T T

%!) +%n , a = const · N & n a = const · N "L  HV%5HB<=ŠA "%!%5HW ) +%M1 &v4456*R ` ´ ` ´ d nS a3 d nB−L a3 3 nB −L = −δ · − cΓtot · neq . S · a · dt dt nq i

B −L

S

S

S

3

S

B −L

B −L

3

3

B −L

34"9K5?' "623% v& %5

gMb@!A& "N R%5 ˆ"% = 2

z=

− nq ∝ T 3

3

mS T

T˙ T2 = H(T ) = H(T = mS ) 2 . T mS

& )!4  2F ™  5™ )  / %C4

dNS = −Kz (NS − NSeq ) , dz dNB−L dNS ˜ · NSeq · NB −L , = −δ · − Kz dz dz

Mb*. !) ! +%5) K  )6*R34X v

 5  5

3

˜ = cT · K ∼ K . K nq

=! / v&™&g% %5 X !"Y :%5S )K%5"5 %X""AA * !!FH ) K!4 451"  (!2! A&/ %5%5) )!"E)2 $ " *%!!"4%2%X) !"4*A! *! H " ?47vLH%5 /45) %!%45 )WH & T SP \N!%5m)7V N47vz "$7v4'1! T

S

neq S =

„

mS T 2π

«3/2 

m

− Ts

,

 ]  "! #%$&"' (*)+ ,.-/01324"- 53&,6-",67'") &")389 :;-<>= '") _ @?A'")>0&")B81 :C:%81-")>,6-# -D '")+E _ '"+"!7-<&'a  0F'/  ") "0>DG'")>0H-IH<JG'",6-> :7,62489$40>= >KML  >0"N81$#E%(O-PJG (B '1,6$ −L) 17Q _ ,R%,6 C0&'1)+: 08""-:>75+& ST$(4  _ : $>Q  "-U:  _  #) E _ -"/ _ >DO  =>'")B2*-HMQ+V _ 4: _  '",681-")>,6-VS 0#,'")+:> :> -eq LBWG'"/>")+10>X'")>0H-Y,6Z:C")+[4 ,0#+1! #\$#&'>(O">, -/"324 --]7$#V- B_ −L P:)+=  ,@0Q ,@nLS:U=0: nHS/-^G_ _ -",F'") /N81 cc L `"L

 

   I    



 

W    

,

%  "%5 "%K,B ( " -C"%45$ H8 " $ HY"%2AH cˆ& NSeq = cˆz 3/2



−z

  5 

.

/!  = !  9) R !?". v &)4'!L'4*2 -2F . 4O )5!4  /M1O)2 =  N&U  + %# 4v4  7V)$  7V2 4  







NS (z) =











Z

z



“ 0 ” −K z 2 −z 2 2

Kz 0 NSeq (z 0 )dz 0 +



−K z 2 −zi2 ) 2 (

NS (zi ) ,

MbK*%5 M1zE)≡ = 28c 4* ! !& K%5 OL)?$*!  8*!) *! +K CH ?*z!&T )=A %)%  "%? !@!* *!I!"%8%97:""%'%5K )ESP?1 !!9"7v%4'! %5N"$2 (z^4%5) )? " !M !" + :97:"%5)"%5") /=%5 "$2= C !K  ',$& $ 1(K!%5  + & )  K  1 "4HO4 !S>P 4N %5$M1) !(@!R%N9*!"%K z 9?9L z N (z) „ «    1 . N (z) = N + O K C"O95"!:9)) K$!% %9  LNHU"M4[)H!""% !4'!]@"!A&14%!) = %5  S[P  * !$"4%7L7*•!9L51  K 2=%"$ 2]K4."""%5%452A%5"W%4&'9%[ ?9  8K ==%5H#)4 )2 %* ! /"% )"A /.")"!4.4 )!4" !"*, ! A %7v94')!!,.%-=  B"62X %5 9) K0• F ) !4  ,
   1 '" !23) !4F )A2A ! K &@M10 !* "!!% + + ))  !%(42 454) 452  "$ %52@"6U2L&*&

)* *45!C454  "KRN2-=<=N%(  F? M $ B  2 )9%5!) "62>!?L"# v!45"$•  )L' + + +%2I %5A) BF^ W9&A "%5!) M1!%@45! )A%52 2F"S%4P  *)!4"% /& "  ? $”g  =  z$&)%!!4Y  %5 #2F" %A !™ 2O +4 %( 4 )!4 4"H82 4 2 %5"62J Z     a5 dN N (z) = −δ · N (z ) , dz + dz Mb* zi

i





mS T

S



0



eq

i

S

eq

S































S

−1



eq

B −L

eq

S

B −L

S









z

B −L

eq

−I(z 0 ,z)

S

0

zi

I(z, z 0 ) =

Z

z0

0

−I(zi ,z)

B −L

i

˜ 00 dz 00 . NSeq (z 00 )Kz

 SG!   L !5 $K  &&U %5"N !*#!# *!*!K * !!2LK %5!2C /  ) %!% )"%! K c SP  *!) "2'%*U !>&*! *Š!  K "!A2#!K"A3  + +%)K  =2C %5m<=[ 9%7:!L"%5 )%5 ?1!N97v4'(z!)%5"$ 2OO)"# =9A9K P . d*e6fgehA s6su x‘'qGkJee ewm }~*m*i0lx5|lOj:sb~C{$… m*s1fUe5|xC}YhAlbm@qGmeu@et=zl us6T‘(uqGfJƒgemJ„ wlnk }yguŠm@i ez~x||5x1lbk@q+x„@qVlnkJm@ee|hA…ekJweO}m@€'xe~ v(Bs1l6w*−lbmL)m*x“‰yesbk@u eq+€qTm@el|1Tk uƒW~Fu €FKoxw}t#1 i lby T‡Js$lb€5mx5uJlby |}0hAqGese|z u “ " !M!ZG  /7:% !2J48&g  %58 *a5!™ X! ?1K!*!2=%5  /

)23!!%"  ) +!0 +*%5") %!U%5&  " %F!,.9-= ?H "$!X2E ) m3? 9 LE9%5 K /=  ) m!%&g))! L 9)<=R 45%5)7: Z    ˜' dN N = −δ · dz , dz Mb* Z z



eq

S

S



i

i

i







S



∞

B −L

−Iˆ(z)

S

eq



S



0

∞

ˆ I(z) ≡ I(z, ∞) =

˜ 0 dz 0 . NSeq (z 0 )Kz

+ NK4 %5% ) ,] M12@) !&8U9  )!?>?  ˜54'!* *!5!9™2#  ) ) +!  %"$ z2C∼ ) ) 18C 9%* 45 K T9∼KL mC9U9?1K !."$ L R%.z %) 4%M1[ /   $)*!!]% 4O)  !4L@& !&7M?!L# /7v4'!") %5!"642# ) (,""m$ $ •',.!"-= H P 





z













S



S

z

   /\N!!  %5 )49) •U! ?97v v45&T!,[-C $7vM1 L%\NM1!) !%5K) ! 8 47:)  !?X4A2%5 "$2%5)) !? 456* *! "% L 4U0&WL"K %"$4 %5"%4*!K&WK17M\ L \.%!NM1$ )! (B − L) "[)"%5 z &'% 5 97v45! %5 /) !% )7[gY)!"%-=SPM#\N!%5) !   "*!9 P dNSeq = −ˆ cz 3/2 dz





−z





−z



= −N eq (z) .





−Iˆ(z)




  
         

,




 

)456* ""1 !=4 7: /7v7v4'%5!M1 )23!9!K" +M1 (%547:)) Y!



%5%O / / I "%M!%5"62> )

ˆ +z . f (z) = I(z)

z = z∗

&%!A  %5

  5 ST" / !2  & + : )39K 9L K  " v   !44 (# 45 0) =2Y ) !4 $HY%!AM1#% !5 “ ”    ˜ + O ln ln K ˜ = ln K + O (ln ln K) . z = ln K SK%5H8%5 AN !Y"1! %5M1)! I(z ˆ ) &%!08M1O)45!2E34%5)!2Y) ?417: c ) 2'* !=$ 7& %!Y %5 ˜ ∗=1. NSeq (z∗ )Kz

















∗

∗

f (z∗ ) = z∗ + O(1) ,



d2 f = O(1) . dz 2

"$ $ N"%  = =?$*! *!%&A %(%5M1)!  ˜'v*!9 )!%5"62>4>9*!"%84) AM z )!? +)! 





∗

∆z ∼ 1  z∗ .

S  %M1. ! 8A !%5$K&".A R% V  5 

NB−L = δ · const · NSeq (z∗ ) ,



NB −L = const ·

  5

δ δ , = const · ˜ K ln K Kz∗





Mb*NA9""% ! %! c  A )A2A 7v!=HF$)?K7%! %0L%5" ),™ *!=4$X*"$R $ 'C4%37v  "$ 1  JST ) 47ML@&J%5 / )!% )! &@)LA%5)H T M1))'%5"$2E!" + +%5)  2@&  )$*$ 2 %5"62Y\[) / HI  &%5#"% K P    5 m T = . ln K ) *L!"K%%0H8%5A / ) !%" %)4C(47:L0 +)7v!"45 !!* E4O!45"$" RC + +-C%5R.) 4 <=K# O#4" =UA&  K1\N\[%4I I(z ˆ ) ∼ 1 $&J!0  %5)! 2 SP 









∗





S



∗



∗

NSeq '

dNSeq 1 ∼ . dz K ln K

 + 5N" )? P !?145!$K%5M14=&@)"?$L9K4%8 ST) ",[ -(! 2O!HL"  +" +%%5M) ! *2#!"K3[%" / =A)! I% ) !! *!N@&@!%<= ]@ )%5 /E 9)9!K % )! L & K "(A!9 K5A&AE) ?$K KC%!!2 % T m ) !%@ SG)"$HU2X& !" + +%5)  20M1 ) ) '%5"62E )8K%5 ˆ4>%5)45!C% / )!% )W& ! >") !4, "("! +HL%5 / P 













∗

S

∆z 1 ∆T = = . T∗ z∗ ln K

545% 99K?X9")%!R=X%) ! F4%54) 452A G*"!! "K>M1TE* ?!]*\.!"  )$$ !  2@,.-2XST +EH#%5"\N%5 v!O&*)% !SŠ    ="""  +4E9 K5) ?! "O4'*!! )1@ !# LX" "!97Q /)!O %"  M1>)A4<() !" L45 !! ,[@)%Y!!84A^Y 345 +%5) "!%5 =/"N, 4[S *P / *H@!"™"%%K4%5  P  ) +4 M% %O7M9L 7:%SK &  *S!¯ )&g )  ! +"" +)W&'%)) 7M""$LL74F*!) ) ! ?6*H8!  M' ' 2 g BEY>) +








K1











   '-   0"-[O#, =>+"!>[O-I? &"'-=+1-I24 - '1)+=  _ )+ _ &I"3?"Q  81H ") ^_ _ R= $ 81 _ K z∗ = ln K 0$#(O _ >: 1L

 

/        



 

 






lα N1



yγα

N1

+

y1α

     ! 


   O

lα





Nγ + ¯lβ

N1

   ∗ y1β



yγβ

h

,
+

lα

¯lβ

 

h ∗ y1β



Nγ   

  

yγβ

yγα

h

h

h

 & + + & + 1 [ .=M> ; >$ > 31/O='"
+PD >
 >;  
+PC>F;+ ? ? N1 → lα h "9UO
> 
+9J>+F;" & S
"@> 'J;#=" ;
8 C
@ 


#;+\

 !P β  
>;  D#=' 8  31 &T[ > 
 95 >+F#;+""9RB  >$EB .>E;%"  ;E B γ (
  D  9 &$[ 
%
M  ; =.  D  9UA(FO=;;!=E<<@ 'J EH  C
 . 9 & 

!

"$#&%('*)+*+,#.-

#/0'*12143657%*'8-

'

12#/9/;:

+*36<*57%*'*+*)*=>?36@BA

57)C;36+*39D6'(/

^"9IH J#= C>KFMLONQP4;RFDO>OSTLOSVUWUWXZYX\[!B9]^S_J`H*J';RF]^S_]^N^SaLOS`bdcHW]^]^H+F`H 7dN^JTLKF'BeF D`]RFM7^NdcfN BgHh  STcC 99c AiJ`Pkj!DlFk]^HhJ7HcmLON!7^N^SfHci]!=lLRj nFJ_J7]^S#Pdc ;^Nd]^H^XOo.HWp_cH ]=Bb!;^S@>KJTcMFP? (B − L) LRjdS_cWJqjfb^ ; NdPkLOS`GFkcSTLA ]9/$b^HWb9 9 cMFMc WA JqjfH  rsjdJ`]^Ndc+A WF+;^N^HW]d]!=EtFkJ`N"gSTc ;^NEuPX;RF+gGF2J cWH+` F H v  SwbdH*>x J Hy > F!zB GzHcWH+;9/B P N^J_b^HkLA+D#W= STcWJqj8bd ; N{H k rsj^J`]dS`]^N^N\nFJ`Je]^S#d P c d ; N^]^H^X|.J_cWH+^ 7 ]^N^GzH

 F+d x ; N^HW]^]^H+v P FJ_N"gS_c+^ ; N^NvPp_cWH+}JTL_=R 7 FkSgH FO= c~JTL_= {  Ndc ATR ; FJ`b^F>9 ?R 7 FkJ_cWN!VIB^ ; FJ`J#gHc ^ ; S_]'? N ]9 J P STLRS7UWUWXZYX\[dXM€p_cN'JSR ; FkJ`bRFK > F2J gHh  STc7H +R ; FmD HP 6 9 PxFkc+WA JqjLOS_bdcWHW]d]RFkjiFJ_N"gS_c+^ ; NdjB GzHcWH+R ; Fkj mD FkcWS# R 7 FJ_cN^ 7 ]^HabdS^ ; S#R ; F+x Fkc+6 9 PxFSTcWJqjVP x F+d ; N^HW]^]!=‚ E FJ`N"gSTc ^ ; N!E lD F?J#^ 7 _capƒLOS`G? c d ; HWJTLKFW H `F H gSOxJ F]^N!DnF!kB H+W J#= Sy > FPS`F HJqjP R ; FD@K > STLOSwUWUWXZYX„UWX€}LONdcWS#R ; Fkc='d ; S9p_cHc7J#^ V S`]RF^ ; NP F S_]^S^ ` ; F+^ V N^N x F+^ ; N^H]^]^H , P FJ`N!"gS_c d ; N^N?]RF2D*6 9 PxF+w E c…LRS`bdcH `F S_]^S#`D N^J`H 6X o.H† ? JO = S`JTcWP+=+B gSOxJ F]^N!D _F S`]^S#R ; F+^ V N^NvLOS_bdcWH]^]^H i P FJ_N"gS_c+^ ; N^Nv‡ˆcHc h  SBd 7 cH.PTR ; F2DOK > SƒLOS UWUWX\[!` B cHkLWA GzHecWS_b^S; A P gS`JTcWH ?R 7 FJTcWNC V A&NF#=^ ; N!;!= E c`F H]^G^Nb^F+R ; F+gSTc ^ ; HWP!lB LRS`bdcHW]^]RFkjvFJ`N!"gS_c d ; NdjvH +R ; FD= S_cJqjvP R ; FαJ`b^Fy > F2J ` Œ Š  ‹ k  R Ž h    ‚ ‘ ! N < D 7 ^  F _ J  c  N V “ ^ ’ k F ` J  N " gS T

c ; ^ d N  j B ^ ^ b l H ^ j  P ^ N  P h  M F ^ j q J B j K P ; R  F ` J ^ b  F > y 2 F \ J  W k H R L ` S  S l c j  h T  _ L  9 J 7 R  F T J W c NVB ‰ ‡‚J`F+"9/8 P LO_F G^NP D F"96PxFS_cJqj?P…b^ l ; H+^ V S`J_JmF2( J JR=R 7 FJTcWN^S# W HkLOS_S7LO_F G^N'J R 7 FJTcWN“ V XKo<W= JTc A N1 cWNdb ?R 7 FJ_cN“ V XRE&FGQ"9”Hc gS#R 7 FMLON*P R ; F2DOK > SƒLKF2? J UWUWXZYXZ[NaUWUWX\[!zB P >^ ; S_PWS`J_]^H Vb^ ; NM LON h  S_]^N^N N bRF+;!^ V NRFMLA ]6 9 SN^ ; N^]!9 PWS#^ ; Hljdc]^HWJ_cNO•/R ; FJ`b^FK > HWP E&FG

N

N1 → lh

 

UWUWX

N

•

N UWUWX  U• N1 → ¯lh N ;RFP]9 gSS>=(J`H WH+P“X [ >K]^FGzH8b;^N =7^_cShb^S_cmLRS`P 9J–PGLKF>KHPBp_cH8]^ScMFGIByS`JƒLONaGzHW]^J_cMFk]'? c 92J`Pj'DmNIBA&NF#=;^N!;!=EvNdShPiLKFF;RF] —NRFk]^S UWUWX B cX˜SNgS_S_cQgS`J_cHB]RF+;'=? 1  •0GzH gbLOS`GzJ`]!9Ry N vS_]^N^S X™€šH>K]^HWb^STcmLOS`PH 4b!;^NkLON h  S`]^NdNaJ#z = vS_J_cWPS`]^]6 9 S K > NRFFR ; F+""9RK B b^ ; NdPWHO > jdN^SvG CP ;RFDTLON7^N!E›bRF+;VdNRFMLAW]!9J?N;^N^]XR ; FJ_bRFK > HP UWUX   •eN UWUWX  U• y B b!^ ; N^PWS K > S`]œ 9 ]RFQd ; N^JXUWUWX„U 1 X N o&F+;V^N^FMLAW]RFMjBvN;dN^]RFS;RFJ_bRFy > F ]RFhHK > ]^HWbdS_cmLON S_PWH =^ ; HPW]^S y > FTcWJqj^ B cMFG^NfH  R ; F2? N1 → lh DmH+/BRP 9/;RF hS_]^N^S#

2 X  M1  X ∗ D Γ(N1 → lh) = const · · y1β yγα yγβ , y1α + Mγ α

β,γ

N

UWUWX



YW•

Y


  
          Y



 

FN>KS ‡

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vH ;]=lLKF R > LRjBbRF+;^ V N^FMLAW]^H+PBvN;^Nd]9 ;RFkJ`bRF>KF b^HkL_=7RFSTcWJqj(N!D UWUWX  YW• DmF+gSO? N1 → ¯lh N ]^H P~ E GFPJ`G^N'8 J GH]^J_cMF]c8]RFhJ`HWb!z; j h  `]d]6 9 S+B X WH+D`]RFM7RFkj  [ ∗ yαβ → yαβ     1 M1 M1 Im D = , f Mγ 8π Mγ NgS_S4>^LRjgN^G^ ; HWJ_GzHWb^N^ 7 S_J`GzH , P LRS`bdcHW]^]^H ? P FJ`N!"gS_c d ; N^N P

Γ(N1 → lh) − Γ(N1 → ¯lh) Γtot
P   ∗ 2 Im 1 X M1 α y1α yγα P = f · . 2 8π γ=2,3 Mγ α |y1α |

δ≡

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Γtot =

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δ , g∗ K ln K



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−3

nM (t) = const . s(t) {` rdgf d [ ' ` ] '

JH '*5 R ) Y*2 '3)mK9' 1>+rd*2 ' / ) RR ) n|7 ) / ) 2 ;6) + ) /B- ' 32 K kh9'*) Y*f ) h61(7 59' d‹2 ) /*e918R4187976fC' / : +.- ] R ) 7 5#)3^8;65 ) +=K 1'|5 / ; + )3) ' 7 )*) ;62 )' d & 76185 e ^8565'35 / )ƒ) % )3281v;6+[) 18767 ) F9st) <; 5 e61snC' 761(RIp ; e61(7618Y3e61 ^ - K / ) f8R ) nC7 ) ' -3767 + h 1'‹R 7 +=1' -37 R 7 +=1 Y3p 18R R41 d nM ,Tc T 6 g∗T 3 √ · s0 = 4.57 g∗ · mM c3 0 30 ρM ,0 = mM nM ,0 = mM s(Tc ) M P l Tc  3 r  Tc g∗ mM −3 o , ∼ 1012 16 10 o 1016 o 102 ^w] m ] ^ n ' ' ^ _^ ] {` &% % '  rd Tc ∼ m M ∼ ` ^ ] ] d ,% 1016 o −3 b d ρc ∼ 10−5 o '{z ` [ C' sf rd f ' ] ] ` ^ ] [ ' ,% n ' ' ] ^ ` _ ^ f ' Tc ∼ 1016 o ] ] ] rd ' \f ' ] g^ ] f n d '  rd ^ [  f n ' h ' ' f ] ` ] f u% f ' ^ ] ] ' n daf   ' `] ^ [ ' rd d f f n &% % ` ` ^ [ h b f b gf ] ' h f ' n h d f ∼ ` ^w] ] [ n(T ) · σ · ∆p n(T ) d f ' b % h f ` ] {` ` 2 gf ' rd C' σ ∼ αgm /T 2 ∆p ∼ T vM `] ^ ` `^w] n dgb vM  1 f f = −κT 2 vM >  ] [ 2 ' f d_b κ ∼ αgm g∗ ∼ g ∗ ^ ` ' dvM = −κT 2 vM , mM dt ] ` 2 t ∼ m /(κT ) g' f rd b d M M ] t] _^ ^ ^w] _^ [ f gf ' ' f #d _^ ` ] J% MP∗ l /T 2 gf ' b {` ] `C]  f ' r d n ' _ d b v ∼ v M T = p ] _^ ^ f T /mM r mM 1 . lM ∼ v T · t M = κT T

“— uF — – U &8% 2(R &8% &V% h 5 5  uF – — K ; )0' ) '*5& ' )*;6565 8) K6h % ) Fi‡ K '*591L;656R4h fS76f p + & 761(e/*fi1 e['*- 56Ah 1(182 7 :65 1 RT-32(D U ' -3+ Y ) / +' 1 ) e7 5 2 ) + 7 3D e ) 8)_) Y3jk1e .5 761 187 5uK 281v+[18767 + &8% Ž;6) +[p h65 /0Dd- K 2 K; + )3' 7 ) 2 ' & 761(e 8595 Y ) +=181 h 18R!76- — 'r;6) e K .:,) / ; e918/ ) 2 ) .59' : 5u'*56h : ; )0' ) ' &V% F % &v'*) ) '3)*59'y; ) ) )*;6) ;6)e +6p h 18761(762 f p €e61 8+ pm+ 7 ' - 2 '|) V7[- h -01 ' [h9'3) + 5 Y 2()dR ;6)d' 1(R 5 + 5]R5 2 76fi2 R ; e 56h6e 5 7[Y0-3+=R}18RT/d-re[R -3767 761 % +=1281 +=' 1(767 ) Q79.1 ) ; e '*)*5 5 2 ) 5 .5 + ) h e ) 5 187 5 1@R ) 7 )3;6)&V+=% 1  7[5 - ;)Ae '*5 )*R4;61() e ) 7[85u- KX() ; e K6) h 1' Q2 ' - ' =565Q' 1(: R ; 18e[) M pqpqe67 fŒ-376R41 1(7 ' 2-\+ *7 -\) + ' 81 7[) - e 56185 7 791 ;6) e[- 8pqR418/B-01 U ' + 2vp Y ? 182 ' / ) +/*-37 59K R e ) 7 2 )*;6e[) -0+=761 2 /BF - st/*) - 2v+=p,1 pqR7 KuK/ / ) 8R ) h6Cnv' 7 ) 2 ) ' )761W2 )) /*R4182 '*55 8RT5 -Z2K 5 565 181 ) )*) ;9+ ) 3D ) 8)5 ) Y3jk'315 =5 ) 7618)*7 ;959) KaF 5 K ' : h '2 /*18797 ) R / pqeVR ) /*76|1E7 /*fi€/ -3) 767 - )|'*+ ) R .h9'*R )XM 7 - ) /*+=fi1 1 ; 18-3e67 1 ) R f 7 2 ) Y3+=e[1 - r) 76/B1 -37 5VR41(R 1(7 R )1 7 )*76;9- ) +=1- ‹1 / 8) e K9h 1 % 281>+=18797 ) =) + |76fŒY3f ' m- ; e91 ? 1(76f F Y*1 .5 R42 K / &v'*) R F K9'35 / 5 2 ' 2 :95 ]R ) 7 )*;6) + / 5 - K 2 E/ ; +.- 8R41 V-3e K d1(7676f |e61v+ K9'*5 / 5 2 ' 2 :65 h 8 1 6 e > 1 + 25 '35 ) i5 2 ' ; f 5 ' fi/*-31 h'w&8MW'*5 M 1 :9'355 /*; 79p 2 5 F +6%p ' e95618h67 5 59K &(/*'*f ) V/*-3765 79p & ) +[1 :9' )ye )) RT- 597 '59' 79: fiR : / mR 1 62 / 18R 2 -02 [-3R ; )0' +.- ) VR4' f 1>+5 ) 79p ' w?E2 5 +=f R ) |)*7 ;6) 6187 h '*5 - ; 1 h P 5 + 7 5 2 o) / m- ' R 5uK 1I65 2 /3p _2ZR 7 5 ; +=1(R )3-0' 2 5 +.R4f P2(1 1(7 1]/ m- R 1 62 / Fˆ - :65 R ) Y*e[P - ) R R pm+ *2 Y e[-018R4f 1 p{R ) 7 )*;6) + K / 5 F p ? ;91 5 8) 2 K 2 '*) ) 2 ):,) e ) 2 :,' ) - 2V-3/X/ e V- 7 '3) 7[- + K R ) 7 )*;6) + K  e6-32(D 5 e6187 5 18R % 281 ) 6 ; * ) : ; +=1(767 - e6187918Y*e61 -318R U 7[- .h6n n(R ' ahu'*)5 m) -Q/*e918'R K 5 ; F.HJ'*) R ) 7 )*K;6) + Q) 2() p ? 18) 2 8' )Z/*18; 797 ) ) R4187 K 1 ' 2(/ ) ) 2 (:,) ) e ) 2 ' m-d2 / V) - R:,) () 1 92 / ‹5 / +=- VR41 9;6)*&('3/*) e618R ) 28/ )*;6Y ) 7 5]&(MNM e :9'*Y*51 -d) /*2(1 5 -ZR4) 7 ' R4187 ' DA1 ; B-3Y*Y0+ ) /*2 5 /*.e9) 18R4'187 ) 5 ' K RIpXR ' 7 ; + ' ) 1  :,/3) 7 ) / ' V- R 1 62 /3p 2 +.- 8R Fu‡ 5 2>+=1 ) /B- ) 1>+ ) *87 ) ; ) / |p 2 ) )*
2 ;6) 18K R 1(; e[- pqe67 2 e 2 + 76-Z2(/ Y 7 e Y31 -ZR 7 + / +.- 8R41ƒe[-0/*7[-



             – —jp   %



"!   # $ %z5 ] 7 )#`[h9'3)"] + 5 7[-Z28/ ) Y )t] 7 )_^8) ; e ) Y*1 ^ -ZR ) 7 )*;9) + K RT-\+.- ; e 5 /*fi2 )*:65 % ' 18R ; 1(e[- ' pue6-\% F HzMWM 1 :9'*5 /*7[- K -3767 5 ^85 + K h 59K R ) 7 )3;6) +=1' 5 -37 '*5 R ) 7 )*;6) +=1'R )_^ +=- ; e )*5 2,% )t]=59' d_76e[; -3797 5 % &v' - ; -\% & / ) F +r b6h 565wK9% '3285 1>+=5 1879' 7 ):9' 5 `4:,)_) ^w] )*- ;6]) + 5 7[5 -y28/ '*) 5 Y )t)] 7 )*)_;6^8){) ; e ) Y*1 5 ^ -LR ) '7 )*K ;6) +=1'^8)/ [ +.\- KfVR415 Y*fz+=& -ART:9-\' +.- ) 18^ e61>59+ ' )_^8)/ ; 2 259'mK 1ER 5u7 K{` +rd 5 -37 )] R 7 ' 5 +rdd;28Y0+ nZ) -gb ;92 ) Y0) +=- K ' ] -3e 2 59+=' 1 +[5 1 )0' e ) RT- 59' 7 7 :qe ) ) nd' 187 5 MN-{) /gfm5 - )R ' 1'62 
/
1€2

+.- fV R ' KfV/ + )1 pq) R418'*7#)md0K D 5 `4d :,)3% '3) ) 7 2 ] 1>+rd*79pb 2 e 2 d 5#^(5 2 5 e6R ' e ) /*- hd h '*5  P 2(/ )_^QfV-3;976) 7 5c1 [ '2 e 2 ) 7 59K 7 U F„1 g R41(767 e)C' 1X- {: / ` / -\r+] d*/*791N1 2 '9' DE- ].1859R5{-0`a765‹7 ; e )*+5 2,% e9)tp3]=1 59' / -3767Y*f5 ^85 76+ fiK 1 h 59K -32 R ) 7 h6)*f ;6) +[-3176'o-\+ /|e[-37676f 1' % 2(1v+=1(767 ) ' F K fV-3767 ) 1t2 ) 2 '*)mK 7 5 1tY3p ] 1 'Z) Y3e[- f ) /B-37 #) ` 182>+ 5]& 761(e ^85uK]& +=1 :9' e ) RT- ^ 7 5u' 7 )_^() /gfV- 5 R )[ / ] ' 59K ) )3;6) [ '35 ) )*;9) ) ; ; 59'];6) 56h95 ' ; ' `a'`F F ; 1e '65 2 r/ . r R =7 g /T+rd F -3y 7 ' 2 b R ] - 7 ;6) +[+rp d*h67 5 R ' ] + -3K‹e6f &(MNM e61 18:9/*'*fi5 2 /37 _) ^8) 28/31 1vh + 187 59K‹761) Y*e[1(\- R f ) 18/Be6-0-7 59pqK e9Rp ) 7 \) 1 [ ;6) + ) 7 59K€{5 ` 2>+=1 ]=) /B- ' 1vr+ d*7 #) ` -3767 #5 ^(5 + K h 565 R ) 7 )*;6) +[1 ' ` σ ∼ r ≡ g /T . HJ' - ) h6187 : -L2 ; e[-0/*1 ] + 5 /B- ; e 5 /*fi2 )*:95 % ' 1(R ; 18e[- ' pqe[-\% `z:,_) ^w] - ] + 5 7[-L2(/ ) Y ) ] 7 _) ^8)L; e ) [ Y*1 ^ - #5 ^8l 5 RTK -\+.59-Kr;65 )].nv2(' e[-3/*79] 187 5 b )62 F r a/ ; e )0'*5 /*7 ) RŒ2>+[p h -31ZR ) 7 )3;6) + ) 7 5 '{761 ) Y*e6- f8p31 ' 2 K `5 -3767 s 5 + h )*:65 ' R41 ; +=1(76' 7 :q) ' 59K ) )*;9) ;6)]#] 5 ' KQ' :q) `9h9'3) e /*) fi2 ) ] _) % ^()X1(; R ) 18e[-^ pqe[) \- % )*;9) #7 h6Ko187;9)Xe[)3g- ' h ) R 7 5 : +=1 ' #5 ^(5 18K
e n 565y/B-3; 1 5 2 - ) ' ) K /*e61(R ) 28/ :,Y) 7 e {5 Y3`[1 '`F-XF R 7 + 7 DA187 b -0767 + h e R41(e67 e[-3/*7 %,-0Y*Y0+ /32 RIpX/*e918R4187 1 

0

2 m

2 0

4 m

2

0

M

σ nM vM ∼

T2 . MP∗ l

% ;6) 5 K{`kh9'*) R ) 7 )*;6) + 55 R41b r' ' 1 ; + ) /3pbx2 :,) e ) 2 ' d )3' 2 2b ] R - 7[-

k;6) +[p h65 R •— uF —tk U n M ·g g √ ∼1, s m T ^w] h {`3h9'*) 5 * F J % ] )#`3h9'*) 5 6 h 1iR4f p + 5 ' s; ∼ g' T FHJ'*7 )3' 5fV-2 5 nv' -0' 767 5#^('*5 )a+ K ].h )d56'5Z)3' ;67 )) DA`[18;67 )*5 : 1 n /s;6) ; - ] K -31 ' ' K 2 pqR418) 7#d05 DA187 18R 18R`„5{1(; e[- 5 pqe6f e615#n ^85 R K 5uR4Ky181 ; R4: 1(2 ? ' 1sKG% Fkg e ;6) -Z] /*f ^8) +=7 ) 1 5u2 K p*;62>) + / h 1 l ' ; r ' e l : ∼ :6r5` -3797 + h e61 e[- -01 2 f 2>+=1 791 p*2>+ / +6p -318R 18R 18e[- pqe9pMfV- -\+ m . T ∼ g κ Št) 7#h61(7 ' e6-gh 5 b R ) 7 )*;9) +=1'€/ &('3)3' R ) R41(7 ' 7[-g' ]=n R 5 f “— uF t— k U ` ∗

M

0

∗ Pl

4 ∗ m

M

vM ∼ v T ∼

p ` T /mM

M

3

M

0

M

M

f

4 2

nM 1 mM ∼ 6 . s κgm MP∗ l g∗ ^ ]

HJ'*)y)3' 7 ) DE187 5 1]2 ) %ue[-07 K 1 ' 2 K ].) 7[-0DE1 () /3e618R41(7 5 ;6) 2 ' -3/0+ KuK g ∼ α ∼ 100 ` κ ∼ `65 ] K ) g ∼ 100 R41818R + 2 /*e618R41(767 ) ' ; + )3' 7 ) 2 '*5 RT-02828f‚R ) 7 )*;6) +=1' `  &8%  m n ρ =m · · s ∼ 10 · 8 & % o 2(R . s 10 o HJ' -/*1v+ 59h65 7[- fV7[- h65u' 1v+ d*7 ) R41(#7 d3DE1 `ih 1(R / : +.- ] R ) 7 )*;9) +=1'~Y*1sfyp h9n(' -}-3767 5#^(5 + K h 565 •j— uF — – U `=) ] 7[- :q) ) 7[-Z/*2 n e[-0/*7 ) 2v+ 5 D :,) R /*1v+ 56: -|/C2>+[p h -31 ' 1 ) e 5 ' ) +rd3D )_^8)|) Y3jk1 ]=5 761 [ 7 59K ’ `^w] 1 ] ” + K  RT"-3!$2(#&%28'f)R( ) # 7  )*'+;6*) ,+ ,K]-.&5-%R4' 18ƒ 1 -' - 2 K‹ ) ,h6/187/%: /- ,-m')01∼*/2 310( *# ” 4o ')&825%c* F # ” ,  #&,678#  */9/9/% ( :01/.5 ( #  #&,-<;#=!>#:%'?)@8#&.5 ” #  */&78-#&,-'W’:#&%#&,,/A<;/B ”  CD   #5, ” /9*25 ( *,-'  /,%$/,-'  EF,F:G- ” * ” 6F: ( *H!>*?)@8-      #&, ”   *,,-.&-%' ƒ --  /,%'I-J*, ” -  /,%' ( G*/ ” - ƒ+K 2 m

∗

M

M ,0

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M

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−1

– —3— g '*) 56h :65{` : K€; ) ) )*;9) 5 ^ '*) ; )3'35 &8: 2 ' e[2 - ;9) e + K h 1(562 5d' 1 )p e 56-fm5a-3^879) 7[e - K9h 1' e % Y\2(+=1v18+=RT1(-Z767 R ) 'd7 / ) Y0+=+.1-3'€2 ' Y*d fz' +.18-dR 2; 1(+re[d*- '76pufie R‰/*-3f†e DEpq1 R4101(7 o R &V%^F*e HJ'3)3'/ -3e ^ pqR4187 ' Y*fz+ O5 2(1' h -32 K /0+ K 1 ' 2 K U )] 7 5 R 5 f ].) / )t]=) /W/ ;6) +rdgf8p 5 7 M + K h 59) 767 ) ' ' 1 ) e 565GF


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 $

     "  g    "    $=!    # $      $    \        j  #  v  $ m ∼ 10  #!  # !      v           \

{  j             

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_ E   \ $           T ∼ 1016        j     !     $

46587

gihj*j1g

∗

M

B=P OHQP&R OH

16

P&I < G 

B|iwmUm ~ s[lxm pw8uUpRp 

r5 7 5 RT-\+rd*76- K R )] 1v+rd ` / :q)0'*) e ) ' 5 R41b ' 2 Kƒ'3)*;6) + )_^856h 182 :95 1 ] 1 M 1 :9' f_e[-f8R418e67 ) 2 '35E)].5 7 P :,) 28R 56h 182 :95 1E2 ' e9pq76f ` P &('*) -3Y*1>+=18/*- R )] 1>+rd 5#^^ 2V- ^2+.- ^ e6-37 n 5 -37 ) R   “— uF — m U v 1 L=D φ D φ− F F −λ φ φ− ,



µ ∗

^w] ^

µ

µν

µν

∗

2

2

4 2 Dµ φ = ∂µ φ − ieAµ φ , Fµν = ∂µ Aν − ∂ν Aµ , ` ^ Aµ U (1) rd &%

1 φ P 5 :,) •R —;qF+[—1 m : 2857 ) 1X2 : 5 -\+ K ' e67 ) )01 ' ;9) ) +=159' P ) : \- + 5 Y*: e ) / 5 )3h 7 ) ) 1 )3h ;6) +=1 ; 9e p ;6) ; f U) (1) F 5 - [ 6e -37 n -37 U 76/B-3e -37 187 7 2 1>+ *7 \- + Y*e / 76f e61 *Y e[-\f /B-07 ' φ → φ|

iα(x)

,

φ∗ → φ ∗ |

−iα(x)



1 Aµ → Aµ + ∂µ α(x) . e

,

% ' 1 ) e 565 •—uF —m U 5 R4181 ' R41(2 '3) 2 ;9) 7 ' -3767 ) 1]7[-3e9p,DA1(7 5 1r2 5 R4R41 ' e 565 z /*- : p,pqR4f ' 1 ) e 565` )*; e91 ] 1>+ K 1(R4fi1 5 f{p32v+ ) / 59K R 5 7 5 RIpuRT- 2 : -`+ K e67 )_^8)};6)3' 187#h 5 -\+.- `i)*;95 28fi/B-b ' 2 K e[-0/*187 [ 2'/)R “— uF —  U v |hφi| = 5 76-3e9p,Ddg- b ' 2 5 R4R41 ' e 5 b G = U (1) ;6) +[7 ) 2 ' d_b 2 G → H ` H = I U F 7 _) ^8)*) Y*e[- f 5 1 &(: / c5 [ /B)*:-\+[187 ' 76) f&%' /B- : p,Opq&(R '*)) / )3' )31 ') e 56' 5_' )*;65 28' fi/B)3-3)31 ' ' 2 )K pue6-35 /*761(7 5 18R “— uF —  U ` 5; e61 ] 2 ' -3/0+ K 1 ' 2 ) Y ) ' eup n|7 2 d S 2 /*1 2 /3p31 2 7 DA187 b G/H = U (1)U 2

2

1

v hφi = √ eiα , α ∈ [0, 2π) . 2 #` ] ^  m ] [ n \f ^ g^ ` = 0 f ' φvac = hφi Avac µ √ ] ] [ φ = v/ 2 + h ' rd h f ]

; 1 :9' e ' 1 ) e 565 7[-g' ]=n R `I: - :) Y*f h 7 ) ; e61 2 ' -3/ 5 /{+.- e[-37 5 -37 “— uF — UW/{/ 5 1Xe[- (+ ) 59K / e K]‚ˆ 1'u+ ) e6-_/ )*: e9p /B- : pqpqR47 ) 8) V7[- h 1(7 59K ;6) +[1 nd1(7 / : -\+ 5 Y3e ) / : 1 Imφ = 0 `.'`F 1 F ' 9 5 ' a F Š 3 ' 6 5 h y K 6 ; ) ) 2 1 62 / 1v+ *79fiR /*- e[- 7[- / RIp ? 1(7 59K R h -32 ' dZ+=- ^ e[-07 n 5 -37[- 5 R4181 ' / 5 

1 1 1 L2 = ∂ µ h∂µ h − Fµν F µν − λv 2 h2 + e2 v 2 Aµ Aµ . 2 4 2 ` _^ f rd ] b √ MV = ev Mφ = 2λv ,

ˆ - :65 R ) Y3e[-f ) R /^e61 (pm+ ' - ' 1‘2 ;9) 7 ' -3767 ) 8) 7[-0e9p,DA187 59K 2 5 R4R41 ' e 565 /*281 ;6) + K 2 ' -37 ) / K9' 2 K RT-32(2 5 /*79fiR 5 z3/R ) 1>+ 5d5 R41 ' 2 K /31 ? 1(2 ' /*187679fi1^/*1 :9'*) e67 ) 1 5 2 : -\+ K e97 ) 1 ;6) + K 2JRT-32828-3R 5 5 ’  "!$#&%'   .%$/9*%6/,/A *9#:%# ( A .& C /A &-  # ” -- ( 2  EF, 9/*/2& ( *,-#.%/9 *H%6,   ”  C , K 



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"!   # $ 2 )3)3' % /*1^8') 2 ' K9/*h 1(767 %)6F ) ; 5A' ; ' : :=] 1'62 ' / 59' 1v+rd*76- K{`B' - :Z5 R47 5 RT- K e ' 1 ( 2 v 1 = + ( 1 6 7 7 ' e 8 1 R ( 1 [ e q p [ e % T  v :,) ;6) ' :,) ; : )_^()C;6) K φ ; e 5 7 5 RT-gb ' e[-f876fi1AfV7[- h 1(7 59K /de[-fv+ 56h 79f&% '*)*h6: -\% ; e R ) 2 ' e679-318767 2 ' f /B- zŽ2vR +[p +=h 1 -'62(7[7 - KM -fm+ - α(x) `G)*; ] n 767[- K / : -n ].) ' '3)*h6: 1 MW) e6RT-`+rd*7 )]: - : 6 e 1 v 1 + ` ; ] ) )6FT% R ) R4187 'LM -f ) / )_^(){; 18e61u% )] - ; e 5 761 :,)[ arctg[Reφ/Imφ] e6-32 e91 1>+=187[-Le[-3/*7 R418e67 '*) e ) ' ' 18R ; 18e[- ' pqe61 T % 5#^^ 2 ) /32 :,) 1 ;6) +=1 ; e 59) Y*e61 ' -31 ' 761879pm+=1(/ ) 1 28e61 ] 761(1 `Ž5 f87[- h 187 59K MW56: 2 5 eucp b ' 2 KGF y 7 5C)*: - f8fi/Bg- b ' 2 {K ` / )*) Y ? 1 ^8) / ) e K{` e[-f876fiR 5 /ƒe[-fv+ 56h 76f&% '3)*h6: -\% ` α(x) ) ] 7'*) [- ;6:,))ƒ;6) 2] :,) + ?Ad : 5 p / ) ;9) +=79p ] b & 5 761(e ^8; 5 b :,) 7 MW:,5#) ^ pqMWe[5#-^ h 565 ] 59- Knv' /' : +.- ]X59' 5 K ^ e[- ' ].5 ' 1(7 ' ) 79fC' )h ] +=187)[ ` 2 2v+=1 p b R %,+.- n 187 18R +=- fVR4f _) ^ 7'o;6) pqe[) g- h ' 2 e918R K9' 2 K 2 - ) ddY n +=1()1 n 7 e^8) ) ] 5 7 ) ) ' F ' %` :,t) ) ] 761 h :,7 )L) R ' 59:q'3g) _)^8)^ 1 M ) - fVf 'α(x) R ; )3p 5 ) '*+[5 7 ; 2 d_5}bx) 2(e[) -3] /*7 ;6)Ld32 )*: / /*2 ) R '35Y3Mj R41 5 ;6) e + K f φ7 R - ) nd1 '7[- 5 fVR41(7 - 59' d*2 KaRF y nd] 1 7 ) fV76- 76h 1 7 ) 2 e ' d f ;6) ' + Kwz ' e618e Y3p31 ' Yt% + 5 eD 1 d ` h9'3) e9Y*fpn|; 7 e 5‰2 ;6) +=7 -) fVR ) tY % ) ] 1@/ ].) r+ dr+ bcY g) ^8) fm-3R : 7up '3_) ^8)ƒ:,) 7 ' pqe[- C M - fm- ;6) + K φ 5 fVR4187 K +.-32 dEY*fw7[-t/*1v+ 56h65 79p ` : e[- ' 79p b 2π ` I •j— uF — ' U dα dθ = 2πN , ∆α = dθ ^w] 5 ' ^8) W M 5 6 5 h , : ) ; ) ' ' F†Št) MN5 ^ 595 ;6) 1 +6p θh95 P+ 5 - f 7[-RIf8p/B-3-\7 + 5 d*1C762 Cf ' '‰e9pqp 7 ` -+‰)*/ ;65 28-3f 7676fC18'2 R4R 1u%,-37 e 5 fV2 R e[5-3%792 ) Y*/3e[1 -\f ) /B7-07 59KLpqK e[/0g- + h K 1 ' 2 2K N761 6=h 18R 0 5 76fiR `T: - : R4s1 %B-37 5 f8R ) R Št5 Y*Y0+.- F4Št) 28R 56h 182 : - K 2 ' e9pq7[-/*7 ) _/ dyR ) nd1 ' Y*f ' d ; e )35 +6r+ b [ 2].' 56e : 5 e ) K /B-37[- K e 5' 2 Fc— qF` _— `J) ]' 7[- :,:6)_5w' 51 ; )18e d ; + ) 2 :,' ) 2 ' :ud}'*e) 5 2vpu7 : ] -P &v'*) ) ; + ) 2 :,)' 2 ' d `i; 1(e ;; 187)[[ 2 eupu791 -oF 2 &(e9'31v5+ '*)*f h Y*e[- nZ5 - b /*1 5 e[-{ŠW/ 5 /3pqR41(e67 ) R/*5 79p e61(76' 7651(R ' e K '2 e[-3p`76+ 2 ' e6/*7[1 - (Reφ, Rp 79187 1(R!R4s1 %B-37 f8R Y*Y0+.-]/*7 g/ d +9r+ bc2 e eup*1 2 Imφ) e) 5 2 F4n — uFi ) F 7 ; e 5 / ) ].59'r:y) Y*e[-\f ) /B-07 5 b ;6) e K ].: - ) ] 7 _) ^8) p h -32 '3: -r2 ' e9pq76f ] + 5 76f l / Y3j  R4;1 l ) ' ;6) K ^ '*5 ' `6hu'*) ' ]=) ' 5 ) : ' {5 ` 5+ Y ) 1 Y*18e62 1(:,e6) fi76/*1 h7 76fi2 R d 5 P + )3'3φ: e6-3f e['-3fi7 R 5 e9p32 1 ' e9pq7[-3R 2 5 e92 pq76:,f) 7 h[-3+ R n|5{`76f
p % Y*)f]=K9?EdN5+ R Y 5 fmfV- -3R ^() 79e p5 f fi) 7 R '
mF  ; 1 e918e6fi/*7 ' ) 2 5L' d :q);9) '+ K φ ^ -3`„e[;6-3)X7 '*:,5 )3e9'*p3) 1 '€) ' - : nd].=1 nvfm' -379KLpm+=5 1(7 ' 5 ^1 φ5 7[) -C761 5:q)0'*) e •) — '{uF —+ ' 5 7 F56HJ{5 '3`„)]; e) ) % \) [[ ]=Ku? h -31 ' 1`4'yh9'*/*) 7up / ]=e ) r+ d &(7 '3) pq'_e[-+ 5 C7 565};6) +[1 e φ RI;6po) [<;/31e61 n|762 1(RIp 7 7[1 \- % e t) ]=e 59' /B2 -0K 7 /{1 76/ 1876-3e9p,DA1(U 767 ) ' M fV\- 7[fV- 1 ` ` - MWfV5#7[^ - h65u' ` 59+ K )*: -\) r+ d*7 ) K /*Y0' + 5 )f 5€&('3) '€+ ; 5 7 )356' 5 )  28' 1(e ] & h618/ 5 ^(7656f!5{` 2 ' e9pq76f@F U@e[-02828R )0' e61(767[- K hφi = 0 " € 5 , : ) 7[-3R g' {: 7 ` pue6g- h ) 2 ) %u] e[-37 5 1 #5 Y ^^ r+ d0DE•— p qFb — m + ) 7 2 d ' 76Kr18e ) ] ) E ∼ v '*)*;6) _) ^859h :65 - /C-0Y*' 1v+=1(/ '€F R y 51v+ ;  ] ' 28- K ' ) U ) Y*e[- f(p ' b 5 2 5 7 R41(e6;676) fi1 :,+ ) MW#5 18^ 2 1 [ ] M :9' 1565 ` 1 ]=f) P' 2 ) e9pqK 76f ?A5 7 ; e61 ) 2 ' -3/\+ ' b 2 ) Y '76:,1 ) e h / ) -\+r'3d*5 76fi1 ) 5 +=1(/*fi51 5 7 pqe[: - [ h K+ e67 _)p ^()A/0;6+=)31 ' 18/ #7 h e 5 -\b +.- •j—1EuF7[!— -  U ` e 7 )E2 5 e[R4-0176b 2 ?E/*518761 7 ] + '{K Y*φ182 5 A761 797 1 ' 2e 5 / 5 p*-\2>+ + d*79/ p bbfmR -3/ 7 5 2 RI5 R pqRT) 2 -|' d2 )0- ' p ].^ + ) ) /*&f ) % 5 ; 1(e6`Ž18R4;6) 1(7676&f % KF:,%~) h MW-32 #5 ' ^ 7 ) R 2>59+[K p h -31ƒY*182 5 :q) 5976'o1 h )37 ' ) ' ; 5e K5 R ) '‹' 2 ' e9' pq765 f ` 5 76- ; e[-3/\+=187675 ) ' [ /;9'3)3r+'*56d :95 2 z ; ) +='1(/B- ' 7 ) pqe[- h :q) 7681h fm) -3/ '35 2 ; ) z:,) '*5R41(1 761  e ] / -\r+ 5 d376] fi1]-32 R y) f 182d -\+=1(1rR4f ` )*; p32 : -31(R 7[; - e )35 e fV/ 2 ) r+ e[d*-37979p 2 b /318;676) 7 2 '*' )mK Y*76187u2 cp b 76M 1 - 7 f8p‹2 W/ fV7[/ - h 1(+ 7 562 5 /B2 - : p,(x, _) ^() ] ^()r;6) K q p 4 R 7 ( 2 6 e 1 6 7 1 + φ ;6) 2 :,) r+ d : p]1 n /*281 ^w] - R ) n|7 )Z) Y*79pm+ 5u' d ; 18e61 )*; e61 ] 1v+[187 5 1(R ;6) +=1 '=U ` “— uF
—  U v N φ→ √ | , A → ∂ θ, e 2 ^w] ;6) K ^8) 5 5 ] 56h :,) 5 ' :,)*) ].5 ' `q) 5 '*5 ) ) ].) ) 2 51 θz ` P - N+ P e6h9761vCf + ' ) 1 p h65 W+ 2>+ h )#`=+ )*; 7 e91 e ] 1>+ 1(K 2 b ? 'X181 2 h62 5 2v18+ R4) f 7[- : e9e p '*)*7[: -  ) Y ) e e )318'*7 ) /,U†e / )/B-3/*76797 p ' 'Qe61(/ 76761(+ R d  JB ” A -G-,#  ” 0    -G#5:0-   ”  C ,<; #5=%$- /,- #5 ” 6 ( W’ &#:%#5,,A<; !$ ( %$6/, ”  C !$, -2&9* ( - ” 64' K c

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 UT-32 5 R ;9` '*)0'*;656): - ;6K ) + ' K K A /3fiY 5 e[-31 ' 2 KX' - :95 R ) Y*e[-f ) R `qh9'*) -32 5 R ;9'*)0'*56h 1 [ 2 e x + y → ∞ /*f +=7 1 2 1

µ

2

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Dµ φ → 0 ,

Fµν → 0 , `

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 U ` “— uFigp U v N x φ = √ (1 − f (ρ)) | , A =− (1 − a(ρ)) , i, j = 1, 2 , eρ ρ 2 ^w] 1) ρ =5 p(x;6) ) K + '(x ) P ) e[- :,].) 5 -`MWr+ 5#d*7[^ - K€:,)*56) 5~e ].].5) 7[- ' -|76- ; + )t) ]=2 59:,' ) 2 '*5 (x,5 y) F[;9st'3)3) '*2 56:,:9) 5r+ d •: —jp uF ; — e 5 ` U '*Y e) + ] d3+ D K€M % pqρ7 : h + 5 ' 2 f (ρ)e9pq76` 7 a(ρ)' Y3p 7 ] 18R 5pqR4e[- 1 h' d + n|76f~/*&f % d 7[-A-32 R 

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` ` φ = vϕ x = (ev)−1 ξ Ai = vAi Dj ϕ = (∂j − iAj ) ϕ , Fij = ∂i Aj − ∂j Ai .

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F e ^8595d#) 5 `0R h9'*; )pm+rd*'28- :,•) —juF}$59' U 5 76/B56- 5 [ e; -07 76? f 5 7 2 1vr+ :,d3) 7 MW+ #5 ^ e6187 h61856/*5 &f %d: Y3:p*2 ]./W5 / g) ^8)Wr+ dt) 2 e9:9pq'76f  ].28)7 ; / - 'Z592K ' p*-gh 1(e618R41 187 1c/*2(1 ' 7 pue6g- h - “1 7 Y3jk1 - U / r+ dt7[- e[-3/0+[187 2 e9pq76f 761 MN% 5 f ] 56h 7 )6F -\+ d*761 'u5 DA18'3R ) 7[?A-05 2Y3p ] p ' ' 5 7 ' FT1(e6s 182 5) /B5 - ' d h RT-0592(5}D ' ' -0Y*:9f 5 e[-3282 '*)m'K 7 5 ) ' l'  ) (ev)& `i^859'`Fc5 1 [ F _) ^() ) R45 7 ; Y +r•d0— D uFi1 + ) 76)dfx;6) 2 e9) pq7659f ' e f(p 187 - %RT-32vD -3Y / 187 f e 761(e R pmr+ d32VUzR n|7 + n d|e[-0/*76fiR •j— uF}' U T = µ · diag (1, 0, 0, −1) δ(x)δ(y) , ^w] 1)3' µ K /0+ K 1 ' 2 K 7[- ; 'mK nA187 5' 1(R ) 5 2 5 '*)*h 7 ) 2 ' )3d_; b ]=] )Lh9K 5 2v' +=K1(767 _) ^() R47 ) 5 n 59' 1v“+— uFK ik fV-3F./ s 5 2 Ku5 ? 1 ^8c5 ) [ Y*51f8e[•-j— fVuFR4}'1(e676f&% -3e[-0R4']1 ; e 5 / N5 λ/e5 U e6:,1 ) 1v+ h 1 )X2 '*) /3fi:,) e[-nd' 1(7 18R `T5 ] K{U 5 e h Y0+ 59K nd1(7 1 ) 2(7 ) /*76&f % ^ e6-3U@/ 597['\- g-fVhfi56/B) g- 76b 76&f % e 5dY\:q+ ) 28R nd) 1(+ 7 g) ^81856RŒh 1(Y*2 18:62 5 % 76&81 MWM 7 1 :9'3) 7 / ` 28'Lg/ fV2 -376e976pq&f 76%Af 2 ) 2 ' e9+ pq7[-3f8R p {5 `318&(7 '*\) [ ^8) ; 5 e Y0_+ 5 Ind 1(7 59K7[-3 R}Y3p ] 1 '  / ;9) +=76 1 ].) 2 g'  - '* )3h 7 )6F !A   t   # %  $#   #     \    #s  &% # .

  c       •j— uF}'  & a         t  #       $   j $ t   t    #   U       _   "~  % #$     µν

µν

µ

∗

ν

µλ

νρ

λρ

µν

−1

µν

2

  c T          $    $        r             \   #u  t    t     ?  \   # t  #s &% #> % ' 187 f ) e61 & 7618e ^(565c[<5 R ; pm+rd*28-_2 ' eupu79f “—uFi' Uƒ+ 5 DFd ] /31 :,) R ;6) 76187 ' f )3' + 56h 79f )0'

9;7 pm+] K z ; '+ ;6)3') 7 ) 2 ' 59d h6& 5 761(e ` ^8565)LT)*: 5 ] -3/0+[' 187 K 5 1ƒ/ ].) )_+^(d|) 2 ' e9pq: 76f FJTHJ'*)3FG'H M 7918e :9^(' 59Ko: 5 ] -3/0+=1('7 5 1ƒ2 )'*/)#[ ` h9'3- ) -gR4b f 5 R41(18/*R 1v+ ] 1>+ ) 761 2Ne67 1>+ K9'35 -/ f85 fi2 /B' -g2 b :95 2R ) e[Y3-\jkfV1 7 :9'3) R f8` 7[] - + Kr- 5 282>+=1 ]=) /B-3- 7 5uKTp ^ e[-\-3fV/ fi59/B' -3-1 h 56) 7[76- 79f&% 28/ ) '62 ' / :,)3) '3) e )_)A^()] 761>+r' dgf `,K}h9'3)_)^ e[-3] 7 59h65C5 /B)3-' ' d*' 2 K + 5 DedL' 7#d_56b : '*) ; 7 ) )/*2 ':65 R ; ' e 5 Y\[ + 5 nd1(7 5 5 18R F 5 ˆ 1(: R 76:L1@; R418) 761(:q1c) R `5yn|7;6)*&('3pu) RT- d ) )€/ 5 -`+;6) ) 2 e9' pq76f~R41 e 5 - ] e K 2 e6' -3'*76562 h /B- :,/*)_e6^(18) R41(: 7 KY0+ f)_^()+ '*) 2 ) ' ;6)3' 5 RIpyR nC`a)*7 ; 2] +n dgf )g/*^8- ) d]) pqe[-3/*761(7 1 + ' 562 : - r5 182 :,) :,)_2 ^8) -\+ )*e6)07 ' )[ 7#d_b 5 7 /B- 18#7 h -\+.- Φ e61 1v+ 767 / /*761vDA761'R41 e 1 7 /32 2 7 DA1(7 1(R HJ'*) pqe[-3/*79187 5 1 5 R41(1 ' / 5 ] 28R F EF —*g— U =`ƒη F —3+—tk U•2ΦU .   “— uFi U 1 = 4πG (T + T + T + T ) . ∆Φ = 8πG T − η T 2 ‡ + KA5 2 '*)3h 7 56: - •— qF}' Uk#7 dgb '*) 7 ) / ;6)3' 18#7 h 5 -\+ e[-3/3187d79pmr+ b `3;6) 2 :,) r+ d : pNpqe[-3/*79187 5 1 “— uFi U ; e 5 7 5 RT-31 ' / 5 ]{) ] 7 ) e ) ] 7 _) ^() pqe[-3/376187 5uK - ; +=-32V- z •— uFi+ U ∆Φ = 4πG (T + T + T + T ) = 0 . 00

33

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N X

δ(x − xi )δ(y − yi ) .

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 !\   4!:%+*
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9 R ∼ v X !  %:"% 6!".< %H8 #

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−2

nl (R, t) ∝ a(t)−3 ∼ t−3/2 .

8 /E?A")B!5 9

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?  nI ?@ f %f?
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6 '  5 5 ? nl (R, t) ∼

1 (Rt)3/2

∼

 µ 3/2 , Et .&#&,#&* ƒ -- ( 

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!%' B/.5  #  *,-2  * #&9 C ? /4' 2"*  0, C   #I C ,  */2  #&$* R ∼ 1  0  -2  K  =%#:! C #;3G ( #  ' EF-2&,- -  #5,,*/0-   C , -  #&# '1!>/0 ( 2&*H* : ( #  #5,,A :#&%#&,,/A K 

16

2










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ρl (t) ∼

Emax Emin

!


  

J

 =lk +L J

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Z dnl (R, t)  µ 3/2 Emax dE EdE = . 3/2 dE t Emin E ] ` 8 n


` !:!
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I

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 µ 3/2 t

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µ

H

dR · f (R/lH ) . lH


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! , // %0. / #%  V5?

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H(t) = 1/(2t)



µ dρl (R, t) + 3H(t)ρl (R, t) = 4 f (R/lH ) , dt lH

lH = 2t

9\ V /  >O" %i15 / / G7 ρl (R, t) =

/ VCH? /  ( 5   / ( 

R/t → 0

µ 1 16 (Rt)3/2

Z

∞ R/t

p ξf (ξ/2)dξ ,

.?N%` xC%- / O"5  , ; Y  , % / %! O / "  ,* 2  . 9  69

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Rmin (t) ∼ Cgw

y

µ t, MP2 l

Emin (t) ∼ Cgw


5%:!'rd

 ]  X= !

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 5 8 5%? √ µMP l 1 ρl (t) ' p , Cgw t2

µ2 t. MP2 l

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46587


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HCGOG 

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P&IJHCGOR

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 =i m _L 1 λ L = ∂ φ∂ φ − φ −v . J 2 4 ^   = i m _L ^ (    nI %^  8 J    !L' E    !

   "   ! '  rd 0 cf? 

 f % 3" 5 


 Q :%   /   f 8 r d!5:!X" !5

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d 5^%8 $ 5 j n
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D@ %g 8 h !  :
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2

A I ,  _5)<(  5%?N / ") ( ?+ VF! , %5G[! %%)% ,*%/ !" ?N%# // + ( ! / -]9F! *Q, ' / &V
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W ! , 15 C / %  1 W / %%)%C" >5F%-.15

M

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/   "!'` )*?\"' 8 5 c"'!5:!^ 8 5 " g' 9  (" 

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? 8 % (




 =im L dφ −λ φ −v φ=0, J dz "` ^ % 
5
C?^8 8 "'  (  ? φ(z( ] → ±∞) = ±v JZ 

 L = φ(z → ±∞) = ∓v J  !   L' ! 6

D@ % h < ? bO! z  =i m  L 2 z  { z J φ(z) = v th , ∆ = ∆ λv  =i mkL z  =!  

{ z φ(z) = −v th ∆ . J 2

2

2

2

2

2

&*2 C :%#+*2& ( .&F#5#  "!>* ,#F& ( $*!>*/#; ( /9/@8# .& ( / '<;  € #  #&*  C ,# :#:!$,#&# /%' ( */0 C1C ,   25,*G#&,-#  v ;"!>,*01 =2 φ!$#":6 B !$%'I,$*/ ,#+ C @8#" ( #&,, K 



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c

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sc Tµν = ∂µ φ∂ν φ − Lηµν ,

DW Tµν =

λv 4 diag (1, −1, −1, 0) . z 2 ch4 4∆

J

 =im_mL

( ] ? ` 5!&! f >#

 ^ [ ? 8 rd"


Ef ( " !
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η≡

Z

√ 2λ 3 2 DW T00 dz = v 3



Z

DW

T11 dz =

Z

DW T22 dz = −η


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t  +m  $$i* + " + .•& , l #i!) 3d y # m4{zh - | o l y # 4Œz— |=   , #„$ .  + #$ o  Y $ `b $#u ,„+3i , &th& w $#u -&$ & $ # $ &o -&" )un Φ 4Œzhf| = 0.35 ± 0.07 # y Φ 4Œz — | Φ # = 1.0 ± 0.2 y Φ +&  t 1 _ -& &) *  r+ & ,l ## y # 4 ™ | $'& d i #  )ƒ"  ce &  ##  y # 4 f| y d h)0# 

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x | iooŒm L‡wmxj} swot3t }‚o t x Lo t3r3o / m}  swu Yo) s 3ot xwxj}q3o } } |+t } x m swp x o } m x r  †    x x } w } x V swr `t /  s qLo {| o)} q m p x 8  |ν2 i x † x }wx V > 0 † }‚ }‚ }    B q o m  ~h m } m ~3|νt3er3io x %  | w} t3o… Yx { }jx  s | |ν32 ir †  m m ~Lt3rL~ ‡ { +o z£q m mz q3rLs šs `r ‡ r 3om  z  x w } x x  rŒt  ‚p } swoq Lt x p3q3rL‡ |+r ”ot3r3r  t3o … q3r3t swm o x }wx t `r m ~ sŽm  m ~L t3} r3r  } |ν2 (L)i xs m r |+t t† x ‡j{ ` o sv} m m x ~3tLr3r   x }  q x ož}wm x swp o mžm m ~3 tLr3o x z }s a{E{EzP x o  x 3x r3m o.z} m m x s }wo!x m m ~3|νt32rL(L j x j } x x x   } x w } x )i |ν i o p  z${ p3q3r | wt3o… ioz¡q m pLq m q t3o2t3r3r sYs L{<{EzP† o }jt x t3r˜}ws x  t3x o0pLx o q3o x }w`x r  3r3x z }jx ‡wq  z A|A~˜swo q ~ t m  r›qLx o }rLm  q 3r3r  q x m p +o‡ q x  } px } |3{ o z }  t  eo z|Aocx |+o x q tLt x }jx t3o … q3r3t x p }4~L† swr3 s Yo x m ~hsŽ  } x x x } x x x   x  x  

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c l t o c ` c i -w o k x o P = |hν |ν i| = sin θ . y † z | / t zPoq3o t3rLoŒp xj}wx   m x |+t3o kLt3ud |Ao  } q x t3t3ud¤tLo … } q3r3t x p x  t us  o } o k }wx {2c x | mzPoi ` rw s tLrL~¤m |+tLo k3t3ud t3o… qLr3t |”m m s |A~3o k q  x i!k3o qL3x t‘ z o k }jx t  ‡ | } u i  o x zPuy ν… w “ν˜   }yx }w}‚x  c xo k }w} x θr t ∼zPo 32q3ot3t3−u37…žp x o$}w}‚x   ‡ x }jq3x tLuPd ν t 




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q3r3m tg 2!θp3q3o3= +i m 0.3s − |Ao 0.6 t3u † t | c rk3t3uo i t3t3uoQp z m oq3t3uzvt3o… q3rLt r ( } (  † † }‚   x  x. } x “ } x †

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 t } r3 z ) u0t3x o t x s ‡wt sšqa|+{2o gš ra} |’q x  t3tLr ud o m  lLt ra} |3r3|At3~o … l3} r3q3… r3t x3i+† t  }w 0 swm‘M} ” g o0o p x }w x Yx x m }‚x  |  m _jk  m } ”_ p3q x m }} q  tLm } s Žx m l3ra|L|Ax ~l3r x t3t3udžp }  q  zPx o } q x s o iA|3~›oa xxj}wx q  x …x ‡  <m p3oq3r3zPot t3o›pLq rLs q3ok i`qL{2cƒi`qa{ c{!xP }wx |+rL‡ x ∆m = (0.2 − 1)  l o x  }‚ xj} |+x rL‡  } q m |_wL{  l sin 2θ = (0.3 − 3) · 10 o m {wm m Ll ra|3|A~3l r3… ~Lx s |+∆m o tLt3u¡ d E'mp37o qLr3zPo o sin t z 2θ &) ' p 4 · 10  † txk x o|_jt  p  p3q3oga`i oswq3ozPo t3t }jx `i o| X t3}‚o ‚} pLq3o3ν¯`i o|↔ ‘ t ν¯ x t r d m t oOsjto› _  o 3r3x o |+rLx ‡ x m oq _} ‘ t3ux o o t3o0}wx zPo t3 oo m x r x zPxo r  _ k o m|ArŒqLo {x |_  u } &)  p w i swx o qi+~ m t~ swus x `i u o z $  x ‡j{`i o † t k _ k o }jzx }Ž m m xj} swu¤ d } m } x m o}jx ~L}jt3x r3… s‰mo  t }jx q3oh}‚ tL} o … } q3r3t x p x x 3} q  …3t3o…£} zPo+o }jqLxo }3k jo x uq3o Y }¢o x…3t m  swr  } o|o j_ t }wx sq  t x m ~2ž d 3s `i q sYz m m ∆m o ∆m r ∆m swr2`i t  † r3oq } qad rL} ~ l xo ” m rE|} w_ t ~‰ g ošt sjq3x o z$} ~ om|+rŸz  }jm mx swu ‰ d m x m }wx ~3t3rL… }wx |j_  x)} q3r Co }wx m{EzPz    † w } x o  x q t m o …‰3s +i q sŒz m m s ~ uX d mŽm swo m } sj{ uGv rLz t t   x z o i x |'šg t  ‡wu } Œ_ q  swt  tL{| t x Ÿ }m t ‡ |  i  o }wz x …Žr3 oq o qad t r3o } …  c xwi xj} m r} ky u q†‘d t3o  … q3u r3t  m x t2i q  t …z }j`i x o|Ax r x 3k o } s‘ _ q t3o o”s xtLt o z … x ˜g q3r3t t x3† i } zP|˜g o } t r3z ‡wo uk }wx _ s xj} & |+rLk %r3o   xj}ž } o t3oB} {2k m x s s }‚ 0  } |+ox  q x m| ‡wuo  … r3s (  tLrL~ _ sg d s } rLz }w+ix o …3m sw} rL~2d x 5Pm|+x r m x k‘ m } zPoY ` † zPrt3 o … } q3x r3t  } 3k o s ‘ q } o x oq3 z |+r t  s t x s›q } t3t3o…¢† lnmo|+ot3t t … } m +'† ‡wuo }4Lk † t3u t z ” g o  q3zPtr3x q  s x t3rLonm  qL{E {Eq r”t  x q or3}wtLx’{ x t3x r x q }  x us Š _ m{ o m } swot3t x o!s } |+ra~3t3}wx r3oQt} xp3r3rt q3o|+ r3}  s v c } r |A{23k x ot3rL~ “ ot3x r3o”q3o x {| _ } shEm} p3o qLr3zPo  t  } &)   z t`i xjo} |_ x-w ‡j‹W~Lm Ct3y m oq3z |+r }j{ x x r3x zPm t ~vm o q3rE|w_ † t3uz€t3o… qLr3t t 3q3u }‚ }jx r t ‡ | `i ot3r3}‚…Œ m qL{E {Eqvx r t3r u } q pL r3rhq3t o|+uGr3v  s c } r |A{23k o tLrL~ } xƒt  }jx t  u } }  w tx } 5Px m|+rŽq3o {|_ }jx u£x  x <m p3t oq3r3zPot † &)  t3onp i swoq+i ~ m ~ t3o ‡d `i r3z m r sws `i r _ 3k o s‘ q o!z t mm }‚s  } o!m m ~3t3r3ogsym }‚o  q˜t3o… q3rLx t } tLo!‡ {h+i } o o c}wx |A~ x p3rLm x t3ra~Žx q3} o {4|'_ x } s swmo} › d <}wm x p3oq3r3 zPo x t }wx x sšp x x }jr3x m m|+o3i x s  t3r }wu x t3o … } }q3r3t3tLxud x m 3l rE|3}4|A† ~Ll r3… o x 3q x zPo &) t o swp }‚x‚|+}wxtLo i x m }‚ }wx 3k t x£} q‘‘ d z  mm x swu“ d m x m }jx ~3t3r3… t3o … } q3r3t x |ν i o i = 1, 2, 3 † m xwxj} swo } m } sj{ uGv” ~ z q3rl m zPoY ` r3s  tLrL~¤t3o … } q3r3t x t x mr } t t s  t3r3o›z  } q3r3l u q x t } o x q3s x‚œ ) 3r œ 3 × 3  }  ) & | † 5 y

 c s m  ‘ {i x ‡wt x p3q3o `i m }‚ sjr } ž_ sŽm|+o3+i { unv o z swr`i ox    œ   }‚ y +

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    iδ /2   1 c21 s12 0 1 0 0 c13 0 s13 iδ 0 0  iδ2 /2 0 1 0  −s c12 0  0 0 U = 0 c23 s23    −iδ 0 −s23 c23 0 c13 0 0 1 −s13 0 0 1 

c i`o s = cos θ o c = cos θ o i, j = 1, 2, 3 r θ ∈ [0, π/2] o p3q3rk‘ zXp x m|+o3i`tLr3…šzPt x gšr } o|_ y c“mz t mu |+o δi+{ o δ |ŠrLr3… zPq o o } i`m ozP| u|ˆ m| }jr3x m |E|_w x k3oiAt3|3r3~‰ozXz   … xr q d  t x swm   x |+…‰rk3z o m m m s x s x p …Xq z  zP} o q3rq l3u s.s!t3z o… } qLq3r3rtl3xo m y zPoi `† r3s  t3rLuG~¢v t3o … } q3tr3t x)}‚ t  x oƒg”o uo   ¢r‰s)z }  } q3“crl3tošm x zPo`ir3s } t3xrL~‰ 3s   q3 } x s x y mz †  q } t i`o| — | † n‡ <m p3oq3r3zPo t }‚ |_wt3uoq3o t {|_ }‚ } u x p  q  zPo } q  d x m l3rE|3|A~lLr3… o p3qLo3i`m }‚ s |+ot3t3uoYt  † q3r3m  r 2 m swo m sj{ u }‰t t  k3o t3ra~3z„z x i+{|+o…Ÿ|+ozPo t }jx s z  } q3rl3u m zPo`Yr3s  tLrL~ ’ † rLt † oq3s |+† u o i xwm x } swo} q3t} m 1 y 3σw }  x }jx x } r|3x ij

ij

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 0.79 − 0.88 0.47 − 0.61 < 0.18 |Uij | = 0.19 − 0.52 0.42 − 0.73 0.58 − 0.82 0.20 − 0.53 0.44 − 0.74 0.56 − 0.81

ra|+r A|A~’{ |   s m zPo ir3s t3rL~ x  θij

x

0.28 ≤ tan2 θ12 ≤ 0.60 0.5 ≤ tan2 θ23 ≤ 2.1 tan2 θ13 ≤ 0.043

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 }  m{ o m swo tLt |+rk o o ‘ t | crk3t …Œz q3rl3u‘m zPo`Yr3s tLrL~Œ3s q3 s  z } q3rl3u x v ‡wr3‡j} ‡ xž‡ xj} ~`ir  } m xjs }š  fx }| zPox gji+{€ } |+o zPot zPr„  q …“r3 zPoox m ~  zP} o t ~ y œ h   y † o }‚ xj}jx x } t }  r3oq qad rLx‚~ œ Šx   ‡wq † r3z ” g oisjt3r3z  t3rLoYt š}jxo k }wx mc{ k‘ }wx z x cq  t3rk3o t3rL…vt  {2c x |vm zPo`Yr3s  tLrL~ t  } qL{` }‚ r3og o  u s.m o q3ot3o … q3r3t gšr2i m ~”mra|_wt p i s |Ao t3t3uzPr θ }` wx r3s tLrL~ } xrcx qLo {|_ u!} s p3q3r3xsjo3i!x ‘t3 t3udt q3r3m † q zPw r3 z |+ounzPv o t s.“pz “ q3r} 3l u¢m zPoY  r x † 2 x x pLq3o3`i o|A~ }ju!x } m ~  }‚}  ”g ocq t t x m } rŒ 3s  `iUq  }wx t s z t  mm y}‚3σ }jw x r3t o } o q3s  |+u i x m }wx sw o q3t xw † Ž † m r|x } l  l 7.3 · 10 ≤ ∆m ≡ ∆m ≤ 9.3 · 10 f| † l  l y ≤ ∆m ≡ ∆m ≤ 3.6 · 10 1.6 · 10   3r3o t t  3k ot3rL~hz x šg t x p x |A{23k r } _0Ai |3~7`i s {)d q -t |+rLk t3u)d r3o q  qfad r3…vz  m mct3o … } q3r3t x y m z † q3r3m |x † †

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¯ µ ∂µ ψ − ψM ¯ ψ, L = iψγ

o

)

(

js a|1 # o s m o ‡~•z m m{ yo q3zPr t r„s r3z o…3m swr3o¡mŸ‡ t3tLuzPr„p |A~LzPr  }  x   x '  } x x t p3q3r3zPoq sX|+o q rLt zPr3
x }  } `  }‚ 4 } † n x x x  p  'm #3r 0šswt3o t3r3zPr)r yr3<m rLq s tLt3uzPr x  #3r u}‚ s } ~ # < p3r x o  z¤| q †t&%˜r t

87 +   s swr o † 0 ¯ }  2 }wx ψ ≡ ψ γ 2  <  $    (('Y†  M

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L = iψ † ∂0 ψ − H , (

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H = −iψ † γ 0 γ i ∂i ψ + ψ † γ 0 M ψ .

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4

sjr t # swum {Ep o s› #3o m swo  x 2 w } x pψ = iψ †  ra|10 } t3x r ‡ ‡ .t3t …’ q rLt o }  } zP l x,$xš |+r x #3rLx o ‡ x tLt3xwux Š p |+ o …} yoψ23q3 zPH r t3tLu$ oBp |A~ }jx ‡ | j x } j x } x  x x 2 x x m t o tLrL~3zPr p3qLrhq sjt3u’swq3ozPo t  x jx xj} x <  

+ Y ' † t3t

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x

m 3p q ~! x t3t3uzPr

 {ψ(x, t), ψ(x0 , t)} = ψ † (x, t), ψ † (x0 , t) = 0 ,  ψ(x, t), ψ † (x0 , t) = δ(x − x0 )

m|Ao t3oo q swo t3m s  3swrLs |+ot t  t t3r #Lo m  z${ m t o tLr  x   } x swswo m } rx p3q m q  t3m x swo tLtL{ x q3o . L{xjrv xj}  xt3< o@#3t3u…  m + |  r {ψ(x, t), pψ (x0 , t)} = iδ(x − x0 ) † ‡j‹>.zƒp3q m q t3m s zPu•p3qLr . z€h3s } t s x …)} zPo"  t3} r3




n

 

o o n † ˆ† , ψn = 0 , ψˆm , ψˆn = ψˆm n o † ˆ ˆ ψm , ψn = δmn ,

r3m3q3o r 3 rL~ zPra|10 Lt r t

}   $  }wx   +( 'Y†

-

)  <”/ 5 #  t 1| 0wt r3t o x  * mm x,z $x q3r3} z  xj}

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r  zPrE|10 3t r tL{ Lr p x' } $  }wx  } 

ˆ = ψˆ† hmn ψˆn . H m & s t … rhp3qLo m s |+ot3r3o |A~  … o q3r3r s swr ˆ  }  ‚ }  ‚}  x } x Tr −β H

o Š{Et3 3 r

q | o  qLLr † •m t3r3z yo q3zPr t3t3uz p3o q q z ˆ r›m p3q ~!%/. t3tLuz£o z${ ˆ† } x x '  x x

 w } x x ψ x ψ † t3r’{ s |+o s q ~& m t o t3rL~Lz +x } x } xwxj} x <



$

n

Q}wx %i- o t3 rLx ~ zPz${ { }‚ o3z r

o n o † ˆ† ˆ ˆ ˆ ψ, ψ = ψ , ψ = 0 , n o ˆ ψˆ† = 1 . ψ,

t3t3uoym t o t3ra~ |A~ yo q3zPr x |3~ p3q3xwo x } o|1x.< t3t m r‰ m'#3r x 0 }m q x 3t m s † m m  ~3tLr3 … r3zP j x oo  s ‡ x r3} m t3uŒsj}‚o ‚ } q ψ }  } x x w} x }    / }jx  #





ψˆ† |0i = 0 , ψˆ† |1i = |0i ,

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ˆ = |1i , ψ|0i ˆ =0. ψ|1i



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m q t3m s m m L~ t3r3…  xs }Œ …  pLo q3} ozPx o t3x t }w… x mt x x x ψ 2Lx



”p3q m  swt o.x m s } x x







q Lt m s Š{Et3 3 rL… t rL z

x  }c t3} Er |10wx p …3 m s } x  q … x } x jx }wx x w} x x

ψ·ψ =0.

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+

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o

r

Ψ(ψ) = α + βψ ,

-

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Z

Z



o q3o r3t

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dψ = 0 ,

dψ · ψ = 1 .

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x } x Ψ(ψ) † {E‡wo r 0wm ~hp3q ~3z Œ … p m t ws  … # sw u p |+ta~  m ~ m|+o {=ir3om t o t3rL~ x  } x x' }‚ x x 2 }wx x }  xwxj} x <

tQ}

Ψ(ψ) = ˆ ψΨ(ψ) = ψˆ† Ψ(ψ) = ˆ†

ˆ ψ ψΨ(ψ) =

Z

Z

˜ ψ˜† dψd

Z

˜ ψ˜† dψd

Z

˜ ψ˜† dψd ˜ ψ˜† dψd

& −ψ˜

†

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†

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†

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†

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(ψ−ψ˜) Ψ(ψ) ˜ , (

(ψ−ψ˜) ψΨ( ˜ ψ) ˜ , (

(ψ−ψ˜) ψ˜† Ψ(ψ) ˜ , (

(ψ−ψ˜) ψ˜† ψΨ( ˜ ψ) ˜ , (

+

+'Y†



+'Y† 0 +'Y†

+

!

ocswm ocp3oq3o zPot3t3uoYr r yo q3ot 3r |+u m #3r  m ~ t Lr  zPz${ r3qL{  '  }‚ }  } x } ψ 2 ψ˜ 2 ψ˜† 2 dψ˜ 2 dψ˜† Yr3zPrhzPo"% {žm ‡ …   x x † , op3o q 0 t3o } qL{  t x  pLr3m  } 0 Š{Et3 3r x t  1| 0jt3u…hr3t } o $ q  |  A| ~Œswo|Ar #33r t3u

r  



|A~ zPra|10 3t r t ws r ˆ = cψˆ† ψˆ † }jx   5 H p3r o$ z

& −βHˆ

<

l m p 1| 0 s s x x x  &

=

r3m@0 <

ˆ −β H

x

 Ψ (ψ)

Z Y n

k=1



 Ψ (ψ)

m {Ep3rLz  i % o  hrŒsŽ‡ 3t t z m|A{# o r x } }‚ 2  x x x  2 9

   ˆ ˆ Ψ = 1 − H∆τ1 . . . 1 − H∆τn · Ψ .

qLz${4| P z r  +( 'Y†

dψk dψk†

ˆ −β H



+

r

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1

1

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x

&

o

$-

ˆ −β H



Ψ (ψ) =

(β)

SE =

Z

β 0



ψ

Z

DψDψ

† ∂ψ

∂τ

&

(β)

† −SE

†

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S=

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&

o

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|?0wt … zPo t … xm |  'm 0swx uR~3tmt3→ r −iτ 0  2 iS  3 → r3z −Sq E † t3rt3uz {jm| swrL~3z p3q3r3s r s ~ r3o’m|Ao  o z x }‚ x } 2  $  x x }  } 5 L† z

<

dt iψ † ∂t ψ − H

ˆ −β H



Ψ (ψ) =

Z

dψi U (ψ, ψi )Ψ(ψi ) ,

Z

D 0 ψDψ †

U (ψ, ψi ) = −

& −S

(β) E

+'Y† 7! (

,

q3r! t # o # p t # |10wt $ z { r3t o qLr3q s 3t r3o!t3op3q r 
x

U (ψ, ψi ) = u0 + u1 ψ + u−1 ψi + u2 ψψi .

|A{#3r3z

Z

Z

dψi U (ψ, ψi ) · 1 = u−1 − u2 ψ , dψi U (ψ, ψi ) · ψi = u0 − u1 ψ .

) ”x p3o q  }wx Lq t x z£~  u
4



|+o

s o|10jt `x  } x 

|1i = u0 |0i − u1 |1i .

2

Tr

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(

dτ .

 zPo r3z # Š{atL 3r t |10wt3u…¢r3t o q |¤s

+ swa|1# } t 2 # |1}w0wx t3u…
?z zPx o t   pL} q3$ r #  .z (('Y† ! r } † s˜ +   x } ψi τ =0 ( ψi = ψ(τ = 0)  %”o q s t3r3ošp r † s  tLo'#3t3u…?z zPo t  x p |A{# o m ~ x r r3} m  τ t = β † ,o…3m swrL~ 2 o sjx a|+ r s x ψo …3m swψr3o `x / x  }  }  x' x,$x  } SE x

Z

+'Y† 7

Ψ(ψi ) ,

 zPu‘p A| {#3ra|Ar 2 x

&

Tr

ˆ −β H

&



ˆ −β H

= u−1 − u1 =



=

Z

Z

ψ(β)=−ψ(0)

dψi U (−ψi , ψi ) .

DψDψ †

& −S

(β) E

,

s

r m ~ x't  sw}  u   x

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 } x x  x x † ‚ }  2 x 'x    tLo q rL~ . m ~hr3t o q | z $ ` } } $  x 4E

B

>

4

& −βF

o s |+o {wm|

=

s q<~ } x

Z

†

DφDψ Dψ

& −S

(β) E

+'† 7,7

≡Z,

+

(

o sja|+r s o …3m swrLo o q3rLršt q3o 
(β)

-

SE

Q=

Z

¯ 0ψ . d3 xψγ

#&. t3otL{|+o s m q3o tLo pa| t m r ‡ s |+ot3r ¯ 0 m s r m ~v `  x  rLswt3} u… zPra|1x,0 $ x tLr  thm| $ x›x } oz xj} x } n =o ψγ ψ 'x  } } $ }wx  $ x,$ x (−µQ) 23}4† †

o

$-

r





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:

q m m z q3r3s o z z   }  x

sž

yo !

('Y† 7

Hef f = H − µQ ,

ar3zPr 3o m3r3…“p ot 3r | l zPo t3ot3r ˆo swa|Ar s xjo} …3m swrL ~ † `x  } µ



x

(

+ )

q3z +| r zPov p3q3r3s r    }jx x }

2

(β)

(β)

SE → S E − µ

s

Z

β

dτ 0

Z

¯ 0ψ d3 xψγ

+'Y† 7 (

 +

q3z${|+o 7,7 + lŸ zXm|A{# o |A~”swo|+r 3# r3tLu r3mp |10 { oq3zPr3t >‡ 1| 0 … x (('Y† † j } x   x  y } } x
/


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"

7hp  'Y † 2 Y ' † ' x x 3r |

m |+o o…3m

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p +| o o t |A~¢swu #3r3m|+o tLrL~ m s ‡ t …Ÿt3o q x !%” x' ošmx swx swo  . t3t  s<| swo # ro" x$ ax |+r3¤m  r3m ozPu 2CŒt }‚ t3o t3 r3Vo ef f (T, q  φ)o2 q3r !{   sw}‚rL m } ~&ioo xj} s o qLr3r¡} mšLs t  swuz•  swr3o )  p3 q3r3} zPoq  pL{wm}¡0v p |Ao z ˆ Ψ †  2 } } x  }jx x 

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swswo . tYm r #3o m3r3… r3m 3# t3rL t # o # zPra|10 3t r t”z r cr&3r3qL{jo m ~ J(x) † Q}wxŠx   }82 j} x $  }jx  x' } m|Ao  {Y}‚r3 z} w‡ q z }jx  x x Z o t3r3 Ψ(x) &2 $-

-

3 ˆ J(x)Ψ(x)d x ≡ HJ ,

H→H−

<

qLo r3t o q s p +| o 3r m r 3# om  ~¤m{EzPz s)p3qLr3m{ m swrLr¡ Lr m  $ x x x 4 † Q}‚ } }   } } }wx$ x w} x ZJ =

& −βF

= Tr

J

& −βH

J




#!

,

p3q3o m s |A~3o m ~‰sžswr o Š{atL 3r t |10wt 3r t o q | 7,7 + o'.iq | ” % ot3r3oip m o  }‚ }  x  x  $ x } $  (+'† 2 ; x  x } p3otL~3z Lr zPo o m s r3zPrŒ  yr 3r3o t zPr m r #Lo m Lr3o Š{Et3 3r3 r Wq3r3t J } x x ‚}  }‚ }  G(x1 , . . . , xn ) = Z 1 × β

−1

Z

Z β

0

Dφ

& −S

(β) [φ]

1 dτ1 φ(x1 , τ1 ) × · · · × β

Z

β 0

t 3q zPr3q sw m m m{azPz … ‡wo r3m #Lt3r3 r  q3u 1 swswo o tLu A| ~’{ ‡wm ( x x  x ‚ }

 } x  w } x   oEm s ‡ t #3o t3rLr |3~)oswa|+r s o…3m swrL~ }wo x m@0šrŒβs |1 0jt3o …  o z pL`{wx m  E x x   9Šq m …¢pLq3r3zPo q  m q3o t3oo˜p `x |Ao  s t3r3}  o o  s)mq3o oš5‡ |  q < ~Xswswx o o tLr x j } x  x 2  x <  $ x'5   #3om  r3m #3t3rL & x $ x , }jx 
−βH

-

l t3r

hφ(x)iJ =

'<

Tr (

o zƒp q ~ 
|+r #3rLo } swo m sj{wo }  } zPo } oq3o m} t3u } }swq3ozPo t   m ~ †  ‡ ‡  x 

Tr

hφ(x)iJ =

Z

&

J

φ(x)

−βHJ )

,+

+'Y† 7

dτn φ(xn , τn ) (

s } o z 

+ 2

r3t!

  m † r! }‚ }

.

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