Фотон-нейтринные процессы во внешнем магнитном поле и плазме(Диссертация)

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Author: Чистяков М.В.

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Y)

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8 7;

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$

$ 80 ;

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(AB)⊥ = (AΛB) = A1 B1 + A2B2 ,

µν = (ϕ˜ϕ) Λµν = (ϕϕ)µν = ϕµρ ϕρν , Λ ˜ µν = ϕ˜µρ ϕ˜ρν ϕµν = Fµν /B µν Λµν 1 ˜µν = 1 εµνρσ ϕρσ 4 Λ $ $ ϕ 2

-

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V πµνρ

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8

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)

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K $1& -$ 8 "; $ -$ 5 γ -$ 1 $ Π± 8 % J;

Sp{γµ γν . . . Π+ } = Sp{γµ γν . . . Π− }, Sp{γµ γν . . . γ5Π+ } = −Sp{γµ γν . . . γ5 Π−}, -$ $ 8 "; 1 < β q2 , q⊥2 1 5 $1& & $ $1 ˆ $1& 6 ' S(X)

ˆ S(X) = Sˆ− (X) + Sˆ+(X) + Sˆ⊥ (X),

8

);

8

;

∞ dτ d2 p i ˆ S±(X) = − [(pγ) + m]Π±(1 ∓ thτ ) × 4π thτ (2π)2 0 τ (m2 − p2 ) βX⊥2 − − i(pX) , × exp − 4 thτ β

))

∞ dτ d2 p β (Xγ)⊥(1 − th2τ ) × Sˆ⊥ (X) = − 2 2 8π (2π) th τ 0 τ (m2 − p2 ) βX⊥2 − − i(pX) . × exp − 4 thτ β

8

,;

! $$ M Sˆ±(X), Sˆ⊥ (X) $ $ $6 ; Sˆ± Sˆ± Sˆ±L ); Sˆ± Sˆ±Sˆ⊥ L ; Sˆ± Sˆ⊥ Sˆ⊥ L ,; Sˆ⊥ Sˆ⊥ Sˆ⊥ K $

Sˆ− (X) τ ∼ β/|p2 − m2 | 1 > thτ 1 τ 1 Sˆ− (X) $1 $ $ 8 =; 5 Sˆ+ (X) Sˆ⊥ (X) τ ∼ 1 ! ' $ 8 ; 8 ,; $ Sˆ+ (X) Sˆ⊥ (X) $ $1& 6

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$ $1& ' exp(−φ⊥ /2β)

1 − φ⊥ /2β + O(1/β 2) + $ $ $ 1 ,$ 1 2 $ M $ $1G $$

MR (q , q ) = M(q , q ) − M(q , 0) − M(0, q ) + M(0, 0),

8

7;

$ 1 $ 8

7; $

E $ & $ $ $ %$ ! 6

MR (q , q ) = M(q , q , me ) − M(q , q , M),

8

=;

M ? ' - $ $ 1& $ 1 β M 2 m2e , q 2 , q 2 5 - 1 + - $ $ $1& Sˆ−Sˆ⊥ Sˆ⊥ $ 1 + $ γγ → ν ν¯

),

$1& 6

GF M√ jµ εν ερ {gV ΠVµνρ + gA ΠA µνρ }, 2e ΠVµνρ = −

8

";

ie3 1 V {(q ϕq ) π + (q I ) ϕ + ((q − q )I)µ ϕνρ + ν ρµ µνρ 2 2π 2

+ (q I )ρ ϕνµ − Iνρ (q ϕ)µ + Iµν (qϕ)ρ + Iµρ (qϕ)ν − − Iνρ (q ϕ)µ − Iµν (q ϕ)ρ − Iµρ (q ϕ)ν },

ΠA µνρ

8 )0;

ie3 1 V = − 2 {(q ϕq ) ϕ˜µσ πσνρ + (q ϕI ˜ )ν ϕρµ − ((q − q )I ϕ) ˜ µ ϕνρ + 2π 2 + (q ϕI ˜ )ρ ϕνµ − (ϕI ˜ )ρν (q ϕ)µ + (ϕI ˜ )µν (qϕ)ρ + (ϕI ˜ )µρ (qϕ)ν − − (ϕI ˜ )νρ (q ϕ)µ − (ϕI) ˜ µν (q ϕ)ρ − (ϕI) ˜ µρ (q ϕ)ν }.

8 ) ;

K $ M $ 6

Iµν

i ≡ Iµν (q) = − π

d2 p Sp{γµ S (p − q)γν S (p)}, (2π)2

V πµνρ $ 8

8 ));

0; +

$ <

Iµν $1& 6 4m2 q µ qν e ˜ µν − , Iµν (q) = Λ H 2 2 q q H(z)6

z 1 8 ); arctg √ − 1, z ≥ 1, z−1 z−1 √ 1 − z − 1 z z 1 − 2 + iπ √ √ ln √ , z < 1. H(z) = 2 1−z 1 − z + 1 1−z H(z) = √

)# V $1& 6 ! πµνρ V πµνρ =

1 q 2 q 2 q 2

)ν (Λq )ρ H [(q ϕq ˜ ){(ϕq) ˜ µ (ϕq ˜ )ν (ϕq ˜ )ρ π⊥ + (ϕq) ˜ µ (Λq

)ρH − (Λq) µ (Λq )ν (ϕq µ(ϕq ˜ )ν (Λq ˜ )ρ H } −(Λq) µ (ϕq ˜ )ν (ϕq ˜ )ρ (H − H ) +(q q )(Λq) )ρ (H − H ) ˜ µ (ϕq ˜ )ν (Λq +(qq )(ϕq) )ν (ϕq ˜ µ (Λq ˜ )ρ (H − H)], +(qq )(ϕq)

8 ),;

π⊥ = H + H + H q2 q2q2 − 2m2e [q2(q q )H − q2(qq ) H − q2(qq ) H ] + 2 . q2q2 q2 − 4m2e [q2 q2 − (q q )2 ]

8 )#;

H ≡ H(4m2e /q2 ) ! $ $ ΠVµνρ ΠA µνρ $1& - 6 A V, A qν ΠV, µνρ = qρ Πµνρ = 0.

C 8 )0; 8 ) ; 1 ' 5 - 6 V qν πµνρ = Iµρ − Iµρ ,

V qρ πµνρ = Iµν − Iµν

8 )/;

C $ γγ → νν $ ' $ (,)* 5 ' 2 4m 2 2 q me $ H q2 e

6

2 q2 4me H , q2 6m2e

8 )7;

)/ V ' 6 πµνρ

1 {(ϕq ˜ )ρϕ˜µν − (ϕq ˜ )ν ϕ˜µρ − (ϕq) ˜ µ ϕ˜νρ }, 8 )=; 6m2e % 8 )7; 8 )=; 8 "; $ V πµνρ

$ $ $ (,)* gV =

gA = 1 N (,)* ! & + ' $ & & $ exp(φ⊥ /2β) D ' $ $ 5

(,)* - $ $1 1& K $

$ $ ! ' $ $$ $& $ $ $ $ $ ! 1 ' $ $ 8

";

$ ' 6

GF igae Maγγ = M(gV = 0, gA √ jµ → qµ ), 2me 2 $ & γ → γγ 6 GF Mγγγ = M(gA = 0, gV √ jµ → εµ ), 2e

)7 gae '

εµ γ → γγ

% γγ → ν ν¯ $1 ' 1 B ! $ ' 5 ' $ 5

$ & 8

";

$ 1& 6

Mλ1 λ2 = M(ε(λ1) (q ), ε(λ2 ) (q )).

8 )";

(λ)

K εµ (q) λ C

(qϕ)µ ε(1) µ = 2 , q⊥

(q ϕ) ˜µ ε(2) µ = 2 , q

8 0;

1& $ 6

q 2 − P (λ) (q) = 0,

8 ;

P (λ) (q)

!

" # $$% " &

)= % 8 0; &

8

";

$ $1& $6

M11

˜ ) 2α GF (q ϕq )(q ϕq √

=i [gV (j ϕq) ˜ + gA (jq) ] H, π 2 q2 q⊥2q⊥2

M12 = −i

8 );

1 2α GF √

π 2 q⊥2q2

) (q q )⊥ (j ϕq)(qq ˜ × gV [(jq )⊥(q ϕq ˜ ) + (j ϕq ˜ )(qq )⊥ ]H − H q2 (jq)(qq ) (q q )⊥ H , 8 ; +gA [(jq ) (qq )⊥ − (jq )⊥ (q q ) ]H − q2 (q ϕq ) 1 2α GF √

M22 = −i ((q ϕq ˜ )[gV (j ϕq) ˜ + gA (jq) ]π⊥ 2 π 2 q2q2 q +(q q )[gV (jq) + gA (j ϕq)](H ˜ − H )) − (jϕq )H [gV (q q ) + gA (q ϕq ˜ )] −(jϕq )H [gV (q q ) − gA (q ϕq ˜ )] < ' $ $ B $1& 6

QB γγ→ν ν¯ = Q11 + Q12 + Q22 ,

4

Qλ1 λ2 = (2π) gλ1 λ2 ×

3

8 ,;

|Mλ1 λ2 |2 Zλ1 Zλ2 (Ei + Ei) δ 4 (q + q − k − k )

i

dq d3k d3 q d3 k f (ω ) f (ω ) . (2π)32ω (2π)3 2 ω (2π)3 2 Ei (2π)3 2 Ei

8 #;

K Ei, Ei '

i = νe , νµ, ντ L ω , ω ' L f (ω) = [exp(ω/T ) − 1]−1 $ $

)"

T L gλ1 λ2 = 1− 12 δλ1 λ2 $ + $ 8 #; $ A $ $ - $ Zλ1 , Zλ2 H $ $ T me ! ' P (λ) (q) 1 6

P (1) (q) −

α 2 q , 3π ⊥

P (2) (q) −ξ q2 ,

$ ξ =

α B 3π Be

8 /;

$

< 10Be ÷ 104 Be ξ 7.7 · 10−3 7.7 8 ; 8 /; $

$$ q 2 0

ω 2 = q⊥2 /(1 + ξ) + q32.

8 7;

$ ξ - $$ A

' $ $& $ )6

ε(2) µ

√ → Zε(2) µ ,

Z=

∂P (2) 1− ∂q2

−1

1 . 1+ξ

8 =;

0 + ' $ $ 8 7; 6

d3q = (1 + ξ) ω 2 dω dϕ dy,

y = cos θ

1 + ξ/ 1 + ξ cos2 θ,

ϕ θ $ $ ! $ - eB m2e /α ξ 1 Qλ1 λ2 $1& 6

Q11 = + Q12 = + Q22 = +

2 8 α2 G2F 13 2 2 g ¯ 7π T π ζ(5) + 330 ζ(7) + V 8505 π 3 m4e 4 2 2 g¯A 9π ζ(5) + 70π ζ(7) + 420 ζ(9) = ' 8.5 · 107 T913 , 8 "; · 3 32 α2G2F 13 2 2 2 T 24π ζ(5) + 285 ζ(7) + g ¯ 14π V 127575 π 3 m4e 2 2 2 2 g¯A (309π + 20)π ζ(5) + (1410π + 315)ζ(7) + 38430 ζ(9) = ' , 8 ,0; 6.9 · 108 T913 · 3 8 α2G2F 13 2 2 2 T π − 50)ζ(5) + 10110 ζ(7) + g ¯ (609π V 127575 π 3 m4e 2 2 2 2 g¯A (879 π + 50)π ζ(5) + 5490π ζ(7) + 56700 ζ(9) = ' , 8 , ; 3.3 · 108 T913 · 3

T9 $ 109K 0.17me ' 2 g¯V2 0.93, g¯A 0.75 $ $

νe , νµ, ντ % ' $ 9 13 QLT γγ→ν ν¯ 1.1 · 10 T9

' . · 3

8 ,);

% $1 γγ → ν ν¯ $ < $

$ $1 ()"*6 ν Qm γγ→ν ν¯

1.4 · 10

−4

T911

' mν 2 . · 3 1 '!

8 ,;

+ $ γγ → ν ν¯ ()*6 L −10 13 QN T9 γγ→ν ν¯ 9.9 · 10

' . · 3

8 ,,;

' $

γγ → ν ν¯ $ $ - $ B Be $ (7*

'

6

9 13 QLT γγ→ν ν¯ 0.3 · 10 T9

B Be

2

' , · 3

8 ,#;

$ $ B 0.1Be 8 ,#; $ $1 $ D Qγγ→ν ν¯

$ (,)* $ $1&$1 $ 8 13 QLT γγ→ν ν¯ 0.7 · 10 T9

' . · 3

) "* +* " *

(

8 ,/;

,' -% *% * "+ 4π " * ,!.- ) % +* ,'- + T = m e *

) ! $ 8 ,); - 8 ,/; % - $ 1 ' $1 (,)* 1 $ $ 1& ' ' C- $ 8 ,); $ $

+ (,)* $ $ $

ξ 1 $ $ $& $ 2 $ H , ξ =

α B 3π Be

$1&

$ $ $

!"

# T me# ξ " ! $ $ " %! # & " ! & " Q11 , Q12 , Q22 ' & ( ) 8 −13 Q0 = 10 T9 · 3

+ &

1& γγ → ν ν¯ H γ → ν ν¯γ % $ $ < γγ → ν ν¯ - $ C γ → ν ν¯γ

$

%'$ $1 ' γγ → ν ν¯ 5 8 #; $ f (ω) → (1+f (ω)) q → −q $ γ1 → ν ν¯γ2 K H # H , H #

* Qγ→γνν¯ ξ !" # T me $1 γ → ν ν¯γ 1 γγ → ν ν¯ γγ → ν ν¯

, 1 γγ → ν ν¯γ C

$& $ $ γγ → ν ν¯ %'$ $ $ ! γγ → ν ν¯γ (,"* $ $1& / 6 2 B ' Qγγ→ν ν¯γ 1.7 · 104 T917 . Be · 3

8 ,7;

$ $ $ T = 109 K γγ → ν ν¯ - γγ → ν ν¯γ B 250Be !

γγ → ν ν¯γ γγ → ν ν¯ + ' $ γγ → ν ν¯γ $ 1 $ $ γγ → ν ν¯ $ $& $ ! $ T me $$ $ $

m2e q2 β ! ' $ P (λ) (q) 6

P (1) (q) −

α 2 q , 3π ⊥

P (2) (q) 6 ξ m2 .

8 ,=;

< ξ 1 0 +

'. +* " * ,(1-% *% * 2 ννγγγ 2" % "+ * % "

% +* ,'3-

# $$ 5 $ $1&$1 $6

QHT γγ→ν ν¯

2.7 · 10

18

T me

9

' . · 3

8 ,";

D $ 8 ,"; γ → ν ν¯ ! ' (# #)* A γ → ν ν¯ - ,$

q 2 > 0 ! ' $ $ q2 > 4m2e C $ $ exp(−2me/T ) $ ! T me 1 $ 1 q2 4m2e +

$ γγ → ν ν¯ $1 γ → ν ν¯ m2e q2 β ! ' $ $ $ T 2 eB, m2e 6 5 2 ' T B HT 18 Qγ→ν ν¯ 0.40 · 10 . me Be · 3

8 #0;

< - $ $1&

6

2 QHT B γ→ν ν¯ R = HT 0.15 . T2 Qγγ→ν ν¯

8 # ;

< $ 8 ,"; 8 #0; -

/ - 8 # ; $ γ → ν ν¯ $ - γγ → ν ν¯ $ < $ T = 2 4'! B = 100Be $ R 5.86 <

' $ $ $ γγ → ν ν¯ γ → ν ν¯ 1 < - T me $ γγ → ν ν¯ $1& ! $ 1 γ → ν ν¯ γγ → ν ν¯ D $ ' & 1 OPQI < OPQI $

1/3 ' ρ 27 6 QD , U RCA ∼ 10 T9 ρ0 · 3 OPQI 2/3 ' ρ M 21 8 QU RCA ∼ 10 T9 , ρ0 · 3

8 #);

8 #;

ρ0 = 2.8 · 1014 /3 8 (#* ; %$

7

$ OPQI B < 1019 3 8 (#* (#,*; 8 #); 8 #; OPQI $ + OPQI $& $ $ < $ $ $ $ - - OPQI (#* + ' $ ' 1

! ' $ 1& D $ $ $1 $ $

! & $ ! $ ' ννγγ ! $ % $

- $ % $1&

γ → γγ ! " " !! #" γ → γγ !$ $ %&' (' )**+*,- ( ! $ . ( )*+%&- $ ! ' ' !' ' (' / ! $0 ' ' /ω me 0 1!

( )**+*, %&- )* *2- ' " ! ! 3 )**+*, %&- ! $ (' )* *2 )* *2- 4 ( 5 " " "$ !! ! " ' !("

2 6 ( )%7- (" ! ( )**+*, %&- "

" ' !"

8 ( 9 ( )%7- : CP ( !

( !"

' ( " ! $

/ω ≤ 2me 0

" " ;

' <

! '

( = 9 " " (

(' )%>+%?- . γ → γγ

(' )%* %%- : !! $ . " (

(@ (

' ! )%,- ! !" ! (! ' " ! " A A B! /CBD0 ' /EFG0 )%- ' "' (@ " " $ " " $ " 1014 − 1016 4 . "

γ → γγ !

?&

γ → e+ e− ! " e+ e− (

' ' ( ($ !" ( ( )%2- ! " ( (" ! ! H! I !' )%2-

(

),&-

" ! ' " ' )* *2 %7 %* %%- γ → γγ ( 9 ),7- (

)%2 ,&- " " "

!" )%2 ,&- !

' '

! (" $ J" !$ γ → γγ " !

.( " (

)%, ,>+,?-

! !" ' )* *2 %7 %* %%- (' ),> , - ( " !

γ → γγ B Be

( !" "' (@ / )%, %- ),*-0 . " '

),>, -

! " ' " ' 1 → 2 2 /γ → γ⊥ γ⊥ (" ' 9 )%7-0 I (B Be ) ),% " 5 ( " ( ),,-

?7 ( ( ! γ → γγ !" ),+ - > ! !" . " ! ( !" " ! ! 1 → 1 2 I" ( ' !" ' !

! :

!

' !

" !" !

! ! ! !

),> , - 1 ? "$ ' !" .!" " K L 1 → 1 2 ( ' $ ω m 8 '

$ " (' ! 1 → 2 2 1 → 1 2 * (!$ "

?>

γ → γγ 6 !! . Pαβ < %

Õ

Õ

! " (λ)

# Pαβ !( ( bα

Pαβ =

(λ) b(λ) α bβ λ

(b(λ) )2

P λ,

P λ = ∆P λ (q, F ) + Π(q 2),

/>70

b(1) = (qϕ)α, α = (q ϕ) ˜ α, b(2) α

/>>0

b(3) = q 2 (qϕϕ)α − (qϕϕq)qα, α ? qα (!$ ( ? M N( " !! !$ " Π(q 2 ) !!$ " ∆P λ ( ( ! ∆P λ " ( ),% ? *-O

∆P

(λ)

α = − 2π

1

∞ du

0

0

2 dt βt −iΦ −iΦ0 q 2 e vλ − e (1 − u ) /> 0 t sin βt 2

?

v1 = q2

u sin βut cos βt cos βut − sin βt

2

2

2

v2 = q cos βt(1 − u ) − q⊥ v3 = q 2

− 2q⊥2

cos βut − cos βt ; sin2 βt

u sin βut cos βt cos βut − ; sin βt

u sin βut cos βt cos βut − ; sin βt

2 1 − u q 2 cos βut − cos βt + ⊥ Φ = t m2 − q2 4 2β sin βt 2 q Φ0 = Φ(B = 0) = t m2 − (1 − u2) ; 4

/>?0

!$ ∆P λ $ !$ !O

α 2 q⊥ − q 2 Λ(β, q 2), 3π 2 4me 2α β (2) − q 2 Λ(β, q 2), H P (q) − 2 π q

/>*0

P (3) (q) − q 2 Λ(β, q 2),

/>,0

P (1) (q) −

/>%0

α [1.792 − ln(B/Be)] , 3π H(z) ! 4 P ! /7>&0O Λ(β, q 2) =

z 1 />0 arctg √ − 1, z ≥ 1, z−1 z−1 √ 1 − z − 1 z z 1 − 2 + iπ √ √ ln √ , z < 1, H(z) = 2 1−z 1 − z + 1 1−z H(z) = √

?? . !" ! ( />>0 ( !$ O

Gαβ =

(λ) (λ) 3 bα bβ λ=1

−i , (b(λ) )2 q 2 − P (λ)

/>20

! !$ !

q 2 − P (λ) = 0

(λ = 1, 2, 3).

/>7&0

1 ! />7&0 $ " !' λ = 1, 2 ),%-

(qϕ)α , ε(1) = α q⊥2

(q ϕ) ˜α ε(2) = , α q2

/>770

< λ = 3 " 8 P (3) (q) ! />7&0 !( " ! q 2 = 0 5 ! " !

( ! !$ (" ( (1)

(" ' ( )%7- 7 εα /γ 0

(2)

> εα " /γ⊥ 0

Q ( ! !$ K!L q 2 = 0 " ! : , " (

7 > ! q2 q⊥2 6 K L

q 2 = q2 − q⊥2 ,

?* ¾ ¾

¾ ¾

# $ ! % % % & ' q 2 = 0 # ( , √ 7 q2 < (m + m2 + 2eB)2 " q 2 < > " ( √ 4m2 < q2 < (m+ m2 + 2eB)2 " q 2 5 " " !" γ → γγ ' " ( " ! N ! " " ! ! 1 → 2 2 ( " / K L" "

!

KL" ' "0 ; ! "

> "!$ ( q2 < 4m2

( !$ " q 2 !" ! ! "

?% 6 ! 1 → 2 2 !" ),> , - " " # ( !$ " q2 = 4m2 q 2 !$ 5 (! !! ! ( ),%- "'

2 q2 = Enn m2 + 2neB + m2 + 2n eB)2, (B) = (

/>7>0

n, n = 0, 1, 2, · · · ,

! % ! ' H! q02 eB !

n = n = 0

q2 = 4m2 N! " ( q2 = 4m2 ' ( (! $ ! ! $ ),%- N ! " ( ! (' !" ' !

ε(λ) α

→

ε(λ) α

Zλ ,

Zλ−1

∂P λ =1− . ∂q2

/>7 0

!

>

?, . ( I ' (! " ! ! $ ! "!$ α ln2 (B/Be) ' !$ )%- ( B

Be /α "!$ α ln(1/α) ln(B/Be) ),-

γ → γγ

. ( ! Þ Õ

Ü

Õ ¼¼

Ý

¼

¼¼

Õ ¼

( ) % * N!$ S " ie3

S(γ → γ γ ) = √ ε(k)S(x, y)ˆ ε(q

) × d4x d4y d4z Sp{ˆ 2ωV 2ω V 2ω

V × S(y, z)ˆ ε(q )S(z, x)}e−i(qx−q z−q y) + (ε(q ), q ↔ ε(q

), q

), />7?0

qα = (ω, q) ? ! εα ˆ − y) S(x, y) = eiΦ(x, y) S(x /! /C>0 . 90 ; ! M S "

i(2π)4δ 4 (q − q − q

) M, S(γ → γ γ

) = √ 2ωV 2ω V 2ω

V

/>7*0

? ( !$ O ˆ )ˆ ˆ ˆ M = e3 d4X d4Y Sp{ˆ ε(q)S(Y ε(q

)S(−X − Y )ˆ ε(q )S(X)} ×

× e−ie (XF Y )/2 ei(q X−q

Y)

+ (ε(q ), q ↔ ε(q

), q

),

/>7%0

X = z − x, Y = x − y N />7%0 ! /7*0 ! " ! M ( O

M = εµ (q)εν (q

)ερ (q ) ΠVµνρ ,

/>7,0

ΠVµνρ = −

ie3 1

V

((q − q )I)µ ϕνρ + {(q ϕq ) π + (q I ) ϕ + ν ρµ µνρ 2 2π 2

+ (q

I )ρ ϕνµ − Iνρ (q ϕ)µ + Iµν (qϕ)ρ + Iµρ (qϕ)ν −

− Iνρ (q

ϕ)µ − Iµν (q

ϕ)ρ − Iµρ (q ϕ)ν }.

/>70

: "

π⊥ = H + H

+ H q2 q 2q

2 − 2m2e [q2(q q

)H − q 2(qq

) H − q

2(qq ) H

] + 2 , q2q 2 q

2 − 4m2e [q 2 q

2 − (q q

)2 ]

/>720

H ≡ H(4m2e /q 2 ) 8 " !( ' ( />7%0 ! $ />770 $ O

Mλλ λ = M(ε(λ) (q), ε(λ ) (q ), ε(λ ) (q

)),

λ, λ , λ

= 1, 2.

/>>&0

?2 M

!" !$ ' !

"$ " O(1/B)O

M111 = 0,

/>>70

M112

/>>>0

M122

M222

α 3/2 (q ϕq

)(q ϕq ˜

) 4m2 , = i4π H π (q

)2 [(q )2 (q

)2⊥ q⊥2 ]1/2 α 3/2

) (q Λq = i4π × π [(q )2 (q

)2 q⊥2 ]1/2 2 4m 4m2

× (qΛq )H + (qΛq )H . (q )2 (q

)2 α 3/2 (q ϕq

)(q ϕq ˜

) = −i 4π π⊥ π [q2 (q )2 (q

)2 ]1/2

/>> 0 />>?0

I " " ' ' ! ( " " ?!

q, q , q

6 />>>0 />>?0 qα ∼

qα ∼ qα

!

M112, M222 ( $

! M122 " ! ! ( ),> , - I " " (@ 1 → 2 2 ! ! ),> , - " "

1 '"" ! " ! !! ! ? ! " .! " !$

*& !

(@O

g Wλλ λ (γ → γ γ ) = 32π 2ω

|Mλλ λ |2 Zλ Zλ Zλ ×

× δ(ωλ (q) − ωλ (q ) − ωλ (q − q ))

d3 q

, ωλ ωλ

/>>*0

g = 1 − 12 δλ λ !" " Zλ !" $ ( KL

! ω = ωλ (q) N !" ! />>70/>>?0 γ →

γγ > $ 1 → 2 2 1 → 1 2 2 → 1 2 2 → 2 2 I " > " (

q2 = ω 2 − q32 > 4m2 I ( K L γ → e+ e− )- ! ( $ 2 W (γ → e+ e− ) 1 eB ∼ 2 1. />>%0 W (γ → γγ) α ω2 N ! 7 !$ ! √ ω < m + m2 + 2eB "$ ( N " " ' "' )%,- " 1 → 2 2 / 0 1 → 1 2 / 0 6 />>*0 " " " M

" "

*7

+ $ * γ → γγ , - ,% .ω sin θ 2me/ 1a, 1b 0 % 1 *2 ! 1 → 12 , B = 102 Be 103 Be 3 2a, 2b 0 % 1 2 ! 1 → 22 , B = 102 Be 103 Be 3 3 0 % 1 2 ! 1 → 22 ! - , - " - 4 % 4 ! $% W0 = (α/π)3 me

!

ω 2 sin2 θ eB /θ +

! ! ! $ q $ B0

27> "

' ' ( $ ! " K L 1 → 1 2 ! " m2 ω 2 sin2 θ eB 1 → 1 2

*>

&5 $ * γ → γγ ,% 1a, 1b 0 % 1 *2 ! 1 → 12 , B = 102 Be 103 Be 3 2a, 2b 0 % 1 2 ! 1 → 22 , B = 102 Be 103 Be !" " O

α3 m2 1 − x · W [1 − x + 2x2 + 2(1 − x)(1 + x)2 ln(1 + x) + 2 4ω x 2 2m 2 2−x ln x], x= 1. />>,0 + 2x 1−x ω sin θ : 77 ! ( ! 4 " ! $ ! B !"

* ¼

½¼

½¼

½¼

½¼

½¼

½¼

½¼

½¼

&& $ * 1 *"2 ! " 1 → 12 4 4 , B = 100 Be . 1 /3 103 Be . 2 /3 104 Be . 3 / 6 4 - , " ! " .''#/ O

(ω − ω )2 − 4m2 dW α3 · , dω

2 ω + (ω − ω)2 − 4m2 ω 2m2 − < ω < ω − 2m, 2 ω

ω, ω " "

/>>0

7

*? ¼

½

½¼

½¼

½¼

½¼

½¼

½¼

½¼

½¼

¾

¿

&' $ * 1 "2 ! " 1 → 22 4 , 4 , &&

!"# $ $ 6 ! " " ! " " !" (@ ! !" ! (! ' " !" A A B! /CBD0 ' /EFG0 )%- :( ! $ (@ ($ ' '

** "' (@ "

" 1014 − 1016 4 ( !$ ! γ → γγ '

e+ e− (' ' ( !" . " e+ e−

$ " γ → e+ e−

" q2 > 4m2 9

e+ e− !" ( ( ),*- Q " e+ e− ! ( !" 1 → 2 2, 1 → 1 1, 2 → 1 2

CP $ 9

"$ "

" ( 1 → 2 2 " ! ( I "

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Π± γ µ = γ µ Π± ,

Π± γ⊥ µ = γ⊥ µ Π∓

Π± γ5 = γ5 Π± .

<4 <"

=! !! <. <4 ! !2 !

Π± γµ Π± = γµ Π± ,

Π∓ γµ Π± = −γ⊥µ Π± .

<6

> 1!5 ! !2 ;

Π±γ⊥µ γ⊥ν Π± = (−Λµν ± iϕµν )Π±,

<?

!! 1! γ # 15 5 ;; ( ( ! ! 7& ! ! ( 5 5 1 1! γ #

Sp{Π±} = 2,

µν , Sp{γµ γν Π±} = 2Λ

ρσ + Λ νρ − Λ νσ ), µν Λ µσ Λ µρ Λ Sp{γµ γν γρ γσ Π± } = 2(Λ

<,

Sp{γµ γν γ5Π±} = ±2ϕ˜µν , ρσ ). µν ϕ˜ρσ + ϕ˜µν Λ Sp{γµ γν γρ γσ γ5 Π±} = ±2(Λ * 1 1! γ # & '1 1 5 ! !

. > 1! ! 12! ! 3

(ϕγ) ˜ µΠ± = ∓γµ γ5Π± γµ γµ = 2,

γµ γν γµ = 0,

γµ γν γρ γµ = 2γρ γν ,

<

˜ µν γρ + Λ ˜ νρ γµ − Λ ˜ µρ γν . γµ γν γρ = Λ @ ( 1 ! 5 γ # & 5 ! ' 1 1! γ # '2 ! (

A!( . '2 !1 1! ( #!! ( 1 : ( ! ( !& >

!( #!' !( ( ! 1! & $ 5& 1 ! τ ; 6 '! ( τ ∼ β/|m2 − p2 | ! β/|m2 − p2 | 1 8 1 1 th τ 1 τ 1 1 !2 ! 1! ˆ & S(X) 3 2 βX d2 p (pγ) + m iβ ⊥ ˆ ˆ S(X) Sa (X) = exp(− ) Π−e−i(pX) . 2 2 2 2π 4 (2π) p − m =! ! β/|m2 − p2 | 1 ! ' ! " (! 1! # 1! 25 #!! ! ! 1 : 1 & ! !

4 # 1! ( 52! @

' (! ! $ ! @ A ! ( & ; 1

n 115 5 1! ! 2( ! 1 ! ! 1 $! ( ! ! ! 2 ! n ; (eB)n 1! 1 ! ! (eB)n−1 δ ; # 2 !5 (! #!! B! ! ( & ' 1 ! 2 ! & '1( !! '& eB qi2

qi2 " (! 5 1! # 1! 25 # 8 ' 5 #!! !( ! '! 1 ; # 1

! Φ(x1, x2) n :!' ! ( 1 !# 5 & Φ(x1, x2) 4 &

S(x1, x2 ) . 1! n 115 5 1! ! %!

!2 ;; ννγγ

! ! 5 11

" < !! ∂µ Kν − ∂ν Kµ = 0 ( x

y " !1 5 ; ! ;

Φ(x, y) + Φ(y, x) = 0. !1 n ; n ≥ 3 ! ; n !# '1 $ ! ( ! " # ! ( '

1 Aµ (x) = xν Fνµ + ∂µ χ(x), 2

C

χ(x) - ( ; #

1! ∂µ Aν −∂ν Aµ =

Fµν 1! " 1 e Φ(x, y) = − (xF y) − e [χ(y) − χ(x)]. 2

C.

1 ! ; ! !&2 ; # χ ! ( ! 2 ! > ; n 11 '& !

& ! 1

! 3

Φtot

n n−1 l−1 e ef = − (xiF xi+1) = (Zk F Zl ), 2 i=1 xn+1 =x1 2

C4

l=2 k=1

Zi = xi − xi+1.

$ 5

e Φ(x1, x2) + Φ(x2, x3) + Φ(x3, x1) = − (x1 − x2)µF µν (x2 − x3)ν , 2

C"

6

Ü Ü

Ü

Ü

Ü

n

" #

A 1 X⊥ Y⊥ &

M C D 6 ! ! 1! 1 5 ''25 !!5

3 !

( 1!

5 ! 2! 2 ! ! ! ; ( " Z " (! Q 1 ! ! 2 X⊥ Y⊥ (! ; q⊥ q⊥ !2 ' 3 q⊥ µ X⊥µ , Qµ = , Zµ = Y ⊥µ −q⊥µ

$

5 Z Q ! !! 2 1! M ! ! !2 3

β − 4 (ZRZ) − i(QZ) . J⊥ = d4Z e $. 9 # E & ' ( ! !

(2 × 2) # #! 5 1 ! ! 3

Rµν = R1 δµν + iR2 εµν ,

$4

δµν ! F εµν ! ( ! 1 ! ! (1, 2)3 ε11 = ε22 = 0, ε12 = −ε21 = −1

?

1 R1 = t1 t2t3

t1 t3 t3 (t1 + t2 ) t1 t3 t1 (t2 + t3 )

,

R2 =

0 1 −1 0

.

ti = tanh τi τi !'! A

J=

dn Z e− ((ZGZ) − i(QZ)) ,

$"

G ! 1 # 1! ! !2 ; 3 π n/2 − 1 QG−1Q , J=√ $6 e 4 detG ( ! ( 1 ' ! 1 # & ' ( ( ! ! 2 ( ' 8 $" ' ! ''25 !!5 1! ! $ ' ' 1 1 # ! ( ' 1 ( $6 8 ' 1 M ! ! 3

J

⊥

Jµ⊥

β 4

(ZRZ) − i(QZ)

−1

(16π 2/β) t1t2 t3 − QRβ = e = t1 + t2 + t3 + t1 t2 t3

β − 4 (ZRZ) − i(QZ) ∂J⊥ = d4Z Zµ e = −i ∂Qµ =

Jµν =

4

2i −1 [R Q]µ J ⊥ , β

⊥

− d Ze

− d Z Zµ Zν e 4

β 4

(ZRZ) − i(QZ)

∂ 2J ⊥ =− ∂Qµ∂Qν

Q

,

$?

,

Jµνρ ⊥

2 −1 2 −1 −1 = R − [R Q]µ [R Q]ν J ⊥ , β µν β

β − 4 (ZRZ) − i(QZ) =i = d4Z Zµ Zν Zρ e

∂ 3J ⊥ ∂Qµ∂Qν ∂Qρ

4i −1 −1 −1 −1 [R−1Q]ν + Rνρ [R−1Q]µ Rµν [R Q]ρ + Rµρ 2 β

2 −1 −1 −1 − [R Q]µ[R Q]ν [R Q]ρ J ⊥ , β =

$,

R−1 # ' R −1 ˜ 1 δµν + iR ˜ 2 εµν , Rµν =R

1 ˜1 = R t1 + t2 + t3 + t1 t2 t3

−t1 t3 t1 (t2 + t3) −t1 t3 t3 (t1 + t2 )

,

˜2 = R

0 −1 1 0

.

# +0 EGHHIJK LL MKGNO GO PGQRNGKRNSIO HRN TUVWGXIVKGJYY Z[\OS]O ^[S]G_R3 `VSaINOSK\ RH ^[S]G_R ZNIOO ? ??" b +.0 c[JRbRa deU ^ROXRbGNKS]JI Z[\OS]OYY MSV_GbRNI3 fRNJW M]SIVKSHS] ZNIOO 6? b +40 FF 5! C g' F ! ; 5 1! #YY 93 % # & hi!5 ;1! 5 j ./// "? ! +"0 kG[]GJJ lm mIUKNSVR GOKNRb[\OS]O YY ^GXQNSW_I3 ^GXQNSW_I `VSaIN OSK\ ZNIOO +60 MKSn d o[I MUV - pV DVKNRWU]KSRV YY kINJSV3 MbNSV_IN +?0 oUN]q^[SIrI M IK GJ o[I ORJGN SVKINSRN YY Z[\O EIbK 4 s .4/ t ." Z 6,.46 +,0 p[XGW uE IK GJ Mmv ]RJJGQRNGKSRV dIGOUNIXIVK RH K[I NGKI RH

nue + d → p + p + e− SVKING]KSRVO bNRWU]IW Q\ k ORJGN VIUKNSVRO GK K[I MUWQUN\ mIUKNSVR vQOINaGKRN\ YY Z[\OEIaPIKK .// s , Z/,4/ + 0 p[XGW uE IK GJ Mmv ]RJJGQRNGKSRV wSNI]K xaSWIV]I HRN mIUKNSVR TJGaRN oNGVOHRNXGKSRV HNRX mIUKNGJ^UNNIVK DVKING]KSRVO SV K[I MUWQUN\ mIUKNSVR vQOINaGKRN\YY Z[\OEIaPIKK .//. s Z/4/ +0 p[XGW uE IK GJ Mmv ]RJJGQRNGKSRV dIGOUNIXIVK RH wG\ GVW mS_[K mIUKNSVR xVIN_\ MbI]KNG GK Mmv GVW ^RVOKNGSVKO RV mIUKNSVR dSnSV_ ZGNGXIKINOYY Z[\OEIaPIKK .//. s Z/4/.

+/0 kG[]GJJ lm ZSVORVVIGUJK dy GVW kGOU M MRJGN dRWIJO3 ]UNNIVK IbR][ GVW KSXI WIbIVWIV]IO

VIUKNSVRO

GVW [IJSROISOXRJR_S]GJ

bNRbINKSIO YY pOKNRb[\O l .// s 666 Z //. +0 wUV]GV E^ o[RXbORV ^ TRNXGKSRV RH aIN\ OKNRV_J\ XG_VIKSrIW VIUKNRV OKGNO3 SXbJS]GKSRVO HRN _GXXGNG\ QUNOKO YY pOKNRb[\O l .s 4. t Z PP4 +.0 o[RXbORV ^ wUV]GV E^ mIUKNRV OKGN W\VGXRO GVW K[I RNS_SVO RH bUJOGN XG_VIKSOX YY pOKNRb[\O l 4s "/

t Z ".,

+40 cRUaIJSRKRU ^ IK GJ pV zNG\ bUJOGN {SK[ G OUbINOKNRV_ XG_VIKS] HSIJW SV K[I ORHK γ NG\ NIbIGKIN MLE /? ./ YYmGKUNI s44 Z.46| +"0 cRUaIJSRKRU ^ IK GJ wSO]RaIN\ RH G dG_VIKGN pOOR]SGKIW {SK[ K[I MRHK LGXXG EIbIGKIN MLE //}" YYpOKNRb[\Ol s6/ ZP6| +60 cRUaIJSRKRU ^ IK GJ dUJKS{GaIJIV_K[ vQOINaGKSRVO RH K[I MRHK LGXXG EIbIGKIN MLE //}" WUNSV_ DKO .// pbNSJ p]KSaGKSRV YYpOKNRb[\Ol .// s66 ZP", +?0 DQNG[SX p D MGHSyGNQ M M{GVq ly ZGNqI f ~GVI M oUNRJJG E wSO]RaIN\ RH ^\]JRKNRV EIORVGV]I TIGKUNIO SV K[I MRHK LGXXG EIbIGKIN MLE /?./ YY pOKNRb[\O l PIKK .//. s6," P6 +,0 DQNG[SX p D M{GVql y ZGNqI f mI{ xaSWIV]I HRN ZNRKRV ^\]JRKNRV EIORVGV]I SV G dG_VIKGN MKNIV_K[ TSIJW HNRX MLE /? ./ YYpOKNRb[\O l.//4 s6 " P, + 0 <! F C = : 5 22! !5 YY,/8 ", ^ 4

// +0 pNWIJGV

m

s

kSOVRaGK\ScR_GV

L

M

dRSOIIVqR

M

L

dG_VIKRNRKGKSRVGJ XI][GVSOX3 .w OSXUJGKSRV YYZNR] Dp` ^RJJ mR ?? o[I JR]GJ QUQQJI GVW QI\RVW xWO w kNISKO][{INWK dl TNI\QIN_ GVW l oNUXbIN PI]KUNI mRKIO SV Z[\OS]O s6/? Z"6 +./0 pNWIJGV

m

s

kSOVRaGK\ScR_GV

L

M

dRSOIIVqR

M

L

mRVOKGKSRVGN\ XG_VIKRNRKGKSRVGJ bNR]IOOIO SV G NRKGKSV_ XG_VIKSrIW ]JRUW YYpp .///s 466Z +.0 C : A : : & ! #!! 1 '1 !5YY >!( 8) 8 ? t 6 = 44,4". +..0 <! = % (! YY i)7 8 ? t ^ ? +.40 < 9 > i!(! ( ! ) ! ; YY8) 6 s4? ^?6?? +."0 yx ^[SU Z dRNNSORV mIUKNSVR IXSOOSRV HNRX QJG]qQRW\ NGWSGKSRV GK [S_[ OKIJJGN KIXbINGKUNI YYZ[\O EIa PIKK ?/ s6 Z6,46,6 +.60 LIJJdGVV d o[I NIG]KSRV γγ → ν ν¯ YY Z[\O EIa PIKK ? s ? t . Z ,/, +.?0 @ @$ : !! 5 ; YY$7 ===% " 8 ?/ = ./, +.,0 eGV_ ^ m MIJI]KSRV NUJIO HRN K[I WIXGKINSGJSrGKSRV RH G bGNKS]JI SVKR K{R b[RKRVO YYZ[\O EIa 6/ s ,, Z .".."6

/ +. 0 ^NI{K[IN El TSVRNW l dSVqR{OqS Z o[I GVVS[SJGKSRV bNR]IOO ν ν¯ →

γγ {SK[ XGOOSaI VIUKNSVR SV ]ROXRJR_\ YY mU]J Z[\O . s k./, t . Z .?. , +.0 wRWIJORV M TISVQIN_ L mIUKNSVR K{Rb[RKRV aINKIn YY Z[\O EIa s w"4 t 4 Z 4./ +4/0 PIaSVI dl o[I bNR]IOO γ + γ → ν + ν¯ YY mURaR ^SX ?, s p" t Z ?,, +40 wS]UO wp MKIJJGN IVIN_\JROO NGKIO SV G ]RVaIN_IVK K[IRN\ RH {IGq GVW IJI]KNRXG_VIKS] SVKING]KSRVO YY Z[\O EIa ,. s w? t " Z " " +4.0 wS]UO wp EIbqR ff Z[RKRV VIUKNSVR O]GKKINSV_ YY Z[\O EIa 4 s w" t Z 6/?6/ +440 EROIVQIN_ P xJI]KNRXG_VIKS] SVKING]KSRVO RH VIUKNSVRO YY Z[\O EIa ?4 s . t ? Z ., ?., +4"0 ^UV_ sc eRO[SXUNG d xJI]KNRXG_VIKS] SVKING]KSRV RH VIUKNSVRO SV _GU_I K[IRNSIO RH {IGq SVKING]KSRVO YY mURaR ^SX ,6 s p. t " Z 66,6?" +460 cUrVIKORa ps dSq[IIa ms ^RXbKRVJSqI SVKING]KSRV RH XGOOSaI VIUKNSVRO {SK[ aSNKUGJ b[RKRVO YY Z[\O PIKK 4 s k. t 4" Z 4?,4? +4?0 F# 95 7 #!! νi γ ∗ → νj γ ∗ !

( ; 1 !!

! YY ) 4 8 6? t ? = / "

/. +4,0 M[GSOUJKGVRa E Z[RKRV VIUKNSVR SVKING]KSRVO SV XG_VIKS] HSIJWO YY Z[\O EIa PIKK s / t Z 6 ?6 , +4 0 wS]UO wp

EIbqR ff Z[RKRV VIUKNSVR SVKING]KSRVO YY

Z[\O EIa PIKK , s , t " Z 6?6, +40 ^[\S oc y{GV_ ^f cGR fT IK GJ mIUKNSVR b[RKRV O]GKKINSV_ GVW SKO ]NROOIW bNR]IOOIO SV G QG]q_NRUVW XG_VIKS] HSIJW YY Z[\O PIKK s k"?? t ." Z .,". / +"/0 ^[\S oc y{GV_ ^f cGR fT IK GJ o[I {IGqHSIJW InbGVOSRV HRN bNR]IOOIO SV G [RXR_IVIRUO QG]q_NRUVW XG_VIKS] HSIJW YY Z[\O EIa ./// s w?. t / Z /6/" 4 +"0 wS]UO wp EIbqR ff mIUKNSVR b[RKRV O]GKKINSV_ SV G XG_VIKS] HSIJW YY Z[\O PIKK ./// s k" . t 4 Z """ +".0 @! 9 = ' $5; & !( YY ! 9Ci3 ; ! 8 .. t " = /4 +"40 p s cUrVIKORa m s dSq[IIa Z[RKRV bGSN ]RVaINOSRV SVKR VIUKNSVRO SV G OKNRV_ XG_VIKS] HSIJW YYdRWZ[\OPIKK .// sp? t .6 Z?6 +""0 ^[SOK\GqRa d s cUrVIKORa ps dSq[IIa ms o[I KNGVOSKSRVO

γγ → ν ν¯ GVW γ → γγ SV G OKNRV_ XG_VIKS] HSIJW YYDV ZNR]IIWSV_O RH ESV_QIN_ xUNR]RVHINIV]I hmI{ oNIVWO SV mIUKNSVR Z[\OS]Oj oI_INVOII LINXGV\ Z ."6

xW Q\ k cVSI[J IK GJ fRNJW M]S MSV_GbRNI

/4 +"60 ^[SOK\GqRa ds dSq[IIa ms Z[RKRV VIUKNSVR SVKING]KSRVO SV OKNRV_ XG_VIKS] HSIJW YY dRWZ[\OPIKK .//. sp, t 4 Z.664 .6?. +"?0 95 7 ! 9 >#!! γγ → ν ν¯ !( YY A!! 5 1! #

! "3 ' !' !

! ! ; 1! ; Ci ! ! !( .//4 ! ?",. +",0 = ' ># ; !5!( YY A ) ,6 t / = "."4 +" 0 PROqUKRa eUd

MqRQIJIa ss mRVJSVIGN IJI]KNRW\VGXS]O SV G

OUbINOKNRV_ XG_VIKS] HSIJW YY Z[\O PIKK ,? s p6? t 4 Z 6 6. +"0 @! 9 = ' ;; & A3 (ν ν¯) ! #!! γγ → γ(ν ν¯) '& YY 89) , 8,/ =4/4 +6/0 F# 95 7 ) & 5 !( YY >!( 8) .//. o ,6 t = 6464" +60 cUrVIKORa ps dSq[IIa ms sGOOSJIaOqG\G Pp Z[RKRV WI]G\ γ →

ν ν¯ SV GV InKINVGJ XG_VIKS] HSIJW YYZ[\OPIKK s k"., Z/6 +6.0 F# 95 7 !! @ A# ; ννγ ! YY)8 ?.=,6

/" +640 eGqRaJIa wL cGXSVqIN pw LVIWSV ve yGIVOIJ Z mIUKNSVR xXSOOSRV HNRX mIUKNRV MKGNO YYZ[\OEIbK .// s 46"Z +6"0 C : A= >#!! ! ! '1 !2 ! !( YY8) .//. 8 . t ? ^ ..4" +660 = ' C %! ; ; YY 8) 6 846 =46 +6?0 dSV_UrrS Z[RKRVO SVKING]KSRV {SK[ [RXR_IVIRUO ]RVOKGVK XG_VIKS] HSIJW YY mRUaR ]SX ? s Z ", +6,0 = == : ! ; YY 8) ?, 86. =4/4 +6 0 pWJIN MP kG[]GJJ lm ^GJJGV ^L EROIVQJUK[ dm Z[RKRV ObJSKKSV_ SV G OKNRV_ XG_VIKS] HSIJW YY Z[\OEIaPIKK ,/ s.6 Z/? +60 kSGJ\VS]qGkSNUJG ~ kSGJ\VS]qGkSNUJG D mRVJSVIGN IHHI]KO SV UGVKUX IJI]KNRW\VGXS]O Z[RKRV bNRbG_GKSRV GVW b[RKRV ObJSKKSV_ SV GV InKINVGJ HSIJW YY Z[\OEIa ,/ sw. Z.4" +?/0 C(# $ = ' %!2 ; # &! 1 (! YY >!( 8) , 84 =,4 +?0 pWJIN MP Z[RKRV ObJSKKSV_ GVW b[RKRV WSObINOSRV SV G OKNRV_ XG_VIKS] HSIJW YY pVV Z[\O me, s?, Z6

/6 +?.0 > : % ! A ># !2 ; ! YY 8) , 8? =..4 +?40 > : % ! A 85; ! ! ! ' ! (YY 8) ,4 8?6 =,6? +?"0 > : % ! A 85; ! ! YY 8 )A7 ? 8? =./ +?60 MKRVI[GX El Z[RKRV ObJSKKSV_ SV XG_VIKSrIW aG]UUX YY lZ[\Op , s. Z. , +??0 < 7 9( A *! % %!2 ;

!( YY 8) ? 8?4 =??6 +?,0 yGNWSV_ p^ kGNSV_ dL LRVK[SIN ZP Z[RKRV MbJSKKSV_ ^GO]GWIO SV LGXXGEG\ ZUJOGNO GVW K[I MbI]KNUX RH ZME6/6

YY

pOKNRb[\Ol , s",? Z."? +? 0 kGNSV_ dL yGNWSV_ p^ EGWSRuUSIK ZUJOGNO {SK[ `JKNGOKNRV_ dG_VIKS] TSIJWO YY pOKNRb[\OlPIKK s6/, ZP66 +?0 dIVKrIJ d kIN_ w fUVVIN w Z[RKRV ObJSKKSV_ SV OKNRV_ XG_VIKS] HSIJWO YYZ[\O EIa " s w6/ Z.6 +,/0 fUVVIN w MGV_ E kIN_ w Z[RKRV MbJSKKSV_ SV MKNRV_J\ dG_VIKSrIW ^ROXS] vQI]KOEIaSOSKIW YYpOKNRb[\OlPIKK 6 s "66 ZP6 +,0 pWJIN MP ^RXXIVK RV hZ[RKRV MbJSKKSV_ SV MKNRV_J\ dG_VIKSrIW vQI]KO EIaSOSKIWj YYGOKNRb[Y?/6?

/? +,.0 kGSIN sm dSJOKISV pD M[GSOUJKGVRa E~[ Z[RKRV ObJSKKSV_ SV G aIN\ OKNRV_ XG_VIKS] HSIJW YY Z[\OEIaPIKK ? s,, Z? +,40 < 7 9( A *! % %!2 ;

!5!( YY 8) , 8 =6. +,"0 pWJIN MP

M][UQINK ^ Z[RKRV ObJSKKSV_ SV G OKNRV_ XG_VIKS]

HSIJW3 NI]GJ]UJGKSRV GVW ]RXbGNSORV {SK[ bNIaSRUO ]GJ]UJGKSRVO YY Z[\OEIaPIKK ? s ,, t Z ?6? +,60 kGNSV_ dL yGNWSV_ p^ Z[RKRV MbJSKKSV_ GVW ZGSN ^NIGKSRV SV yS_[J\ dG_VIKSrIW ZUJOGNO YY pOKNRb[\Ol ./// s 6", t . Z. +,?0 *' px ># ! ! YY 8 )A7 ===% h># ; ; 5 '15 5j 93 7

8 .

^ 66. +,,0 i! *' p x : ! ' 1 ' 5! (!YY >!( ! & 4 8 ^ "/"/" +, 0 F# d5 7 ! 9 %!2

( ; ; !( YY

( ' ; =' 1 !(3 ! ! , = ... +,0 ^[SOK\GqRa ds cUrVIKORa ps dSq[IIa ms Z[RKRV ObJSKKSV_ GQRaI K[I bGSN ]NIGKSRV K[NIO[RJW SV G OKNRV_ XG_VIKS] HSIJW YYZ[\OPIKK s k"4" Z?,

/, + /0 ^[SOK\GqRa d s cUrVIKORa p s dSq[IIa m s Z[RKRV MbJSKKSV_ SV G MKNRV_ dG_VIKS] TSIJWYY DV ZNR]IIWSV_O RH K[I /K[ DVKINVGKSRVGJ MIXSVGN huUGNqO j

IWSKIW Q\ TP kIrNUqRa

sp dGKaIIa

sp EUQGqRa pm oGaq[IJSWrI M soNRSKOq\ dRO]R{3 DVOKSKUKI HRN mU]JIGN EIOIGN][ RH EUOOSGV p]GWIX\ RH M]SIV]IO s D Z . 4/ + 0 F# 95 7 ! 9 %!2 ; ; !( YY) 8 ?. ^?4 + .0 ^[SOK\GqRa ds cUrVIKORa ps dSq[IIa ms Z[RKRV ObJSKKSV_ SV G OKNRV_ XG_VIKS] HSIJW YYMUNaI\O SV yS_[ xVIN_\ Z[\OS]O .// s 6 Z. + 40 ^[SOK\GqRa d s cUrVIKORa p s dSq[IIa m s mIUKNSVR b[RKRV GVW b[RKRVb[RKRV bNR]IOOIO GO XGVSHIOKGKSRV RH K[I K[NII aINKIn JRRb SV OKNRV_ XG_VIS] HSIJWYY DV bNR]IIWSV_O RH K[I zDK[ SVKINVGKSRVGJ O][RRJ ZGNKS]JIO GVW ^ROXRJR_\IWSKIW Q\ x m pJInIIa s p dGKaIIa c[ M mSNRa s p EUQGqRa dRO]R{3DVOKSKUKI HRN mU]JIGN EIOIGN][ RH EUOOSGV p]GWIX\ RH M]SIV]IO .//4 Z.,,. ? + "0 oOGS f xNQIN o o[I bNRbG_GKSRV RH b[RKRVO SV [RXR_IVIRUO XG_VIKS] HSIJWO3 SVWIn RH NIHNG]KSRV YYZ[\OEIa ,6 s w. Z4. + 60 dIJNROI wk

MKRVI[GX El

sG]UUX bRJGNSrGKSRV GVW b[RKRV

bNRbG_GKSRV SV G XG_VIKS] HSIJW YYmURaR ^SX ,? s p4. Z"46 + ?0 @! 9 @! < = ' > # ! 1! '(5 5 YY89 ) . 864 = "?

/ + ,0 @! 9 = ' :; 1! ! !! YY! 9! i = 43 )! 4 8." =6 +

0 F 7> A1 ; 5 YY 8) 6" 8 .? t = 4"

+ 0 MKUNNR]q Zp p XRWIJ RH bUJOGNO YYpOKNRb[\Ol , s ?" Z6. +/0 oGWIXGNU x vV K[I xVIN_\ MbI]KNUX RH EIJGKSaSOKS] xJI]KNRVO SV K[I ^NGQ mIQUJG YYpOKNRb[\Ol ,4 s 4 Z?.6 +0 EUWINXGV dp MUK[INJGVW ZM o[IRN\ RH bUJOGNO3 bRJGN _GbO ObGNqO GVW ]R[INIVK XS]NR{GaI NGWSGKSRV YYpOKNRb[\Ol ,6 s ? Z6 +.0 <! = %& !( YY! ; . 8 ="4 +40 wGU_[INK\ lc

yGNWSV_ pc

ZGSN bNRWU]KSRV SV OUbINOKNRV_

XG_VIKS] HSIJWO YYpOKNRb[\Ol 4 s .,4 Z,? +"0 95 7 ! 9 B 5 ; ( & !( YY >!( 8) .// 8 ,4 . ! ,.?,4/ +60 kR\GVRaOq\ w WI sI_G yl m_ el PII wM fGV_ Me' TINXSRV WGXbSV_ SV G HINXSRVO]GJGN bJGOXG YYZ[\OEIa s w6 Z/6// +?0 @ @$ @# 9 = ! 1! ; 1 . YY 7 9!

/ +,0 ; @ YY8) 6, 8 44 = 4, + 0 lRS][S D dGKOUXRKR M[ eRO[SXUNG d uUGVKUX WSOOSbGKSRV GVW WI]G\ SV G XIWSUX YYZ[\OEIa s p6, Z, +0 7 g 1 8) ?" s ^ ? +//0 wvJSaR l mSIaIO l ZGJ Z ^[INIVqRa NGWSGKSRV Q\ XGOOJIOO VIUKNSVRO YYZ[\OPIKK? sk4?6 Z, +/0 yGNW\ M l dIJNROI w k PGV_XUSN {GaI IXSOOSRV Q\ VIUKNSVRO SV G XIWSUX YYZUQJ pOKNRV MR] pUO ? s4 Z"" +/.0 C(# $ 7 7= ) #!! !( YY 8) ,. 8 ?. = .// ./. +/40 = ' : #5 γ → ν ν¯ ν → γν !( YY 8) ,? 8 , t "/ = .?4.?, +/"0 DRGVVSOSGV pm

EGHHIJK LL ^[INIVqRa NGWSGKSRV Q\ XGOOJIOO

VIUKNSVRO SV G XG_VIKS] HSIJW YY Z[\O EIa , s w66 Z ,/4 ,/"4 +/60 LaRrWIa pp dSq[IIa ms sGOOSJIaOqG\G Pp mIUKNSVR SVVIN QNIXOOKNG[JUV_ SV G OKNRV_ XG_VIKS] HSIJW YY DV ZNR]IIWSV_O RH K[I K[ DVKINVGKSRVGJ MIXSVGN huUGNqO?j IWSKIW Q\ sp dGKaIIa pp ZIVSV sp EUQGqRa GVW pm oGaq[IJSWrI dRO]R{3 DVOKSKUKI HRN mU]JIGN EIOIGN][ RH EUOOSGV p]GWIX\ RH M]SIV]IO , sRJ D Z 444"?

/ +/?0 LaRrWIa pp dSq[IIa ms sGOOSJIaOqG\G Pp EIORVGV]I VIUKNSVR QNIXOOKNG[JUV_ ν → νγ SV G OKNRV_ XG_VIKS] HSIJW YY Z[\O PIKK , s k"/ t ." Z ..6 +/,0 d5 7 ! 9 %# 5

ν → νγ YY =' ( ' ; ! .3 =' 15 5 15 ! ! ! ! !( = 4.4 +/ 0 ^[SOK\GqRa ds dSq[IIa ms EGWSGKSaI VIUKNSVR KNGVOSKSRV ν → νγ SV OKNRV_J\ XG_VIKSrIW bJGOXG YY Z[\OPIKK s k"?, b .4..4, +/0 ^[SOK\GqRa ds dSq[IIa ms EGWSGKSaI VIUKNSVR KNGVOSKSRV ν → νγ SV OKNRV_J\ XG_VIKSrIW bJGOXGYY MUNaI\O SV yS_[ xVIN_\ Z[\OS]O ./// s 6 Z .4."? +/0 ^[SOK\GqRa d s dSq[IIa m s Z[RKRVVIUKNSVR SVKING]KSRV SV OKNRV_J\ XG_VIKSrIW bJGOXGYYDV bNR]IIWSV_O RH K[I SVKINVGKSRVGJ {RNqO[Rb MKNRV_ dG_VIKS] TSIJWO SV mIUKNSVR pOKNRb[\OS]O IWSKIW Q\ p s cUrVIKORa m s dSq[IIa p eG ZGNq[RXIVqR eGNROJGaJ ./// Z "/" +0 ^[SOK\GqRa d s dSq[IIa m s Z[RKRVVIUKNSVR DVKING]KSRVO SV MKNRV_J\ dG_VIKSrIW ZJGOXGYY ZNR]IIWSV_O RH K[I K[ DVKINVGKSRVGJ MIXSVGN uUGNqO./// IWSKIW Q\ L k ZSaRaGNRa sp dGKaIIa pp ZIVSV sp EUQGqRa GVW pm oGaq[IJSWrI dRO]R{3 DVOKSKUKI HRN mU]JIGN EIOIGN][ RH EUOOSGV p]GWIX\ RH M]SIV]IO .//. Z /66 +.0 A 5 < <! # F YY 7 9!

+40 < <! # 9 @;# @> > ! F

YY 7 9! +"0 <! # = ! > ) 5 5 1 ! i)7 , s?, ^ ." +60 fIJWRV yp MSXbJI NUJIO HRN WSO]RVKSVUSKSIO SV HSVSKI KIXbINGKUNI HSIJW K[IRN\ YY Z[\O EIa 4 s w. Z .//, +?0 oOGS fe sG]UUX bRJGNSrGKSRV SV [RXR_IVIRUO XG_VIKS] HSIJW YY Z[\O EIa ," s w/ Z .? .,/. +,0 ) %' @3 A @Ci 6, + 0 A# ! F B' < F Y > ] 8 93 9 " "" = +0 M][{SV_IN l vV _GU_I SVaGNSIV]I GVW aG]UUX bRJGNSrGKSRV YY Z[\O EIa 6 s . Z ??"?, +./0 * ! # ! 1 Y > ! 8 93 9 ,4 6/" =| 8 . 93 9 ,? ", =

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