July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i

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1 July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac Picture) e iht e iht Ψ H S Φ(t) O S Ψ(t) S H Φ O H (t) Ψ H D Φ(t) O D (t) Ψ(t) D Ψ(t) D e ih t Ψ(t) S e ih t e iht Ψ H U(t, ) Ψ( ) D U(t, )e iht e iht e ih e ih e iht e ih(t t) e ih U(t,t )U(t, ) U(t, ) U (t, )U (t, ) U(,t) U(t, t) U(t, ) i t U(t, )e ih t ( H + H)e ih(t ) e ih H int (t)u(t, )

2 H int (t) e ih t H int e ih t U(. ) t U(t, ) i H int (t )U(t, )dt n U(t, ) i ( i) n t t dt H int (t )dt +( i) t dt tn t dt H int (t ) t dt H int (t )+ dt n H int (t )H int (t ) H int (t n ) () T t i >t i > >t in T (H int (t )H int (t ) H int (t n )) H int (t i )H int (t i ) H int (t in ) n n! t,t, t n n () ( i) n n! t dt t dt t dt n T (H int (t )H int (t ) H int (t n )) () U(t, ) + ( i) n n T exp ( i T exp ( i n! t t dt dt H int (t )) t t dt n T (H int (t ) H int (t n )) d 4 xh int (x)) (3). S ( ) (t ) H int (t) H a in a out S a in

3 S S U(+, ) (4) a out S a in b b in in b S a in S in b S a in.3 i f f S i δ fi + i(π) 4 δ (4) (p f p i )T fi (5) p f,p i n P p + p + + p n, (6) P + P p + p + + p n (7) (p V ) M fi T fi T fi P V p V p nv M fi, (8) P V P V p V p n V M fi. (9) f (S ) i (π) 4 δ (4) (p f p i ) T fi (π) 4 δ (4) () () (π) 4 δ (4) () d 4 xe i x V T () () T w fi (π) 4 δ (4) (p f p i ) T fi V. () L V L 3 3

4 n (8) () Γ f w fi P p V p nv (π)4 δ (4) (p f P ) M fi. (3) f V d 3 p p p V (π) 3 d3 p (4) (4) (3) V d 3 p V p (π) (5) 3 p Γ d 3 p P (π) d 3 p n 3 (π) 3 p p n (π) 4 δ (4) (p f P ) M fi. (6) Γ M P M τ τ Γ (7) (6) M fi τ P v P M/ v /c τ v /c (8) flux σ fi flux w fi (9) flux ρ v rel () 4

5 ρ v rel V ρ V () ()() (9) σ fi V w fi (π)4 δ (4) (p f p i ) T fi V. () v rel v rel T fi (9) T fi P V P V p V p n V M fi (3) (4) V (5) (4)(3) () σ fi P P v rel d 3 p (π) 3 p d 3 p n (π) 3 p n (π) 4 δ (4) (p f p i ) T fi. (4) P P v rel P P P + M,P M v rel v P (5) P P P P P v rel M P M P M (M + P ) M M (P P ) M M (6) (4) Mφller σ 4 d 3 p d 3 p n (P P ) M M (π) 3 p (π) 3 p n (π) 4 δ (4) (p f p i ) T fi (7) 5

6 (λ) u(pλ)ū(pλ) p/+m λ v(pλ) v(pλ) p/ m (8) λ Trace Tr (γ µ γ ν )4g µν, Tr (γ µ γ ν γ ρ ) Tr (γ µ γ ν γ ρ γ σ )4(g µν g ρσ g µρ g νσ + g µσ g νρ ) Tr (γ 5 γ µ γ ν ), Tr (γ 5 γ µ γ ν γ ρ γ σ )4iɛ µνρσ (9) γ µ γ ν γ µ γ ν, a/γ µ a/a µ a/ a γ µ, γ µ γ ν γ ρ γ σ γ µ γ σ γ ρ γ ν (3) Trace Trace Tr (AB) Tr(BA) (3) γ µ γ 5 + γ 5 γ µ, (γ 5 ) Tr (γ µ γ µ γ µn ) Tr((γ 5 ) γ µ γ µ γ µn ) Tr (γ 5 γ µ γ 5 γ µ γ µn ) ( ) n Tr (γ 5 γ µ γ µ γ µn γ 5 ) ( ) n Tr (γ µ γ µ γ µn ) (3) Tr (γ µ γ µ γ µn )Tr(γ 5 γ µ γ µ γ µn ) n (33).4 d 3 p d 3 p n Φ n (P ) (π) 4 δ (4) (p (π) 3 p (π) 3 p + p + p n P ) (34) n n Phase Space M P 6

7 δ (3) (p + p ) p Phase Space d 3 p Φ (E,m,m ) (π) πδ( p + m + p + m E) 3 p + m p + m (35) p + m + p + m E p p (E (m + m ) E )(E (m m ) ) (36) dxδ(f(x)) f (x ) p f(x ) Φ (E,m,m ) 8πE (E (m + m ) )(E (m m ) ) (37) m m, m Φ (E,m,) 8π ) ( m E (38) W ± (Charged Current) H CC H CC G F J µj µ (39) Charged Current J mu (4) J µ J µ lepton + J µ quark J µ lepton ēγ µ ( γ 5 )ν e + µγ µ ( γ 5 )ν µ + τγ µ ( γ 5 )ν τ (4) J µ quark 3 ( ) d i γ µ ( γ 5 )u j V KM ij (4) i,j V KM 7

8 . Pion Charged Current Pion (π ± ) (39) S- (4) S i d 4 xh CC i d 4 x G F J µ (x)j µ (x) (43) i G F ( ) d 4 x J lepton,µ J µ lepton + J lepton,µ J µ quark + J quark,µ J µ lepton + J quark,µ J µ quark i π (P ) f µ (p ), ν µ (p ) (43) µ(p ), ν µ (p ) S π (P ) i G F d 4 x µ(p ), ν µ (p ) J quark,µ J µ lepton π (P ) i G F d 4 x J quark,µ π (P ) µ(p ), ν µ (p ) J µ lepton (44) ū d π π π Pion f π π ūd J quark,µ π Axial-Vector ūγ µ γ 5 d f π ū(x)γ µ γ 5 d(x) π (P ) P V if πp µ e ip x. (45) J quark,µ π (P ) V ud P V if πp µ e ip x (46) V ud KM (, ) (44) Pion µ µ(p ) µ(x) p V ū(p ) e ip x ν µ ν µ (p ) ν µ (x) p V v(p ) e ip x µ(p ), ν µ (p ) J µ lepton µ(p ) µ(x) γ µ ( γ 5 ) ν µ (p ) ν µ (x) p V p V ū(p )γ µ ( γ 5 )v(p ) e i(p +p ) x (47) 8

9 u(p ) v(p ) (45) (47) (44) µ(p ), ν µ (p ) S π (P ) i (π)4 δ (4) (p + p P ) G F V ud P V p V p V if π P µ ū(p )γ µ ( γ 5 )v(p ) M fi i G F V ud f π (p,µ + p,µ )ū(p )γ µ ( γ 5 )v(p ) i V udg F f π m µ ū(p )( γ 5 )v(p ) (48) ū(p )(p/ m µ ) p/ v(p ) (48) (6) (8) Γ(π µ + ν µ ) G F V ud f π m µ 4M π Tr (p/ ( + γ 5 )(p/ + m µ )( γ 5 ))Φ (M) (49) Tr (p/ ( + γ 5 )(p/ + m µ )( γ 5 )) Trp/ (p/ + m µ )) 8p p 4 ( (p + p ) p p ) 4(M π m µ) (5) Γ(π µ + ν µ ) G F V ud fπ ( ) m µ M π m µ. (5) 8π Mπ π e + ν e Γ(π e + ν e ) G F V ( ) ud fπ m e M π m e (5) 8π Mπ Γ(π e + ν e ) Γ(π µ + ν µ ) m e m µ 4. (53) 9

10 3 3. φ(x) φ (+) (x)+φ ( ) (x) φ (+) (x) ωk V a ke ik x k φ ( ) (x) k ωk V a k eik x φ (+) (x), φ ( ) (x) (54) [φ (+) (x),φ ( ) (y)] i (+) (x y) [φ ( ) (x),φ (+) (y)] i ( ) (x y) [φ (+) (x),φ (+) (y)] [φ ( ) (x),φ ( ) (y)] i (+) (x y) i e ik (x y) ω k k V d 4 k i (π) 3 θ(k )δ(k m )e ik (x y), ( ) (x y) i d 4 k (π) 3 θ( k )δ(k m )e ik (x y) (+) (y x) d 3 k (π) 3 e ik (x y) ω k (55) (56) (+) (x y) ( ) (x y) Jordan-Pauli (x y) (+) (x y)+ ( ) (x y) (57) 3. (Normal Product) φ ( ) φ (+) : φ(x)φ(y) : φ(x)φ(y) ( φ (+) (x)+φ ( ) (x) )( φ (+) (y)+φ ( ) (y) ) : φ(x)φ(y) : φ (+) (x)φ (+) (y)+φ ( ) (y)φ (+) (x)+φ ( ) (x)φ (+) (y)+φ ( ) (x)φ ( ) (y) φ(x)φ(y) : φ(x)φ(y) :+[φ (+) (x),φ ( ) (y)] : φ(x)φ(y) :+i (+) (x y) (58)

11 A µ (x)a ν (y) : A µ (x)a ν (y) : g µν id (+) (x y) ψ α (x) ψ β (y) : ψ α (x) ψ β (y) :+is (+) αβ (x y) S (+) (x y) {ψ α (+) ( ) (x), ψ β (y)} (iγµ µ + m) i (+) (x y) D (+) (x y) (+) (x y) : φ(x)φ(y) : : φ(y)φ(x) : : A µ (x)a ν (y) : :A ν (y)a µ (x) : : ψ α (x) ψ β (y) : : ψ β (y)ψ α (x) : 3.3 Feynman Propagator Dyson S T- T- T- (58) Tφ(x)φ(y) θ(x y )φ(x)φ(y)+θ(y x )φ(y)φ(x) (59) : φ(x)φ(y) :+ F (x y) (6) F (x y) θ(x y )i (+) (x y)+θ(y x )i (+) (y x) (6) x >y y x y >x x y (56) y >x y x T- (55) F (x) θ(x ) πi θ( x ) πi + πi + πi dτ τ i eiτx dτ τ i e iτx dτ τ i eiτx dτ τ i e iτx d 3 k e iωkx+ik x (π) 3 ω k d 3 k e iωkx ik x (π) 3 ω k

12 k ω k τ k k { } d 4 k e ik x F (x) i (π) 4 ω k ω k k i + ω k + k i d 4 k (π) i 4 k m + i e ik x (6) (6) F (x) F (x) Tφ(x)φ(y) (63) φ ( + m )φ(x) d dx θ(x )δ(x ) ( + m ) F (x y) δ(x y ) [ φ(x),φ(y)] iδ (4) (x y) (64) F (x) 3 F (x) TA µ (x)a ν (y) g µν D F (x y) d 4 k (π) ig µν 4 k + i e ik (x y) (65) Tψ α (x) ψ β (y) S F (x y) αβ (i / x + m) αβ F (x y) d 4 k (π) i(k/+m) αβ 4 k m + i e ik (x y) { } d 4 k (π) i e ik (x y) (66) 4 k/ m + i T- (59) αβ Tφ(x)φ(y) Tφ(y)φ(x) TA µ (x)a ν (y) TA ν (y)a µ (x) Tψ α (x) ψ β (y) T ψ β (y)ψ α (x) T ψ β (y)ψ α (x) Tψ α (x) ψ β (y) S F (x y) αβ (67) 4 4. H int (x) e ψγ µ ψa µ (x) (68) 3 (6)

13 J µ (x) e ψγ µ ψ ψγ µ ψ H int (x) e : ψ(x)γ µ ψ(x) :A µ (x) (69) e e 4π 37 S S () (ie) d 4 x d 4 yt{: ψ(x)γ µ ψ(x) :A µ (x) : ψ(y)γ ν ψ(y) :A ν (y)} (7) ψ A µ T T S () (ie) d 4 x Wick d 4 yt{: ψ(x)γ µ ψ(x) :: ψ(y)γ ν ψ(y) :}T {A µ (x)a ν (y)} (7) T {A µ (x)a ν (y)} : A µ (x)a ν (y) :+A µ (x)a ν (y) (7) T {: ψ(x)γ µ ψ(x) :: ψ(y)γ ν ψ(y) :} : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : + : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) :+: ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : + : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : (73) d 4 k A µ (x)a ν (y) g µν D F (x y) (π) ig µν 4 k + i e ik (x y) (74) ψ α (x) ψ β (y) ψ β (y)ψ α (x) S F (x y) αβ (i / x + m) αβ F (x y) d 4 k (π) i(k/+m) αβ 4 k m + i e ik (x y) { } d 4 k (π) i e ik (x y) (75) 4 k/ m + i (73) ψ(x) ψ(x) Wick (7)(73) (7) 8 4 S eγ () (ie) d 4 x d 4 y : A µ (x)a ν (y) : ( ) : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) :+: ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : αβ. (76) 3

14 ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) :: ψ(y)γ ν ψ(y) ψ(x)γ µ ψ(x) : (77) (76) µ ν x y (76) S () eγ (ie) d 4 x d 4 y : A µ (x)a ν (y) :: ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : (78) S 4 S () ee (ie) d 4 x d 4 y A µ (x)a ν (y) : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : (79) S () e (ie) d 4 x d 4 y A µ (x)a ν (y) : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : (8) S () γ (ie) d 4 x d 4 y : A µ (x)a ν (y) : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) (8) ψ α (x)(γ µ ) αβ ψ β (x) ψ γ (y)(γ ν ) γδ ψ δ (y) αβγδ (γ µ ) αβ ψ β (x) ψ γ (y)(γ ν ) γδ ψ δ (y) ψ α (x) αβγδ Tr (γ µ S F (x y)γ ν S F (y x)) (8) 4

15 S () γ (ie) d 4 x d 4 y : A µ (x)a ν (y) :Tr(γ µ S F (x y)γ ν S F (y x)) (83) S vac () (ie) d 4 x d 4 y A µ (x)a ν (y) ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) (84) (74)(8) S vac () (ie) d 4 x d 4 y ( g µν D F (x y)) Tr (γ µ S F (x y)γ ν S F (y x)) (85) S () (ie) d 4 x d 4 y : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) ::A µ (x)a ν (y) : (86) S S () S () eγ + S () ee + S () e + S () γ + S () vac (87) 4. p λ ψ(x) e (p,λ) ψ(x)b pλ, {ψ(y),b pλ }, (88) p V u pλ e ipx (89) 5

16 e (p,λ) ψ(x) bpλ ψ(x), (9) {b pλ, ψ(x)}, p V ū pλ e ipx (9) ψ(x) e + (p,λ) ψ(x)d pλ, { ψ(y),d pλ }, (9) p V v pλ e ipx (93) e + (p,λ) ψ(x) d pλ ψ(x), {d pλ,ψ(x)}, (94) p V v pλ e ipx. (95) k e (), e () e (), e (), k/ k z e (), e () (96) ( ) ( ) ɛ µ (k, ), ɛ µ (k, ) e () e () (97) ( ) ɛ µ (k,l) i k µ i k, (98) k k k ɛ µ (k,s) ikµ k k µ k µ k µ kµ,k µ kµ k ( i k k k ) (99) k µ ɛ µ (k,h) (h,,s), () ɛ µ (k,l)ɛµ (k,l)ɛ µ (k,s)ɛµ (k,s), () ɛ µ (k,l)ɛµ (k,s)ɛ µ (k,s)ɛµ (k,l). () 6

17 g µν ɛ µ (k,h)ɛ ν (k,h)+ɛ µ (k,s)ɛ ν (k,l)+ɛ µ (k,l)ɛ ν (k,s) g µν. (3) h, e (), e () e (+) (e () + ie () ), e ( ) (e () ie () ) (4) ( ) ( ɛ µ (k, +), ɛ µ (k, ) e (+) e ( ) ) (5) g µν g µν h,h ɛ µ (k,h)η(h, h)ɛ ν (k,h) (6) h± ɛ µ (k,h)ɛ ν (k,h)+ɛ µ (k,s)ɛ ν (k,l)+ɛ µ (k,l)ɛ ν (k,s). (7) η(h, h) (for h, or h ±), (8) η(l, S) η(s, L), (9) others A µ (x) k,h,h k V [ a(k,h)η(h, h )ɛ µ (k,h )e ik x + a (k,h)η(h, h )ɛ µ(k,h )e ik x] () [a(k,h),a (k,h )] δ k,k η(h, h ) () γ(k,h) a (k,h) () A µ (x) γ(k,h) A µ (x)a (k,h) (3) [A µ (x),a (k,h)] ɛ µ (k,h) e ik x k V 7

18 h η(h, h )η(h,h )δ h,h (4) γ(k,h) A µ (x) a(k,h)a µ (x) (5) [a(k,h),a µ (x)] ɛ µ(k,h) e ik x k V null γ(k,s) γ(k,l) null A µ (x) γ(k,s) A µ (x)a (k,s) (6) ɛ µ (k,s) e ik x k V (7) ik µ k k V e ik x ( ) µ k k V e ik x (8) S 4.3 S S () eγ ψ ψ (p,λ), (p,λ ) e (p,λ ) : ψα (x)ψ β (y) : e (p,λ) b p λ : ψ α (x)ψ β (y) :b pλ (9) e (p,λ ) ψα (x) ψ β (y) e (p,λ) b pλ ψ β(y) b p λ ψ α (x) ψ β (y)b pλ {ψ(y),b pλ } b pλ ψ β(y) b pλ (9) b pλ (9) b pλ ψ(y) 8

19 b pλ ψ(x) (88)(9) (9) (k,h), (k,h ) (3)(5) γ(k,h ) : A µ (x)a ν (y) : γ(k,h) a(k,h ):A µ (x)a ν (y) :a (k,h) γ(k,h ) A µ (x) A ν (y) γ(k,h) + γ(k,h ) A ν (y) A µ (x) γ(k,h) () (78) S fi e (p,λ ),γ(k,h ) S () eγ e (p,λ),γ(k,h) (ie) d 4 x d 4 y γ(k,h ) : A µ (x)a ν (y) : γ(k,h) e (p,λ ) : ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) : e (p,λ) ( (ie) d 4 x d 4 y γ(k,h ) A µ (x) A ν (y) γ(k,h) ) + γ(k,h ) A ν (y) A µ (x) γ(k,h) e (p,λ ) ψ(x) γ µ S F (x y)γ ν ψ(y) e (p,λ) () x, y S F (x y) y x x, y ieγ µ x () (89)(9)(3)(5) (75) S fi (ie) d 4 x d 4 y k V k V p V p V (ɛ µ(k,h )ɛ ν (k,h) e ik x e ik y + ɛ ν(k ),h )ɛ µ (k,h) e ik y e ik x d 4 q (π) ū i 4 p λ γµ q/ m + i γν u pλ e ip x e ipy e iq (x y) () x y (π) 4 δ(p + k q)(π) 4 δ(q k p) 9

20 e (p,λ ) γ(k,h ) e (p,λ ) γ(k,h ) p + k p k e (p, λ) γ(k, h) e (p, λ) γ(k, h) : (π) 4 δ(p k q)(π) 4 δ(q + k p) () q S fi (ie) k V k V p V p V (π)4 δ(p + k p k) ( ɛ µ(k,h i )ɛ ν (k,h)ū p λ γµ p/+k/ m + i γν u pλ +ɛ ν(k,h i ) )ɛ µ (k,h)ū p λ γµ p/ k/ m + i γν u pλ (3) q m q m N V N P N L N P (N V ) 4 N L T fi f S i δ fi + i(π) 4 δ (4) (p f p i )T fi (4) 4 N V

21 (p V ) M fi T fi k V k V p V p V M fi. (5) (3) µ ν im fi (ie) ɛ µ (k,h )ɛ ν (k,h) ( ū p λ γ µ i p/+k/ m + i γν + γ ν i ) p/ k/ m + i γµ u pλ. (6) () im ieγ µ i p/ m + i ig µν k + i (7) (8) u pλ (9) ū pλ (3) v pλ (3) v pλ (3) ɛ µ (k,h) (33) ɛ µ (k,h) (34) N L N P (N V ) d 4 k (π) Ward-Takahashi null state Ward-Takahashi (6) null state

22 (6) (6) ɛ ν (k,h) k ν k ν M ν k ν M ν (ie) ɛ µ (k,h )ū p λ ( γ µ p/+k/ m + i k/+k/ p/ k/ m + i γµ ) u pλ (35) p + k p + k p k p k k/ k/ (p/+k/ m) (p/ m) ( ) k/ (p/ m) (p/ k/ m) ( ) (36) (p/ m)u pλ, ū p λ (p/ m) k ν M ν. (37) (6) ɛ µ (k,h ) k µ 4.5 e + e + µ + µ + ψ (e) (x) ψ (µ) (x) H int e ( ψ(e) γ λ ψ (e) + ψ (µ) γ λ ψ (µ) ) Aλ (x) (38) 4 p 4 p 4 p 3 4 p 4 µ (p 3,λ 3 ) µ + (p 4,λ 4 ) p + p e (p,λ ) e + (p,λ )

23 λ,λ,λ 3,λ 4 µ (p 3,λ 3 ),µ + (p 4,λ 4 ) S e (p,λ ),e + (p,λ ) M fi i(π) 4 δ (4) (p + p p 3 p 4 ) p V p V p 3 V p 4 V M fi ū(p 3 λ 3 )γ µ v(p 4 λ 4 ) e g µν (p + p ) + i v(p λ )γ ν u(p λ ) (39) (ū(pλ)γ µ v(p λ )) v(p λ )γ ν u(pλ) (4) 4 λ,λ,λ 3 λ 4 M fi e 4 v(p 4 λ 4 )γ µ u(p 3 λ 3 )ū(p 3 λ 3 )γ ρ v(p 4 λ 4 ) 4 λ 3,λ 4 g µνg ρσ [(p + p ) ] ū(p λ )γ ν v(p λ ) v(p λ )γ σ u(p λ ) λ,λ u(pλ)ū(pλ) λ v(pλ) v(pλ) λ p/+m p/ m λ 3,λ 4 v(p 4 λ 4 )γ µ u(p 3 λ 3 )ū(p 3 λ 3 )γ ρ v(p 4 λ 4 ) Tr[γ µ (p/ 3 + m µ )γ ρ (p/ 4 m µ )] 4(p µ 3 pρ 4 + pρ 3 pµ 4 gµρ p 3 p 4 g µρ m µ ) λ,λ ū(p λ )γ ν v(p λ ) v(p λ )γ σ u(p λ ) Tr[γ ν (p/ m e )γ σ (p/ + m e )] Trace 4(p ν pσ + pσ pν gνσ p p g νσ m e ) Tr (γ µ γ ν )4g µν, Tr (γ µ γ ν γ ρ ) Tr (γ µ γ ν γ ρ γ σ )4(g µν g ρσ g µρ g νσ + g µσ g νρ ) (4) 3

24 4 λ,λ,λ 3 λ 4 M fi 8e 4 (p + p ) {(p 4 p 3 )(p p 4 )+(p p 4 )(p p 3 ) } +m µ(p p )+m e(p 3 p 4 )+m em µ (4) σ d 3 p 3 d 3 p 4 4 (p p ) m 4 (π) 3 p e 3 (π) 3 p 4 (π) 4 δ (4) (p 3 + p 4 p p ) 4 M fi (43) λ,λ,λ 3,λ 4 p (E,p), p (E, p) E m e + p p + p (E,) (43) δ (3) (p 3 + p 4 ) p 4 p 4 p 3 p 3 (E, p ), p 4 (E, p ) E m µ + p e (p ) µ (p 3 ) θ µ + (p 4 ) e + (p ) s (p + p ) 4E E s p p m e p p { (p + p ) m e } s m e (44) 4 (p p ) m 4 e s(s 4m e ) 4

25 δ(e s) d 3 p p z p p θ p d 3 p (π) 3 (E ) πδ(e s) p dp dφd(cos θ) π δ(e s) (π) (E ) d(cos θ) p (45) E p (E ) dp de de δ(e s) (45) E m µ (46) dp E de p d(cos θ) p 3 p 4 (s m µ ) s 4m µ 6π s p p 3 p p 4 p p 4 E p p E pp cos θ p p 3 E + p p E + pp cos θ (43) dσ 4πα s 4m µ d(cos θ) 8s s 4m e { + 4(m µ + m e ) + s 4πα 8s ( + cos θ) ( 4(m µ + m e ) ) } + 4m µ m e cos θ s s s α e 4π σ total 4πα 8s ( + m µ s ) 4m µ s m e m µ (47) (48) 5

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