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3 19 QED 19.1 {γ µ, γ ν } g µν {γ µ, γ µ } δ µ µ 8 γ µ γ µ 4. γ µ γ α γ µ g µα γ α γ µ γ µ γ α 4γ α γ α. γ µ γ α γ β γ µ 4g αβ, γ µ γ α γ β γ γ γ µ γ γ γ β γ α n : trγ µ 1 γµ γµ n n 0 n trab trba trγ µ γ ν 1 tr{γµ, γ ν } g µν tr 1 4g µν. n 4 trγ µ γ ν γ ρ γ σ 1/ tr{γ µ, γ ν γ ρ γ σ } 1/ tr{γ µ, γ ν }γ ρ γ σ γ ν {γ µ, γ ρ }γ σ + γ ν γ ρ {γ µ, γ σ } g µν trγ ρ γ σ g µρ trγ ν γ σ + g µσ trγ ν γ ρ 4g µν g ρσ 4g µρ g νσ + 4g µσ g νρ 3

4 γ 5 trγ 5 γ µ 1 γµ γµ n n 0 trγ 5 0, trγ 5 γ µ γ ν 0 trγ 5 γ µ γ ν γ ρ γ σ 4 ɛ µνρσ 4 trγ 5 γ µ γ ν γ ρ γ σ ɛ µνρσ trγ 5 γ 0 γ 1 γ γ 3 iɛ µνρσ trγ 5 γ 5 4iɛ µνρσ /p mu s p 0, /p + mv s p 0, u s pū s p /p + m, v s p v s p /p m s /p γ p s 19. : eē µ µ QED e 19.1 M igµν k 1 +k v ieγ µ u 1 ū 3 ieγ ν v 4 ie k 1 +k v γ µ u 1 ū 3 γ µ v 4 u i u si k i, v i v si k i M e 4 v k s 1 s 1 +k 4 γ µ u 1 ū 3 γ µ v 4 ū 1 γ ν v v 4 γ ν u 3 4 s 1 s 4 e 4 trγ k 1 +k 4 µ u 1 ū 1 γ ν v v trγ µ v 4 v 4 γ ν u 3 ū 3 s 1 s 4 e 4 k 1 +k trγ µ/k 4 1 γ ν /k tr γ µ /k 4 mγ ν /k 3 +m. 4

5 19.1: m 106 MeV µ 0.5 MeV m s 1 s 4 M e 4 4k1µ k k 1 +k 4 ν + 4k 1ν k µ 4g µν k 1 k 4k µ 3 kν 4 + 4k ν 3k µ 4 4gµν k 3 k 4 4g µν m 3e4 k 1 +k 4 k1 k 3 k k 4 + k 1 k 4 k k 3 + m k 1 k. k µ 1 k, k, kµ k, k, kµ 3 k, k, k µ 4 k, k k k θ s 1 s 4 M 3e4 k k k cos θ + k + k k cos θ + m k k 4 4e4 k k + k cos θ + m dσ dω 1 k 64π k k 1 4 M e4 k k + m + k cos θ 56π k 5 s 1 s 4 5

6 k k m π σ π dθ sin θ dσ dω e4 k m k +m 96π k : ee ee : A, B A ie k 1 k 3 ū 3γ µ u 1 ū 4 γ µ u, B ie k 1 k 4 ū 4γ µ u 1 ū 3 γ µ u. B M A + B A + B + AB + A B s 1 s 4 s 1 s 4 s 1 s 4 s 1 s 4 s 1 s 4 s 1 s 4 A e 4 ū3 γ k 1 k 3 4 µ u 1 ū 4 γ µ u ū 1 γ ν u 3 ū γ ν u 4 e 4 k 1 k 3 tr γ 4 µ /k 1 +mγ ν /k 3 +m tr γ µ /k +mγ ν /k 4 +m e 4 4k1µ k k 1 k 3 4 3ν + 4k 1ν k 3µ 4g µν k 1 k 3 + 4m g µν 4k µ kν 4 + 4k ν k µ 4 4gµν k k 4 + 4m g µν 6

7 A 3e4 k1 k k 1 k 3 4 k 3 k 4 + k 1 k 4 k k 3 m k 1 k 3 m k k 4 + m 4. m B k µ 3 kµ 4 B 3e4 k 1 k k 1 k 4 4 k 3 k 4 + k 1 k 3 k k 4 m k 1 k 4 m k k 3 + m. 4 AB e 4 ū3 γ k 1 k 3 k 1 k 4 µ u 1 ū 4 γ µ u ū 1 γ ν u 4 ū γ ν u 3 e 4 k 1 k 3 k 1 k 4 tr γ µ /k 1 +mγ ν /k 4 +mγ µ /k +mγ ν /k 3 +m tr /k 4 γ ν /k 1 + 4mk 1ν +k 4ν m γ ν /k +mγ ν /k 3 +m 3 3 k 1 k k 3 k m k 1 k 4, 16m k 1 k + k 1 k 3 + k k 4 + k 3 k 4, 16m k k 3 3m 4 AB 16e 4 k 1 k k 1 k 3 k 1 k 4 k 3 k 4 + m 4 m k 1 k + k 1 k 3 + k 1 k 4 + k k 3 + k k 4 + k 3 k 4. A B k, θ k 1 k k 3 k 4 k + m, k 1 k 3 k k 4 k 1 cos θ + m, k 1 k 4 k k 3 k 1+cos θ + m A 8e 4 k +m + k 4 1+cos θ + 4m k cos θ + m 4 k 4 1 cos θ, B 8e 4 k +m + k 4 1 cos θ 4m k cos θ + m 4 AB A B 8e 4 4 k 4 m 4 k 4 sin θ 7 k 4 1+cos θ,

8 M 16e4 k 4 sin 4 9 k k 4 cos θ + k 4 cos 4 θ θ + 4m k + 1m k cos θ + m 4 + 3m 4 cos θ 16e4 sin 4 4 sin θ + m m +4 k 4 3 sin θ. θ k 4 dσ dω 1 64π k +m e 4 64π k +m sin 4 θ 1 M 4 4 sin θ + m m +4 k 4 3 sin θ k : eē eē : A, B A ie k 1 k 3 ū 3γ µ u 1 v γ µ v 4, B ie k 1 +k v γ µ u 1 ū 3 γ µ v 4. B 8

9 k i m m A spins B spins AB spins e 4 k 1 k 3 tr γ 4 µ /k 1 γ ν /k 3 tr γ µ /k 4 γ ν /k 3e4 k 1 k 3 4 k1 k k 3 k 4 + k 1 k 4 k k 3, e 4 k 1 +k tr γ 4 µ /k 1 γ ν /k tr γ µ /k 4 γ ν /k 3 3e4 k 1 +k 4 k1 k 3 k k 4 + k 1 k 4 k k 3, e 4 k 1 k 3 k 1 +k tr γ µ /k 1 γ ν /k γ µ /k 4 γ ν /k 3 3e 4 k 1 k 3 k 1 +k k 1 k 4 k k 3. k, θ k 1 k k 3 k 4 k, k 1 k 3 k k 4 k sin θ/, k 1 k 4 k k 3 k cos θ/ A 8e cos4 θ/ sin 4 θ/, B 4e cos θ, AB + A B AB 16e 4 cos4 θ/ sin θ/. dσ dω π k 4 e 4 18π k A + B + AB + A B 1 + cos 4 θ/ sin 4 θ/ cos4 θ/ sin θ/ cos θ 9

10 19.5 : eγ eγ : M ie ε µ λ q ε ν λq ū s p 1 γ µ /p+/q m γ 1 ν + γ ν /p /q m γ µ u s p 4 p µ, p µ q µ, q µ ε µ 0 q + εµ 3 q qµ / q : 1 /p+/q m /qu sp u s p, ū s p /q 1 /p /q m ū s p M M λ0 +M λ3 0, M λ 0 +M λ 3 0. d λ d λ M M λ,λ λ1, λ 1, 10

11 d λ λ 0 1, +1 M : M d λ d λ M phys s,s λ1, λ 1, s,s,λ,λ e 4 d λ d λ ε µ λ q ε ν λq ε ρ λ q ε σ λq s,s,λ,λ ū s p 1 γ µ /p+/q m γ 1 ν + γ ν /p /q m γ µ u s p 1 ū s p γ σ /p+/q m γ 1 ρ + γ ρ /p /q m γ σ u s p 1 /p+/q m /p+/q+m p q, phys M λ d λ ε µ λ qεν λq g λλ ε µ λ qεν λ q gµν λ,λ 1 /p /q m /p /q +m p q e4 4p q tr /p +mγ µ /p+/q+mγ ν /p+mγ ν /p+/q+mγ µ + e4 4p q tr /p +mγ ν /p /q +mγ µ /p+mγ µ /p /q +mγ ν e 4 tr /p +mγ 4p q p q µ /p+/q+mγ ν /p+mγ µ /p /q +mγ ν e 4 tr /p +mγ 4p q p q ν /p /q +mγ µ /p+mγ ν /p+/q+mγ µ 4 tr /p m/p+/q+m/p m/p+/q+m 4 p µ p µ + q µ q µ p m q q p q p q 4 16m 4 +16m p q+16m p q +3 p q p q, 48m 4 +16m p q 16m p q, 64m 4 3m 4 + 3m p q + 3 p q p q. q q, q q 3m 4 3m p q + 3 p q p q. 11

12 3 tr /p +m /pγ ν /p+/q+4mp ν +q ν m γ ν /p /q +mγ ν 3 16m 4 16m p q+3m p q, 64m 4 +3m p q 3m p q, 16m 4 16m p q 3 3m m p q 16m p q. q q, q q 4 3m m p q 16m p q. M 8e 4 p q p q + p q p q m phys 1 p q m 4 p q p q 1 p q p µ E, p, q µ p, p, p µ E, p, q µ p, p E p + m θ phys M 8e 4 E + p cos θ E + p + E + p m sin θ E + p cos θ E + p cos θ dσ dω π E + p 4 e 4 3π E + p M phys E + p cos θ E + p + E + p m sin θ E + p cos θ E + p cos θ e 4 E σ 16πE + p E + p + 4m p E p m E log E + p p 3 E p 1

13 m p dσ dω e 4 3π m sin θ, σ e4 6πm q q p q p q v p /E ω, ω 19.5 ω ω 1 + v cos θ 1 v 1 + v p p E + p, 1 v m p p m E + p cos θ 19.5: Θ cos Θ cos θ + v 1 + v cos θ sin Θ m sin θ E + p cos θ, E cos θ + p E + p cos θ cos Θ cos θ 13 m E + p cos θ

14 1 dσ cos Θ dcos Θdφ dσ cos θ dω e 4 ω ω 3π m ω ω + ω ω sin Θ ω, cos Θ p ω 1 + ω/m, E m + ω 1 + ω/m, cos θ m + ω cos Θ ω m + ω1 cos Θ ω ω ω 1 + ω/m1 cos Θ * λ λ + π/m1 cos Θ 19.7 : eē γγ : M ie ε µ λ q ε ν λq v s p 1 γ µ /p /q m γ 1 ν + γ ν /p /q m γ µ u s p. 14

15 M phys e4 4p q tr /p mγ µ /p /q+mγ ν /p+mγ ν /p /q+mγ µ + e4 4p q tr /p mγ ν /p /q +mγ µ /p+mγ µ /p /q +mγ ν e 4 + tr /p mγ 4p q p q µ /p /q+mγ ν /p+mγ µ /p /q +mγ ν e 4 + tr /p mγ 4p q p q ν /p /q +mγ µ /p+mγ ν /p /q+mγ µ 1 3m 4 + 3m p q + 3 p q p q. 3m 4 + 3m p q + 3 p q p q m m p q + 16m p q M 8e 4 p q p q + p q 1 p q + m p q + 1 p q phys 1 m 4 p q + 1 p q p µ E, p, q µ E, p, q µ E, p, E p p + m θ M 8e 4 E+ p cos θ E p cos θ + E p cos θ E+ p cos θ phys + m E 1 E p cos θ + 1 m4 E+ p cos θ E 1 E p cos θ + 1 E+ p cos θ 16e 4 E4 p 4 cos 4 θ + m p sin θ E p cos θ. dσ dω 1 64π E E p 1 4 M phys e 4 64π E p E 4 p 4 cos 4 θ + m p sin θ E p cos θ. 15

16 θ 0 π/ e 4 σ 1 m E 3πE p E + p + m E p m4 log E + p E 3 p E p : L I G F f π µ π lγ µ 1 γ 5 ν : π l ν. l µ ν τ τ π G F f π π 19.7: 19.7 M ig F f π ik µ ū s pγ µ 1 γ 5 v s p G F f π ū s p/p+/p 1 γ 5 v s p G F f π m l ū s p1 γ 5 v s p. 16

17 m l 0 M G F f π m l s,s ū s p1 γ 5 v s p v s p 1+γ 5 u s p s,s G F f π m l tr /p+m l 1 γ 5 /p 1+γ 5 G F f π m l tr /p+m l /p 1+γ 5 4G F f π m l p p. p µ E, p, p µ p, p, E p + m l M 4G F f π m l p E + p s,s Γ T p 3π m π M 4π G F f π m l πm s,s π p E + p m π E + p p p m π m l /m π Γ T GF f π m l m π m l 8πmπ 3 m e MeV, m µ MeV, m π MeV, f π 18 MeV, G F MeV µ Γ/T MeV, Γ/T MeV µ π µ ν µ π τ π MeV sec f π µ π 19.9 s, s s s

18 M ss G F f π m l s p1 γ 5 v ν p ū l s v s p iγ u sp, u s p Rθ, φ 0 f s p 0 Rθ, φ Rπ θ, φ ± π ±irθ, φσ 1, R θ, φɛr θ, φ ɛ det R θ, φ ɛ M ss ±i G F f π m l f s l p σ f ν s p k0 + k 0 f + k k0 k, 0 0 f k0 k k i 0 k0 + k s s +1 0 M ++ ±i G F f π m l p E + p, E p + m l π l ν 19.8 π + lν 4 CP 18

19 19.8: µ e ν e ν µ 19.9 M ig F ū 1 γ µ 1 γ 5 v ū 3 γ µ 1 γ 5 u s k 19.9: µ m M G F tr /p 1 γ µ 1 γ 5 /p γ ν 1 γ 5 tr /p 3 γ µ 1 γ 5 /k+mγ ν 1 γ 5 s,s 1 s 3 G F 8 p µ 1 pν + p ν 1p µ gµν p 1 p iɛ µνρσ p 1ρ p σ 8 p 3µ k ν + p 3ν k µ g µν p 3 k iɛ µνλτ p λ 3k τ 18G F p 1 p 3 p k. 19

20 ɛ µνρσ ɛ µνλτ δ ρ λ δσ τ δ ρ τ δ σ λ s,s 1 s 3 M 18G F m p 1 p p 3 1 cos θ. θ p 1 p 3 3 dγ <f S i> <i i> d 3 p 1 π 3 p 0 1 d 3 p π 3 p 0 d 3 p 3 π 3 p 0 3, <i i> k 0 V, <f S i> π 4 T V δ 4 p 1 +p +p 3 k M dγ T δ4 p 1 +p +p 3 k 16π 5 k 0 p 0 1 p0 p0 3 1 s,s 1 s 3 M d 3 p 1 d 3 p d 3 p 3 4G F π 5 δ4 p 1 +p +p 3 k1 cos θ d 3 p 1 d 3 p d 3 p 3 G F 8π 5 δ p 1 + p 3 + p 1 +p 3 m1 cos θ d 3 p 1 d 3 p 3 d 3 p 1 d 3 p 3 8π p 1 p 3 d p 1 d p 3 sin θdθ p 1 m s, p 3 m t, cos θ x Γ T G F m5 I I 3π 3, 1 dsdt s t dx 1 x δ s +t +stx + s + t 1 y s + t + stx I I 1 s+t dsdt dy y s + t y δy + s + t s t dsdt s + t s + t 1 Σ Σ s t < s t < s + t s < 1 t < 1 s + t > 1 I 1 0 ds 1 1 s dt s + t s + t

21 Γ T G F m5 19π MeV µ T/Γ sec : n pe ν e M ig F ū 1 γ µ 1 γ 5 v ū 3 γ µ a bγ 5 u s k. a, b 19.10: M G F tr /p 1 +m e γ µ 1 γ 5 /p γ ν 1 γ 5 s,s 1 s 3 tr /p 3 +m p γ µ a bγ 5 /k+m n γ ν a bγ 5 G F 8 p µ 1 pν +p ν 1p µ gµν p 1 p iɛ µνρσ p 1ρ p σ 4 a +b p 3µ k ν +p 3ν k µ g µν p 3 k +m n m p a b g µν iabɛ µνλτ p λ 3k τ 3G F a+b p 1 p 3 p k+a b p 1 k p p 3 m n m p a b p 1 p. m e, m p, m n 1

22 p 1 p θ p 3 m p M 3G F m n m p p 0 1p 0 a +3b +C cos θ, C a b p 1 p p 0 s,s 1 s 3 1 p0 dγ T δ4 p 1 +p +p 3 k 16π 5 m n m p p 0 1 p0 1 s,s 1 s 3 M d 3 p 1 d 3 p d 3 p 3 G F π 5 δ4 p 1 +p +p 3 ka +3b +C cos θ d 3 p 1 d 3 p d 3 p 3 G F π 5 δ m e + p 1 + p +m p m n a +3b +C cos θ d 3 p 1 d 3 p d 3 p 1 d 3 p 8π p 1 p d p 1 d p sin θdθ dγ T G F π 3 a +3b p 1 m n m p m e+ p 1 d p1. Γ T G F m5 ea +3b mn m p J, Jk π 3 m e k 1 a 0.98, b 1.5, G F MeV, 0 ds s k 1+s m e MeV, m n MeV, m p MeV J T/Γ MeV sec 9 10 sec 19.1 π 0 : π 0 γγ

23 19.11: A, B A d A ε µ λ 1 p 1 ε ν 4 k λ p π 1 4 i/k /p + m tr g N γ 5 k p m +iɛ ieγ i/k + m ν k m +iɛ ieγ i/k + /p 1 + m µ k+p 1 m +iɛ ig N e ε µ λ 1 p 1 ε ν λ p d 4 k π 4 tr γ 5 /k /p + mγ ν /k + mγ µ /k + /p 1 + m k m +iɛ k+p 1 m +iɛ k p m +iɛ. m 4im ɛ µνρσ p ρ 1 pσ 4 p µ 1, pµ m p µ 1, pµ 0 d A 4g N e m ɛ µνρσ ε µ λ 1 p 1 ε ν λ p p ρ 4 k 1 1 pσ π 4 k m + iɛ 3 d 4 k 1 d 4 I : π 4 k m + iɛ k 1 3 π 4 k0 k m +iɛ 3 d 3 k dk 0 α k π 4 k 0 + α 3 k 0 α 3, + m iɛ k + m k 0 3πi/8α 5 ɛ +0 d 3 k 3πi I π 4 8 k + m i 5/ 3π m 3

24 A A ig Ne 8π m ɛ µνρσε µ λ 1 p 1 ε ν λ p p ρ 1 pσ γ 5 B A p µ 1, pµ λ 1, λ A M A + B A ig Ne 4π m ɛ µνρσε µ λ 1 p 1 ε ν λ p p ρ 1 pσ p 1 p p p p z M M λ1 λ M 11 M 0, M 1 M 1 g Ne p π m Γ T π p 3π M λ 1 1, λ 1, M λ1 λ g N e4 p 5 3π 5 M m g N e4 M 3 10 π 5 m M π 0 p M/ g N 10, e 0.303, M 135 MeV, m 938 MeV π 0 T/Γ sec π 0 γγ π 0 γγ 0 π 0 γγ

25 k θ, φ 0 cosθ/ k 0 + k sinθ/ sinθ/ k 0 k k u + k 0 + k cosθ/ k 0 k sinθ/, u cosθ/ k k i 0 k sinθ/ k 0 + k k 0 k cosθ/. k 0 + k k 0 k + m m : v s k iγ u s k k, ±1 a ± k a 1 k 1 a + k + a k a ± k 1 a 1 k ia k a k ε µ λ ka λk λ1, λ± i a + k a k ε µ λ ka λk ε µ ±k 1 ε µ 1 k ± iεµ k k θ, 0 ε µ 1 k 0, cos θ, 0, sin θ µ, ε µ k 0, 0, 1, 0 µ ε µ ±k 1 0, cos θ, ±i, sin θ µ. ε µ λ k εµ λ k ε µ λ k 5

26 19.1: : eē γγ 19.1 A A ε µ λ q ε ν λ q v s p ieγ ν i /p /q m ieγ µu s p πiα p q v s p /ε λ q /p /q + m/ε λ qu s p. m α e /4π θ, 0 p µ E, 0, 0, p µ, q µ E, E sin θ, 0, E cos θ µ, E p + m, p p, q q B A q µ q µ, λ λ M A + B dσ dω 1 E 64π E p M M σ π dθ sin θ dσ 56π Ep, dω θ λ λ 0 π/, 0 π σ σ λλ p ss σ λλ ss p σλ λ ss p. 6

27 QED C P σss λλ p σλλ s s p, σss λλ p σ λ λ s s p σ ++p, ++ σ++p, σ ++p, + σ + p, ++ σ + p σ + p : * Excel VBA / for VBA URL: 7

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