1 12 CP 12.1 SU(2) U(1) U(1) W ±,Z [ ] [ ] [ ] u c t d s b [ ] [ ] [ ] ν e ν µ ν τ e µ τ (12.1a) (12.1b) u d u d +W u s +W s u (udd) (Λ = uds)

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1 1 1 CP 1.1 SU() U(1) U(1) W ±,Z 1 [ ] [ ] [ ] u c t d s b [ ] [ ] [ ] ν e ν µ ν τ e µ τ (1.1a) (1.1b) u d u d +W u s +W s u (udd) (Λ = uds) n + e + ν e d u +W u + e + ν e (1.a) Λ + e + ν e s u +W u + e + ν e (1.b) W ± W ± (d,s,b ) d s b d = U s (1.3) U 3 3 [ ] [ ] ψ d d cosθ C sinθ C d = cosθ C d + sinθ C s s = Uψ, U = sinθ C cosθ C s = sinθ C d + cosθ C s - θ C 1/3 ψ d +/3 ψ u b (1.4) ψ u = U 1 ψ u ψ u ψ d = (U 1ψ u ) (U ψ d ) = ψ uu 1 U ψ d ψ u(uψ d ) = ψ uψ d (1.5) (u,d) g W cosθ C (u,s) g W sinθ C

2 1 CP (1.a)(1.b) g W cos θ C,gW sin θ C cosθ C.973, sinθ C., θ C 13 Λ (u,s) s u +W d u +W (ds) FCNC: J NC = g z i=u,c,d,s q i (I 3 Qsin θ)q i Z J NC = g z i=u,c,d,s q i (I 3 Qsin θ)q i Z = (u,c ) + g z ψ d U (I 3 Qsin θ)uψ d Z = (u,c ) + g z ψ d (I 3 Qsin θ)ψ d Z = J NC (I 3 Qsin θ) d, s U U = 1 (FCNC=Flavor Changing Neutral Current) GIM (Glashow-Illiououlos-Maiani) s (u,c) d K L (ds) µ µ + ( 1.1) d s u,c GIM (1.6) 1.1: GIM u c m u = m c u c Γ(K L µ µ + ) Γ(K L all) = (7.15 ±.16) 1 9 (1.7) 3 µ ν µ + e + ν e (1.8a) τ ν τ + e + ν e τ ν τ + µ + ν µ (1.8b) (1.8c) g W ν e µ ν µ e π,k µ ν e ν µ + e + X Br(µ e + γ) < Br(τ e + γ) < Br(τ µ + γ) < (1.9a) (1.9b) (1.9c)

3 1 CP 3 ( 1.) γ 1.: µ e + γ J CC = gw ψ ul γ µ ψ d L, u ψ u c t U ud U us U ub ψ d = Uψ d, U = U cd U cs U cb U td U ts U tb d ψ d s b (1.1a) (1.1b) (N N) N N C = N(N 1)/ θ i φ i N ψ u Aψ u = e iφ u e iφ c U A UB e iφ t ψ u ψ d Bψ d = e iφ d e iφ s e iφ b ψ d (1.11) U jk e i(φ j φ k ) U jk (1.1) (N-1) N N(N 1) (N 1) = (N 1)(N ) (1.13) N U N 3 N=3 3

4 1 CP 4 U 1 c 13 s 13 e iδ c 1 s 1 U = c 3 s 3 1 s 1 c 1 s 3 c 3 s 13 e iδ c 13 1 c 1 c 13 s 1 c 13 s 13 e iδ 13 = s 1 c 3 c 1 s 3 s 13 e iδ 13 c 1 c 3 s 1 s 3 s 13 e iδ 13 s 3 c 13 s 1 s 3 c 1 c 3 s 13 e iδ 13 c 1 s 3 s 1 c 3 s 13 e iδ 13 c 3 c 13 c i j = cosθ i j, s i j = sinθ i j (1.14) (s 13 < ) V us = s 1 c 13 s 1, V ub = s 13, V cb = c 13 s 3 s 3 (1.15) s 1 = λ, s 3 = Aλ, s 13 = Aλ 3 (ρ iη) λ 1 O(λ 4 ) U = 1 λ λ Aλ 3 (ρ iη) λ 1 λ Aλ Aλ 3 (1 ρ iη) Aλ 1 (1.16) (Wolfenstein) - λ =. ±.6, A =.8 ±.4, (1.17a) ) ) ρ = ρ (1 λ =. ±.9, η = η (1 λ =.33 ±.5 (1.17b) CP 9 CP L WEAK = gw ] [ψ 1 γ µ (1 γ 5 )ψ W µ + h.c.{= ψ γ µ (1 γ 5 )ψ 1 W µ} CP gw ] [ψ γ µ (1 γ 5 )ψ 1 W µ + h.c. (1.18a) (1.18b) CP 3 ψ 1 u i (= u,c,t), ψ d j (= d,s,b) V i j - CP L WEAK u i V i j d j W + d j V i ju i W + CP d j V i j u i W + + u i V i jd j W (1.19) CP V i j = V i j - CP CP 1967 K 3 CP u,d,s 6 - (1973) - 3 V ud V ub +V cdv cb +V tdv tb = (1.)

5 1 CP 5 1.3: (a) (b) V cd V cb = 1 (1.16) Aλ 3 (ρ + iη)(1 λ /) + ( λ)aλ + Aλ 3 (1 ρ iη)(1) Aλ 3 [(ρ + iη) 1 + (1 ρ iη] = ρ = (1 λ /)ρ, η = (1 λ /)η 1.3 φ 1 = Arg [ V cdvcb ] V td Vtb Arg[V td ], φ = Arg [ V tdvtb V ud Vub ] [, φ 3 = Arg V udv ub V cd V cb ] (1.1) (1.) 1.1 J/ = (1/)I(V ud V cb V cd V ub ) (1.11) KM CP CP CP - J = c 1 c 13c 3 s 1 s 13 s 3 sinδ O(λ 6 ) 1 4 (1.3) CP * 1) 1. V ud V cb, V ub B n(udd) (u +W )(ud) (uud) + e + ν B (bd) (c +W )(d) D + (cd) + l + ν B (bd) (u +W )(d) π + (ud) + l + ν V us, V cd, V cs 1.3 * ) 1 * 1) * ) N N /N N < 1 4

6 1 CP 6 (1) () CP (3) 3 (1967) ( ) X X B 1 (B 1 ) B (B ) b(b) b(1 b) Γ Γ(X B 1 ) Γ(X all) = b, Γ(X B ) Γ(X all) = 1 b Γ(X B 1 ) Γ(X all) = b, Γ(X B ) Γ(X all) = 1 b (1.4a) (1.4b) CPT CP ( I) X X CPT : Γ(X all) = Γ(X all) (1.5) B = (b b)b 1 + {(1 b) (1 b)}b = (B 1 B )(b b) (1.6) B 1 B b b CP CPT ( ) r Γ(B r) = r Γ(B r) = (B r) (1.7) r CPT 3 r r CPT r Γ(r B) = Γ(r B) (1.8) r r (1) () - CP ( ) letogenesis ( ) GIM (d s),(u c),(b d,s) W ± K (ds) K (sd), D (uc) D (cu), B d (db) B d (bd), B s (sb) B s (bs) B d,s B, B B d,s ( 1.4) B, B B = 1, 1 CP ψ(t) = a(t) B > +b(t) B > + c i (t) f i > (1.9a) B > e iφ CP B > (1.9b)

7 1 CP 7 1.4: (a) (b) B B f i B,B φ = B,B H H Γ H = M i Γ, M = M, Γ = Γ (1.3) ψ(t) >= α(t) B > +β(t) B > (1.31) [ ] [ ][ ] i α M 11 M 1 α = t β M 1 M β M i j =< i H j >, i, j = 1, = B,B (1.3) CPT M 11 = M CP M 1 = M 1 ( I) CPT B L, B H B L > = B > +q B > B H > = B > q B + q = 1 (1.33) > λ L,H = m L,H iγ L,H / L,H ( ) CP B L, B H CP = ± = q = 1/ CP q/ λ H λ L = m i Γ = M 1 M 1 q M = 1 M 1 (1.34a) (1.34b) 1.4. B B B t = B, B t > B L, B H B L B L ()e iλ Lt, B H B L ()e iλ Ht

8 1 CP 8 B (t) > = 1 B (t) > = 1 q f ± = 1 [ B L > e iλ Lt + B H > e iλ Ht ] = f + (t) B > + q f (t) B > (1.35a) [ B L > e iλ Lt B H > e iλ Ht ] = q f (t) B > + f + (t) B > (1.35b) [ e iλ Lt ± e iλ Ht ], λ L, H = m L, H i Γ L, H (1.35c) ( ) m B 5GeV m K,m π Γ Γ = Γ H Γ L Γ L + Γ H, Γ L = Γ H = Γ (1.36) P(B B ) = P(B B ) = f + (t) Γt 1 + cos mt = e P(B B ) = q f (t) = q Γt 1 cos mt e P(B B ) = q f (t) = q Γt 1 cos mt e (1.37a) (1.37b) (1.37c) T = π/ m B τ 1 1 cτ 3µm * 3) B s B (db),b (bd) b (c,u) +W (c,u) + l + ν, b (c,u) + l + ν (1.38a) b (c,u) +W + (c,u) + l + ν, b (c,u) + l + ν (1.38b) e e + B + B ll + X ( 1.5) Asymmetry t (s) 1.5: B d B d e e + B d B d + X m d =.494 ±.1s 1 (Belle : PRL 89,5183()) * 3) K τ 1 1

9 1 CP 9 m B = ±.5MeV (1.39a) m Bd =.5 ± h/s (1.39b) m Bs > h/s 95%CL (1.39c) Γ Bd / h Γ Bs / h = (1.536 ± sec) 1 (1.39d) 1.5 CP CP A f = Γ(B f ) Γ(B f ) Γ(B f ) + Γ(B f ) (1.4) CP ( /q 1) CPT ( I ) A(B f ) =< f T B >= D i e iδsi = D i e iφ W e iδ Si i A(B f ) =< f T B >= D i e iδsi = D i e iφ W e iδ Si i (1.41a) (1.41b) i φ W δ S CP φ W CP Γ(B f ) Γ(B f ) = D 1 D sin(φ W1 φ W )sin(δ S1 δ S ) (1.4) φ W1 φ W δ S1 δ S CP CP f >= f >= ± f >= ξ f f > (B f B B f ) ( 1.4) CP (1.35) A(B f )(t) = f + (t) < f T B > + q f (t) < f T B >=< f T B > [ f (t) + λ f f + (t)] A(B f )(t) = q f (t) < f T B > + f + (t) < f T B >= ξ f q < f T B > [λ f f + (t) + f (t)] λ f = q < f T B > < f T B > = ξ q < f T B > f < f T B > (1.43a) (1.43b) (1.43c) Γ(B f ;t) < f T B > [ f + (t) + λ f f (t) + R(λ f f+ f )] (1.44a) = < f T B > e [cos Γt m t + λ f sin m ] t + I(λ f )sin mt (1.44b) Γ(B f ;t) q < f T B > [ λ f f + (t) + f (t) + R(λ f f + f )] (1.44c) = q < f T B > e [ λ Γt f cos m t + m ] sin t I(λ f )sin mt (1.44d)

10 1 CP 1 q/ 1 CP A f (t) Γ(B f ;t) Γ(B f ;t) Γ(B f ;t) + Γ(B f ;t) = S f sin mt C f cos mt (1.45a) S f = I(λ f ) 1 + λ f, C f = 1 λ f 1 + λ f (1.45b) (a) (1.16) V ud Vub (1 λ )Aλ3 (ρ + iη), V tdvtb Aλ3 (1 ρ iη) (1.46) - CP ( φ 1 ) V ub, V td V ub B V td t V td 1.3 m(b d ) = m BH m BL = R( ( ( M 1 M 1 ) = R M 1 i Γ )( ) ) 1 M1 iγ 1 (1.47a) M 1 =< B H B >, M 1 =< B H B > (1.47b) M i, j ( m i /m W ) i = j = t - M 1 Vtb V td, M 1 = M 1 V tb 1 m(b d ) AB m(b d ) V td (1.48) 1.6: φ 1 (B J/ψ + K s ) (a) (b) φ q M = 1 = V td (1.49) M 1 V td

11 第 1 章 世代混合と CP の破れ 11 次に 混合の干渉に基づく CP の破れを 終状態 f >= J/ψ(cc) + KS (sd) > (ξ f = 1) に適用して考察しよう (図 1.6) この反応には (a) のトリー図と (b) のペンギン図が寄与する ペンギン図は強い相互作用を含む過程で V に比例するのでほぼ実 ある これらの反応の虚数部は小林-益川行列要素で与えられる トリー図の寄与は Vcb cs 数である ペンギン図はループを含むので VibVis に比例するが i = t の寄与が優勢なので やはり実数である 従って式 (1.45) に現れる ℑ(λ f ) は Ã q < f T B > ℑ(λ f ) = ℑ ξ f < f T B >! µ µ V q (1.) ℑ = ℑ td === sin φ1 Vtd (1.5) となり 頂角 φ1 が測れる 現時点では純粋な B, B ビームを作ることは難しく s = MB 1GeV の B-ファク トリーの e + e+ ϒ(4S) B + B 反応において B ℓ+, B ℓ により 片方の B もしくは B を同定した 上で 他方の崩壊モードを検出する この場合 B, B に相関が生じるが ϒ(4S)(J PC = 1 ) のように CP が正で 軌道角運動量が奇数の状態を通して対生成を行い B, B の崩壊時間の差 t = (t1 t ) を測定し t1 + t について は積分して消去すれば t t とするのみで 式 (1.45) がそのまま成立することが知られている 図 1.7(a) は qξf =+1 1/N dn/d( t). qξf = t (s) 図 1.7: (a) B J/ψ + Ks 等の崩壊の非対称 qξ f = 1 は親が B で ξ f = 1 の崩壊 (Belle : PRD 66, 711(R)) (b) 小林-益川モデルの整合性 PDG : Phys.Lett.59(4)133 そのようにして得られた B J/ψ + Ks の時間分布である 非対称が明瞭に見られ CP の破れを検証したことに なる 図 (b) はこれらの非対称から得られた頂角 φ1 と それまでに得られたデータから作ったユニタリー三角形 の整合性を表したものである 全てのデータが同じ (ρ, η) に収束した 従って 現在までの所 観測された全て の CP 非保存現象は 小林-益川理論で説明可能である 演習問題 1.4 Bs Bs 系では q/ 1 であることを示せ 演習問題 1.5 ペンギン図を無視してトリー図のみ考慮するとき B π+ π, Bs (bs) ρ + Ks 崩壊反応で φ, φ3 が測れることを示せ

7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±

7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ± 7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α

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