59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ 0 ψ (7-3)
60 7 ψψ ψγ 0 ψ ψψ = ψγ 0 ψ = i F i i F i ψ i (x)ψ i (x) (7-4) ψ i (x)γ 0 ψ i (x) (7-5) i F σ(r) ψ i ψ i (x) = ψ nκjm = f nκm(r)y κm (ˆr) (7-6) g nκm (r)y κm (ˆr) Y κm (ˆr) = (lm l 1/2m s jm)y lml (ˆr)χ ms (7-7) κ > 0 j = κ 1/2 l = κ κ < 0 j = κ 1/2 l = (κ + 1) j l f nκm (r) g nκm (r) 3 L = ψ(iγ µ µ m g σ σ g ω γ µ ω µ g ρ γ µ τ a ρ a µ)ψ + 1 2 µσ µ σ 1 2 m2 σσ 2 1 4 F µνf µν + 1 2 m2 ωω µ ω ν 1 4 Ga µνg aµν + 1 2 m2 ρρ a µρ aµ (7-8)
7.1. σ-ω 61 7.1: -
62 7 7.1.1-2-2 σ ω 1. 2.
7.1. σ-ω 63 7.2: 3. MeV 4. 5.
64 7
65 8 Quantum Chromodynamics = 8.1
66 8 m π 138 MeV J P = 0, I = 1 (π 1, π 2, π 3 ) (π +, π, π 0 ) π ± = 1 2 (π 1 ± iπ 2 ), π 0 = π 3 (π ±, π 0 ) ( 2 + m 2 π)π a (x) = 0. (8-1) ( q 2 + m 2 π)π a (q) = 0. (8-2) J a (π, ρ, N,, etc) ( 2 + m 2 π)π a (x) = J a (π, ρ, N,, etc) (8-3) π, ρ, N, 1934 m π 1 fm 1 200 MeV (8-4) 1936 Anderson Anderson 1947 1948 Anderson 1930 Wenzel, Tomonaga, Oppenheimer, Schwinger 1942 Pauli Dankoff
8.2. 67 20 Skyrme Skyrme 20 1980 Witten QCD QCD Skyrme Pauli Dankoff m π ± = 139.57 MeV, m π 0 = 134.89 MeV (8-5) 138 MeV 138 MeV 0.5 MeV (938 MeV) up, down 8.2 x x x f(x) f(x) ψ R (x) f( x) ψ L (x)
68 8 ψ + (x) = f(x) + f( x) = ψ R (x) + ψ L (x), ψ (x) = f(x) f( x) = ψ R (x) ψ L (x) 1 ψ γ 5 ψ 2 σ p ψ R 1 + γ 5 ψ = N 1 + E + M 2 2 σ p χ 1 + E + M σ p ψ L 1 γ 5 ψ = N 1 E + M 2 2 σ p χ (8-6) 1 + E + M N χ 2 2 ψ L,R γ 5 γ 5 ψ L,R = ψ L,R (8-7) ψ L,R γ 5 ψ L,R (Weyl (8-6) ψ R = N 2 1 + σ ˆp 1 + σ ˆp χ, ψ L = N 2 1 σ ˆp 1 + σ ˆp χ (8-8) ˆp = p/ p ψ L,R Σ ˆp 1 Σ σ 4 4 1
8.2. 69 : Σ = σ 0 0 σ. (8-9) Σ ˆp ψ L,R M 0 ψ L,R (8-6) ψ L,R M Σ ˆp ψ = N 1 σ p E + M χ (8-10) χ σ ˆp σ ˆpχ = ±χ. (8-11) γ 5 u, d u L, d L u R, d R u L u R d L d R 4 QCD
70 8 ψ L ψ R ψ γ 5 ψ (ψ L, ψ R ) (ψ, γ 5 ψ) ψ γ 5 ψ ψ L ψ R 8.3 n φ = (φ 1, φ 2, ) φ U(α)φ = exp(iαt)φ (8-12) U(α) n n M t i (i = 1,, M) α i (i = 1,, M) t i n n αt M M αt = α i t i. (8-13) i=1 (1) U(1) φ M = 1. t = 1, U(α) = exp(iα). (8-14)
8.3. 71 L = 1 2 µφ µ φ 1 2 m2 φφ. (8-15) (2) xy φ (1) t = 0 i cos α sin α, U(α) = exp(iαt) =. (8-16) i 0 sin α cos α L = 1 2 ( ) ( µ φ 1 ) 2 m 2 φ 2 1 + 1 ( ) ( µ φ 2 ) 2 m 2 φ 2 2. (8-17) 2 (3) SU(2) φ M = 3. t = τ, U(α) = exp(i α τ) = cos α + i τ ˆα sin α. (8-18) α = (α 1, α 2, α 3 ) τ L = 1 2 ( ( µ φ φ) 2 m 2 φ φ 2). (8-19) (8-12) L(φ, φ ) = L(φ, φ), φ = U(α)φ. 0 = δl L(φ, φ ) L(φ, φ) L φ iαtφ + L ( µ φ) iαt µφ. (8-20) L x µ ( µ φ) = L φ (8-21) L 0 = x µ ( µ φ) iαtφ + L ( µ φ) iαt µφ = L φ iαtφ + L ( µ φ) iαt µφ = ) L α i µ (i ( i µ φ) ti φ. (8-22)
72 8 α i M µ Jµ a = 0, Jµ a L = i ( µ φ) ta φ, (a = 1,..., M). (8-23) 0 φ φ (8-12) g φ gφg (8-24) g Q i g = exp(iα i Q i ) (8-25) (8-24) α 0 φ iα i [Q i, φ] (8-26) Q a d 3 x J0 a = d 3 L x i ( 0 φ) ta φ = i d 3 x π(x)t a φ(x), (8-27) φ(x) π(x) = L ( 0 φ). (8-28) [φ a (x 0, x), π b (y 0 = x 0, y)] = iδ( x y)δ ab. (8-29) = i[q a, φ(x)] [ ] d 3 y π(y)t a φ(y), φ(x) = it a φ(x) (8-30)
8.4. 73 8.4 7-2 ψ γ 5 ψ ψ m L = ψ(i / m)ψ, (8-31) ψ Dirac / = µ γ µ ψ e iv ψ. (8-32) v ψ Dirac ψ = ψγ 0 V µ = ψγ µ ψ (8-33) γ 5 γ 5 ψ e iaγ 5 ψ ψ ψe iaγ 5 (8-34) a v ψ /ψ ψe iaγ 5 /e iaγ 5 ψ = ψ /ψ ψψ ψe iaγ 5 e iaγ 5 ψ ψψ (8-35) ψ /ψ ψψ A µ = ψγ µ γ 5 ψ, µ A µ = 0. (8-36)
74 8 µ A µ = m ψψ. (8-37) ψ L,R g V (1+iv), g A (1 + iγ 5 a) ψ L,R ψ R g V ψ R (1 + iv)ψ R ψ L g V ψ L (1 + iv)ψ L ψ R g A ψ R (1 + iaγ 5 )ψ R = (1 + ia)ψ R ψ L g A ψ L (1 + iaγ 5 )ψ L = (1 ia)ψ L γ 5 ψ R,L = ±ψ R,L ψ R 1 2 (g V + g A )ψ R = (1 + i(v + a)/2) ψ R g R ψ R ψ L 1 2 (g V + g A )ψ R = (1 + i(v a)/2) ψ L g L ψ L (8-38) v = a = r/2 v = a = l/2 ψ L,R g L,R g L,R g V,A U(1) U(1) L U(1) R (8-39) SU(2) L SU(2) R, SU(3) L SU(3) R (8-40)
8.4. 75 1. 2. γ 5 3. 4. 5. 1. P L = 1 γ 5 2, P R = 1 + γ 5 2. (8-41) P L + P R = 1, P 2 L,R = P L,R, P L P R = P R P L = 0. (8-42) 2. ψ(x) γ 0 ψ(t, x) ψ ψ γ 5 ψ ψ ψ L,R 3. (8-14), (8-16)
76 8 4. m L = ψ(i / m)ψ. (8-43) ψ L,R ψ L, ψ R
8.4. 77
79 9 6 Walecka σω global 9.1 SU(2) SU(2) U(1) U(1) (J P = 0 ) γ 5 L πnn = ig ψγ 5 πψ = ig(ψ L ψ R ψ R ψ L)π. (9-1) i ψγ 5 ψ (9-1) ψγ 5 ψ 2ψ L ψ R 1
80 9 i ψγ 5 πψ = i(ψ L ψ R ψ R ψ L)π (9-2) 1 ψ L e il ψ L, ψ R e ir ψ R (9-3) (l = r) ( l = r = a/2) i(ψ L ψ Re ia ψ R ψ Le ia )π. (9-4) ψ L ψ R a (9-1) γ 5 π i qγ 5 q (9-5) (9-1) ψγ 5 ψ πnn z N = 2ψ L ψ R z q = 2ψ L ψ R Im[z N ]Im[z q ] + Re[z N ]Re[z q ] (9-6) Re[z N ] Re[z N ] = ψ L ψ R + ψ R ψ L = ψψ. (9-7) Re[z q ] q L q R + q R q L = qq σ. (9-8)
9.1. 81 9.1: ψ L ψ R γ 5 L int = g( ψσψ + i ψγ 5 πψ) = g ψ(σ + iγ 5 π)ψ. (9-9) πnn σnn ψ e iv ψ, ψ ψe iv σ π ψ e ia ψ, ψ ψe ia (9-10) σ π cos a sin a sin a cos a σ π (9-11) 9.1 σ π σ 2 + π 2 V (x) σ 2 + π 2 σ 2 + π 2
82 9 L σ = ψ (i / g(σ + iγ 5 π)) ψ + 1 (( µ σ) 2 + ( µ π) 2) 2 µ2 2 (σ2 + π 2 ) λ 4 (σ2 + π 2 ) 2 (9-12) 9.2 L σnn = g σ ψψσ, (9-13) L ωnn = g ω ψγ µ ψω µ, (9-14) L P S πnn = g P S i ψγ 5 ψπ. (9-15) i ψγ 5 ψ (Pseudo-scalar = PS) (PS) (Pseudo-vector = PV 1 ) L P V πnn = g P V ψγ µ γ 5 ψ µ π. (9-16) ψ(x) = d 3 p ( s (2π) 3 u(p, s)a p,s e ipx + v(p, s)b p,se +ipx), (9-17) d 3 p 1 σ(x) = (2π) 3 (a p e ipx + a 2ωp pe +ipx), (9-18) ω µ (x) = d 3 p 1 (2π) 3 (a p ɛ (λ) µ e ipx + a 2ωp pɛ (λ) µ e +ipx). (9-19) λ 1 PV
9.2. 83 ψ(x) u(p, s) p, z s Dirac v(p, s) p, z s Dirac A p,s, B p,s a p ɛ (λ) µ, λ = 1, 2, 3 E + M 1 u(p, s) = 2E σ p χ s E = p 2 + M 2 (9-20) M + E χ s (s = ±1/2) Pauli u u = 1 2 1/ 2ω p (ω p = m 2 σ + p 2 2 v = p/m (9-13)-(9-16) 9.2 p L σnn pq g σ 1 2ωq exp( ipx iqx + ip x)χ s 1χ s. (9-21) χ s 1χ s 1 ) p L ωnn pq = g ω (ūγ 0 uω 0 ū γ ωu exp( ipx iqx + ip x) [ g ω χ s χ s ω 0 1 ( 2M χ s ( p + p ) ω + i σ q ω ) ] χ s 1 2ωq exp( ipx iqx + ip x). (9-22) χ s 1χ s 1 1 ūu ūγ 0 u = u u
84 9 p' p' 1 p' 2 p' 1 p' 2 q q q p p 1 p 2 p 1 p 2 9.2: v = p/m q σ p L P S pq g P S χ σ q s 2M χ 1 s exp( ipx iqx + ip x). (9-23) 2ωq σ q q exp( iqx) = exp( iq 0 t+i q x) χ s χ s exp( ipx + p x) 9.2 V σ ( q) = n 0 V n n V 0 E n E 0
9.2. 85 g2 σ = 2 1 ( ) 2 gσ = ω q 2ωq q 2 + m 2 σ (9-24) 3 V σ ( x) = g2 σ e mr 4π r (9-25) V ω ( q) = λ ɛ (λ) µ ɛ (λ) ν g 2 ω q 2 + m 2 ω (9-26) µ, ν µ = ν = 0 (µ = ν = 0) λ ɛ (λ) µ ɛ (λ) ν = g µν (9-27) g2 ω V ω ( q) = + q 2 + m 2 ω (9-28) 5 1. ψγ5 ψ ψψ 2 ψ L ψ R γ 5
86 9 2. 3. 1. ψψ, ψγ5 ψ, ψγ µ ψ, ψγ µ γ 5 ψ ψ R, ψ L 2. E + M ψ(x) = 2E 1 σ p M + E e( ipx)χ s, φ(x) = 1 e( ipx) (9-29) 2ω E = p 2 + M 2, ω = p 2 + m 2, M, m 3. πn PS PV g P S, g P V 4. E 1 + E 2 + ω q (E 1 + E 2 ) ω q
9.2. 87
89 10 -Goldstone 600 MeV 10.1 (9-12) L σ = ψ (i / g(σ + iγ 5 π)) ψ + 1 2 (( µ σ) 2 + ( µ π) 2) µ2 2 (σ2 + π 2 ) λ 4 (σ2 + π 2 ) 2. (10-1)
90 10 -Goldstone V (σ, π) = µ2 2 (σ2 + π 2 ) + λ 4 (σ2 + π 2 ) 2 (10-2) (10-1) (10-1) µ 2, λ, g µ 2 /2 λ µ 2 µ 2 1 H σ = d 3 x [ ψ ( iα + g(σ + iγ5 π)) ψ + 1 2 (Π2 σ + ( σ) 2 ) + 1 2 (Π2 π + ( π) 2 ) + µ2 2 (σ2 + π 2 ) + λ ] 4 (σ2 + π 2 ) 2. (10-3) Π σ, Π π σ, π 1 1
10.1. 91 Π 2 σ,π ( σ) 2, ( π) 2 x φ σ 2 + π 2 V (φ) = µ2 2 φ2 + λ 4 (φ2 ) 2 (10-4) φ 0 φ = φ 0 + ϕ φ 0 ϕ φ 0 = 0 φ 0 0 ϕ 2 V (φ) = V (φ 0 + ϕ) = V (φ 0 ) + V (φ 0 )ϕ + 1 2 V (φ 0 )ϕ 2 +. (10-5) V (φ 0 ) = 0 V (φ 0 ) 0 φ 2 µ 2 /2 µ 2 /2 < 0 φ 0 0 µ 2 /2 µ 2 µ 2 > 0 (σ, π) = (0, 0) σ π m 2 /2
92 10 -Goldstone µ 2 µ 2 V (φ) V (φ) σ π σ π 10.1: µ 2 > 0 µ 2 < 0 σ σ, π µ 2 Wigner (σ, π) = (0, 0) (9-11) µ 2 < 0 φ 2 = σ 2 + π 2 = fπ( ) 2 10.1 f π 9-4 φ 2 = fπ 2 c a c a c = c (10-6) c c c = exp(ca ) 0 (10-7)
10.2. 93 9-2 0, 1, 2,... π n π π σ (9-11) σ (σ, π) = ( µ 2 /λ, 0) (f π, 0) -Goldstone -Goldstone -Golstone -Golstone σ f π + σ σ π L = ψ (i / gf π g(σ + iγ 5 π)) 1 4 µ2 + fπ 2 λfπσ 2 2 λf π σ(σ 2 + π 2 ) λ 4 (σ2 + π 2 ) 2 (10-8) µ 2 > 0 σ π (σ, π) = (f π, 0) (9-11) 10.2 2.6 10 8 2 (9-10) (9-11) 2 8.4 10 17
94 10 -Goldstone ν e σ π π 10.2: V µ = ψγ µ ψ, A µ = ψγ µ γ 5 ψ σ µ π + π µ σ. (10-9) (10-9) L W I = G 2 pγ µ (1 g A γ 5 )nē(1 γ 5 )γ µ ν. (10-10) n, p G = (1.026 ± 0.001) 10 5 m 2 p g A g A 1.25 n p 10.2 n p, σ π (10-9) 10.2 0 A µ (x) π(p) = 0 σ(x) µ π(x) π(p) ip µ f π e ipx (10-11)
10.2. 95 f π f π = 93 MeV f π πn πn g 10 M N = gf π 900 MeV f π 1. µ 2 2. 3. 4. 1. H L H = Π φ (x) φ(x) L (10-12) φ Π φ L/ φ (10-3)
96 10 -Goldstone 2. c = exp(ca ) 0 c a c = c c 3. (10-8) 4. (10-9) 5. (10-9) 0 σ 0 = f π 0 A µ (x) π(p) ip µ f π e ipx (10-13)
10.2. 97
99 11 - Nambu-Jona-Lasinio) µ 2 µ 2 > 0 Wigner µ 2 < 0 -Goldstone µ 2 µ 2 -Jona-Lasinio NJL Jona-Lasinio -Jona-Lasinio QCD Ginzburg-Landau -Jona-Lasinio BCS(Bardeed- Cooper-Schrifer) BCS Cooper Cooper QCD
100 11 - Nambu-Jona-Lasinio) 11.1 NJL (x) L NJL = ψi /ψ + g 2 ( ( ψψ) 2 + ( ψiγ 5 ψ) 2). (11-1) g g > 0 QCD u, d order parameter ψψ σ i ψγ 5 ψ ψψ (11-1) g 2 ( ψψ) 2 g ψψ ψψ. (11-2) m = g ψψ (11-3) ψψ ψψ ψ Λ 10-1 ψψ Λ = 2 0 d 3 p (2π) 3 m p 2 + m 2 (11-4)
11.1. 101 m/ p 2 + m 2 u pu p = 1 ū p u p d 3 p/(2π) 3 Λ m (self consistent) 11.1 m Λ m = 2 0 d 3 p (2π) 3 m p 2 + m 2 (11-5) = = 11.1: m g m = 0 m 0 (11-5) m 1 = g 2π 2 ( Λ Λ 2 + m 2 m 2 ln Λ + ) Λ 2 + m 2. (11-6) m m 11.2 Λ g Λ
102 11 - Nambu-Jona-Lasinio) Λ QCD Λ 1 Λ m m 11.2: m (11-3) ψψ -Jona-Lasinio ψψ ψ L ψ R + ψ R ψ L
11.2. -Goldstone 103 H = i ψ γ ψ g 2 ( ψψ) 2 = ψ( i γ + m)ψ g 2 ( ψψ) 2 m ψψ (11-7) ψψ ψψ = m/g H = ψ( i γ + m)ψ + m2 2g Λ d 3 p = 2 p 0 (2π) 3 2 + m 2 + m2 2g V (m) (11-8) m m (11-5) 11.2 -Goldstone -Goldstone -Jona-Lasinio m ( ψiγ 5 ψ) 2 (11-1) p H 0 = 2 p 2 + m 2 (11-9)
104 11 - Nambu-Jona-Lasinio) (11-1) p 1, p 2, p p 2 p 1 p 1, p 2, p p 2 p 1 q(p 1) q(p 2) g( ψiγ 5 ψ) 2 q(p 1 ) q(p 2 ) = 4g (11-10) g d 3 q V ( x) = ( 4g)ei q x (2π) 3 = 4gδ( x). (11-11) ( ) 2 2 + m 2 4gδ( x) Ψ( x) = EΨ( x). (11-12) 3 (11-12) 2 p 2 + m 2 d 3 q Ψ( p) 4g Ψ( q) = EΨ( p). (11-13) (2π) 3 d 3 q (2π) 3 1 V n (11-14) V N Ψ = a 1 a 2.. (11-15)
11.2. -Goldstone 105 (1/V ) n a n 2 d 3 p/(2π) 3 Ψ( p) 2 = 1 2m 2 a 1 1 1 a 1 a 1 m 2 + 2 a 2 1 1 a 4g 2 a = E 2 (11-16)...... n 2 m 2 + (n ) 2 a n 4g V N a n = Ea n (11-17) n (11-17) K = (1/V ) a n a n a n = 4gK 2 (n ) 2 + m 2 E (11-18) a n 1/V K K = 4gK V n 1 2 (n ) 2 + m 2 E (11-19) K 1 = 4g V n 1 2 (n ) 2 + m 2 E (11-20) E 0 (11-5) -Goldstone 11.3 (11.3) f(e) E f(e) = 1 f(e) 2m E m 2 + (n ) 2 < E < m 2 + ((n + 1) ) 2 E < 2m f(e) = 1 a n
106 11 - Nambu-Jona-Lasinio) n n E = 0 E > 0 a n m a (m) n m n 1.... 2m E.... 11.3: f(e) 1. µ 2 -Jona- Lasinio 2. ψψ 3. BCS 4. 5.
11.2. -Goldstone 107 1. (11-4) dp (11-6) 2. Λ = 600 MeV m = 300 MeV g g [ ] 2 MeV 2 fm 2 3. ϕ( x) ϕ ( x)ϕ( x) = 1. a x a x E(a) = ( d 3 x 1 2m ϕ ( x) 2 ϕ( x) ϕ ( x) 1 ) r ϕ( x) (11-21) a E(a) a V ( x) = gδ( x) g
108 11 - Nambu-Jona-Lasinio)
109 12 12.1 10 12.1 1 70 80% 12.1 σ -Goldstone 1
110 12 12.1: L σ = ψ (i / g(σ + iγ 5 π)) ψ + 1 (( µ σ) 2 + ( µ π) 2) µ2 2 2 (σ2 + π 2 ) λ 4 (σ2 + π 2 ) 2 (12-1) µ 2, λ, g L σω = ψ (i / g(σ + iγ 5 π) g ω ω µ γ µ ) ψ + 1 (( µ σ) 2 + ( µ π) 2) µ2 2 2 (σ2 + π 2 ) λ 4 (σ2 + π 2 ) 2 + ɛσ 1 4 F µνf µν + g ω ω µ ω µ (σ 2 + π 2 ) (12-2)
12.1. σ 111 σ σ, ω µ ωδ µ0, π a = 0 σ ω 12.2 12.2: σ-ω
112 12 12.3: σ-ω 1.
12.1. σ 113 2. 3. 4. 1. (12-2) σ σ = f π = 93 MeV m = gf π, m 2 σ = 2λfπ, 2 m 2 π = ɛ/fπ, 2 m 2 ω = g ω fπ 2 2. m = 938 MeV, m σ = 550 MeV, m π = 139 MeV, m ω = 780 MeV g, µ, λ, ɛ, g ω
114 12
115 13 1/2 13.1 -Goldstone
116 13 13.2 10 80 4 He 4 He 11 Li 50 10MeV 40 60 1934 100 13.3
13.3. 117 50MeV 1. 2. 3. 4.
118 13 5. 6. 7.
13.3. 119