note01

γ 5

J, M α J, M α = c JM JM J, M c JM

e ipr p / M p = 0 M J(J + 1) / Λ p / M J(J + 1) / Λ ~ 1 / m π

m π ~ 138 MeV J P,I = 0,1 π 1, π, π 3 ( ) ( π +, π 0, π ) ( ), π 0 = π 3 π ± = m 1 π1 ± iπ ( ) π ±, π 0 ( + m π )π a F.T. ( x) = 0 q ( + m π )π a q ( ) = 0 J a (π, ρ, N,,etc) ( + m π )π a = J a (π,ρ, N,,etc ) π, ρ,n, π + ~ ud, π 0 ~ uu + dd, π + ~ du q q

π = qq + qq qq + qq qq qq +... F π ( q ) γ γ ρ 0 e π ± e π ± ρ ρ π = 6 F π q ( q = 0) = 6 m ρ ρ m ρ = 770 π = 0.63 fm (exp) = 0.65 fm π

Space-like egion Pion fom facto F π ( q ) Time-like egion 4m π q [GeV ] q [GeV ] f π = 93

q µ, q µ = (ω,q i ), ( ω, q i ) a, b p µ, p µ = (E, p i ), ( E, p i ) α, β ( ) π b ( ω, q ) π a ω, q N α ( ) ( E, p ) N β E, p S-wave P-wave N,, etc

S = exp(iδ ) =1 + it = 1 + i q K 1 i q K K = 1 q tanδ a = lim q 0 q (l+1) tanδ = lim q 0 q l K l =1 l = 0 a u Tu = 4π s M χ fχ f u χ s = ( p + q) M q i q i q σ, q σ, q q, σ i q q σ f = B + C q q + id σ q q f q, q τ 1 = 0 1 1 0, τ = 0 i i 0, τ 3 = 1 0 0 1 0 0 0 0 0 i 0 i 0 t 1 = 0 0 i, t = 0 0 0, t 3 = i 0 0 0 i 0 i 0 0 0 0 0 t τ

f = b 0 + b 1 t τ + c 0 + c 1 t τ ( ) q q + i d 0 + d 1 t τ σ q q ( ) b 0,1, c 0, 1, d 0,1 S - wave P - wave b 0 b 1 c 0 c 1 d 0 d 1 Spin - - a a f f Isospin a f a f a f a:, f: a I = a 1, a 3 a I,J = a 11, a 13, a 31, a 33 b, c, d a b 0 1 3 b 1 1 3 c 0 c = 1 d 0 d 1 3 1 3 1 3 3 1 3 3 1 3 1 3 1 3 1 3 3 4 3 1 3 3 3 3 1 3 1 3 a 1 a 3 a 11 a 13 a 31 a 33

0.1 m π 1 10 m π 1 b 0 = 0.010 m π 1, b 1 = 0.091 m π 1 a 1 = 0.173 m π 1, a 3 = 0.101 m π 1 c 0 = 0.08 m π 3, c 1 = 0.175 m π 3 a 11 = 0.081 m π 3, a 13 = 0.030 m π 3 d 0 = 0.190 m π 3, d 1 = 0.069 m π 3 a 31 = 0.045 m π 3, a 33 = 0.14 m π 3 b 0 a 33 P 33 S-wave ρ P-wave

L πnn = ign τ π γ 5 N N g γ 5 γ 5 = 0 1 1 0 N(p) = E + M 1 σ p E + M 1 χ exp(ipx) M σ p χ exp(ipx) M L πnn = ign (p ) τ π g γ 5 N( p 1 ) M M χ σ i τ a χ ( ) ( π i π a ) σ q q σ q p 1 p σ 1 q τ 1 σ q τ

V(q) = g M = 1 g 3 M σ 1 q σ q q + m τ 1 τ q q + m σ 1 σ q + q + m S 1 ( q ˆ ) τ 1 τ FT 1 g 3 M 1 m e m 4πδ 3 () σ 4π 1 + m + 3 m + 3 e m S 1 (ˆ ) σ τ 1 τ V στ () = 1 g 3 M 1 e m 4π m σ 1 σ τ 1 τ σ 1 σ τ 1 τ = ( S(S +1) )( I(I +1) ) = S I L 9 0 0 odd SO 3 1 0 even TE 3 0 1 even SE 1 1 1 odd TO

1 q + m Λ q + Λ 1 q + m Λ Λ q + Λ 1 q + m ~ Λ q + Λ 1 = ~ Λ 4 Λ Λ 1 A q + m + Λ q + Λ 1 q + m (If Λ 1 ~ Λ ) 1 q + Λ 1 1 q + Λ B q + Λ 1 + C q + Λ 1 q + m Λ 1, Λ g 4π ~ 14.9 g ~ 13.7 ; Λ = 1400 MeV a q 1 b q p 1 p T = g u ( p )iγ 5 τ b i / p 1 + q / 1 M τ a iγ 5 u(p 1 ) + g u (p )iγ 5 τ a i p / 1 q / M τ b iγ 5 u( p 1 ) At theshold a=b + i g M u (p )u( p 1 )

p 1 ~ p ~ (M,0), q / 1 ~ q / ~ γ 0 m π m π σ = (π) 4 T 4 ( pq) m M dφ m dφ u ( p )u( p 1 ) ~ M dφ = 1 q (π) 5 m + M (angle integated ), T ~ 4g σ ~ g4 4πM ~ 15 fm =150 mb

γ 5 m L = ψ ( i / m )ψ, ψ = u d u, d ψ exp(iτ v )ψ ; ψ ψ exp( i τ v ), ( ψ ψ exp( iτ v )) τ v = 3 i=1 τ i v i v i (i =1,,3) ψ / ψ ψ e i τ v / e i τ v ψ = ψ / ψ (invaiant ) ψ ψ ψ e i τ v e i τ v ψ = ψ ψ ( " ) γ 5 γ 5 ψ e i τ a γ 5 ψ ; ψ ψ e i τ a γ 5, ψ ψ e iτ a γ ( 5 ) ψ / ψ ψ e i τ a γ 5 / e i τ a γ 5 ψ = ψ / ψ (invaiant ) ψ ψ ψ e i τ a γ 5 e i τ a γ 5 ψ ψ ψ (noninvaiant )

γ 5 ψ = 1+ γ 5 + 1 γ 5 ψ ψ R +ψ L P R 1+ γ 5, P L 1 γ 5 P R = P R, P L = P L, P R P L = 0, P R + P L = 1 γ 5 ψ R = γ 5 P R ψ = +ψ R, γ 5 ψ L = γ 5 P L ψ = ψ L ψ R, ψ L P 1+ γ ψ R γ 5 0 ψ = 1 γ 5 γ 0 ψ = 1 γ 5 ψ = ψ L ψ R ψ L c =1 p = (0,0,1) s z = +1 / 1 ψ + = σ p χ = 1 0 1 0 = 1 1 χ

χ = 1 0 1 0 ( ) ( 0 1) ψ = 1 σ p ˆ χ = 0 1 0 1 = 1 1 χ P R ψ + = 1 + γ 5 ψ + = P L ψ = 1 γ 5 ψ = P R ψ = P L ψ + = 0 1 1 1 1 1 1 1 χ = ψ + 1 1 1 1 1 1 1 χ = ψ ψ R ψ L exp(iτ v ) ~ 1 + iτ v +... g V exp(iτ a γ 5 ) ~ 1 + iτ a γ 5 +... g A g V ψ R = (1 + iτ v )ψ R g V ψ L = (1 + iτ v )ψ L g A ψ R = (1 + iτ aγ 5 )ψ R = (1 + iτ a)ψ R g A ψ L = (1 + iτ aγ 5 )ψ L = (1 iτ a)ψ L γ 5 ψ R,L = ±ψ R,L

g V + g A ψ R = 1 + iτ v + a ψ R g R ψ R g V g A ψ R = iτ v a ψ R g V + g A ψ L = 1 + iτ v a ψ L g L ψ L g V g A ψ L = iτ v + a ψ L v = a v = a l g R ψ R = (1+ iτ )ψ R, g L ψ R = ψ R g L ψ L = (1 + iτ l )ψ L, g R ψ L = ψ L ψ R, ψ L g R, g L g R, g L g V, g A g R, g L g A, g B g A ψ R, g B ψ L g A ψ L, g B ψ R g A g B g B g A (D A, D B ) g A g B ψ R ~ (D A, D B ), ψ L ~ (D B, D A ) D A A

g L(φ, µ φ) φ = ( φ 1,φ,L,φ n) ( ) G φ = φ 1,φ,L,φ n φ a g ( D(g)φ ) a D(g) ab φ b b ~ 1 + i ε m T m φ φ + δφ m D(g) T m φ n n ε m 0 = δl(φ, µ φ) L(φ + δφ, µ φ + µ δφ) L(φ, µ φ) ~ L φ δφ + L ( µ φ) µ δφ L = i µ ( µ φ) T m φ ε m µ J m m m ( ) µ ε m ( J m ) µ = i L ( µ φ) T m φ ε m m µ ( J m ) µ = 0 V µ a = ψ γ µ τ a ψ, A µ a = ψ γ µ γ 5 τ a ψ

V µ a = ψ R γ µ τ a ψ R +ψ L γ µ τ a ψ L R µ a + L µ a A µ a =ψ R γ µ τ a ψ R ψ L γ µ τ a ψ L R µ a L µ a V µ a ψ (1 iτ v)γ µ τ a (1 + iτ v)ψ A µ a ~ ψ γ µ τ a ψ + i ψ γ µ [ τ a, τ v]ψ a c = V µ ε abc v b V µ A µ a c ε abc v b A µ V µ a V µ a ε abc a b A µ c A µ a A µ a ε abc a b V µ c a a c R µ R µ ε abc b R µ a a L µ L µ a a c L µ L µ ε abc l b R µ R µ a R µ a φ π a L (x) = ( 0 φ a )

[ π a (x),φ b (y)] = iδ ab δ 3 (x y) ( J m ) µ = i L ( µ φ) T m φ µ =0 i πt m φ Q m d3 x J m 0 (x) = i d 3 x πt m φ [ ] = d 3 y [ π(y)t m φ(y),φ(x) ] i Q m, φ(x) = i T m φ(x) e i Qm ε m φ(x)e i Q m ε m = D(g)φ(x) Q a b [ V, Q V ] = iε abc Q c V, Q a b [ V, Q A ] = iε abc Q c A, Q a b c [ A, Q A ] = iε abc Q V Q R a = 1 Q a a a ( V + Q A ), Q L = 1 Q a a ( V Q A ) Q a b [ R, Q R ] = iε abc Q c R, Q a b [ L, Q L ] = iε abc Q c L, Q a b [ R, Q L ] = 0

SU() SU() SU() SU() J(J +1) J H ghg 1 = H

g g A = a 0, B = b 0 B g A E A = A H A = B ghg B = B H B = E B GeV 1.5 1.0 0.5 Meson spectum 1 + 1 σ(~ 600) 0 + a 1 (160) ρ(770) 1 + 1 f 1 (185) ω (780) 0 0 + π (1300) η(195) a 0 (980) f 0 (975) 1 + 1 K 1 (1400) K 1 (170) K * (890) 0 π(139) GeV 1.5 Bayon spectum 1 N (1535) Λ (1670) Λ (110) Σ (1750) Σ (1190) 3 3 + (1700) (13) 1.0 1 + N (939)

A g A = ga 0 = ga g g 0 = b g 0 g 0 = 0 g SU() SU() SU() Isospin SU() Isospin SU() SU() SU() SU() SU() Isospin ~ U = exp(iτ π / f π ) 1.8 10 8

ν π + u µ d 0 A µ π d A µ u W + ν µ L int ~ J µ J µ J µ ~ J µ (h) + J µ (l) J µ (l) J µ (l) = l γ µ (1 γ 5 )l v µ a µ µν µ L int (x) π(q) ~ µν µ J µ (x)j µ (x) π(q) ~ µν µ J µ(l) (x) 0 0 A µ (x) π(q) q µ A µ A a µ = σ µ π a π a µ σ f π f π µ π a f π

µν µ J µ (l) 0 0 A µ π A µ a (x) 0 A µ a (x) π b (q) = iq µ δ ab f π exp( iqx) f π q f π exp( iqx) 0 µ A µ a (x) π b (q) = δ ab m π f π exp( iqx) 0 π a (x) π b (q) = δ ab exp( iqx) 0 µ A µ a (x) π b (q) = m π f π 0 π a (x) π b (q) µ A µ a (x) = m π f π π a (x) m π m π = 0

p(p f )eν e L int n( p i ) = µν e J µ (l) 0 p(p f ) (V µ A µ ) n(p i ) p(p f ) A µ a n( p i ) p(p f ) A a µ (x) n(p i ) = u p ( p f ) γ µ g A (q ) + q µ h A (q τ [ a )]γ 5 u n ( p i ) exp(iqx) q = p f p i g A (q ) h A (q ) ν e f π = + g π NN n p g A tem h A tem µ A µ a (x) = 0 m π = 0 0 = p( p f ) µ A a µ n( p i ) = iu p ( p f ) q / g A (q ) + q h A (q τ [ a )]γ 5 u n (p i ) h A (q ) = M N g A (q ) q h A (q ) 1 / q A µ a = f π µ π a +...

x = 0 p(p f ) A µ a n( p i ) 1π = if π q µ p(p f ) π a n( p i ) m π = 0 p(p f ) π a n(p i ) i q g πnn (q ) u p ( p f )γ 5 τ a u n ( p i ) p(p f ) π a n(p i ) = 1 q p( p f ) J a n(p i ) i q g πnn (q ) u p ( p f )γ 5 τ a u n ( p i ) 1 / q J a A µ a p(p f ) A µ a n( p i ) 1π = f π q µ q g πnn (q ) u p (p f )γ 5 τ a u n (p i ) p(p f ) A µ a n( p i ) 1π = u p (p f )q µ M N g A (q ) q γ 5 τ a u n ( p i ) g πnn M N = g A f π g πnn = 13.7, g A =1.5, M N = 938 MeV, f π = 93 MeV g πnn f π

a q 1 b q p 1 p π b (q )N(p ) π a (q 1 )N(p 1 ) I = π b (q )N(p ) π a (q 1 )N(p 1 ) ~ d 4 xd 4 y e iq 1 x e +iq x N(p ) Tπ b (x)π a (y) N( p 1 ) I ~ d 4 xd 4 y e iq 1 x e +iq x N(p ) T µ A b µ (x) ν A a ν (y) N(p 1 ) I = I 1 + I + I 3 I 1 = d 4 xd 4 y e iq 1 x e +iq y δ(x 0 y 0 ) N(p ) A b 0 (x), ν A a ν (y) I = d 4 xd 4 y e iq 1 x e +iq y µ x ν y N(p ) A b 0 (x),a a ν (y) I 3 = i d 4 xd 4 y e iq 1 x e +iq y q µ δ(x 0 y 0 ) N(p ) A b 0 (x), A a µ (y) [ ] N(p 1 ) [ ] N(p 1 ) [ ] N(p 1 )

I I 1 I 3 O(m π ) O(m π ) O(m π ) I 1 I 3 I m π I 1 ν A a ν (y) ~ m π O(m π ) O(m π ) O(1) I 3 O(m π ) T = i f π I π I N, I N = τ, (I π a ) bc = iε abc a = m π 8πf π = m π 8πf π 1 + m π M 1+ m π M 1 I π I N 1 [ I(I +1) I N (I N +1) ] a 1 = 0. m π 1, a 3 = 0.1 m π 1 I N = 1 / I = I π + I N =1 / o 3 / a 1 = 0.173 m 1 1 π, a 3 = 0.101 m π

ψ,σ,π L SU() R SU() L 1 (, 0 ) ( 0, 1 ) 1 ( ), 1 1 (, 0 ) SU() R SU() L SU() R ~ Isospin 1 /, SU() L ~ Isospin 0 ( ) SU() R ~ Isospin 0, SU() L ~ Isospin 1 / 0, 1 1 ( ) ~ ψ R 0, 1 B α, 0 ( ) ~ ψ L B α ψ R τ α ψ L

B 0 B B B 0 B 0 B + v B B 0 + i a B B + i a B 0 B = τ α ( ψ R τ α ψ L ) = ψ L ψ R U B U ψ R τ α ψ L ψ L ψ R Paity ψ R τ α ψ L ψ L τ α ψ R S α = ψ R τ α ψ L + ψ L τ α ψ R =ψ τ α ψ ( ) = iψ τ α γ 5 ψ P α = i ψ R τ α ψ L ψ L τ α ψ R i S 0 S 0 S S + v S, P P 0 P 0 P + v P S 0 S 0 + a P P 0 P 0 a S, S S + a P 0 P P a S 0 S 0 + P P 0 + S S 0 + P, P 0 + S : Invaiant

S 0 + P (S 0, P ) (σ, π ) L σ = 1 ( µ σ) + ( µ π ) ( ) V (φ ) V(φ ) φ = σ + π V(φ ) = µ φ + λ 4 φ 4 λ µ H = 1 p α + 1 i φ α ( ) + V (φ ), p α = L φ = φ α α µ µ 1 0.1 m π 10 1 mπ

φ φ π π σ 0 σ µ µ µ > 0 0 0 (σ, π ) = (0, 0 ) µ < 0 φ = µ λ

0 µ λ, 0 f π, 0 ( ) SU() SU() SU() SU() R SU() L SU() V χ = ( χ 0, χ 1, χ, χ 3 ) φ = φ vac + χ χ V(φ vac + χ) = V(φ vac ) + χ α α V(φ vac ) + 1 χ α χ β α β V (φ vac ) +... V(φ), φ = φ 0 + φ 1 +φ + φ 3

α V(φ) = φ V (φ ) = φ α V (φ) φ ˆ φ α φ α V (φ ) ( ) = P αβ α β V(φ) = ˆ β φ α V (φ) P αβ δ αβ ˆ φ α ˆ φ β, φ V (φ) + φ ˆ ˆ α φ β V (φ ) P αβ ˆ φ α = P αβ ˆ φ β = 0 φ vac = ( f π, 0,0,0) L σ ~ 1 ( µ χ ) 1 V''(φ vac ) χ 0 ( ψ ψ, iψ τ γ 5 ψ ) σ, π ( ) σψ ψ + π iψ τ γ 5 ψ = ψ ( σ + iτ π γ 5 )ψ L = i ψ / ψ gψ ( σ +iτ π γ 5 )ψ + 1 µ σ ( ) + ( µ π) V(φ) ψ i ψ L / ψ L + i ψ R / ψ R gψ L ( σ + iτ π )ψ R gψ R ( σ iτ π )ψ L gσψ ψ gf π ψ ψ

M = gf π g A g A g A σ ψ ψ = ψ R ψ L +ψ L ψ R ψψ ψψ = ψ R ψ R +ψ L ψ L + 1 µ σ L = i ψ / ψ gψ ( σ + iτ π γ 5 )ψ [( ) + ( µ π) ] λ 4 σ + π ( f π ) m σ = λf π M = gf π

σσσ, σππ σσσσ, σσππ, ππσσ a q 1 b q π π p 1 p σ (3d tem ) = g u u i q m λf π i g σ M k 0 L PS = iψ γ 5 τ π ψ ψ ψ (x) = u n ( x )exp( ie n t)b n + v m ( x ) exp(+ie m t)d m E n >0 E m <0 L PS n L PS n v ~ p / E m L PS n

n Bon tem n ~ n iψ γ 5 τ π ψ iψ γ 5 τ π ψ n ~ u n γ 5 0 T(ψψ ) 0 γ 5 u n ~ u n (x)γ 5 ~ dω π dω π u n (x)u n (y) ω E n + iε + v m (x)v m (y) γ ω + E m iε 5 u n (y) u n (x)γ 5 u n (x)u n (y)γ 5 u n (y) ω E n + iε + u n (x)γ 5 v m (x)v m (y)γ 5 u n (y) ω + E m iε γ 5 O(1) L PV = 1 f π ψ γ µ γ 5 τ µ π ψ O(q) µ = 1,,3 n L PV n n L PV n n L PV n ~ O(q) ( n L PV n ) / (Enegy denominato ) O(q) O(q ) σ q

1 (, 1 ) f 0 (600) o σ ( σ,π 1,π,π 3 ) O(4) f = σ +π 1 +π + π 3 P Q P(x,y, z) Q( x, y, z ) δx δy = δz 0 ε 3 ε ε 3 0 ε 1 ε ε 1 0 x y z (x, y, z) (ε 1,ε,ε 3 )

x = sinθ cosϕ, y = sinθ sinϕ, z = cosθ δθ = δϕ sinϕ ε 1 cosϕ ε cot θ cosϕ ε 1 + cot θ sinϕ ε ε 3 (θ,ϕ) sinϕ φ µ ~ ( σ,π 1,π,π 3 ) σ = f cosθ 1, π 1 = f sinθ 1 sinθ cosθ 3, π = f sinθ 1 sinθ sinθ 3, π 3 = f sinθ 1 cosθ. ( σ,π 1,π,π 3 ) (θ 1,θ,θ 3 ) L = 1 ( µ ) φα = 1 m θ m x µ φ α θ m = 1 m,n g mn (θ ) θ m x µ θ n x µ g mn (θ ) (θ 1,θ,θ 3 ) g mn (θ ) = δ mn + θ m θ n f π θ L = f π ( ) ( µ σ) + µ θ, σ = 1 θ

g mn (θ) = θ m θ n θ + δ mn θ θ m θ n θ 4 sin θ L = f π 4 t µ UU µ UU = f π 4 t µ U µ U ( ) U = exp i τ θ U U U U g L U g R g V U g V g A U g A V µ a A µ a = i f 4 tτ a U, µ U = i f 4 tτ a U, µ U [ ] ~ ε abc φ b µ φ c { } ~ f µ φ a L = i ψ / ψ gψ ( σ +iτ π γ 5 )ψ 1 + µ σ ( ) + ( µ π ) V(φ) σ + iτ π f U, U = exp( iτ φ / f π ) σ + iτ π γ 5 f U 5, U 5 = exp iτ φ γ 5 / f π ( ), ψ ( σ +iτ πγ 5 )ψ = f ψ U 5 ψ = f ψ U 5 U 5 ψ f N N

ξ 5 ψ = N, ψ = ξ 1 5 N U 5 = exp iτ φ γ 5 / f π ( τ ) ( ) ξ 5 exp( i φ / f ) ξ π ψ N N L = i N ξ 1 5 / ( ξ 1 5 N) gf N N + f 4 t µ U µ U + 1 ( µ f ) + V ( f ) N ξ 5 1 L K = i N ξ 1 5 / ( ξ 1 5 N) = i N / N +i N ξ 1 5 / 1 ( ξ 5 )N i N ξ 1 5 ( µ ξ 5 )γ µ N = i N 1 ξ 5 1 1 ( ( µ ξ 5 ) + ξ 5 ( µ ξ 5 ))γ N + i N 1 ξ 5 1 1 ( ( µ ξ 5 ) ξ 5 ( µ ξ 5 ))γ N φγ 5 γ 5 φγ 5 1 ( ( ) + ξ 5 ( µ ξ 5 ) v µ = 1 ξ 5 1 µ ξ 5 a µ = 1 ξ 5 1 1 ( ( µ ξ 5 ) ξ 5 ( µ ξ 5 ))γ 5 τ a L = i N D / N + i N a / γ 5 N gf N N + f 4 t µ U µ U + 1 ( µ f ) + V( f ) 1 ξ 5 / 1 ξ 5 ( ) = ξ 1 5 γ µ µ ξ 1 5 = ξ 1 5 µ ξ 5 γ µ γ 1 µ ξ 5 1 ξ 5 ξ 5

D µ = µ iv µ ξ U U U = ξ g L Ug R = ( g L ξ) ( ξg R ) U U = ξ ( ) h ξ = g L ξh = hξg R h ξ (g R, g L ) ξ g R = g L = g V h h = g V ε δφ = ε φ h h ~ 1+ iε φ τ + L

SU() R SU() L SU() R SU() L SU() R SU() L SU() V π ~ SU() R SU() L / SU() V SU() V U SU() R SU() L SU() R SU() L τ R a ~ SU() R, τ L a ~ SU() L τ R a τ L a τ a (R) 1+ γ 5 τ a = P R τ a, τ a (L) 1 γ 5 τ a = P L τ a τ a γ 5 P R,L τ V a = τ R a + τ L a = τ a τ A a = τ R a τ L a = γ 5 τ a τ V a = τ a τ A a = γ 5 τ a γ 5 H G G G

g Π = G / H a H h g = ha a i i a h ha i i { ha 1,ha,ha 3,L} G Π = G / H SU() SU() / SU() ( ) ξ 5 (φ) = exp iφ τ γ 5 / a i ξ 5 (φ) φ γ 5 τ a = τ R a τ L a = τ A a ~ SU() SU() / SU() ξ 5 ( ) γ 5 ξ 5 ξ = exp iτ φ / γ 5 ξ 5 (φ) G ξ 5 (φ) g G ξ 5 ( φ ) h ξ 5 (φ) g = h(g,φ ) ξ 5 ( φ ) ha i h h (g,φ) h ξ 5 (φ) φ g ξ 5 ( φ ) = h(g,φ)ξ 5 (φ )g φ φ

N h N = h(g,φ)n i N / N i N h / (hn) = i N h ( / h)n +i N / N v µ h v µ h h ( / h) a µ h a µ h i N D / N i N ( / + v /)N i N D / N, N a / γ 5 N N a / γ 5 N L = i N D / N + i g A N a / γ 5 N gf N N + f 4 t µ U µ U + 1 ( µ f ) + V( f ) g A =1 g A g A N = ξ 5 ψ g A

λ 1 = 0 1 1 0 0 0, λ = 0 i 0 i 0 0, λ 3 = 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 λ 4 = 0 0 0 0 1 0, λ 5 = 0 0 i 0 0 0 1 0 0 i 0 0 λ 6 = 0 0 0 0 0 1, λ 7 = 0 0 0 0 0 i, λ 8 = 1 1 0 0 0 1 0 0 1 0 0 i 0 3 0 0 U = exp( iθ a λ a ), θ a = θ 1, θ,lθ 8 I 3 Y U + Y V + I I + I 3 V U I ± = 1 ( λ 1 ± iλ ), V ± = 1 ( λ 4 ± iλ 5), U ± = 1 ( λ 6 ± iλ 7) 3 3 = 1 + 8, 3 3 3 = 1 + 8 + 8 +10

π + I + + π I + π 0 I 0 + K + V + + K V + K 0 U + + K 0 U + η λ 8 π 0 + η π + K + = π π 0 + η K 0 = λ a φ a φ K K 0 η 6 exp( iφ / f π ) U U Peudoscala meson octet K 0 ( ds) K + (us) Vecto meson octet K *0 (ds) K *+ (us) π ( d u ) π 0 (uu dd) π + (ud ) ρ ( du) ρ 0 (uu dd) ρ + (ud) η(uu + dd ) ω ( uu + dd ) K (su) K 0 (sd) K * (su) K *0 (sd ) n(udd) Bayon octet p(uud) (ddd) Bayon decuplet 0 (udd) + (uud) ++ (uuu ) Σ (dds) Σ 0 (uds) Λ (uds) Σ + (uus) Σ * ( dds) Σ *0 (uds) Σ *+ (uus) Ξ (ssd ) Ξ 0 (ssu) Ξ (dss) Ξ *0 (uss ) Ω (sss)

Σ 0 + η Σ + p B = Σ Σ0 + Λ n Ξ Ξ 0 Λ 6 p n 8 8 = 1 + 8 + 8 + 10 +10 + 7 1 (8 8) 8 = (1 + 8 + 8 + 10 +10 + 7) 8 8 8 + 8 8 1 +1 8 + 8 8 8 L = f π ( ) + t B i / 4 t µ U µ U ( D B) + m 0 tb B + F tb γ 5 [ a /, B] + D tγ 5 { a /, B}

R 3 SU() ~ S 3 x = (x 1, x,x 3 ) R 3 R 3 π = (π 1,π, π 3 ) S 3 S 3 N

N = 1 V dπ = 1 V (π ) (x) dx V S 3 dπ x (π) / (x) V N π 3 (S 3 ) = Z (intege ) x π ( x ) ( ) U( x ) = exp iτ π ( x ) / f π f π π µ = ( µ U)U N = ε ijk 4π t ( ) U = exp iτ π / f π d3 x i j k L = f π 4 t µ µ E = f π d 3 x 4 t i i U( x ) x ax a

E E ae a U( x ) 1 E 4 E 4 / a E = E + E 4 a L 4 = 1 3e t [ µ, ] ν e ρ π L Skyme = f π [ ] 4 t µ µ + 1 3e t µ, ν E = E + E 4 E E 4 a

( ) U H exp( iτ ˆ F() ) U = exp i τ π / f π F() K = I + J K I = τ / U H U H Isospin : U H AU H A A SU() J = i Spin : ( ) R O(3) U H exp iτ Rˆ F() B R R ab = i t [ τ a B τ b B] A = B K U H E E = kq ˆ

1 / B = 1 E ~ Y 1, e [ ] 0 0 Y 1 e E = d 3 x f π 4 t i i + 1 3e t i, j [ ] = 4π d f π F + sin F + sin F e F + sin F F + sin F F + e f π sin F F + sinf F sin F sin F = 0 B = 1 π d 3 x sin F F = 1 π ( F(0) F( ) ) F() F(0) = π F( ) = 0 F( ) =1 / 3 F( ) 1 0 0.0 0.5 1.0 1.5 Radial distance [fm].0

e f π = 93 MeV e ~ 5 E ~ 1.4 GeV ρ π R 0 e ~ 4.5 E ~ 1.4 GeV I = J K = 0 (I, I z ) = (J, J z ) I = J U H φ c ϕ φ = φ c + ϕ φ c L( φ c + ϕ) = L(φ c ) + L φ (φ c ) ϕ + 1 L φ (φ c ) ϕ + L L / φ ϕ ϕ ϕ L φ (φ c ) = 0 φ c ϕ L φ (φ c ) ϕ = 0 ϕ t ϕ = Hϕ

H H ϕ(t, x ) = n a n e iω n t ϕ n ( x ) φ Gφ φ Gφ G L φ (Gφ c ) = L φ (φ c ) = 0 ε T G ~ 1 + iεt 0 = L φ (Gφ c ) ~ L φ ((1 + iεt)φ c ) ~ L φ (φ c ) + i L φ (φ c ) εtφ c L φ (φ c ) Tφ c = 0 Tφ c 0 Tφ c = 0 Tφ c Tφ c ω = 0 Tφ c 0 φ c φ c

U H AU H A A SU() A A A(t) A A SU() ~ S 3 (α,β,γ ) ψ ~ exp(ip α α)exp(ip β β)exp(ip γ γ ) ψ ~ D t, s (α,β, γ ) J J K = I + J = 0 I = J J J = SU(N f > ) N c N c N c 1/ p ~ D 1/, 1/ (α,β,γ ) = e i α cos β i e γ A / A U(x) U(x) AU H A d 3 x A A A / A

H = d 3 x L = M H + Λ π Ω Λ π Ω = (i / )t[ τ A A ] (~ A ) A F() Λ π = 8π 3 f π R d sin F 1 + 1 e F + sin F f π A A / A H = M H + 1 Λ π J(J +1) J(J +1) M N M f π 1/ I =0 1/ M,I =1 µ p µ n µ p / µ n g A g πnn g πn µ N

1 (,0 ) ( 0, 1 ) 1 (, 1 ) 1 (,0 ) ( 0, 1 ) g A =1