< qq > (Quark Gluon Plasma,QGP) QGP (< qq >= ) < qq > π - π K + Nambu-Goldstone K + S = + S = K K + K + - K + t free ρ K + N K + N next-to-leading ord

Transcription

1 -K +

2 < qq > (Quark Gluon Plasma,QGP) QGP (< qq >= ) < qq > π - π K + Nambu-Goldstone K + S = + S = K K + K + - K + t free ρ K + N K + N next-to-leading order (NLO) NLO (low energy constant,lec) χ I = I = K +

3 K QCD Nambu-Goldstone Gell-Mann-Oakes-Renner Nambu-Goldstone QCD K K + n K + n LO NLO LO+NLO

4 3 5.3 K + p K + p LO NLO LO+NLO s (l = ) p (l = ) K

5 4 (Quantum Chromodynamics,QCD) QCD SU(3) c QCD ( ) ( ) ( ) QCD - π K QCD QCD QCD QCD ( ) ( ) < qq >. (Quark Gluon Plasma,QGP) QGP (< qq >= )

6 5 (< qq >= ) Nambu-Goldstone (NG ) (NG π, K, η ) π (π - ) 3% ] (subthreshold) (threshold) π ± - π ± π ±.5MeV/c b free /b =.69 ] π ± - K - K + - K + N( ) K +. K + - K + - K + 3] K + N K + 5fm K + P LAB 8MeV/c σ(k + A) Aσ(K + N) (.) K + K + N K + S = + K + -N K + S = K K Λ(45) K + K +

7 6 C d R R = σ(k+ C)/ σ(k + d)/ > (.) 4] d σ(k + d) = σ(k + N) (.3) R > C σ(k + C) > σ(k + N) (.4) K + Σ free = m N M(K + N) free ρ (.5) M(K + N) free = 8π sb free (.6) m N ρ s K + N b free K + - Σ fit = m N M(K + N) fit ρ (.7) M(K + N) fit = 8π sb fit (.8) K + N b fit E.Friedman A.Gal ] Re b fit Re b free 4% 5% Im b fit Im b free % 5%.3 (.5) (.7) NG K + NG Z Σ = Z M(K + N)ρ m N (.9) Z + Σ E (.)

8 7 Z ρ.4 Z K + - M(K + N) M(K + N) vector meson exchange QCD (Chiral Perturbation Theory,ChPT) QCD NG.5 QCD QCD - K + N

9 8 QCD (Quantum Chromodynamics,QCD). QCD SU(3) c / u, d, s, c, b, t QCD SU(3) c L free = A= f= α,α = q α,f,a (γ µ α,α i µ m f δ α,α )q α,f,a (.) A, f, α q f = (q f,, q f,, q f,3 ) t SU(3) c : q f (x) q f (x) = exp i 8 a= θ a (x) λ a ] q f (x) U(x)q f (x) (.) q f (x) q f (x) = q f (x)u (x) (.3) U SU(3) c (λ a : θ a (x) : ) µ D µ L free A µ D µ q f ( µ + iga µ )q f (.4) A µ = A aµ λ a A µ = UA µ U + i g ( µu)u (.5)

10 QCD 9 A aµ (a =...8), g g QCD L matter = q f (i D m f )q f (.6) f=u,d,s,c,b,t L QCD ( ) ( ) λ a F µν F aµν (.7) F µν = D µ A ν D ν A µ = µ A ν ν A µ + iga µ, A ν ] (.8) F aµν = µ A aν ν A aµ gf abc A bµ A cν (f abc : SU(3) ) (.9) F µν F µν = UF µν U (.) L gauge = 4 F aµνf µν a QCD L QCD = f=u,d,s,c,b,t = T r(f µνf µν ) (.) q f (i D m f )q f T r(f µνf µν ) (.). γ 5 ± γ 5 γ = ( ) (, γ j σ j = σ j ) (j =,, 3) (.3) γ 5 = iγ γ γ γ 3 = ψ ξ, ζ ( ) ξ ψ = ζ ( ) (.4) (.5)

11 QCD ( ξ ψ = ) ( + ζ ) = ψ R + ψ L (.6) ψ R,ψ L γ 5 ψ R = ψ R (.7) γ 5 ψ L = ψ L (.8) γ 5 γ 5 + ψ R, - ψ L P R = + γ5, P L = γ5 (.9) P R + P L = (.) (P R ) = P R, (P L ) = P L (.) P R P L = P L P R = (.) P R ψ = ψ R (.3) P L ψ = ψ L (.4) ψ L,ψ R ψ L = ψ P L γ = ψp R (.5) ψ R = ψ P R γ = ψp L (.6) γ 5 = γ 5, {γ5, γ µ } = QCD ψψ QCD L QCD = f ψψ = ψ R ψ R + ψ R ψ L + ψ L ψ R + ψ L ψ L = ψp L P R ψ + ψp L P L ψ + ψp R P R ψ + ψp R P L ψ = ψ R ψ L + ψ L ψ R (.7) ψγ µ ψ = ψ R γ µ ψ R + ψ L γ µ ψ L (.8) q L,f i Dq L,f + q R,f i Dq R,f m f ( q L,f q R,f + q R,f q L,f ) 4 F aµνf a µν (.9)

12 QCD m f ( ) QCD U(N f ) L U(N f ) R = SU(N f ) L SU(N f ) R U() L U() R SU(N f ) L SU(N f ) R ] 8 q L U L q L, U L = exp i θl a t a a= ] 8 exp i θl a t a SU(N f ) L (.3) a= q R U R q R, U R = expi exp i 8 a= θr a t a ] 8 θr a t a ] a= SU(N f ) R (.3) (θ :, t a : SU(N f ) ) t a, t b ] = if abc t c (f abc : SU(N f ) ) (.3) T rt a t b ] = δ ab (a = Nf ) (.33) t = /N f (.34) N f = : τ =, τ,,3 (Pauli (.35) N f = 3 : λ = /3, λ, 8 (Gell Mann ) (.36) QCD L QCD = f ( q L,f i Dq L,f + q R,f i Dq R,f ) 4 F aµνf µν a (.37) L QCD SU(N f ) L SU(N f ) R L µ a = q L iγ µ ta q L, µ L µ a = (.38) R a µ = q R iγ µ ta q R, µ R a µ = (.39) L µ a, V µ a Q a L = d 3 xl a µ=, Q a R = d 3 xrµ=, a dq a L = (.4) dt dq a R = (.4) dt

13 QCD SU(N f ) L SU(N f ) R Q La, Q Lb ] = if abc Q Lc, Q Ra, Q Rb ] = if abc Q Rc, Q Ra, Q Lb ] = (.4) SU(N f ) L SU(N f ) R U L U R (θ V U R (θ A θ R = θ L ) Q a V, Qa A : q V q, V = exp : q Aq, A = exp i i a= θ R = θ L ) U L ] 8 θv a t a ] 8 θaγ a 5 ta a= (.43) (.44) V µ a = R µ a + L µ a = qiγ µ ta q, µv µ a = (.45) A µ a = R µ a L µ a = qiγ µ γ 5 ta q, µa µ a = (.46) Q a V = Q a L + Q a R (.47) Q a A = Q a R Q a L (.48) Q a V, Q b V ] = if abc Q c V, Q a A, Q b A] = if abc Q c V, Q a V, Q b A] = if abc Q c A (.49) Û V = expiθ a V Q a V ], Û A = expiθ a AQ a A] (.5) Û V qûv + iθv a t a ]q (.5) Û A qûa + iθaγ a 5 ta ]q (.5)

14 3 3 QCD < qq > < qq > Nambu-Goldstone (NG ) π K η 3. N f = q = (u, d) t m f QCD H QCD tq a V =, tq a A = H QCD, Q a V ] =, H QCD, Q a A] = (3.) Û V H QCDÛV = H QCD, Û A H QCDÛA = H QCD (3.) ( SU() V ( ) I = : m π ± = 4 MeV, m π = 35 MeV I = : m p = 938 MeV, m n = 939 MeV ( ) I =, J P = + : I =, J P = : m a = 6 MeV m ρ = 77 MeV SU() V ( )

15 3 4 SU() L SU() R SU() V Û V >= >, Q a V >= > (3.3) Û A > >, Q a A > > (3.4) Q a A ( ) qq < qq > : µ j µ = (3.5) : C V (t) d 3 xρ(x, t), C = C (3.6) V : C, H] =, t C = (3.7) C = (3.8) (3.8) well-defined C() = C() C() = d 3 x ρ(x, ) C() = d 3 x ρ(, ) C() V (3.9) V V 3 C V well-defined well-defined S t (x) = lim C V (t), P (x)] V = lim d 3 x ρ(x, t), P (x)] (3.) V V ds t dt = (3.) S t (x) well-defined x µ A(x), y µ B(y) A(x), B(y),

16 3 5 (x y) < (space-like), x y A(x), B(y)] =, (x y) < (3.) (x y) > (time-like) V (x y) = (light-like) bound bound well-defined well-defined S t (x) = lim d 3 x ρ(x, t), P (x)] (3.3) V order parameter field H Ω Ω order parameter V Ω S t (x) Ω = lim d 3 x Ω ρ(x, t), P (x)] Ω V V = lim Ω C V (t), P (x)] Ω (3.4) V H Ω, C Ω ] = Ω C V Ω S t (x) Ω = (3.5) H Ω, C V ] = Ω H Ω C V C V Ω S t (x) Ω = (3.6) order parameter < qq > C V = Q a=,,3 A = V d 3 xq iγ 5 τ a q (3.7) P a (x) = qiγ 5 τ a q (3.8) Q a A(t), P b (x)] = iδ ab qq (3.9) < qq > (3.)

17 Nambu-Goldstone Q a A (SU() ) masless Ω S t (x) Ω = Ω S t (x) Ω = lim Ω e iht C V ()e iht n n P (x) Ω Ω P (x) n n e iht C V ()e iht Ω ] V n = lim Ω CV () n n P (x) Ω e ient Ω P (x) n n C V () Ω e +ie nt ] V n = lim E n Ω CV () n n P (x) Ω e ient + Ω P (x) n n C V () Ω e +ie nt ] V n t (3.) E n (3.) Ω C Ω () n n P (x) Ω = (3.3) Ω S t (x) Ω E n = Nambu-Goldstone Nambu-Goldstone a SU() NG (π, π ± ) (3.) 3.3 Gell-Mann-Oakes-Renner ˆm (m u + m d )/ µ V µ a ] = i q M, λa q (3.4) } q (3.5) µ A µ a = i qγ 5{ M, λa ( ) ˆm M = ˆm (3.6) SU() V PCAC(Partially-Conserved-Axial vector-current) ˆm

18 3 7 A a µ(x) π b (p) = if π (p )p µ e ipx δ ab (3.7) π b (p) f π f π = 9.4MeV (3.5) G (3.9) µ A µ a = ˆmi qγ 5 τ a q = ˆmP a (x) (3.8) µ A a µ(x) π b (p) = f π m πe ipx δ ab = ˆm < P a (x) π b (p) > = ˆme ipx δ ab G (3.9) f π m π = ˆmG (3.3) V m Bogoliubov limit iδ ab qq = d 3 p Q a (π) 3 E A π c (p) π c (p) P b (x) P b (x) π c (p) π c (p) Q a A ] c p (3.3) Q a A π c (p) = if π p µ= δ ac d 3 xe ipx = if π p µ= δ ac (π) 3 δ 3 (p) (3.3) π c (p) P b (x) = e ipx δ cb G (3.33) (3.3) d 3 p (π) 3 if π p µ= δ ac (π) 3 δ 3 (p)δ cb G e ipx + e ipx] = if π δ ab G (3.34) E p c Glashow-Weinberg relation qq m = f π G m (3.35) qq m f π m G m qq G m = m (3.36) f π m m π = ˆm m π = ˆm G f π (3.37) qq fπ m + O( ˆm ) (3.38) Gell-Mann-Oakes-Renner

19 8 4 QCD Nambu-Goldstone QCD - K + - 5, 6, 8] 4. Nambu-Goldstone NG QCD 4.. G G H NG ϕ i M 4 NG Φ = (ϕ, ϕ,..., ϕ n ) t R n M NG M {Φ : M 4 R n ϕ i : M 4 R} (4.) (g, Φ) G M φ(g, Φ) M φ φ φ(e, Φ) = Φ, Φ M (4.) φ(g, φ(g, Φ)) = φ(g g, Φ), g, g G, Φ M (4.3) G M G M M M φ G φ(g, λφ) λφ(g, Φ) Φ = M (origin) NG Φ =

20 4 9 H h H φ(h, Φ = ) = (4.4) Φ = h H G G H {gh g G} G/H G/H {gh h H} G G = eh + ah + bh + a, b, G (4.5) + G/H φ. gh Φ = R n φ(gh, ) = φ(g, φ(h, )) = φ(g, ), g G, h H (4.6) φ(g, ) G/H φ(g, ) G/H NG. G/H φ g, g G, g / gh φ(g, ) = φ(g, ) = φ(e, ) = φ(g g, ) = φ(g, φ(g, )) = φ(g, φ(g, )) = φ(g g, ) (4.7) g g H g gh g φ(g, ) φ(g, ) ( ) / gh G/H NG Φ gh Φ G/H gh Φ NG R n gh x NG g G Φ f = gh gh NG Φ = φ(f, ) = φ( gh, ) (4.8) φ(g, Φ) = φ(g, φ( gh, )) = φ(g gh, ) = φ(f, ) = Φ f g( gh) (4.9) Φ Φ Φ gh g Φ g Φ gh g g( gh)

21 4 4.. QCD QCD ( ) G H G = SU(N f ) SU(N f ) = {(L, R) L SU(N f ), R SU(N f )} (4.) H = SU(N f ) = {(V, V ) V SU(N f )} (4.) g = ( L, R) G SU(N f ) Ũ = R L G (4.) gh = (, R L )H (4.3) U g = (L, R) G g( gh) = (L, R)(, R L ) = (L, R R L )H = (, R R L L )(L, L)H = (, R R L L )H (4.4) U U = R L U = R R L L = RUL (4.5) x U(x) U(x) = RU(x)L (4.6) QCD N f =, N f = 3 NG M N f =: N f =3: M = {Φ : M 4 R 3 ϕ i : M 4 R} (4.7) M = {Φ : M 4 R 8 ϕ i : M 4 R} (4.8) N N A(N) {A gl(n, C) A = A, T ra = } (4.9) M {ϕ : M 4 A(N) ϕ} (4.) M M τ i, λ a N f =: N f =3: ϕ = 8 a= ϕ = 3 i= ϕ a λ a = ϕ i τ i = ( π π + π π ) π + η 6 π + K + π π + η 6 K K K η 6 (4.) (4.)

22 4 { ( ) } ϕ M 3 U : M 4 SU(N f ) U = exp i, ϕ M F (4.3) F exp NG F U SU(N f ) M 3 SU(N f ) SU(N f ) M 3 φ : G M 3 M 3, φ(l, R), U] RUL (4.4) G M 3. U M 3, R, L SU(N f ) RUL M 3. ϕ(, ), U] = U = U 3. g i = (L i, R i ) G g g = (L L, R R ) φg, φg, U]] = φg, R UL ] = R R UL L (4.5) φg g, U] = R R UL(L L ) = R R UL L (4.6) φg, φg, U]] = φg g, U] (4.7) ϕ = U = H = {(V, V ) V SU(N f )} QCD U ϕg = (V, V ), U ] = V U V = U (4.8) QCD ϕg = (A, A), U ] = A U A = A A U (4.9) NG NG 4. U(ϕ) SU(3) L SU(3) R NG

23 4 NG 4 p µ = ( p, p) 4πF π 7MeV p µ SU(3) L SU(3) R NG p µ = ( p + m NG, p) m NG p p µ NG L eff = n L n (4.3) U U = T r(u U) = 3 SU(3) L SU(3) R L eff = F 4 T r( µu µ U ) (4.3) U RUL (4.3) µ U R µ UL (4.33) U LU R (4.34) µ U L µ U R (4.35) L eff F 4 T r(r µul L µ U R ) = F 4 T r(rr µ U µ U ) = L eff (4.36) F /4 µϕ a µ ϕ a / U = + i ϕ F F ϕ + (4.37) µ U = i µ ϕ F F ϕ µ ϕ + (4.38)

24 4 3 (3) L eff = F 4 T r(i µ ϕ)( i µ ϕ)] + F F = T r( µ ϕ a λ a )( µ ϕ bλ b )] + = 4 µϕ a µ ϕ b T rλ a λ b ] + µϕ a µ ϕ a + L int (4.39) 8 NG NG 4.3 QCD QCD L ext QCD = L QCD + L ext (4.4) L ext QCD = qγ µ (v µ + γ 5 a µ )q q(s iγ 5 P )q = q L γ µ (v µ a µ )q L + q R γ µ (v µ + a µ )q R q R (S + ip )q L q L (S ip )q R = q L γ µ l µ q L + q R γ µ r µ q R q R (S + ip )q L q L (S ip )q R (4.4) v µ, a µ, S, P (4.4) SU(3) L SU(3) R l µ = v µ a µ, r µ = v µ + a µ (4.4) q L V L (x)q L, q R V R (x)q R (4.43) S + ip V R (S + ip )V L (4.44) l µ V L l µ V L + iv L µ V L (4.45) r µ V R r µ V R + iv R µ V R (4.46) L ext QCD = L QCD + L ext SU(3) L SU(3) R U(x) U (x) = V R (x)u(x)v L (x) (4.47) D µ U = µ U ir µ U + iul µ (4.48)

25 4 4 D µ U µ (V R UV L ) i(v Rr µ V R + iv R µ V R )V RUV L + iv RUV L (V Ll µ V L + iv L µ V L ) = ( µ V R )UV L + V R( µ U)V L + V RU( µ V L ) iv R r µ UV L + V R( µ V R )V RUV L +iv R Ul µ V L L V R U( µ V L }{{} ) ( µ V R )UV L = V R ( µ U + iul µ ir µ U)V L = V R(D µ U)V L (4.49) r µ,l µ (field strength tensor) F Rµν = µ r ν ν r µ ir µ, r ν ] (4.5) F Lµν = µ l ν ν l µ il µ, l ν ] (4.5) χ = B (S + ip ) (4.5) 3F B = < qq > (4.53) F Rµν V R F Rµν V R, F Lµν V L F Lµν V L (4.54) χ V R χv L (4.55) NG U = O(q ) (4.56) D µ U = O(q ) (4.57) r µ, l µ = O(q ) (4.58) F Lµν, F Rµν = O(q ) (4.59) χ = O(q ) (4.6) building block O(q ) L = F 4 T rd µu(d µ U) ] + F 4 T rχu + Uχ ] (4.6) O(q n ) building block NG : S = M = diag(m u, m d, m s ) (4.6) v µ = a µ = P = (4.63) QCD

26 NG SU() Ψ = ( p n SU(3) (/) + ) B = 8 a= B a λ a = ) Σ + Λ 6 Σ + p Σ + Λ 6 n Ξ Ξ Λ 6 SU(),SU(3) Σ (4.64) (4.65) 4.4. G SU() L SU() R SU(3) L SU(3) R N f = U SU() (4.) U (π, π ± ) 4. U G U RUL u = U(x) K(L, R, U) u(x) u (x) = RUL RuK (L, R, U) (4.66) K(L, R, U) = u (x)ru = RUL R U (4.67) G {(U, Ψ)} φ(g) : ( U Ψ ) ( ) ( U = Ψ RUL K(L, R, U)Ψ ) (4.68) G {(U, Ψ)} ( U φ(g g ) Ψ ) ( = φ(g ) R UL K(L, R, U)Ψ ( R = R UL ) L K(L, R, R UL )K(L, R, U)Ψ ( ) R = R U(L L ) K(L L, R R, U)Ψ ( ) U = φ(g g ) Ψ ) (4.69)

27 4 6 3 (4.67) K(L, R, R UL )K(L, R, U) = K(L L, R R, U) (4.7) SU() (4.68) g = (L, R) G Ψ U R = L = V U (4.66) (4.7) U = V UV = V u V = V uv V uv = u (4.7) u = V uv (4.7) u (x) = V uk (V, V, U) (4.73) V = K (V, V, U) (4.74) V = K(V, V, U) (4.75) U Ψ G = SU(3) L SU(3) R ( ) ( ) ( ) U U RUL φ(g) : B B = K(L, R, U)BK (L, R, U) (4.76) U SU(3) N f = SU(3) V B = V BV B 4.4. SU() L SU() R U() V (4.68) ( ) ( U(x) V R (x)u(x)v L (x) Ψ(x) exp iθ(x)]k V L (x), V R (x), U(x)] Ψ(x) exp iθ(x)] U() V ) (4.77) D µ Ψ(x) = D µ Ψ(x)] = exp iθ(x)]k V L (x), V R (x), U(x)] D µ Ψ(x)] (4.78) D µ K V R,V L U chiral connection Γ µ = u ( µ ir µ )u + u( µ il µ )u ] (4.79) ( D µ Ψ = µ + Γ µ iν (s) µ ) Ψ (4.8)

28 4 7 ( )] D µψ = µ + Γ µ i ν µ (s) µ θ exp( iθ)kψ (4.8) ( ) = exp( iθ)k µ + Γ µ iν µ (s) Ψ (4.8) (4.8) µ exp( iθ)kψ] = i µ θ] exp( iθ)kψ + exp( iθ) µ K] Ψ + exp( iθ)k µ Ψ] (4.83) µ K = KΓ µ Γ µk (4.84) (4.8) (4.84) (4.79) V, µ V V = ( µ V ) V = V ( µ V ) K = u V R u = u u u V R u = u U V R u = u V L U V R V Ru = u V L u (4.85) Γ µ = ( ) ( ) u µ iv R r µ V R + V R µ V R u + u µ iv L l µ V L + V L µ V L u ] (4.86) O(q) building block chiral vielbein u µ i u ( µ ir µ ) u u ( µ il µ ) u ] (4.87) u µ i u ( µ il µ ) u u ( µ ir µ ) u ] = u µ (4.88) ϕ(t, x) ϕ(t, x), U(t, x) U (t, x) (4.89) l µ r µ, r µ l µ (4.9) SU() L SU() R U() V u µ Ku µ K (4.9) (4.79) (4.87) V µ = iγ µ, A µ = u µ (4.9) building block Ψ, Ψ = O(q ) (4.93) u µ, Γ µ, (i D m) Ψ = O(q ) (4.94) χ ± = O(q ) (4.95) L () πn L () πn = Ψ ( i D m + g ) A γµ γ 5 u µ Ψ (4.96)

29 4 8 m, g A m ΨΨ m ΨΨ tree-level m E (E m) SU(3) L SU(3) R N f = building block (leading order,lo) L () MB = T r B (i D M ) B ] + D T r ( Bγ µ γ 5 {u µ, B} ) + F T r ( Bγ µ γ 5 u µ, B] ) (4.97) M D µ B = µ B + Γ µ, B] (4.98) D,F semi-leptonic decay (next-to-leading order,nlo) - L () MB =b DT r( B{χ +, B}) + b F T r( Bχ +, B]) + b T r( BB)T r(χ + ) +d T r( B{u µ, u µ, B]}) + d T r( Bu µ, u µ, B]]) + d 3 T r( Bu µ )T r(u µ B) + d 4 T r( BB)T r(u µ u µ ) b D, b F, b, d, d, d 3, d 4 - NLO 4. (4.99) 4.: (a)weinberg-tomozawa (WT ) (b)born ( ) (c)born ( ) (d)nlo

30 9 5 -K + - K + p K + p K + n K + n NLO M-matrix M-matrix M-matrix s p 5. K + N ( ) p = p CM + m K, p CM ( ) p = p CM + m N, p CM ( ) p 3 = p CM + m K, p CM ( ) p 4 = p CM + m N, p CM (5.) (5.) (5.3) (5.4) m K K + m N p CM 5. s = (p + p ) = (p 3 + p 4 ) (5.5) t = (p p 3 ) = (p p 4 ) (5.6) u = (p p 4 ) = (p p 3 ) (5.7)

31 5 -K : p p = p 3 p 4 = s m N m K p p 3 = m K t p p 4 = m N t p p 4 = p p 3 = m N + m K u (5.8) (5.9) (5.) on-shell p = p 3 = m K, p = p 4 = m N t = m K p p 3 = p CM ( cos θ) (5.) u = m K + m N s t = s + p CM ( cos θ) + (m K + m N ) (5.) (s (mn + m K ) p CM = )(s (m N m K ) (5.3) s (5.) s + t + u = m K + m N M-matrix 5. K + n K + n SU(3) - Born Q =, S = Σ B = Σ n (5.4) Φ = K+ K (5.5) ( 5.)

32 5 -K : K + n K + n Born ( ) 5.. LO (4.97) LO M-matrix L () = + i 4f K ( n K K + n nk K + n) f K (D F )( Σ K γ 5 n + n K + γ 5 Σ ) (5.6) M () W T = 4fK ū(p 4, s 4 )( p + p 3 )u(p, s ) (5.7) M () Born F ) = (D fk ū(p 4, s 4 ) p γ 5 S Σ (p p 3 ) p 3 γ 5 u(p, s ) (5.8) S Σ S(q) = m q iϵ = m+ q m q iϵ (5.9) : p + p = p 3 + p 4 (5.) Dirac : ( p m)u(p, s) =, ū(p, s)( p m) = (5.) WT M () W T = 4f ū(p 4, s 4 )( p + p 3 )u(p, s ) = 4fK ū(p 4, s 4 )( p + p + p p 4 )u(p, s ) = 4fK ū(p 4, s 4 )( p + p + p p }{{} 4 )u(p, s ) m n m n = fk ū(p 4, s 4 ) p u(p, s ) (5.)

33 5 -K + 3 Born Dirac M () Born p k = (kp) k p (5.3) F ) m Σ + ( p p 3 ) = (D fk ū(p 4, s 4 ) p γ 5 m Σ (p p 3 ) p 3γ 5 u(p, s ) (D F ) = fk m Σ (p p 3 ) m Σū(p 4, s 4 ) p γ 5 p 3 γ 5 u(p, s ) +ū(p 4, s 4 ) p γ 5 p p 3 γ 5 u(p, s ) ū(p 4, s 4 ) p γ 5 p 3 p 3 γ 5 u(p, s )] (5.4) m Σ ū(p 4, s 4 ) p γ 5 p 3 γ 5 u(p, s ) = m Σ m n ū(p 4, s 4 ) p u(p, s ) + m Σ (p p 4 ) m K]ū(p 4, s 4 )u(p, s ) (5.5) ū(p 4, s 4 ) p γ 5 p p 3 γ 5 u(p, s ) = (p p 3 ) m n]ū(p 4, s 4 ) p u(p, s ) + m n (p p 4 ) m K]ū(p 4, s 4 )u(p, s ) (5.6) Born ū(p 4, s 4 ) p γ 5 p 3 p 3 γ 5 u(p, s ) = m Kū(p 4, s 4 ) p u(p, s ) (5.7) M () F ) Born = (D fk m Σ (p mσ p 3 ) m n + (p p 3 ) m n m ] K ū(p4, s 4 ) p u(p, s ) (D F ) fk m Σ (p p 3 ) (m Σ + m n )((p p 4 ) m K)ū(p 4, s 4 )u(p, s ) (5.8) M () = M () = W T + M() Born f K (D F ) f K ] m Σ (p p 3 ) ( m Σm n + (p p 3 ) m n m K) ū(p 4, s 4 ) p u(p, s ) (D F ) fk m Σ (p p 3 ) (m Σ + m n )((p p 4 ) m K)ū(p 4, s 4 )u(p, s ) (5.9) 5.. NLO (4.99) L () = B f K ( ˆm + m s )( b b D + b F ) nnk + K + f K ( d + d + d 4 ) nn µ K + µ K (5.3)

34 5 -K + 33 ˆm = m u+m d, m s u d s B = M-matrix M () = m K ˆm+m s B fk ( ˆm + m s )( b b D + b F ) fk ( d + d + d 4 )(p p 3 ) ] ū(p 4, s 4 )u(p, s ) (5.3) 5..3 LO+NLO M (+) = + (D F ) f K (D F ) fk m Σ (p p 3 ) ( m Σm n + (p p 3 ) m n m K) + ] fk ū(p 4, s 4 ) p u(p, s ) m Σ (p p 3 ) (m Σ + m n )((p p 4 ) m K) B fk ( ˆm + m s )( b b D + b F ) fk ( d + d + d 4 )(p p 3 ) ] ū(p 4, s 4 )u(p, s ) (5.3) M (+) = ū(p 4, s 4 ) A n + B n p ] u(p, s ) (5.33) (D F ) A n = fk m Σ (p p 3 ) (m Σ + m n )((p p 4 ) m K) B fk ( ˆm + m s )( b b D + b F ) fk ( d + d + d 4 )(p p 3 ) (5.34) (D F ) B n = fk m Σ (p p 3 ) ( m Σm n + (p p 3 ) m n m K) + fk (5.35) 5.3 K + p K + p SU(3) - Born Q =, S = Λ Σ B = Σ + Λ 6 p Σ + Λ 6 Λ 6 (5.36) ϕ = K+ K (5.37)

35 5 -K + 34 ( 5.3) K + p K + p K + n K + n 5.3: K + p K + p Born ( ) 5.3. LO (4.97) LO L () = i f K + D F f K M-matrix ( pk+ K p pk K + p ) ( Σ K γ 5 p + p K + γ 5 Σ ) D + 3F 3f K ( Λ K γ 5 p + p K + γ 5 Λ ) (5.38) M () W T = fk ū(p 4, s 4 )( p + p 3 )u(p, s ) (5.39) ( ) M () D F Born = ū(p 4, s 4 ) p γ 5 S Σ (p p 3 ) p 3 γ 5 u(p, s ) f K ( ) 3F + D ū(p 4, s 4 ) p γ 5 S Λ (p p 3 ) p 3 γ 5 u(p, s ) (5.4) 3f K 5. M () W T = fk ū(p 4, s 4 ) p u(p, s ) (5.4) ( ) M () D F Born = f K m Σ (p p 3 ) ( m Σm p + (p p 3 ) m p m K) ( ) ] 3F + D 3f K m Λ (p p 3 ) ( m Λm p + (p p 3 ) m p m K) ū(p 4, s 4 ) p u(p, s ) ( ) D F + f K m Σ (p p 3 ) (m Σ + m p )((p p 4 ) m K) ( ) ] 3F + D 3f K m Λ (p p 3 ) (m Λ + m p )((p p 4 ) m K) ū(p 4, s 4 )u(p, s ) (5.4)

36 5 -K + 35 M () = M () W T + M() Born ( ) D F = f K m Σ (p p 3 ) ( m Σm p + (p p 3 ) m p m K) ( ) 3F + D 3f K m Λ (p p 3 ) ( m Λm p + (p p 3 ) m p m K) + ] fk ū(p 4, s 4 ) p u(p, s ) ( ) D F + f K m Σ (p p 3 ) (m Σ + m p )((p p 4 ) m K) ( ) ] 3F + D 3f K m Λ (p p 3 ) (m Λ + m p )((p p 4 ) m K) ū(p 4, s 4 )u(p, s ) (5.43) 5.3. NLO (4.99) L () = 4B fk ( ˆm + m s )(b + b D ) ppk + K + fk (d + d 3 + d 4 ) pp µ K + µ K (5.44) M-matrix M () 4B = fk ( ˆm + m s )(b + b D ) ] fk (d + d 3 + d 4 )(p p 3 ) ū(p 4, s 4 )u(p, s ) (5.45) LO+NLO ( ) D F M (+) = f K m Σ (p p 3 ) ( m Σm p + (p p 3 ) m p m K) ( ) 3F + D 3f K m Λ (p p 3 ) ( m Λm p + (p p 3 ) m p m K) + ] fk ū(p 4, s 4 ) p u(p, s ) ( ) D F + f K m Σ (p p 3 ) (m Σ + m p )((p p 4 ) m K) ( ) 3F + D 3f K m Λ (p p 3 ) (m Λ + m p )((p p 4 ) m K) + 4B fk ( ˆm + m s )(b + b D ) ] fk (d + d 3 + d 4 )(p p 3 ) ū(p 4, s 4 )u(p, s ) (5.46) M (+) = ū(p 4, s 4 ) A p + B p p ] u(p, s ) (5.47)

37 5 -K + 36 ( D F A p = f K ( 3F + D 3f K f K ) m Σ (p p 3 ) (m Σ + m p )((p p 4 ) m K) ) m Λ (p p 3 ) (m Λ + m p )((p p 4 ) m K) + 4B ( ˆm + m s )(b + b D ) (d + d 3 + d 4 )(p p 3 ) (5.48) f K ( ) D F B p = f K m Σ (p p 3 ) ( m Σm p + (p p 3 ) m p m K) ( ) 3F + D 3f K m Λ (p p 3 ) ( m Λm p + (p p 3 ) m p m K) + fk (5.49) 5.4 M-matrix K + n K + n K + p K + p M-matrix m x A n (A p ) B n (B p ) A, B A m x (p p 3 ) = m x u = m x (m K + m N ) + s p CM + p CM cosθ = m x (m K + m N ) + s R(s) + R(s)τ (5.5) R(s) = p CM, cosθ = τ (5.5) V (s) = m Σ (m K + m N ) + s R(s) (5.5) T (s) = m Λ (m K + m N ) + s R(s) (5.53) m Σ u = V (s) + R(s)τ (5.54) m Λ u = T (s) + R(s)τ (5.55) (p p 4 ) m K = m N m K + s p CM ( cosθ) = s R(s) m K m N + R(s)τ = J(s) + R(s)τ (5.56)

38 5 -K + 37 J(s) = s R(s) m K m N (5.57) R(s) = p CM (5.58) cosθ = τ (5.59) ( ) ( ) D F 3F + D C =, E = f K, C = 3f K (D F ) f K (5.6) B J(s) + R(s)τ A n = C (m Σ + m p ) V (s) R(s)τ }{{} Born B f (m + m s)( b b D + b F ) f ( d + d + d 4 )(m K + R R τ) }{{} NLO J(s) + R(s)t A p = C(m Σ + m N ) V (s) + R(s)t E(m J(s) + R(s)t Λ + m N ) T (s) + R(s)t }{{} Born + 4B fk ( ˆm + m s )(b + b d ) fk (d + d 3 + d 4 )(p p 3 ) }{{} NLO (5.6) (5.6) m x m N + (p p 3 ) m x m K = m x m N 3m N m K + s R(s)] + R(s)cosθ (5.63) W (s) = m Σ m N 3m N m K + s R(s) (5.64) X(s) = m Λ m N 3m N m K + s R(s) (5.65) m Σ m N + (p p 3 ) m N m K = W (s) + R(s)cosθ (5.66) m Λ m N + (p p 3 ) m N m K = X(s) + R(s)cosθ (5.67) B n = C W (s) + R(s)τ V (s) + R(s)τ } {{ } Born W (s) + R(s)τ B p = C V (s) + R(s)τ + f K }{{} W T E X(s) + R(s)τ T (s) + R(s)τ } {{ } Born + f K }{{} W T M-matrix A, B s τ (5.68) (5.69)

39 5 -K / / σ f = F (θ) iσ ng(θ) (5.7) p p n = p p (5.7) 3],F (θ), G(θ) F (θ) = G(θ) = (l + )f + l + lf l )]P l (cosθ) (5.7) l= l= f + l f l ] dp l(cosθ) d(cosθ) sinθ }{{} P l (5.73) f + l = eiδ+ l sinδ + l k f l = eiδ l sinδ l k k (5.74) (5.75) (5.77) P l (x)p l (x)dx = l + δ l,l (5.76) P l (x)p l (x)dx = l + l(l + )δ l,l (5.77) P m l Pl m (x)p m l (x)dx = (l + m)! l + (l m) δ l,l (5.78) = ( x ) m d ( dx ) m P l (x) (5.79) m = F (θ), G(θ) f + = F l= (5.8) f + = 3 (F l= + G l= ) (5.8) f = 3 (F l= G l= ) (5.8) f + = 5 (F l= + G l= ) (5.83) f = 5 (F l= 3G l= ) (5.84) / L S j j = l ± f ± j = l ± ±

40 5 -K + 39 L IJ l =,,, 3,... s,p,d,f... L I J f ± l, I = S : f +, P 3 : f +, P : f M-matrix u(p, s) = E p + m p ( σ p E p +m p ) χ (s) (5.85) M (+) = ū(p 4, s 4 ) A + B p ] u(p, s ) (5.86) σ i σ j = δ ij + iϵ ijk σ k (5.87) CM ū(p 4, s 4 )u(p, s ) = p 4 p + i(p 4 p ) σ E p + m p (5.88) ū(p 4, s 4 )γ u(p, s ) = + p 4 p + i(p 4 p ) σ E p + m p (5.89) ū(p 4, s 4 )γ i u(p, s ) = p i + p i 4 + i(p σ) i i(p 4 σ) i (5.9) M =(E N + m N ) AE K + A p CM ] + B + p CM E N + m N E N + m N iσ ( pˆ 3 pˆ ) p CM E N + m N p = p (5.9) p 3 = p 4 (5.9) ( (E N + m N )A + AE K B) τ ] AE K + (E N + m N )A B τ ] (5.93) F (θ) = (E N + m N ) AE K + A p CM ] + B E N + m N + p CM ] AE K + (E N + m N )A B τ (5.94) E N + m N G(θ) = p CM sinθ ((E N + m N )A + AE K B) (5.95) E N + m N

41 5 -K s (l = ) X P l (τ) l X l = l + W + Rτ W dτ = V V + Rτ R W + Rτ V + Rτ dτxp l (τ) (5.96) ( R + V ln R + V ] τdτ = V (V W ) R ln A, B K + n K + n A l= ) + (5.97) ( R + V R + V n = C ( ) (m Σ + m N ) R + V (V J) ln R R + V + B fk ( ˆm + m s )(b + b D b F ) + + C ( (V W ) R + V ln B l= n = f K R R + V V (V J) (A n τ) l= = C (m Σ + m N ) R ln ) + W V R C (m Σ + m N ) ( m K + R ) (5.98) f (d d d 4 ) (5.99) ) C (5.) ( R + V R + V ) + J V ] R + R 3fK ( d + d + d 4 ) (5.) (B n τ) l= = C V (V W ) R ( ) R + V ln + W V ] R + V R (5.) K + p K + p A l= p = C(V J)(m ( ) Σ + m N ) R + V ln + E(T J)(m ( ) Λ + m N ) R + T ln R R + V R R + T C(m Σ + m N ) E(m Λ + m N ) + 4B fk ( ˆm + m s )(b + b D ) + ( fk (d + d 3 + d 4 ) m K + R ) (5.3) Bp l= = C(V W ) ( ) ( ) R + V E(T X) R + T ln + ln C E + R R + V R R + T fk (5.4) ( ) V (V J) R + V (A p τ) l= = C(m Σ + m N ) R ln + J V ] R + V R ( ) T (T J) R + T E(m Λ + m N ) R ln + J T ] R + T R + R 3fK (d + d 3 + d 4 ) (5.5) ( ) V (V W ) R + V (B p τ) l= = C R ln W V ] R + V R ( ) T (T X) R + T E R ln X T ] (5.6) R + T R

42 5 -K p (l = ) l = G(θ) G(θ) Pl m= O Pl m= (τ) l O l = l + dτopl m= (τ) (5.7) l(l + ) l = P m= l= = x 3 τ W + Rτ 4 V + Rτ dτ = 3 4 π πv (W V ) + R ( )] R V (5.8) K + n K + n A l= n = 3C ( ) ] (m Σ + m N ) R + V R V (V J) ln + R(J V ) R + V B l= n = 3C R + R fk ( d + d + d 4 ) (5.9) ( ) R + V V (V W ) ln (A n τ) l= = 3C (m Σ + m N ) R 3 R + V V (V J) ln ] + R(W V ) ( R + V R + V (5.) ) + RV (J V ) 3 ] R3 B fk ( ˆm + m s )( b b d + b f ) ( fk ( d + d + d 4 ) m K + R ) (B n τ) l= = 3C ( ) R + V R 3 V (V W ) ln + RV (W V ) 3 ] R + V R3 + f K ( ) l= 3C A n τ π(m Σ + m N ) = 4 (J V )V + R ( )] R V (5.) (5.) ( B n τ ) l= = 3C π 4 3πB 4fK ( ˆm + m s )( b b d + b f ) 3π 4fK ( d + d + d 4 ) (W V )V + R ( m K + R ) ( R V )] + 3π 6f K (5.3) (5.4)

43 5 -K + 4 K + p K + p A l= p = 3C(m Σ + m N ) R B l= p 3E(m Λ + m N ) R ( ) R + V V (V J) ln R + V ( R + T T (T J) ln R + T ] + R(J V ) ] ) + R(J T ) + R fk (d + d 3 + d 4 ) (5.5) ( ) R + V = 3C R 3E R (A p τ) l= = 3C(m Σ + m N ) R 3 (B p τ) l= = 3C R 3 ] V (V W ) ln + R(W V ) R + V ( ) ] R + T T (T X) ln + R(X T ) (5.6) R + T ( ) R + V V (V J) ln + RV (J V ) 3 ] R + V R3 ) + RT (J T ) 3 ] R3 + 3E(m Λ + m N ) R 3 T (T J) ln ( R + T R + T + 4B fk ( ˆm + m s )(b + b d ) ( fk (d + d 3 + d 4 ) m K + R ) ( ) R + V V (V W ) ln + RV (W V ) 3 ] R + V R3 ) + RT (X T ) 3 ] R3 + 3E R 3 + f K T (T X) ln ( ) l= 3Cπ(m A p τ Σ + m N ) = 4 ( B p τ ) l= = 3Cπ 4 3Eπ(m Λ + m N ) 4 ( R + T R + T + (J V )V R (J T )T + R ( )] R V ( )] R + 6πB 4fK ( ˆm + m s )(b + b d ) 3π 4fK (d + d 3 + d 4 ) ( )] (W V )V + R R V ( 3Eπ )] (X T )T + 4 R R T + 3π 8fK T ( m K + R ) (5.7) (5.8) (5.9)

44 43 6 M-matrix NLO χ K + - m N = MeV m Σ = MeV m Λ = MeV m K = MeV f K =. MeV ħc = MeV fm D =.8 F = M-matrix NLO I = I = χ χ χ = n ( ) yi f(x i ) (6.) σ i= i y i f(x i ) σ i n I = I = b I= b + b d (6.) d I= d + d 3 + d 4 (6.3)

45 6 44 b I= b b f (6.4) d I= d d 4 + d 3 (6.5) I = n = 35 I = n = χ I = I = ( 6. 6.) 6.: I = MeV ] b I= d I= χ /d.o.f (a) (b) : I = MeV ] b I= d I= χ /d.o.f (c) (d) ( 6. 6.) 5 TOTAL CROSS SECTION mb] 5 5 Bugg,et al (968) Bowen,et al (97) Adams,et al (97) Bowen,et al (973) TOTAL CROSS SECTION mb] 5 5 Bowen,et al (97) Bowen,et al (973) Carroll,et al (973) : I = P LAB MeV/c] 6.: I = P LAB MeV/c] (a),(b) (c),(d) 4, 7, 8, 9] 7, 9, ] M-matrix K + n K + n dσ dω = f = 64π s M (6.6)

46 6 45 K + p K + p (I = ) I = K + n K + n f(k + n K + n) = (f I= + f I= ) (6.7) I = I = b n = b I= b I= (6.8) d n = di= d I= (a)(c),(a)(d),(b)(c),(b)(d) ( 6.3) (6.9) DIFFERENTIAL CROSS SECTION mb/sr].5.5 Damerell,et al (975) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =434. MeV/c] DIFFERENTIAL CROSS SECTION mb/sr].5.5 Damerell,et al (975) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =56. MeV/c] cos(x) cos(x) DIFFERENTIAL CROSS SECTION mb/sr].5.5 Damerell,et al (975) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =64. MeV/c] DIFFERENTIAL CROSS SECTION mb/sr].5.5 Giacomelli,et al (973) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =64. MeV/c] cos(x) cos(x) DIFFERENTIAL CROSS SECTION mb/sr].5.5 Damerell,et al (975) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =688. MeV/c] DIFFERENTIAL CROSS SECTION mb/sr].5.5 Giacomelli,et al (973) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =7. MeV/c] cos(x) cos(x)

47 6 46 DIFFERENTIAL CROSS SECTION mb/sr].5.5 Damerell,et al (975) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =77. MeV/c] cos(x) DIFFERENTIAL CROSS SECTION mb/sr].5.5 Giacomelli,et al (973) (a)(c) (a)(d) (b)(c) (b)(d) P LAB =7. MeV/c] cos(x) 6.3: K+n K + n, ] M-matrix P LAB =434 MeV/c P LAB (b)(c) ( ) P LAB =64 MeV/c P LAB =688 MeV/c P LAB K + n K + n 434. MeV/c P LAB 64. MeV/c

48 : I = 4, 7, 8, 9] p LAB MeV] σ mb] ±.4 Bugg, et al. (968) 6.9 ± ± ± ± ±.35 Bowen,et al. (97) ± ± ± ± ± ± ± ± ± ± ± ±.4 Adams,et al. (97) ± ± ± ± ± ± ± ± ±.9 Bowen,et al. (973) ± ± ± ± ± ± ± ±.3

49 : I = 7, 9, ] p LAB MeV] σ mb] ±.75 Bowen,et al. (97) ± ± ± ± ± ± ± ± ± ±.6 Bowen,et al. (973) ± ± ± ± ±.44 Carroll,et al. (973) ± ± ± ± ± ± ± ± ±.43

50 a (5.8) s f + a = 3] f M-matrix a (6.) (6.) lim f + (6.) p CM f = 8π s M (6.) a = 8π(m N + m K ) M( s = m N + m K ) (6.) Born (6.) a > : a < : (6.3) M < : (6.) I = I = s 6.5 M > : (6.4) 6.5: I = I = s fm] WT Born NLO LO + NLO S ±. Cameron et al. 4] S ±.4 Stenger et al. 5] S S (threshold) 6.3 f l f = (l + )f l P l (cosθ) (6.5) l= f l = eiδ l sin δ l k (6.6)

51 6 5 σ l σ = σ l (6.7) l= σ l = 4π k (l + ) sin δ l (6.8) 6] tree-level (6.6) (6.8) e iδ l f l = sin δ l k (6.9) δ l δ l = arcsin(kf l ) (6.) M-matrix ( ) δ l = arcsin k 8π s M l (6.) (6.4) δ l > : δ l < : (6.) (6.) (5.8),(5.8),(5.8) S, P, P 3, S, P, P 3 ( 6.4) PHASE-SHIFT DEGREES] S P LAB MeV/c] PHASE-SHIFT DEGREES] S P LAB MeV/c]

52 6 5 PHASE-SHIFT DEGREES] P P LAB MeV/c] PHASE-SHIFT DEGREES] P P LAB MeV/c] PHASE-SHIFT DEGREES] P 3 PHASE-SHIFT DEGREES] P P LAB MeV/c] P LAB MeV/c] 6.4: I =, I = s, p 7] I = I = P LAB = 6 MeV/c P LAB = 5 MeV/c 7] P LAB S unitalization8] ( ) 6.4 (6.6) I = ( 6.5) 45 MeV/c P LAB 385 MeV/c P LAB =5MeV/c x =

53 6 5 DIFFERENTIAL CROSS SECTION mb/sr] Cameron,et al (974) P LAB =45. MeV/c] DIFFERENTIAL CROSS SECTION mb/sr] Cameron,et al (974) P LAB =75. MeV/c] cos(x) cos(x) DIFFERENTIAL CROSS SECTION mb/sr] Cameron,et al (974) P LAB =5. MeV/c] DIFFERENTIAL CROSS SECTION mb/sr] Cameron,et al (974) P LAB =35. MeV/c] cos(x) cos(x) DIFFERENTIAL CROSS SECTION mb/sr] Cameron,et al (974) P LAB =355. MeV/c] DIFFERENTIAL CROSS SECTION mb/sr] Cameron,et al (974) P LAB =385. MeV/c] cos(x) cos(x)

54 6 53 DIFFERENTIAL CROSS SECTION mb/sr] 3 Cameron,et al (974).5 P LAB =5. MeV/c] cos(x) DIFFERENTIAL CROSS SECTION mb/sr] 3 Charles,et al (977).5 P LAB =698. MeV/c] cos(x) 6.5: I = 9, ]

55 K + χ NLO K + K + ρ K + - E p m K Σ(E, p, ρ) ] ϕ = (6.3) Σ(E, p, ρ) K + E, m K, p K + Σ free = m N M(K + N) free ρ (6.4) M(K + N) free = 8π sb free (6.5) m N ρ s K + N b free K + - Σ fit = m N M(K + N) fit ρ (6.6) M(K + N) fit = 8π sb fit (6.7) K + N b fit b fit K + - E.Friedman A.Gal ] Re b fit Re b free 4% 5% Im b fit Im b free % 5% ( 6.6) K : K + ] P LAB MeV/c] Σ Reb fm] Imb fm] 488 Σ fit -.3(6).7(7) Σ free Σ fit -.96(39).(9) Σ free Σ fit -.(5).6() Σ free Σ fit -.4(53).85(5) Σ free -.6.8

56 6 55 ( ) (6.3) Π(p) = i E p m K Σ(E, p, ρ) + iϵ (6.8) K + p = m K (ρ) m K + Σ(E = m K, p =, ρ) (6.9) ] on-shell E = m K ( ) Σ Σ(E, p, ρ) = Σ(E = m K, p =, ρ) + (E m K ) E + (6.3) E =m K, p= E m K Σ(E, p, ρ) E m K (6.8) Σ(E = m K, p =, ρ) + (E m K ) ( ) Σ = E m K (E m K ) E ( ) Σ = (E m K ) E E =m K, p= E =m K, p= ] ( ) ] Σ E E =m K, p= (6.3) iz Π(p) = E m K + iϵ (6.3) ( ) ] Σ Z = E (6.33) E =m K, p= E = m K ( ) Σ Σ(E, p, ρ) = Σ(E = m K, p =, ρ) + (E m K) E E =m K ( ) ], p= Σ Σ(E = m K, p =, ρ) + E E =m K, p= ZΣ(E = m K, p =, ρ) (6.34)

57 6 56 Z = ( ) ] ( ) Σ Σ E + E =m K, p= E (6.35) E =m K, p= Z ρ N = Z, K + N M-matrix ρ M(K + p) = M(I = ) (6.36) M(K + n) = M(I = ) + M(I = )] (6.37) M(K + N) = M(K + p) + M(K + n) ] = 3M(I = ) + M(I = )] (6.38) 4 ρ p = ρ n = ρ (6.39) Σ = m N M(K + p)ρ p + M(K + n)ρ n ] = 4m N M(K + p) + M(K + n) ] ρ = 8m N 3M(I = ) + M(I = )] ρ (6.4) ( 6.7) 6.7: P LAB MeV/c] Z (WT) Z (Born) Z (NLO) Z (LO+NLO) ρ ρ.9 ρ ρ +. ρ ρ +.3 ρ ρ ρ ρ.8 ρ ρ +. ρ ρ +.9 ρ ρ ρ ρ.3 ρ ρ +. ρ ρ +.7 ρ ρ ρ ρ. ρ ρ +. ρ ρ +.7 ρ ρ 3% 6.6 5% 5%

58 6 57 Born WT Born NLO ( ) Born ( ) χ constraint TOTAL CROSS SECTION mb] WT Born NLO P LAB MeV/c] TOTAL CROSS SECTION mb] Born NLO P LAB MeV/c] 6.6: I = 6.7: I = on-shell E = p + m on-shell E p E p E E p off-shell P LAB

59 58 7 NLO K + N K + N χ NLO I =, I = K + n K + n I = χ K + χ unitalization unitalization χ K + 3% 5 % 5 % on-shell Born χ constraint

60 59

61 6 ] K.Suzuki et al.,phys.rev.lett. 9 (4) 73 ] E. Friedman,A. Gal, Phys.Rept.45:89-53,7 3] W.Weise, Nuovo Cim. A (989) ] D.Bugg et al.,phys.rev.,68,466(968) 5] S.Scherer and M R.Schindler, A Primer for Chiral Perturbation Theory (Springer) 6] A. Pich, arxiv:hep-ph/ ] T.Bowen et al.,phys.rev.d(97) ] C.J. Adams et al.,phys.rev. D4 (97) ] T.Bowen et al.,phys.rev.7d(973). ] A.S Carroll et al.,phys.lett. B45 (973) ] C.J.S Damerell et al.,nucl.phys. B94 (975) 374 ] G.Giacomelli et al.,nucl.phys. B56 (973) ] J.M. Eisenberg and D. Koltun, Theory of Meson Interactions with Nuclei (Wiley, New York, 98) Chapter. 4] W. Cameron et al,, NucI. Phys. B78 (974) 93. 5] V.J. Stenger et al., Phys. Rev. 34B (964). 6] N.Zettili, Quantum Mechanics:concepts and applications nd ed. (Wiley, New York,9) Chapter. 7] K.Hashimoto, Phys.Rev. C9 (984) ] T.Hyodo and D.Jido,Prog.Part.Nucl.Phys.67:55-98, 9] W.Cameron et al.,nucl.phys. B78 (974) 93 ] B.J. Charles et al.,nucl.phys. B3 (977) 7-53 ] W. Weise, Nucl. Phys. A 69 () 98.