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qq U(1)A 1/Nc 1/Nc U(3)L U(3)R U(1)A U(3)L U(3)R 1/Nc n 1/Nc 1/Nc U(1)A U(3)L U(3)R Nf = 3 n U(2)L U(2)R 1/Nc m qq qq 00 +- 1 1 dt W [Y ] = - ln | det D| = (t, 2) e-tD D +. 2 t 0 D DD = M2 - 2 + Y Y t t (t, 2) W [Y ] an (n = 0, 1, 2...) (an) an n = 5 M = (m, m,...) m2 an M = (m1, m2,...) M n ij Jl(m2) - Jl(m2) = Jl+n(m2) - (-1)nJl+n(m2), j i i j 2nn! n=1 ij = m2 - m2 Jn(m2) i j dt 2 Jn(m2) = e-tm (t, 2) t2-n 0 d4p 1 = 162(n + 1) . (2)4 (p2 + m2)n+1 Ji(m2) j Nf 1 Ii Ji(m2). Nf j=1 j d4x W [Y ] = Ii-1 (bi). 322 i=0 W [Y ] n bi Nf = 3 2 Y 12 1 b0 = 1, b1 = -Y, b2 = + 3Y + (13 + 23) 8Y, 2 2 2 3 3 Y 1 1 b3 = - - (Y )2 - 12 (31 + 32) 3Y 3! 12 12 1 + [13(21 + 23) + 23(12 + 13)] 8Y 12 3 1 1 2 2 + (31 + 32) 8Y + 213Y. 4 4 3 m1 = m2 =... = mN f an bn DD (DD) = i[, DD] bn bn = i[, bn] = + 5 Jl l 1 Jl+1(m2) m-2l bl i i 1/m2 U(3)L U(3)R U(1)A SU(2)f mu = md = ms SU(3) SU(3) L8q = L(1) + L(2) 8q 8q L(1) = 8g1 [(qiPRqm)(qmPLqi)]2, 8q L(2) = 16g2 [(qiPRqm)(qmPLqj)(qjPRqk)(qkPLqi)]. 8q U(3)L U(3)R L(1) 8q Z Z = DqDq Da Da exp i d4xLq(q, q, , ) a a + Dsa Dpa exp i d4xLr(, , ; s, p), a a - Lq = q(i - m - - i5)q, G Lr = sa(a + a) + paa + s2 + p2 a a 2 g1 2 + Aabcsa (sbsc - 3pbpc) + s2 + p2 a a 32 8 g2 + [dabedcde (sasbscsd + 2sasbpcpd + papbpcpd) 8 + 4facefbdesasbpcpd]. a = ma - ma Aabc U(3) a a sa pa Lr sa, pa sst = sa(, ), pst = pa(, ) a a Lr Lr = 0, = 0. sa pa a, a sst = ha + h(1)b + h(1) bc + h(2) bc +...





a ab abc abc pst = h(2)b + h(3) bc +... a ab abc h(i) a...

G, , g1, g2 a h(i) a...

ha 3 g1 g2 a + Gha + Aabchbhc + hah2 + dabedcdehbhchd = 0. b 32 2 2 U(3) SU(3) 2 1 g1 > 0, g1 + 3g2 > 0, G > g1 16 1/Nc 8(m2 - m2)2 + 3cq K m2 = m2 -, 0 9(m2 - m2) 0 1 m2 = (4m2 -m2 ) 0 K 3 SU(3) cq cq = 0 SU(3) m0 = 565 m = 547.30 0.12 (m2 - m2 )2 K m 496 cq m = 486 2 + 1 3 + 1 H = 0 H H = 0 f fK G g1 G g1 g1 1. 2. n 1/Nc 3. 1/Nc 4. U(3) SU(Nf )L SU(Nf)R 5. U(1)A U(3)L U(3)R 6. 7. 8. Nf = 3 9. B, H, H, Q1 Q2 a1 a1 SU(3) UA(1) SU(3) UA(1) UA(1) UA(1) UA(1)

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