Norisuke Sakai (Tokyo Institute of Technology) In collaboration with M. Eto, T. Fujimori, Y. Isozumi, T. Nagashima, M. Nitta, K. Ohashi, K. Ohta, Y. Tachikawa, D. Tong, M. Yamazaki, and Y. Yang 2008.3.21-26, JPS and JMS meeting at Kinki University Contents 1 Introduction 3 2 BPS 6 3 U(N C ), N F ( ) 8 4 1/2 BPS 10
5 : 15 6 ( ) 17 7 21 8 Conclusion 24 2
1 Introduction SU(3) SU(2) U(1) 1. (Technocolor) TeV L. Susskind,Phys. Rev.D20 (1979) 2619; S. Weinberg, Phys. Rev.D19 (1979) 1277; D13 (1976) 2. 974; S. Dimopoulos, and L. Susskind,Nucl. Phys. B155 (1979) 237; 0 : m B m B = m F 1 2 : m F m F = 0 S.Dimopoulos, H.Georgi, Nucl.Phys.B193 (1981) 150; N.Sakai, Z.f.Phys.C11 (1981) 153; 3 E.Witten, Nucl.Phys.B188 (1981) 513;
Figure 1: ( ) ( ) α i = gi 2 /4π, (i = 1, 2, 3) U(1), SU(2), SU(3) : (sequestering) 3. (Brane World)= 4 ( ) 4 P.Horava and E.Witten, Nucl.Phys.B475, 94 (1996); N.Arkani-Hamed, SDimopoulos, G.Dvali, 4
y: Phys.Lett.B429 (1998) 263 ; I.Antoniadis, N.Arkani-Hamed, S.Dimopoulos, G.Dvali, Phys.Lett.B436 (1998) 257; Randall, Sundrum, Phys.Rev.Lett.83 (1999) 3370; 4690; = :,,... LHC 5
トンネル周長 27km ( 参考 : 東京 JR 山手線の周長 34.5km) 4
2 BPS φ (λ > 0) : φ + v, L = µ φ µ φ λ(φ 2 v 2 ) 2 φ v ( ) ( ) : ( ) π 0 (M) y = x 2 E = ( y φ) 2 + λ(φ 2 v 2 ) 2 = ( y φ + λ(φ 2 v 2 )) 2 + y [2 )] λ (v 2 φ φ3 3 [ dye 2 )] λ (v 2 φ φ3 3 Bogomol nyi-prasad-sommerfield (BPS) 6
Bogomol nyi, Sov.J.Nucl.Phys. 24 (1976) 449; Prasad and Sommerfield, Phys.Rev.Lett. 35 (1975) 760. BPS (1 ) y φ + λ(φ 2 v 2 ) = 0 φ = v tanh( λv(y y 0 )) : y 0 ( ) ( ) : BPS ( ) 7
: U(1) φ L = 1 4e 2F µνf µν + D µ φ(d µ φ) λ ( φφ v 2) 2 4 D µ φ = ( µ + iw µ )φ, F µν = µ W ν ν W µ : ( ) k = 1 2π d 2 x F 12 : = U(1) :, λ < e 2 : : λ = e 2 : BPS λ > e 2 : (λ = e 2 ) (SUSY) BPS 3 U(N C ), N F ( ) 5 (M, N, = 0, 1, 2, 3, 4), U(N C ) g 8
W M, Σ ( :N C N C ) ( ) : H ra H ra ( :N C N F ) ( r = 1,, N C ; A = 1,, N F ) L = 1 2g 2Tr(F MN(W )F MN (W )) + 1 g 2Tr(DM ΣD M Σ) +Tr [ D M H(D M H) ] V V = g2 [ (HH 4 Tr ) 2 ] c1 NC + Tr [ (ΣH HM)(ΣH HM) ] D M H = ( M + iw M )H, D M Σ = M Σ + i[w M, Σ] F MN (W ) = M W N N W M + i[w M, W N ], (M) A B m A δ A B (8 SUSY) m 1 = = m NF SU(N F ) F Σ m A1 = = m Ak SU(k) 9
U(1) N F 1 F : HH = c1 NC, ΣH HM = 0 ( ),,, 4 1/2 BPS : m A > m A+1 A 1 A 2 A NC : H ra = c δ A r A, Σ = diag(m A1,, m ANC ) 1/2 BPS N F! (N F N C )!N C! en F log(x x (1 x) (1 x)), x N C /N F y x 4, 4 D W M y = 0 10
E : [ E = Tr D y H 2] + Tr [ ΣH HM 2] + 1 ( g 2Tr (D y Σ) 2) + g2 [ (HH 4 Tr ) 2 ] c1 NC = Tr D y H + ΣH HM 2 + 1 ( g 2Tr D y Σ g2 ( c1nc HH )) 2 + c y TrΣ 2 1/2 BPS y H + iw y H = ΣH + HM, (4.1) D y Σ = g 2 ( c1 NC HH ) /2 (4.2) : ( ) A 1 A 2 A NC B 1 B 2 B NC BPS Σ + iw y S 1 (y) y S(y) S(y) GL(N C, C) BPS (4.1) : H(y) = S 1 (y)h 0 e My 11
H 0 N C N F Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys.Rev.Lett.93 (2004) 161601; BPS (4.2) Ω SS ( y Ω 1 y Ω ) = g 2 c ( ) 1 C Ω 1 Ω 0, Ω0 c 1 H 0 e 2My H 0 H 0 Ω(y) S(y) Σ, W y, H y = ± : U(1) : H 0 N.Sakai and Y.Yang, Com.Math.Phys.267 (2006) 783; N.Sakai and D.Tong, JHEP 03 (2005) 019 H 0 g 2 c/ m 1: Ω = Ω 0 c 1 H 0 e 2My H 0 g 2 : (NLSM) 12
V - : (S NC 2 ) (S, H 0 ) (S, H 0 ) H = S 1 H 0 e My S S = V S, H 0 H 0 = V H 0, V GL(N C, C) BPS : M = {H 0 H 0 V H 0, V GL(N C, C)} G NF,N C SU(N F ) SU(N C ) SU(N F N C ) U(1) ( ) N C Ñ C N C (N F N C ) 1 0.8 0.6 0.4 0.2-40 -20 20 40 y Figure 2:. H 0 : 13
U(1) : H 0 = (e r 1, e r 2,, e r N F ), H = S 1 H 0 e My = S 1 (e r 1+m 1 y,, e r N F +m NF y ) i i + 1 Rer i + m i y Rer i+1 + m i+1 y Im(r i r i+1 ) : m 3 m 2 m 1 m NF m NF-1 m NF-2 m 4 m 3 A3 m 2 A2 m 1 A1 A1 1 2 3 B1 BNC BNC-1 B1 Figure 3: U(N C ) : Ñ C N F N C : 1,, N C Ñ C + 1,, N F dim R M 1,,N C Ñ C +1,,N F N F,N C 14 = 2N wall = 2N C Ñ C
M = M 1/1 + M 1/2 = M 0 M 1 M N CÑC D M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, K.Ohta, and N.Sakai, Phys.Rev.D71 (2005) 125006, BPS M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys. Rev. D73 (2006) 125008, 1/2 BPS M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys.Rev.Lett.96 (2006) 161601, 5 : M.Eto, T.Fujimori, M.Nitta, K.Ohashi, and N.Sakai, arxiv:0802.3135, ( ) U(1),N F = 4, M = (m, 0, 0, m) U(2) U(1) 2 SU(2)/U(1) 15
¹Ñ ¼ Ñ ¼ Æ Æ ¼µ µ ½ Æ Æ ½ ½µ Æ Ò............ Æ Ô Æ ¼ Ƶ ¼µ Æ Ô µ Æ Ô µ ÑÓ ËÍ ¾µ ËÍ Æ µä ËÍ Æ µê Figure 4: ( )U(1), ( m, 0, 0, m) ( )U(N),N F = 2N, (m,, m, m,, m) U(N),N F = 2N, M = 1 N N 2 diag( {}}{{}}{ m,, m, m,, m) 2N 2 N SU(N) SU(N) U(1)/SU(N) N 2 (NG) N 2 NG 16
SU(N) SU(N) U(1)/U(1) N 1 2N 2 N + 1 (NG) N 1 NG 6 ( ) : Tong, Phys.Rev.D69 (2004) 065003; Auzzi-Bolognesi-Evslin-Konishi, Nucl.Phys.B686 (2004) 119; Shifman-Yung, Phys.Rev.D70 (2004) 045004; Auzzi-Bolognesi-Evslin, JHEP 0502 (2005) 046; 1/2 γ 123 ε i = ε i x 3 1/2SUSY : γ 12 (iσ 3 ) i jε j = ε i ( + ): 1/4 SUSY γ 3 (iσ 3 ) i jε j = ε i x 3 1/4 BPS D 3 Σ = g 2 ( c1 NC H 1 H 1 ) /2 + F 12, D 3 H 1 = ΣH 1 + HM, 0 = D 1 H 1 + id 2 H 1, 0 = F 23 D 1 Σ, 0 = F 31 D 2 Σ 17
Figure 5: ( ) ( ) BPS E t w + t v + t m + m J m t w, t v, t m t w = c 3 Tr(Σ), t v = ctr(f 12 ), t m = 2 g 2 mtr( 1 2 ϵmnl F nl Σ) : [D 1 + id 2, D 3 + Σ] = 0 ( ) S(x m ) GL(N C, C) (D 3 + Σ)S 1 = 0 Σ + iw 3 S 1 3 S (D 1 + id 2 )S 1 = 0 W 1 + iw 2 2iS 1 S z x 1 + ix 2, and / z. 18
BPS H 1 = S 1 (z, z, x 3 )H 0 (z)e Mx3 H 0 (z): z N C N F Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys.Rev.D71 (2005) 065018 ; Ω SS (Ω 0 H 0 e 2My H 0 ) 4 (Ω 1 Ω) + 3 (Ω 1 3 Ω) = g 2 ( c Ω 1 Ω 0 ) x 2 10 5 10-10 0-5 5 x 1 0-5 -10-20 -10 x 3 0 10 20 Figure 6: (t w + t v = 0.5c) : H 0 (z)e Mx3 = c((z 4 2i)(z + 5 + 8i)e 3/2x3, (z + 8 i)(z 7 + 6i)e 1/2x3 +15/2, z 2 e 1/2x3 +15/2, (z 6 5i)(z + 6 7i)e 3/2x3 ). 19
(g 2 ) M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys.Rev.D72 (2005) 025011 ; 1/4BPS M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys.Rev.D72 (2005) 085004 ; 1/4BPS (6 5 4) M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, J.Phys.A 39 (2006) R315 ; T.Fujimori, M.Nitta, K.Ohta, N.Sakai and M.Yamazaki, in preparation ; 4 2 (det(hh ) = 0) = 20
7 M.Eto, T.Fujimori, T.Nagashima, M.Nitta, K.Ohashi, and N.Sakai, Phys.Rev.D75 (2007) 045010; M.Eto, Y.Isozumi, M.Nitta, K.Ohashi, and N.Sakai, Phys.Lett. B632 (2006) 384; 1/4 BPS : ( ) (6 )2 4, M = 0, 1, 2, 3 5 4 3 2 1 0-10 10-5 5 0 0 x 5-5 y (a) (Σ ) (b) (g 2 ) Figure 7: U(1) (N F = 4) (a): Σ = Σ 1 + iσ 2, (b):. ([m A, n A ] = [1, 0], [0, 1], [ 1, 1], [0, 0]) 21
1/4 BPS H = S 1 H 0 e M 1x 1 +M 2 x 2 H 0 : N C N F ( N C ) 4 2 0-2 -4 4 7.5 2 5 0-2 2.5-40 (a) U(1), (N F = 4) (b) U(2) (N F = 4) Figure 8: R 3. ( ) H 0 = c(1, 1, 1, φ) with φ = e r+iθ L eff = K ij (φ, φ ) µ φ i µ φ j, K(φ, φ ) = K w (φ, φ )+K g (φ, φ ) 22
K w (φ, φ ) d 2 x c logdetω, K g (φ, φ ) d 2 x 1 2g 2Tr(Ω 1 α Ω) 2 U(1),N NF = 4 ds 2 = c [ r 1 ( m12 2 + m 23 2 + m 31 2 )] (m 2 dr 2 +dθ 2 ) α 1 α 2 α 3 g 2 c α 3 α 1 [123] [123] :Σ, α A 1 2 [123] ϵ ABC m B m C α 2 Figure 9: U(2), 23
8 Conclusion 1. 2. ( )U(N C ) N F BPS 3. ( ) H 0 4. (g 2 ) ( ) 5. BPS 6. 7. 1/4 BPS ( ) 8. 1/2, 1/4 BPS 24