Kaluza-Klein(KK) SO(11) KK 1 2 1

Maskawa Institute, Kyoto Sangyo University Naoki Yamatsu 2016 4 12 ( ) @

Kaluza-Klein(KK) SO(11) KK 1 2 1

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1. 標準理論 物質場 ( フェルミオン ) スカラー ゲージ場 クォーク ヒッグス u d s b ν c レプトン ν t ν e μ τ e μ τ e h ヒッグス場が真空で非自明な期待値を持つ クォークとレプトンは湯川相互作用を通して質量獲得する G SU(3) W SU(2) B U(1) 質量獲得 ( 光子を除く ) 電弱対称性 SU(2)xU(1) の電磁対称性 U(1) への 自発的対称性の破れ 3

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U(1) Y SU(3) C SU(2) L G SM SU(5) GUT SU(5) G SM G GUT 10

U(1) Y SU(3) C SU(2) L G SM SU(5) GUT SU(5) G SM G GUT 11

SU(3) SO(5) U(1) 5 SO(5) A M / \ 4 A µ W W A y SM H SO(5) U(1) GHU in RS [1 6, K.Agashe et al 05;Y.Hosotani et al. 08-15] 12

2. SO(11) [7, Y.Hosotani,N.Y. 15] ( ) [7, Y.Hosotani,N.Y. 15] ( ) [8, N.Y. 16] 13

SO(11) [7, Y.Hosotani,N.Y. 15] { SO(10) SO(11) BC SO(4) SO(7) on Planck brane on TeV brane = BC SO(4) SO(6) SU(2) L SU(2) R SU(4) C Φ SU(3) C SU(2) L U(1) Y θh SU(3) C U(1) em VEV 14

SO(11) [7, Y.Hosotani,N.Y. 15] { SO(10) SO(11) BC SO(4) SO(7) on Planck brane on TeV brane = BC SO(4) SO(6) SU(2) L SU(2) R SU(4) C Φ SU(3) C SU(2) L U(1) Y θh SU(3) C U(1) em VEV 15

SO(11) [7, Y.Hosotani,N.Y. 15] { SO(10) SO(11) BC SO(4) SO(7) on Planck brane on TeV brane = BC SO(4) SO(6) SU(2) L SU(2) R SU(4) C Φ SU(3) C SU(2) L U(1) Y θh SU(3) C U(1) em VEV 16

SO(11) [7, Y.Hosotani,N.Y. 15] { SO(10) SO(11) BC SO(4) SO(7) on Planck brane on TeV brane = BC SO(4) SO(6) SU(2) L SU(2) R SU(4) C Φ SU(3) C SU(2) L U(1) Y θh SU(3) C U(1) em VEV 17

SO(11) [7, Y.Hosotani,N.Y. 15] { SO(10) SO(11) BC SO(4) SO(7) on Planck brane on TeV brane = BC SO(4) SO(6) SU(2) L SU(2) R SU(4) C Φ SU(3) C SU(2) L U(1) Y VEV θh SU(3) C U(1) em ( ) [9, Y.Hosotani 83] 18

Venn SO(11) SO(10) (5),(6) (4) BC Planck SU(5) G SM (2) Φ (1) G P S (3) BCs 5D (A M = A µ A z ) A (z=1) µ A (z=z L ) µ A (z=1) z A (z=z L ) z (1) G SM N N D D (2) SU(5) G P S N D D N (3) SU(5) G P S D eff N D D (4) SO(10) (SU(5) G P S ) D eff D D N (5) SO(5)/SO(4) D D N N (6) SO(7)/SO(6) D N N D 19

SO(11) 5 SO(11) A M / \ 4 A µ SM g, W, B A y SM H SO(11) GHGUT in RS [7, 10, Y.Hosotani,N.Y. 15] 20

SO(11) 5 SO(11) Ψ (a) 32 4 ψ (a) SM q (a), l (a) / \ L 4 ψ (a) R SM u (a), d (a), e (a) SO(11) GHGUT in RS [7, 10, Y.Hosotani,N.Y. 15] 21

SO(11) Bulk field A M Zero modes G µ W µ A µ φ G SM (8, 1) 0 (1, 3) 0 (1, 1) 0 (1, 2) 1/2 SL(2, C) (1/2,1/2) (1/2,1/2) (1/2,1/2) (0,0) Bulk field Zero modes G SM (3, 2) +1/6 (3, 1) +2/3 (3, 1) 1/3 (1, 2) 1/2 (1, 1) 1 SL(2, C) (1/2,0) (0,1/2) (0,1/2) (1/2,0) (0,1/2) Ψ (a) 32 q (a) L u (a) R d (a) R l (a) L e (a) R 22

[11, N.Y. 15] SO(11) [7, Y.Hosotani,N.Y. 15] [8, N.Y. 16] SO(11) [A.Furui,Y.Hosotani,N.Y. ] 23

[11, N.Y. 15] SO(11) [7, Y.Hosotani,N.Y. 15] [8, N.Y. 16] SO(11) [A.Furui,Y.Hosotani,N.Y. ] 24

3. (1-loop) [12, E.g., Slansky 81] d dlog(µ) α 1 i (µ) = b1 loop i 2π, α i := g 2 i /4π, T (R):R Dynkin, b 1 loop i = 11 3 Vector (Real) T (R V ) + 2 3 Fermion (Weyl) T (R F ) + 1 6 Scalar (Real) T (R S ) 25

(1-loop) [12, E.g., Slansky 81] b 1 loop i d dlog(µ) α 1 i (µ) = b1 loop i 2π, µ α 1 i (µ) = α 1 i (µ 0 ) b1 loop i 2π ( µ log [11 13, McKay-Patera 81;Slansky 81;N.Y. 15] β µ 0 ). 26

G µ W µ B µ Q u c d c L e c φ SU(3) C 8 1 1 3 3 3 1 1 1 SU(2) L 1 3 1 2 1 1 2 1 2 U(1) Y 0 0 0 1/6 2/3 1/3 1/2 1 1/2 b SM i = 11 3 C 2(G i ) + 2 3 Fermion (Weyl) T (R i ) + 1 3 Higgs (Complex) T (R i ). 27

G µ W µ B µ Q u c d c L e c φ SU(3) C 8 1 1 3 3 3 1 1 1 SU(2) L 1 3 1 2 1 1 2 1 2 U(1) Y 0 0 0 1/6 2/3 1/3 1/2 1 1/2 b SM i = 0 22/3 11 + n F 4/3 4/3 4/3 + n H 1/10 1/6 0 = +41/10 19/6 7. 28

[8, E.g., N.Y. 16] Standard Model 60 50 40 30 20 10 0 100 10 5 10 8 10 11 10 14 10 17 Α 1 (@ µ = M Z 91 GeV) Ref. [14, PDG 14] Μ α 3C 0.118, α 2L = α em sin 2 θ W, α 1Y = 5α em 3 cos 2 θ W, α 1 em 128, sin 2 θ W 0.23. 29

(@ µ = M GUT ) α 3C (M GUT ) = α 2L (M GUT ) = α 1Y (M GUT ) α em (µ) = 3α 1Y (µ)α 2L (µ) 3α 1Y (µ) + 5α 2L (µ), sin2 θ W (µ) = 3α 1Y (µ) 3α 1Y (µ) + 5α 2L (µ). sin 2 θ W (M GUT ) = 3α 1Y 3α 1Y + 5α 2L (M GUT ) = 3 8 0.23 sin2 θ W (M Z ) 30

(1-loop) [12, E.g., Slansky 81] d dlog(µ) α 1 i (µ) = b1 loop i 2π, b 1 loop i KK µ? KK 4 k m KK (k Z) k KK 1-31

5D GHGUT d dlog(µ) α 1 i (µ) = b1 loop i 2π, km KK < µ < (k + 1)m KK β b 1 loop i =b 0 i + k b KK. 32

5D GHGUT { d b0 i dlog(µ) α 1 i (µ) = 2π for µ < m KK b0 i 2π, k bkk 2π for km KK < µ < (k + 1)m KK KK k k µ m KK. km KK < µ < (k + 1)m KK α 1 i (µ) = α 1 i (m KK ) b0 i 2π log µ µ m KK m KK m KK b KK 2π. 33

5D GHGUT b KK < 0 α i (µ) 2π m KK b KK µ b KK > 0 α i (µ) near α 1 i (m KK ) bkk 2π µ m KK 34

SO(11) A M Ψ (a) 32 Ψ (b) 11 SO(11) 55 32 11 5D RS 1 5 4 4 Orbifold BC (, ) (, ) β 1st KK b KK = ( 11 3 Vector + 1 ) C 2 (SO(11)) + 4 6 3 Scalar Fermion (Dirac) T (R F ). 35

SO(11) A M Ψ (a) 32 Ψ (b) 11 SO(11) 55 32 11 5D RS 1 5 4 4 Orbifold BC (, ) (, ) β 1st KK b KK = 7 2 C 2(SO(11)) + 4 3 (n ST (32) + n V T (11)), C 2 (SO(11) = 55) = T (55) = 9, T (32) = 4, T (11) = 1. n S, n V n S = 3, n V = 0. 36

SO(11) A M Ψ (a) 32 Ψ (b) 11 SO(11) 55 32 11 5D RS 1 5 4 4 Orbifold BC (, ) (, ) β 1st KK b KK = 7 2 9 + 4 3 (3 4 + 0 1) = 31 2. 37

SO(11) GHGUT Α 1 b KK 31 2 100 80 60 40 20 0 100 10 5 10 8 10 11 10 14 10 17 m KK 10 6 GeV Α 1 b KK 13 2 100 80 60 40 20 0 100 10 5 10 8 10 11 10 14 10 17 [8, N.Y. 16] 2 (n 32, n 11 ) = (3, 0), (5, 2), (5, 4) Μ m KK 10 6 GeV Α 1 b KK 1 2 100 80 60 40 20 0 100 10 5 10 8 10 11 10 14 10 17 (@ µ = M Z 91 GeV) Ref. [14, PDG 14] Μ m KK 10 6 GeV α 3C 0.118, α 2L = α em sin 2 θ W, α 1Y = 5α em 3 cos 2 θ W, α 1 em 128, sin 2 θ W 0.23. 38

SO(11) GHGUT Α 1 b KK 31 2 70.0 50.0 30.0 20.0 15.0 10.0 100 10 5 10 8 10 11 10 14 10 17 m KK 10,10,10 GeV Α 1 b KK 31 2, 13 6, 1 2 70.0 50.0 30.0 20.0 15.0 10.0 100 10 5 10 8 10 11 10 14 10 17 [8, N.Y. 16] 3 KK β Μ m KK 10 10 GeV (@ µ = M Z 91 GeV) Ref. [14, PDG 14] α 3C 0.118, α 2L = α em sin 2 θ W, α 1Y = 5α em 3 cos 2 θ W, α 1 em 128, sin 2 θ W 0.23. 39

4. Kaluza-Klein(KK) SO(11) KK ( ) [11, N.Y. 15] ( [8, N.Y. 16] ) 40

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