Aharonov-Bohm Effect and Unification of Elementary Particles. Yutaka Hosotani, Osaka University Warsaw, May 2006

Transcription

1 Aharonov-Bohm Effect and Unification of Elementary Particles Yutaka Hosotani, Osaka University Warsaw, May 26

2 - Part 1 - Aharonov-Bohm effect

3 Aharonov-Bohm Effect! B)! Fµν = (E, vs empty or vacuum!= Fµν = ") Aµ = ( ϕ, A in non-simply connected space e P1 e! = B I Aharonov-Bohm 1959!!= B II P2 feels vanishing Fµν Y. Hosotani, Warsaw, May 26-3

4 ! = A! B r! = B Φ I magnetic flux! d2 x Bz Φ= II = II! C! d! x A θ C Aθ = Φ 2πr in region I Schro dinger equation $ 2! " 1 eφ #2 h 1 ψ = Eψ r + 2 i 2m r r r r θ 2πh c Spectrum & scattering amplitudes depend on eφ!c θab Y. Hosotani, Warsaw, May 26-4

5 Gauge invariance Physics is invariant under A θ = A θ α r ψ = exp {i e } c αθ ψ Both ψ and ψ are single-valued. α is quantized. θ AB = eφ c θ AB = θ AB 2πn Aharonov-Bohm phase e iθ AB is physical. It cannot be gauged away. Y. Hosotani, Warsaw, May 26-5

6 Superconducting rings! B r! = inside SC Meissner effect: B Aθ = Φ 2πr To minimize the free energy & " 2 #$! # % 2! # e 1 2 # 2 4 3!! i A Ψ + α Ψ + β Ψ + B FGL = d x # # GL GL GL 2m!c 2 ΨGL! e Φ " = exp i θ Ψ!c 2π ΨGL : single-valued Φ= Cooper pairs e* = 2e hc e n Flux is quantized! Y. Hosotani, Warsaw, May 26-6

7 Tonomura s experiment to see θab (1986) Phys. Rev. Lett. 56 (1986) 792 ferromagnet permalloy e! B SC Nb Φ=!! B! : quantized for T < Tc ds When Φ = hc 2e n, θab = eφ!c interference non-trivial for n = 1, 3, 5, = πn for e. Y. Hosotani, Warsaw, May 26-7

8 n : odd T = 4.5 K T = 15 K (T c = 9.2 K) n : even T = 4.5 K T = 15 K Y. Hosotani, Warsaw, May 26-8

9 If space is not simply connected, there exists the Aharonov-Bohm phase. F µν = does not imply empty or vacuum Y. Hosotani, Warsaw, May 26-9

10 - Part 2 - Dynamical θ AB

11 World on a ring QED on S 1 R2! = B L electrons around a solenoid r Φ θ x x x+l Ax = const = α! L e elα θab = dx Ax =!c!c θ θ + 2π Aθ = Φ 2πr θab = eφ!c θab cannot be gauged away. θab affects the spectrum. θab : dynamical θab : external parameter Y. Hosotani, Warsaw, May 26-11

12 QED on S 1 R 2 Spectrum θ AB is a part of A x (t, x, y, z), and dynamical. { i [γ t + γ 1 ( x ie ] c A x) + γ 2 y + γ 3 z E(n, p y, p z ) 2 = 2 c 2 Energy density of the vacuum E vac (θ AB ) = n = 4 π 2 L 4 dp 2 { 1 } (2π) 2 2 E(n, p y, p z ) cos n θ AB + constant n=1 n 4 } mc ψ = L 2 ( 2πn θab ) 2 + p 2 y c2 + p 2 z c2 + m 2 c 4 zero-point energies Minimized at θ AB = π Y. Hosotani, Warsaw, May 26-12

13 - Part 3 - Standard model of elementary particles

14 Standard model of elementary particles and the Higgs particle Forces γ gluons Z, W graviton φ H? electromagnetic strong weak gravitational Higgs Y. Hosotani, Warsaw, May 26-14

15 Unification of EM & WEAK interactions SU(2) L U(1) Y Higgs symmetry breaking U(1) EM W: 8 GeV Z: 91 GeV photon: massless masses of quarks and leptons Y. Hosotani, Warsaw, May 26-15

16 Issues to be settled : Higgs particle has not been found. Higgs mass & interactions --- to be discovered at LHC. no principles gauge principle How many kinds? Fundamental or composite? Y. Hosotani, Warsaw, May 26-16

17 Possible scenarios Standard model fine-tunig / gauge hierarchy problem in GUT Supersymetric standard model light Higgs Extra-dimensional gauge-higgs unification Y. Hosotani, Warsaw, May 26-17

18 - Part 4 - Gauge-Higgs unification through θ AB

19 Gauge-Higgs unification Higgs field is a part of gauge fields. In higher-dimensional gauge theory A M = ( A µ, A yj ) extra-dimensional components A yj serve as 4-d Higgs fields. Fairlie 1979, Manton 1979 Higgs as Yang-Mills AB phases in non-simply connected extra-dimensions Hosotani mechanism 1983 Y. Hosotani, Warsaw, May 26-19

20 Dynamical symmetry breaking by Yang-Mills AB phases Example: SU(3) gauge theory on M 4 S 1 L = 1 2 Tr F MN F MN + ψ iγ M D M ψ y = y = 2πR QED on S 1 R 2 θ AB = e c 2πR dx A x Yang-Mills AB phases P exp { ig 2πR dy A y } Y. Hosotani, Warsaw, May 26-2

21 P exp { ig 2πR dya y } e iθ 1 e iθ 2 e iθ 3 ( j θ j = ) Yang-Mills AB phases θ AB θ AB cannot be gauged away. affects the spectrum. Energy density of the vacuum depends on θ j in the quantum theory. Y. Hosotani, Warsaw, May 26-21

22 Energy density in SU(3) (θ 1, θ 2, θ 3 ) = (πa, πb, π(a + b)) V C V 2 C a b a b Pure gauge theory With N F fund = 3, β = Minimum at (θ 1, θ 2, θ 3 ) = (,, ), (± 2π 3, ±2π 3, ±2π 3 ) (± 2π 3, ±2π 3, ±2π 3 ) SU(3) unbroken. Y. Hosotani, Warsaw, May 26-22

23 V C a b V C a b With N F fund =, N F ad = 1 With N F fund = N F ad = 1 Minimum at (θ 1, θ 2, θ 3 ) = (+ 2π 3, 2π 3, ) (π, π, ) SU(3) U(1) U(1) SU(3) SU(2) U(1) Dynamical gauge symmetry breaking takes place. Y. Hosotani, Warsaw, May 26-23

24 Hosotani mechanism (1983) F µν = pure gauge Yang-Mills AB phases θ j 1. θ j is dynamically determined. 2. Gauge symmetry can be dynamically broken. 3. Higgs fields are unified with gauge fields. Dynamical gauge-higgs unification Y. Hosotani, Warsaw, May 26-24

25 - Part 5 - Gauge-Higgs unification in electroweak interactions

26 Electroweak unification by Yang-Mills AB phases Higgs = Yang-Mills AB phases Gauge group to start with. SU(2) L U(1) Y U(1) EM non-abelian (Yang-Mills) gauge theory of larger group SU(3), SO(5), SO(5) U(1), Fairlie 1979, Manton 1979 Y. Hosotani, Warsaw, May 26-26

27 Chiral fermions Extra-dimensional space is orbifold Pomarol, Quiros 1998 M 4 (S 1 /Z 2 ) y = y =! R Spacetime Maybe curved as in Rundall-Sundrum warped space y Λ < ds 2 = e 2k y dx µ dx µ + dy 2 Y. Hosotani, Warsaw, May 26-27

28 Higgs boson mass & stability m H > 114 GeV (exp.) Standard model Gauge-Higgs unification Veff b -.5 θ a θ Yang-Mills AB phases Y. Hosotani, Warsaw, May 26-28

29 Standard model unstable against large radiative corrections Supersymmetric standard model predicts light Higgs: m H < 13 GeV Gauge-Higgs unification m H > 114 GeV (exp.) flat R Randall-Sundrum R, kr = 12 M KK 4 8 GeV TeV m H 7.5 GeV 15 GeV GeV Y. Hosotani, Warsaw, May 26-29

30 Hierarchy in fermion masses? Observed fermion masses f e r m i o n m a s s e s ( l a r g e a n g l e M S W ) " 1 " 2 " 3 d s b u c e µ! t T e V G e V M e V k e V e V m e V µ e V Gauge-Higgs unification in RS spacetime gauge coupling + wave function fermion mass u c t e µ τ kink mass / k Y. Hosotani, Warsaw, May 26-3

31 Other consequences: Universality of weak interactions W g ν e ν µ t e µ b W g W Z tiny non-universality? Yukawa couplings φ H g Y Standard model mass = g Y φ H e d Gauge-Higgs unification mass = gauge coupling + wave function g Y : suppressed Y. Hosotani, Warsaw, May 26-31

32 Summary Aharonov-Bohm phase observed in QM There exist Yang-Mills AB phases Gauge-Higgs unification Higgs as Yang-Mills AB phases Higgs search at LHC Higgs mass Non-universality in gauge couplings Yukawa couplings Y. Hosotani, Warsaw, May 26-32