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Weak interaction, interaction Spontaneous symmetry Breaking, and Electro-weak unification Particle Physics Phenomenology 1 guidance: g particle p p physics y p phenomenology1 gy • Text : 前半はスライド、後半は L Ryder, L. Ryder Quantum Field Theory (2nd.ed), ed) Chap8 • Goals ★弱い相互作用の基本事項を理解する。 ★弱い相互作用の基本事項を理解する ★V-A理論を用いて崩壊率などが計算できる。 ★Goldstoneの定理、およびHiggs機構について説明できる。 定 機構 ★電弱統一理論の構造を理解する • 前提となる知識 場 理論と 準量 化 場の理論と正準量子化 S行列と摂動展開、ファインマン規則 相対論的量子力学 guidance (cont.) (cont ) • 使用言語:英語 • 成績評価 レポート 合計4回課し成績判定を行う。 最低でも3回提出しないと不合格になります。 S h d l Schedule 1.Weak interaction & Standard Model: Brief review 2.Phenomenology of Weak Interaction: Fermi’s theorem 3.Spontaneous p Symmetry y y Breakingg and Goldstone theorem 4.SSB with gauge field 1 5.SSB with gage field2 6.Electro-weak theory 1 7.Electro-weak theory2 chapter 1 Weak Interaction: Brief review History: y Puzzle of beta decayy β-decay of the atom or neutron: puzzling energy spectrum A(N , Z ) → A(N , Z ± 1) + e ∓ N. N Bohr Bohr’ss comment ‘At the present stage of atomic theory, however, we may say that h we h have no argument, either i h empirical i i l or theoretical, h i l ffor upholding the energy principle in the case of β-ray di i disintegrations.’ i ’ Pauli’s Discoveryy ((1930)) Dear Radioactive Ladies and Gentlemen, As the bearer of these lines, to whom I graciously ask you to listen, will explain to you in more detail, how because of the "wrong" statistics of the N and Li6 nuclei and the continuous beta spectrum, I have hit upon a desperate remedy to save the "exchange exchange theorem theorem" of statistics and the law of conservation of energy. Namely, the possibility that there could exist in the nuclei electrically neutral particles, that I wish to call neutrons, which have spin 1/2 and obey the exclusion principle and which further differ from light quanta in Neutrino at theofpresent that they do not travel with the velocity light. The mass of the neutrons should be of the same order of magnitude as the electron mass and in any event not larger than 0.01 0 01 proton masses. masses The continuous beta spectrum would then become understandable by the assumption that in beta decay a neutron is emitted in addition to the electron such that the sum of the energies of the neutron and the electron is constant... ‘Weak interaction’ is weak! Decay of particles induced by strong, EM, and weak forces. n (life time = 10n [s ]) 2 1/(life time) ≈(coupling constant) ×(phase space) Fermi’s theoryy ((1932)) construct effective 4-point interaction universal for any y kind of hadrons e.g. Calculated neutrino cross section (Bethe) ( ) N.B. Need 50 light-years of water to stop one 1 MeV neutrino. linear in Energy E ⇒ indicate Fermi’s theory is an ‘effective theory’. Introduction to the Weak interaction in SM Interaction between charged leptons, neutrino, and quarks in the Standard Model 1★Charged Current Interaction by W± exchange (Fermi) 2★Neutral Current Interaction by y Z0 exchange g ((1973)) Gauge boson propagator Weak vertex: V-A (left handed) Physical Processes induced by the Weak interaction 1-A: Leptonic process 1-B:Semi-leptonic process 1-C: Non-leptonic process only tree diagrams are shown for the strong interaction → Need knowledge of QCD and quark structure of hadron to calculate them. 2 Neutral interaction (discovered in 1973 @CERN) Vertex for Z0 depends on the Weinberg angle (see Higgs mechanism) Effective Interaction at the low energy Introduce the effective Fermi coupling constant for the low energy weak interaction. yields the weak coupling constant in SM. Coupling constant itself is not so week! Lepton p Familyy lepton number is conserved in SM Quark Family quark WFs are eigenstates of the strong interaction, but not for the weak and EM interactions interactions. Cabibbo angle : Experimental p evidence of the Cabibbo angle g s→u process is suppressed largely compared with d→s process Leptonic epto c decays of o Kaon ao aandd pion po → More on the q quark familyy Puzzle of K0 decay into muon pairs. calculation is smaller than the data GIM mechanism (1970) add the charm quark contribution, which w c be belong o g to the t e generation ge e at o of o s-quark. s qua . N.B. It was proposed before the discovery the charm quark! q 3-generations: g Cabibbo-Kobayashi-Maskawa y matrix which hi h is i expressedd with i h 3-real 3 l andd 1-imaginary 1i i parameters The imaginary phase produces CP-violating processes! q quiz • Suppose CKM matrix is unitary, show that the CKM matrix is parametrized by 3 real parameters and 1 imaginary phase. chapter 2 C Consequences off V V-A A interaction i i ・V-A structure & left-handed current ・parity violation ・ΔI = ½ rule Current-current Interaction Feynman and Gell-Mann, Sudarshan and Marshak (1958) where ‘Left-handed’ structure of the weak current Define the left- and right-handed spinors as 1±γ5 ψ L ≡ PLψ ψ R ≡ PRψ PR / L = 2 with the gamma5 matrix ⎛0 1⎞ γ 5 ≡ iγ 0γ 1γ 2γ 3 = ⎜ 1 0 ⎟ {γ μ , γ 5 } = 0 ⎝ ⎠ We introduce ‘chirality’ ±1 for right/handed spinors as the eigenvalues i l off th the gamma55 matrix; ti γ 5ψ R/L = ±1 ⋅ ψ R/L ・For the mass-less fermion, chirality coincides with the helicity h h= p ⋅s |p| spin projection along the momentum Proof: chiralityy = helicityy in the mass-less limit begin with the Dirac eq. (α i pi + β m )ψ = Eψ ⎛ 0 α =⎜ i ⎝σ i ⎛1 σi ⎞ β = ⎜0 0 ⎟⎠ ⎝ 0 ⎞ − 1 ⎟⎠ ⎛φ ⎞ ψ = Rewrite with two component spinor ⎜ χ ⎟ as, ⎝ ⎠ σ ⋅ p χ + mφ = E φ σ ⋅ pφ − m χ = E χ Thus, in the mass-less case, one easily finds eigenvalue equation equat o for o tthee helicity. e c ty. σ ⋅p (φ ± χ ) = ± 1 ⋅ (φ ± χ ) E On the other hand, the chirality projection provides 1±γ5 1 ⎛φ ± χ ⎞ PR/Lψ = ψ = ⎜ ⎟ 2 2 ⎝⎜ φ ± χ ⎟⎠ Q.E.D V-A V A interaction plays a role of ‘chirality chirality projection projection’ Weak current-current interaction ue γ μ (1 − γ 5 )uν e u p γ μ (1 − γ 5 )u n only acts on only the left-handed spinors, which is peculiar f t feature off th the weak k iinteraction, t ti becasuse γ μ (1 − γ 5 )u n = γ μ (1 − γ 5 )(u nR + u nL ) 1 + γ5 1 − γ5 = γ μ (1 − γ 5 )( un + un ) 2 =γμ 2 1 − γ5 1 − γ5 L un = γ μ un 2 2 ue L γ μ uν e L u pL γ μ u nL Parityy operation p on Dirac spinor p Parity transformation for the Dirac spinor p The free Dirac eq. eq is invariant under this operation ★Decay rate includes the parity violating term due to the V-A structure of the interaction. Experiment : Parity non-conservation non conservation θ-τ puzzle (1956~) : kaon decays in S-wave final state 1958 Lee Lee-Yang Yang postulate observation of the parity violation parity-violation 1961 Co60 experiment (Wu) Example1: p electron-neutrino elastic scattering g With the neutral current exchange, elastic scattering amplitude p is ggiven byy Ignoring the electron mass m and taking the CM frame, one finds and also for the total cross section Example2: p Neutron beta decayy With the charged current interaction Straightforward calculation yields In the neutron rest frame, Resulting energy spectrum is consistent i to data d ΔI=1/2 rule In strangness changing weak meson and baryon decays, there should exist both ΔI=1/2 and ΔI=3/2 processes. p e.g. Λ baryon (I=0) can decay into Λ 0 → pπ − or Λ 0 → nπ 0 nucleon: II=1/2, 1/2, pion: II=1 1 ⇒ final state has II=3/2 3/2 or 1/2. for total I=3/2, for I=1/2, 2 1 1 1 , − 1,0 + 3 2 2 3 nπ 0 1 1 1 2 , − 1,0 − 3 2 2 3 1 1 , 1, −1 2 2 pπ − 1 1 , 1, −1 2 2 ΔI=3/2 ΔI=1/2 Exp: nπ0 15.6, pπ- 22.4 ★Data indicate complete dominance of ΔI ΔI=1/2 1/2 amplitudes over ΔI=3/2. Amp (ΔI = 1 / 2) ~ 20 ⋅ Amp (ΔI = 3 / 2) Origin of the ΔI=1/2 rule For the hyperon decay, we calculate a weak matrix element < N | d γ μ ((1 − γ 5 )u u γ μ (1 ( − γ 5 )s | Λ > between nucleon and Λ states. In the non non-relativisitc relativisitc limit, we can rewrite < N | d †u † (1 − σ u ⋅ σ d )u s | Λ > spin i projection j ti operator t This operator forces ud-pair to be spin=0 and isospin=0 due to the Pauli principle. Hence, this weak transition must be ΔI=1/2. (Pati-Woo theorem) NB. However,, understanding of ΔI=1/2 rule for kaon decay is still incomplete quiz 1 q • calculate neutrino-electron elastic scattering amplitude and show that the amplitude square with the electron mass being 0 becomes i the in th CM ((center t off mass fframe). ) chapter 3 Spontaneous Symmetry Breaking and Nambu Goldstone boson Nambu-Goldstone