(Fundamental) Physics of Elementary Particles Asymmetry of weak - PowerPoint PPT Presentation

(Fundamental) Physics of Elementary Particles Asymmetry of weak interactions; T e GIM mechanism & anomaly; Weak angle; Feynman rules Tristan Hübsch Department of Physics and Astronomy Howard University, Washington DC Prirodno-Matemati č ki Fakultet Univerzitet u Novom Sadu Wednesday, January 18, 12

Fundamental Physics of Elementary Particles Program Le f -right asymmetry in weak interactions Dirac’s gamma matrices & the Cli ff ord algebra Dirac and Weyl spinors and equations T e GIM mechanism 1st order e ff ect 2nd order e ff ect T e U ( 1 ) A anomaly Two classical symmetries & their anomalies T e Adler-Bell-Jackiw (triangle) anomaly T e interactions of ma t er fermions with weak gauge bosons Cabbibo-Kobayashi-Maskawa mixing Feynman’s rules for weak interactions 2 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra Recall the non-relativistic Hamiltonian ⇣ ¯ h ∂ h ⌘ 2 1 ~ ∂ t = H = + V ( r , t ) ~ i ¯ r 2 m i Using the familiar classical ↔ quantum correspondence h h ∂ p = ¯ ~ ~ p $ ~ r , and E $ H = i ¯ ∂ t . i the Hamiltonian becomes p 2 E = ~ 2 m + V ( r , t ) , ~ which is the well known classical relationship between energy, linear momentum, mass and the potential. 3 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra Reverse-engineering then the relativistic relationship p 2 c 2 + m 2 c 4 = E 2 ~ we obtain c 2 ⇣ ¯ h ⌘ 2 h ∂ ⌘ 2 h + m 2 c 4 i ⇣ ~ Ψ ( r , t ) = Ψ ( r , t ) , ~ ~ r i ¯ h ⇣ ⌘ i ⇣ i r i ∂ t ⇣ mc ⌘ 2 i h ⌘ 2 i ⇣ mc h ⇤ + Ψ ( r , t ) = 0, ~ ) h ¯ the Klein-Gordon equation. Here, h 1 ∂ 2 r 2 i ∂ t 2 � ~ ⇤ : = c 2 is the d’Alembertian (wave) operator. 4 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra Motivated by the rest-frame factorization E 2 − m 2 c 4 = 0 ( E + mc 2 )( E − mc 2 ) = 0, ⇒ Dirac a t empted to factorize the Klein-Gordon equation: p 2 − m 2 c 2 = 0 0 = ( β µ p µ + mc )( γ γ ν p ν − mc ) , ⇒ γ γ − γ µ − β µ ) p µ − m 2 c 2 . γ ν p µ p ν + mc ( γ = β µ γ γ γ γ γ As no term in the Klein-Gordon equation is linear in the linear momentum, β μ = γ μ . Equating the quadratic terms then yields γ ν p µ p ν = p 2 ≡ η µ ν p µ p ν , γ µ γ γ γ γ γ γ Since p μ p ν = p ν p μ , γ µ , γ � γ ν = 2 η µ ν , γ γ γ γ γ T is de fi nes the Cli ff ord algebra Cl(1,3)= — (4). 5 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra Some canonical identities: � b γ µ = 0, γ ) 2 = 1 ; γ 0 γ γ 1 γ γ 2 γ γ 3 : = i b γ : = i γ γ µ γ γ ν γ γ ρ γ γ σ , ( b γ , γ γ γ γ γ γ γ γ γ γ γ γ 4! ε µ νρσ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ ⇤ = 0, ⇥ γ ± ) 2 = γ γ ± : = 1 2 [ 1 ± b γ ] , γ + + γ γ − = 1 , ( γ γ + , γ γ ± , γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ − γ ρσ ⇤ = η µ ρ γ ⇥ γ µ ν : = i γ νρ + η νσ γ γ µ ν , γ 4 [ γ γ µ , γ γ ν ] , γ νσ − η µ σ γ γ µ ρ − η νρ γ γ µ σ . γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ and similarly: γ µ γ γ µ γ γ ν γ γ ρ γ γ ν γ γ ρ , γ µ = 4 1 , γ µ = 4 γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ µ γ γ ν γ γ ν , γ µ γ γ ν γ γ ρ γ γ σ γ γ ν γ γ ρ γ γ σ , γ µ = − 2 γ γ µ = − 2 γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ ρ = η µ ν γ γ ν + η νρ γ γ µ + i ε µ νρσ γ γ ν γ γ ρ − η µ ρ γ γ µ γ γ σ b γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ and completeness: γ µ + 1 γ µ ν + b γ + b f ( γ γ ) = C 0 1 + C µ γ γ µ b C 0 b 2 C µ ν γ C µ γ γ . γ γ γ γ γ γ γ γ γ γ γ γ γ γ 6 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra Finally, we also have γ ν γ γ ρ ] = 0, γ ν γ γ ρ γ γ σ γ γ λ ] = 0, Tr [ γ γ µ ] = 0, Tr [ γ γ µ γ Tr [ γ γ µ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ etc. γ σ ] = 4 ( η µ ν η ρσ − η µ ρ η νσ + η µ σ η νρ ) , γ ν ] = 4 η µ ν , γ ν γ γ ρ γ Tr [ γ γ µ γ Tr [ γ γ µ γ γ γ γ γ γ γ γ γ γ γ γ γ γ ν b γ ρ γ γ σ b γ ] = − 4 i ε µ νρσ . Tr [ b γ ] = 0, Tr [ γ γ µ γ γ ] = 0, Tr [ γ γ µ γ γ µ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ Feynman’s slash-notation / γ µ p µ p : = γ γ γ whereupon p = p 2 1 , p / / p / / q = ( p · q − 2 ip µ γ µ ν q ν ) 1 ; p / / q + / q / p / / q − / q / p = 2 ( p · q ) 1 , p = − 4 i ( p µ γ µ ν q ν ) 1 ; Tr [ / p / q ] = 4 p · q 1 , Tr [ / p / q / r / s ] = 4 [( p · q )( r · s ) − ( p · r )( q · s ) + ( p · s )( q · r )] ; Tr [ / p ] = 0 = Tr [ / p / q / r ] , Tr [ b γ / p / q / r / s ] = 4 i ε µ νρσ p µ q ν r ρ s σ ; γ γ γ γ µ / γ µ / γ µ / p / γ µ = 4 p · q 1 , γ µ = − 2 / p / q / γ µ = − 2 / r / q / q γ p γ p , r γ p . γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ 7 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra Dirac conjugation b b b γ 0 ) † = γ γ i ) † = − γ γ µ ) † = γ γ i , i = 1, 2, 3, γ 0 , γ 0 γ γ 0 . ( γ ( γ ( γ γ µ γ and ⇔ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ 0 = γ Ψ : = Ψ † γ γ 0 γ 0 ( γ γ µ ) † γ γ µ : = γ γ µ . ⇔ γ γ γ γ γ γ γ γ γ γ γ γ γ γ 0 � � i ( γ γ 0 ) † � γ 0 = γ γ 0 ( i γ γ 0 γ γ 1 γ γ 2 γ γ 3 ) † γ γ 3 ) † ( γ γ 2 ) † ( γ γ 1 ) † ( γ γ 0 , γ : = γ b γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ 0 = � i γ γ 0 = � i γ γ 3 γ γ 2 γ γ 1 γ γ 0 γ γ 1 γ γ 2 γ γ 3 , γ 3 γ γ 2 γ γ 1 γ = � i γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ = � b γ ± : = 1 2 [ 1 ± b γ ] = 1 2 [ 1 ⌥ b γ ] = : γ γ , γ ⌥ . ) γ γ γ γ γ γ γ γ γ γ γ γ γ γ so that Ψ ± = γ γ ± Ψ = Ψ γ γ ± = Ψ γ γ ⌥ = Ψ ⌥ . γ γ γ γ γ γ Roughly , Ψ L = Ψ – , and Ψ R = Ψ + : le f /right-handed helicity vs . chirality; identical for massless particles only. 8 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac’s gamma matrices & the Clifford algebra In case you prefer 4 × 4 matrices… T e Dirac basis:  1 �  �  O � σ i O O 1 σ σ σ γ i = γ 0 = b , , , γ γ γ γ γ γ γ γ γ γ = σ i − 1 1 O O O σ − σ σ T e Weyl (chiral) basis:  O �  �  1 �  Ψ + � σ i O O − 1 σ σ σ γ 0 = γ i = b , , , Ψ Dirac = ; γ = γ γ γ γ γ γ γ γ γ σ i O O − 1 O − 1 Ψ − − σ σ σ T e Majorana basis:  O �  i σ �  O �  − i σ � σ 2 σ 3 σ 2 σ 1 O O σ σ σ σ σ σ σ − σ σ σ γ 0 = γ 1 = γ 2 = γ 3 = , , , , γ γ γ γ γ γ γ γ γ γ γ γ σ 2 σ 3 σ 2 σ 1 O O O O i σ − i σ σ σ σ σ σ σ σ σ σ σ  σ � σ 2 O σ σ b in which all components of the Dirac spinor Ψ are real, , γ γ γ = γ σ 2 O σ σ σ However, “canonical computations” are preferable. 9 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac and Weyl spinors and equations T e Dirac equation: p 2 − m 2 c 2 = 0 = ( γ γ µ p µ − mc )( γ γ µ p µ + mc ) , γ γ γ γ leads to the choice : h p µ → ¯ ⇥ ⇤ γ µ ∂ µ − mc Ψ ( x ) = 0, γ i ∂ µ i ¯ h γ γ ⇒ where ∂ c ∂ t , ~ ! ( � 1 ∂ µ : = r ) , ∂ x µ , � T e Weyl decomposition: � � Ψ = Ψ + + Ψ − , Ψ ± : = γ ± = 1 2 [ 1 ± b γ ] . γ ± Ψ , γ γ γ γ γ γ γ γ γ γ µ D µ − mc ⇥ ⇤ i ¯ hc γ Ψ h 1 Ψ γ γ ¯ Ψ − − mc ⇥ ⇤ ⇥ ⇤ γ µ D µ γ µ D µ i ¯ hc γ i ¯ hc γ = Ψ + Ψ + + Ψ − h Ψ − Ψ + . γ γ γ γ ¯ 10 Wednesday, January 18, 12

Left-right asymmetry in weak interactions Dirac and Weyl spinors and equations So, the (Dirac) mass-term mixes fermions of le f - and right- handed chirality. If the mass is zero (or negligible) le f - and right-handed fermions (approximately) decouple b and can satisfy di ff erent boundary conditions. | {z } | {z } Notice that charge conjugation ⇔ Dirac conjugation ( Ψ ± ) c = ( Ψ ± ) T = ( Ψ γ γ ⌥ ) T = ( γ Ψ c : = ( Ψ ) T , γ ± ) T Ψ c γ γ γ γ For a chargeless fermion (neutrino), can Ψ → Ψ c : Ψ ± ( Ψ ± ) c = Ψ γ γ ⌥ ) T Ψ c = Ψ γ γ ⌥ Ψ c , in Dirac & Weyl bases . γ ⌥ ( γ γ γ γ γ γ γ T is allows a Majorana mass-term for chargeless particles, but not for charged particles (quarks, charged leptons). 11 Wednesday, January 18, 12