Flavor Physics: Introduction to C, P and T symmetries David Delepine, Carlos Vaquera-Araujo. Conacyt DCI-Campus Le´ on Universidad de Guanajuato. October 25, 2018 (MSPF-Sonora) Flavor Physics October 25, 2018 1 / 31
Outline 1 Introduction 2 C , P and T in field theory 3 Standard Model 4 Cabibbo Kobayashi Maskawa Matrix 5 CP violation (MSPF-Sonora) Flavor Physics October 25, 2018 2 / 31
EW SM ingredients Generation: 1st 2nd 3rd everyday matter Gauge Bosons unstable matter Scalar Boson R 2 / 3 R 2 / 3 R 2 / 3 Charge / / / 2 . 3 MeV G 1 . 28 GeV G 173 . 2 GeV G 125 . 1 GeV Strong Interaction (color) u / c / / B B B t H Color Electromagnetic Interaction (charge) Mass up top Higgs charm Spin 1/2 1/2 1/2 0 quarks − 1 / 3 − 1 / 3 − 1 / 3 R R R c / / / o 4 . 8 MeV G 95 MeV G 4 . 7 GeV G g l o / s / / r B B B d b down strange bottom glun 1/2 1/2 1/2 1 Weak Interaction (weak isospin) − 1 − 1 − 1 105 . 7 MeV 1 . 777 GeV 511 keV µ γ e τ muon electrn tau fotn 1/2 1/2 1/2 1 leptons ± 1 < 2 eV < 190 keV ν µ < 18 . 2 MeV 80 . 4 GeV 91 . 2 GeV ν e ν τ W ± Z µ neutrino e neutrino τ neutrino 1/2 1/2 1/2 1 1 fermions bosons (MSPF-Sonora) Flavor Physics October 25, 2018 3 / 31
Flavor Physics Flavor physics is the study of different types of quarks and leptons, or flavors, their spectrum and the transmutations among them. Flavor physics is very rich. Check out http://pdg.lbl.gov for the many different transition rates among hadrons with different quark content. We aim at understanding this wealth of information in terms of some simple basic principles. (MSPF-Sonora) Flavor Physics October 25, 2018 4 / 31
C , P and T C , P and T are discrete spacetime transformations A priori , they have noting to do with flavor physics, as flavor has to do with internal symmetries. However, it turns out that in nature, all observations of CP violation happen to come along with flavor violation. (MSPF-Sonora) Flavor Physics October 25, 2018 5 / 31
C , P and T in Quantum Mechanics • Parity: P performs a spatial inversion through the origin x → − x U P ψ ( t, x ) = η P ψ ( t, − x ) Introduced by Wigner in 1927/28 Unitary transformation Applying parity twice restores the original state, U 2 P = 1 up to an unobservable phase. From this the parity of the U P eigenfunctions has to be either even, η P = +1, or odd, η P = − 1. (MSPF-Sonora) Flavor Physics October 25, 2018 6 / 31
• Time reversal: T performs reversal of motion in time (sense of time evolution). That is t → − t with exchange of initial and final states. A T ψ ( t, x ) = η T ψ ( − t, x ) Introduced by Wigner in 1932 Antiunitary transformation A T = U T K (Necessary to preserve [ x i , p j ] = i � δ ij ). Antiunitary: unitary- for conserving probabilities, anti- for complex conjugation (antilinear). (MSPF-Sonora) Flavor Physics October 25, 2018 7 / 31
• Charge Conjugation: C reverses the sign of the electric charge, colour charge and magnetic moment of a particle. Introduced by Kramers in 1937. Requires quantum field theory, as it is better understood as particle-antiparticle interchange (MSPF-Sonora) Flavor Physics October 25, 2018 8 / 31
Maxwell Equations, C , P and T L ( A µ ) = − 1 4 F µν F µν + j µ A µ ; F µν = ∂ µ A ν − ∂ ν A µ Equations of Motion: ∂ µ F µν = j ν , ∇ · E = ρ, ∇ · B = 0 , ∇ × E = − ∂ B ∇ × B = j + ∂ E ∂t , ∂t , E = − ∇ φ − ∂ A ∂t , B = ∇ × A are invariant under: PARITY: x → − x TIME REVERSAL t → − t CHARGE CONJUGATION ρ → − ρ (MSPF-Sonora) Flavor Physics October 25, 2018 9 / 31
P T C P T C t + - + x µ x µ - + + x µ − x µ x ρ + + - j µ − j µ j - - - j µ j µ φ + + - A µ − A µ - - - A µ A µ A - + - E F µν − F µν B + - - F µν − F µν (MSPF-Sonora) Flavor Physics October 25, 2018 10 / 31
Fermion Fields The fermion fields transformation rules under C , P and T symmetry follow from − 1 4 F µν F µν + i ¯ ψγ µ ( ∂ µ − iQA µ ) ψ − ¯ L QED = ψmψ P L QED ( t, x ) P − 1 = L QED ( t, − x ) C L QED ( t, x ) C − 1 = L QED ( t, x ) T L QED ( t, x ) T − 1 = L QED ( − t, x ) ψ = ψ † γ 0 and using with ¯ γ µ γ ν + γ ν γ µ 2 g µν = γ 0 γ µ † γ 0 γ µ = (MSPF-Sonora) Flavor Physics October 25, 2018 11 / 31
ψ P ( t, x ) = Pψ ( t, x ) P − 1 = P ψ ( t, − x ) ψ T ( t, x ) ψ C ( t, x ) = Cψ ( t, x ) C − 1 C ¯ = ψ T ( t, x ) = Tψ ( t, x ) T − 1 = T ψ ( − t, x ) where P and C are unitary operators and T is a anti-unitary operator. We obtain (in the chiral representation): P γ µ P − 1 = ( γ µ ) † = γ µ P = γ 0 → C = − iγ 0 γ 2 = −C T C − 1 γ µ C = − ( γ µ ) T → T γ µ T − 1 = ( γ µ ) T T = iγ 1 γ 3 = −T ∗ → (MSPF-Sonora) Flavor Physics October 25, 2018 12 / 31
Using the 16 Dirac matrices which form a complete basis for the Clifford Algebra, one can build the corresponding bilinear forms: : ¯ s 12 ( x ) = ψ 1 ( x ) ψ 2 : : ¯ ψ 1 ( x ) iγ 5 ψ 2 : p 12 ( x ) = v µ : ¯ ψ 1 ( x ) γ µ ψ 2 : 12 ( x ) = a µ : ¯ ψ 1 ( x ) γ µ γ 5 ψ 2 : 12 ( x ) = t µν : ¯ ψ 1 ( x ) σ µν ψ 2 : 12 ( x ) = where σ µν = i 2 [ γ µ , γ ν ] and γ 5 = iγ 0 γ 1 γ 2 γ 3 (MSPF-Sonora) Flavor Physics October 25, 2018 13 / 31
v µ a µ t µν s 12 ( x ρ ) p 12 ( x ρ ) 12 ( x ρ ) 12 ( x ρ ) 12 ( x ρ ) v 12 − a 12 t 12 P s 12 ( x ρ ) − p 12 ( x ρ ) µ ( x ρ ) µ ( x ρ ) µν ( x ρ ) v 12 a 12 − t 12 T s 12 ( − x ρ ) − p 12 ( − x ρ ) µ ( − x ρ ) µ ( − x ρ ) µν ( − x ρ ) − v µ a µ − t µν s 21 ( x ρ ) p 21 ( x ρ ) 21 ( x ρ ) 21 ( x ρ ) 21 ( x ρ ) C (MSPF-Sonora) Flavor Physics October 25, 2018 14 / 31
CPT Theorem Under CPT, any hermitian local Poincar´ e invariant theory described by L satisfies L ( x ) → ( CPT ) L ( x )( CPT ) − 1 = L † ( − x ) = L ( − x ) Thus, the action is invariant under CPT. ⇒ Particles and antiparticles have equal masses, equal total lifetimes and opposite charges | CPTα > ≡ CPT | α > ≡ | ¯ α > m α = m ¯ α , τ ( α ) = τ (¯ α ) , Q ( α ) + Q (¯ α ) . (MSPF-Sonora) Flavor Physics October 25, 2018 15 / 31
Status of discrete symmetries No evidence for CPT violation m K 0 − m K 0 � � � � < 10 − 18 , � � � m K 0 � � 0 ) � Γ( K 0 ) − Γ( K � � � < 10 − 17 , � � � m K 0 � � � � Q ( p ) + Q (¯ p ) � < 10 − 21 . � � � � e � No evidence for C , P , or T violation in purely electromagnetic or strong interactions. (MSPF-Sonora) Flavor Physics October 25, 2018 16 / 31
P and C maximally broken in weak interactions Violation of CP and T has been observed in weak interactions. The amount of CP and T observed is small (MSPF-Sonora) Flavor Physics October 25, 2018 17 / 31
Standard Model Gauge symmetry is G SM = SU (3) C × SU (2) L × U (1) Y There are three fermion generations: Q Li ∼ ( 3 , 2 , 1 / 6) , u Ri ∼ ( 3 , 1 , 2 / 3) , d Ri ∼ ( 3 , 1 , − 1 / 3) , L Li ∼ ( 1 , 2 , − 1 / 2) , e Ri ∼ ( 1 , 1 , − 1) The scalar representation is given by φ ∼ ( 1 , 2 , 1 / 2) (MSPF-Sonora) Flavor Physics October 25, 2018 18 / 31
The pattern of symmetry breaking is given by: G SM → SU (3) C × U (1) EM The SM lagrangian is the most general renormalizable lagrangian consistent with the gauge symmetry and the given particle content: L SM = L kin + L Higgs + L Yukawa −L Leptons λ ij E ¯ = L Li φe Rj + h.c. , Yukawa −L Quarks λ ij D ¯ Q Li φd Rj + λ ij U ¯ Q Li ˜ = φu Rj + h.c. , Yukawa where ˜ φ = iτ 2 φ ∗ . (MSPF-Sonora) Flavor Physics October 25, 2018 19 / 31
Global accidental symmetry: SM is invariant under the accidental symmetry U (1) B × U (1) e × U (1) µ × U (1) τ Including neutrino masses, the accidental symmetry is reduced to U (1) B × U (1) L . Taking into account the chiral anomaly and the topological gauge structure of the SM which implies that ( B + L ) is significantly violated through instantons and sphalerons at early times of the Universe, the accidental symmetry is reduced to U (1) B − L . P ( C ) explicitly and maximally broken. CP violation not obvious, since both P and C transformations take left- and right-handed fields into one another. (MSPF-Sonora) Flavor Physics October 25, 2018 20 / 31
Cabibbo Kobayashi Maskawa √ 2) T (suppressing flavor indices): After SSB � φ � = (0 , v/ − L m = v 2 u L λ U u R + v 2 d L λ D d R + v √ √ √ 2 e L λ E e R + h.c. Diagonalization (quark sector): Field redefinition (flavor eigenstates → mass eigenstates) u R → V u R u R , u L → V u L u L , d R → V d R d R , d L → V d L d L . V † V † u L λ U V u R = λ ′ d L λ D V d R = λ ′ U , D . Here the matrices λ ′ U and λ ′ D , are diagonal, real and positive, and the transformation matrices V u,d L,R are unitary. (MSPF-Sonora) Flavor Physics October 25, 2018 21 / 31
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