Lecture 4 Mixing and CP Violation Mixing of neutral K mesons CP violation in K decays T violation and CPT conservation Observation of charm mixing d and s mixing CP violation in decays
Mixing of Neutral Mesons A second order weak interaction transforms an initial K, D or into a final K, D or : K and box diagrams contain two W bosons and two u-type quarks D box diagram contains two W bosons and two d-type quarks 2
General Description of Mixing A state that is initially K or K evolves as a function of time: ψ(t) = a(t) K > +b(t) K > i dψ dt = Hψ(t) H is an effective Hamiltonian describing time-dependent mixing: H = M i 2 Γ where M and Γ are 2 2 mass and decay matrices Diagonal elements of H are flavour-conserving, S = Off-diagonal elements of H are flavour-changing, S = 2 They describe the mixing transitions K K If H is diagonal there is no mixing, and the flavour states of neutral mesons are the same as their decay eigenstates 3
The Decay Eigenstates K S and K L The matrix H has eigenvectors corresponding to the weak decay eigenstates K L and K S K S >= p K > +q K > K L >= p K > q K > q p = 2M 2 i 2 Γ 2 m K i 2 Γ K q 2 + p 2 = The flavour eigenstates have equal mass (CPT theorem): M(K ) = M( K ) = 498MeV The weak eigenstates have different masses and lifetimes: m K = m L m S = (3.52 ±.) 2 MeV =.53 s τ L = 5ns τ S =.9ns Γ K =. s 4
Time evolution of K states A S (t) = A S ()e (Γ S/2+im S )t A L (t) = A L ()e (Γ L/2+im L )t τ =.5 m K τ S t τ S is pure K L ψ K (t) 2 = 4 ψ K (t) 2 = 4 ) (e ΓLt + e ΓSt + 2e (Γ L +Γ S ) 2 t cos m K t ) (e ΓLt + e ΓSt 2e (Γ L +Γ S ) 2 t cos m K t 5
CP Eigenstates K and K 2 The combined operation of Charge Conjugation and Parity: CP eigenstates are: CP K >= K > CP K >= K > K = 2 [K + K ] CP = + K 2 = 2 [K K ] CP = If CP is conserved K 2π and K 2 3π CP π + π >= CP π π >= + CP π + π π >= CP π π π >= τ S < τ L explained by more phase space for K 2π than K 2 3π 6
Weak and CP Eigenstates If p/q = can identify K S = K, K L = K 2 K S >= K L >= (p + q) (p q) K > + K 2 > 2 2 (p q) (p + q) K > + K 2 > 2 2 Writing p = + ǫ, q = ǫ, where ǫ is in general complex: K L = + ǫ 2 [ǫk + K 2 ] K S = + ǫ 2 [K + ǫk 2 ] If the weak states are not identical to the CP eigenstates there should be some decays K S 3π and K L 2π ǫ measures CP violation in K decays 7
Observation of CP violation In 964 K L 2π decays were observed: η + = K L π + π K S π + π = ǫ + ǫ η = K L π π K S π π = ǫ 2ǫ ǫ represents CP violation in the mixing amplitude ǫ represents direct CP violation between I = /2 and I = 3/2 After 4 years the magnitudes and phases are now measured: ǫ = (2.232 ±.7) 3 φ ǫ = (43.5 ±.5) ǫ ǫ = (.66 ±.26) 3 φ ǫ = (42.3 ±.5) 8
T violation in Semileptonic Decays K π l + ν and K π + l ν are flavour-specific decays which obey the Q = S rule The semileptonic charge asymmetry in K L decays measures ǫ: Γ(K L π l + ν) Γ(K L π + l ν) Γ(K L π l + ν) + Γ(K L π + l ν) = 2Re(ǫ) = (3.27±.2) 3 Can test T violation in mixing using initial K and K beams: Γ(K K π + l ν) Γ( K K π l + ν) Agrees with expectation if T,CP are violated but CPT is conserved A test of CPT conservation is the comparison of: Γ(K π l + ν) = Γ( K π + l ν) CPT violation parameter Re(δ) = (2.9 ± 2.7) 4 9
Observation of D Mixing (27) aar measurement of decay time distribution of wrong-sign D K + π decays Events/.2 ps 6 Data Mixing fit 4 2 8 6 4 2 a) Random π s Misrecon. D Combinatorial No mixing fit Residuals 5-5 b) ottom plot shows difference from no-mixing -2 2 3 4 t (ps)
Mixing of d mesons Measured by factories (aar & elle) using coherent production e + e Υ(4S) t is the difference between the two decay times A( t) f/ f e t /τ ( ± cos m d t) m d = (.58 ±.4)ps M 2 (V tb V td) 2 τ d = (.53 ±.)ps q p = V tb V td V tb V td
Mixing of s mesons Measured at Fermilab in 26 using p p collisions Plot shows scan of predicted amplitude vs oscillation frequency Amplitude 2.5.5 -.5.5 CDF Run II Preliminary data ± σ.645 σ data ±.645 σ 95% CL limit sensitivity data ±.645 σ (stat. only) 7.2 ps 3.3 ps -2 5 5 2 25 3 35 m s = (7.8 ±.)ps L =. fb m s [ps ] τ s = (.47 ±.6)ps From the ratio of the two mixing results: V td V ts =.2 ±.7 2
CP Violation in Decays There are three types of CP violation that can be observed: CP violation in mixing, due to the weak eigenstates being different from the CP eigenstates, q/p. A SL (b clv) =.2 ±. Direct CP violation in decay amplitudes A( f) and Ā( f), due to A/Ā. This does not require mixing, and is seen in both charged and neutral decays. A CP ( K ± π ) =.93 ±.5 CP violation in the interference between mixing and decay amplitudes A( f) and Ā( f). This requires Imλ, where λ = qā/pa. 3
Raw Asymmetry Events / (.8 ps ) CP Violation in b c cs Decays Top left: J/ψKS λ =.982 ±.25 Im(λ) = sin2β β = 23 is phase of Vtd Raw Asymmetry Events / (.8 ps ) Raw Asymmetry Events / (.8 ps ) 4 2.5 -.5 8 6 4 2.5 -.5 - J/ψK (π + S π ) -5 5 4 2.5 -.5 S ψ(2s)k -5 5 η c K S -5 5 (a) (b) t (ps) (a) (b) t (ps) (a) (b) t (ps) Raw Asymmetry Events / (.8 ps ) Raw Asymmetry Events / (.8 ps ) Raw Asymmetry Events / (.8 ps ) 5 5.5 -.5 4 3 2.5 -.5 5 J/ψK (π π S ) -5 5.5 -.5 χ c K S -5 5 J/ψK* (K π ) S -5 5 (a) (b) t (ps) (a) (b) t (ps) (a) (b) t (ps) ACP( J/ψKS) = ( λ 2 ) cos md t 2Im(λ) sin md t 4
The CKM parameters ρ and η Unitarity triangle: V ub V ud + V cb V cd + V tb V td = Normalised sides are V ub /V cb, V td /V ts and Angles are α, β (phase of V td ) and γ (phase of V ub ) CP violation requires non-zero complex phase η η.7.6.5.4.3.2 excluded area has CL >.95 γ sin 2β ε K m d α m d & m s ε K CKM f i t t e r Summer 8 sol. w/ cos 2β < (excl. at CL >.95) α. α V ub γ β..4.2..2.4.6.8. ρ 5