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: Section 3: Mixing and CP violation in (mostly) neutral mesons

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State evolution in neutral mesons Neutral meson states P o, P o –Could be K o, D o, B o With internal quantum number F –Such that F=0 emag/strong, F 0 for H weak Obeys time dependent Schrödinger eqn M, Hermitian 2x2 matrices, mass matrix and decay matrix, H 11 =H 22 from CPT (mass/decay particle = anti- particle)

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Solve Schrödinger for Eigenstates of H w From characteristic equation If E 1 =M 1 -i 1 /2, E 2 = M 2 -i 2 /2 and M=M 2 -M 1, = 2 - 1 Complex coeff. p,q obey |q| 2 +|p| 2 =1

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Eigenvector Eqn Eigenstates have time evolution: then with

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Finally probabilities… With The oscillations depend upon the parameter x the speed of oscillations c.f. lifetime Where = ( 1 + 2 )/2 Interference term

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Mixing probabilities x in B s not yet measured >19 Probability of finding P o from initial pure P o beam

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CP in decay CP in mixing CP in interference between mixing and decay Types of CP violation P f f P f f P P f f P PP P f f P PP P + +

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1) CP Violation in mixing Indirect CP Violation Mass eigenstates being different from CP eigenstates Mixing rate P o P o is different to P o P o If conserved CP|P 1 > = +1 |P 1 > CP|P 2 > = -1 |P 2 > with If violated Such asymmetries usually small Need to calculate M, , involve hadronic uncertainties Hence, tricky to relate to CKM parameters

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2) CP Violation in decay direct CP Violation Two types of phase –Weak phase: due to weak interactions (complex CKM elements) –Strong phase: contribution from intermediate states, CP conserving (same sign in both) f is final state common to both decays occurs for both charged and neutral states P

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3)CP violation in the interference of mixing and decay Choose state* f, P o f, P o f Two possible decay chains, with or w/o mixing CP can be conserved in mixing and in decay and still be violated ! *Not necessary to be CP eigenstate Interference term depends on Can putand getbut

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: Section 4: Neutral Kaon system

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K o K o system Kaon mesons in two isospin doublets K + = us K o = ds K - = us K o = ds S=+1S=-1 Part of pseudo-scalar J P =0 - meson octet with , I 3 =+1/2 I 3 =-1/2 Kaon production K o : - + p o + K o But from baryon number conservation: K o : + + p K + + K o + p Or K o : - + p o + K o + n +n Requires higher energy Much higher S 0 0 -1 +1 S 0 0 +1 -1 0 S 0 0 +1 -1 0 0

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Kaon oscillations So say at t=0, pure K o, –later a superposition of states d s u, c, t WW WW _ s d _ d s WW WW _ s d _ _ _ _ K0K0 K0K0

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K o Decay In that case CP=+1 CP=-1 K s o o K s + - K L + - o K L o o o CP=+1 CP=-1 K S branching fractions: 69%, 31% K L branching fractions: 21%, 13%, 66% mass eigenstates KSKS KLKL show Assume CP

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Time dependent probabilities for the neutral kaon case. t (1/ )

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K o Regeneration Start with Pure K o beam –After time all K s component decayed Introduce slab of material in beam –reactions 1) Elastic scatttering2) Charge exchange 3) Hyperon production Hence K o absorbed more strongly i.e. K s regenerated

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Discovery of CP Violation K 1 o o K 1 + - K 2 + - o K 2 o o o CP=+1 CP=-1 So if K L =K 1 CP eigenstate, Observe no two pion component But if broken get: Where quantifies degree of CP violation BUT Can one find K L decaying into + - ?

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K L X K L p = p + p = angle between p KL and p If X = 0, p = p KL : cos = 1 If X 0, p p KL : cos 1 cos KLKL m ( ) < m K J.H. Christenson et al., PRL 13,138 (1964) Discovery of CP violation m ( ) = m K m ( ) > m K Mass and angular spectrum

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So CP symmetry is violated in the neutral kaon system. Mass eigenstates (K S and K L ) CP eigenstates Both K S and K L could decay into + - -. Experimentally well known: The majority of K S decays into - and K L into - - . Small but with profound implications In K L

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Decay final state at time t -- Spin( ) = 0 L - = 0 Initial state at t = 0 S = 0 CP( - i.e. CP eigenstate K 0 at t = 0 decays into vs K 0 at t = 0 decays into any difference = CP violation CPLEAR revisited Tag K o /K o from charged pion/kaon

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K0K0 K0K0 _ CPLEAR R +- (t) and R +- (t) _ CP violation

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CPLEAR CP asymmetry A (t) = R +- (t) R +- (t) R +- (t) R +- (t) _ _ large difference! CP violation in mixing The two mass eigenstates are not CP eigenstates

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Kaons: CP violation in Decay CP violation first through existence of certain decay modes ~2.3x10 -3 If CP violation is only in mixing, i.e. independent of decay So, put channel independent term and channel dependent ’ Hence, by measuring only rates: get

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So, two expts in the 80’s did it: NA31 (CERN) E731 (Fermilab) Ambiguous result! So, two expts did it again…….

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NA48

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KTeV

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Measure and at the same time: NA31, NA48 K S is regenerated from K L : E731, KTeV No normalization is required, but efficiencies, acceptances etc. have to be corrected… Normalisation constants

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Effort over 30 years! Note; 3 Re = | / | i.e. Re 0 Not easy to compare with SM theory

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CKM parameters with CP conserving parameters |Vud| : nuclear Beta decay 0.9734 0.0008 |Vus| : semileptonic Kaon and hyperon decay (CCFR) 0.2196 0.0026 |Vcd| : Neutrino and anti-neutrino production of charm off valence d quarks 0.224 0.016 |Vcs| : W decays (LEP/me!) 0.996 0.013 |Vcb| : semileptonic inclusive and exclusive B decays (LEP/CLEO) 0.0412 0.0020 |Vub| end point spectrum in semileptonic B decays(LEP/CLEO) 0.0036 0.0007 B o mixing x d, + lattice gauge inputs |Vtb*Vtd| 0.0079 0.0015 Can use Unitarity constraints

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