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Discrete Symmetries

Discrete Symmetries. Noether’s theorem – (para-phrased) “A symmetry in an interaction Lagrangian corresponds to a conserved quantity.”. Conserved Quantities. Parity P. Particles have “intrinsic parity” =± 1 P |  > = - |  > ; P |q> = +1 ( q is a quark); etc..

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Discrete Symmetries

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  1. Discrete Symmetries Noether’s theorem – (para-phrased) “A symmetry in an interaction Lagrangian corresponds to a conserved quantity.” Brian Meadows, U. Cincinnati

  2. Conserved Quantities Brian Meadows, U. Cincinnati

  3. Parity P • Particles have “intrinsic parity” =± 1 P |> = - |> ; P |q> = +1 (q is a quark); etc.. • We define parity of quarks (ie the proton) to be positive. (ieP=+1) • It is usually possible to devise an experiment to measure the “relative parity” of other particles. • Parity of 2-body system is therefore P = (-1)l 1 2 • Example: parity of Fermion anti-Fermion pair (e.g. e+e-): Whatever intrinsic parity the e- has, the e+ is opposite (actually a requirement of the Dirac theory) So, P = (-1)(l+1) Brian Meadows, U. Cincinnati

  4. Parity Violation • Parity is strictly conserved in strong and electromagnetic interactions • Helicity can be +1 or -1 for almost any particle. • It can flip if you view particle from a different coordinate system • BUT not if the particle travels at c! • Real photons have both +1 and -1 helicities (not zero) • Consequence of conservation of parity in e/m interactions • Not so for neutrinos • In + + +  helicity of + is ALWAYS = -1 (“left-handed”)  The neutrino is LEFT-HANDED (always!) • Parity is “maximally violated” in weak interactions. Brian Meadows, U. Cincinnati

  5. Charge Conjugation C • Operator C turns particle into anti-particle. • C |+> = |-> ; C |K+> = |K-> ; C |q> = |q> ; etc. • C2 has eigenvalue 1 • Therefore C=± 1 • Since C reverses charges, E- and B-fields reverse under C. • Therefore, the  has C=-1 • C is conserved in strong and E/M interactions. • Since 0 2, then C|0> = +|0> • Since 0 2, then C|0> = +|0> • AND 0 cannot decay to 3 (experimentally, 0  3/0  2 < 3 10-8) Brian Meadows, U. Cincinnati

  6. Time Reversal T • This, in effect, reverses the direction of time • It does not reverse x, y or z. Brian Meadows, U. Cincinnati

  7. CPT and Time-Reversal • There is compelling reason to believe that CPT is strictly conserved in all interactions • It is difficult to define a Lagrangian that is not invariant under CPT • T is an operator that reverses the time • No states have obviously good quantum numbers for this, but you can define CP quantum number • e.g. CP |+-> = (-1)L (why?) • Even CP is not conserved • e.g. K0 observed to decay into +- (CP=+1) as well as into -+0 (CP=-1) • B0 decays to J/psi Ks, J/psi KL and +- Brian Meadows, U. Cincinnati

  8. CP Conservation • Recall that P is not conserved in weak interactions since ’s are left-handed (and anti-’s are right-handed). • Therefore, C is not conserved in weak interactions either: + + +  Makes a left-handed + (because  is spin 0) C(+ + +  )  (-  - +  ) makes a left-handed - (C only converts particle to anti-particle). BUT – the - has to be right-handed because the anti- is right-handed. • However, the combined operation CP restores the situation CP(+  + +  )  (-  - +  ) Because P reverses momenta AND helicities Brian Meadows, U. Cincinnati

  9. CP and the K0 Particle • The K0 is a pseudo-scalar particle (P=-1), therefore P |K0> = - |K0> and P |K0> = - |K0> • The C operator just turns K0 into K0 and vice-versa C |K0> = + |K0> and C |K0> = + |K0> • Therefore, the combined operator CP is CP |K0> = - |K0> and CP |K0> = - |K0> • Neither |K0> nor |K0> are CP eigen-states • We can define odd- and even-CP eigen-states K1 and K2: |K1> = (|K0> - |K0>) / \/2  CP |K1> = (+1) |K1> |K2> = (|K0> + |K0>) / \/2  CP |K2> = (-1) |K2> Brian Meadows, U. Cincinnati

  10. CP and K0-K0 Mixing • Experimentally, it is observed that there are two K0 decay modes labeled as KL and Ks: Ks +-(s= 0.893 x 10-10 s) KL  +-0(L= 0.517 x 10-7 s) • The decay products of the Ks have P = (-1)L = (-1)0 = +1 • For the KL the products have P = -1 • It is tempting to assign KL to K1 and Ks to K2 However, this is not exactly correct: V. Fitch and J. Cronin observed, in an experiment at Brookhaven, that about 1 in 500 times, either Ks 3por KL  2p So one defines KL=1/sqrt(1+e2) (K2+e K1) where e is the deviation from CP conservation Brian Meadows, U. Cincinnati

  11. W d s K0 K0 u, c, t u, c, t W d s CP and K0-K0 Mixing • It is possible for a K0 to become a K0 ! • The main diagram contributing to mixing in the K0 system: • This contributes to the observation of CP violation in the K0K0 system. • It generates a difference in mass between K1 and K2 • It is described by a phase in the CKM matrice. Brian Meadows, U. Cincinnati

  12. Strangeness Oscillations • Graph shows I(K0) and I(K0) as function of t • for D mts/ ~ = 0.5 • Experimentally, measure hyperon production in matter (due to K0, not K0) as function of distance from source of K0) • D m ts/ ~ = 0.498. • This corresponds to D m/m ~ 5 x 10-15 ! Brian Meadows, U. Cincinnati

  13. Observation of K0-K0 Oscillations • K0->3p is only 34%, 39% of the decays are leptonic • Observe the asymmetry in the leptonic sector • Use the sign of lepton in decays K0+e-eK0-e+e World Average: Gjesdal et al, Phys.Lett.B52:113,1974 Brian Meadows, U. Cincinnati

  14. Other examples of “Mixing” • Evidence now also exists for mixing in other neutral meson systems: • K0 - K0 (ds) - observed in ~1960 • B0 - B0 (bd) - observed in ~1992 • Bs - Bs (bs) - observed in 2005 • D0 - D0 (cu) - observed in April 2007 ! by BaBar and (almost simultaneously) by Belle Similar mass oscillation versus “flavor observations” are Observed with neutrinos, revealing that neutrino have mass. Brian Meadows, U. Cincinnati

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