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

2. Brian Meadows, U. Cincinnati Conserved Quantities StrongE/MWeak 4-momentumYes ChargeYes Baryon #Yes SpinYes Lepton # (e, ,  ) --Yes Flavour (S,C,B,T) Yes No (CKM) (or  Q =  F) Iso-spinYesNo PYes No CPT, CP, CYes, Yes, Yes Yes, No, No

3. Brian Meadows, U. Cincinnati 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. (ie P=+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)

4. Brian Meadows, U. Cincinnati 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.

5. Brian Meadows, U. Cincinnati Charge Conjugation C  Operator C turns particle into anti-particle.  C |  + > = |  - > ; C |K + > = |K - > ; C |q> = |q> ; etc.  C 2 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 )

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

7. Brian Meadows, U. Cincinnati 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. K 0 observed to decay into  +  - (CP=+1) as well as into  -  +  0 (CP=-1)  B 0 decays to J/psi K s, J/psi K L and  +  -

8. Brian Meadows, U. Cincinnati 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

9. Brian Meadows, U. Cincinnati CP and the K 0 Particle  The K 0 is a pseudo-scalar particle (P=-1), therefore P |K 0 > = - |K 0 > and P |K 0 > = - |K 0 >  The C operator just turns K 0 into K 0 and vice-versa C |K 0 > = + |K 0 > and C |K 0 > = + |K 0 >  Therefore, the combined operator CP is CP |K 0 > = - |K 0 > and CP |K 0 > = - |K 0 >  Neither |K 0 > nor |K 0 > are CP eigen-states  We can define odd- and even-CP eigen-states K 1 and K 2 : |K 1 > = (|K 0 > - |K 0 >) / \ / 2  CP |K 1 > = (+1) |K 1 > |K 2 > = (|K 0 > + |K 0 >) / \ / 2  CP |K 2 > = (-1) |K 2 >

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

11. Brian Meadows, U. Cincinnati CP and K 0 -K 0 Mixing  It is possible for a K 0 to become a K 0 !  The main diagram contributing to mixing in the K 0 system:  This contributes to the observation of CP violation in the K 0 K 0 system.  It generates a difference in mass between K 1 and K 2  It is described by a phase in the CKM matrice. d d s s u, c, t W W K0K0 K0K0

12. Brian Meadows, U. Cincinnati Strangeness Oscillations Graph shows I(K 0 ) and I(K 0 ) as function of t for  m  s  / ~ = 0.5 Experimentally, measure hyperon production in matter (due to K 0, not K 0 ) as function of distance from source of K 0 )  m  s  / ~ = 0.498. This corresponds to  m/m ~ 5 x 10 -15 !

13. Observation of K 0 -K 0 Oscillations  K 0 ->3  is only 34%, 39% of the decays are leptonic  Observe the asymmetry in the leptonic sector  Use the sign of lepton in decays K 0   + e - e K 0   - e + e Brian Meadows, U. Cincinnati Gjesdal et al, Phys.Lett.B52:113,1974 World Average:

14. Brian Meadows, U. Cincinnati Other examples of “Mixing”  Evidence now also exists for mixing in other neutral meson systems:  K 0 - K 0 (ds) - observed in ~1960  B 0 - B 0 (bd) - observed in ~1992  B s - B s (bs) - observed in 2005  D 0 - D 0 (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.