1 / 23

Discrete Symmetries

Discrete Symmetries. Noether’s theorem – (para-phrased) “A symmetry in an interaction Lagrangian corresponds to a conserved quantity.”. Conserved Quantities. Many examples of conservation exist – a few: Partial decay rates ( G f ) to modes (f) may vary greatly because of

aaron
Download Presentation

Discrete Symmetries

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 • Many examples of conservation exist – a few: • Partial decay rates (Gf) to modes (f) may vary greatly because of • Conservation principles (embedded in matrix element Mif) e.g. G(p+ m+nm) G(p+ e+ne) or G(J/y  p-p+p0) only 91.0 § 3.2 keV - “OZI” rule OR • Phase space: e.g. G (K+ p+ + p0) G (K+ p+ + p+ + p-) ~ 104 due to spin (helicity) conservation ~ 3.762 Brian Meadows, U. Cincinnati

  3. u d s or c d for J/y s or c d u d OZI suppressed What is the OZI Rule? • Named after its inventors (Okubo, Zweig and Iizuka) based on observation rather than sound logic: If you can cut a diagram in half by cutting only gluon lines, then it is suppressed (but NOT forbidden). e.g. f p-p+p0 is OZI suppressed relative to K-K+ that is kinematically suppressed - K - s 0 f s K+ + Not OZI suppressed Brian Meadows, U. Cincinnati

  4. Conserved Quantities Brian Meadows, U. Cincinnati

  5. Parity P • The operation that reverses the spatial coordinates is called the parity operation P: Its eigenvalue isP • P=§1 since the eigenvalue of P2 is 1: • In some cases, there is a definite parity and we get e/v eq. • Examples: Brian Meadows, U. Cincinnati

  6. 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 +ve. (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

  7. The Effect of Parity Brian Meadows, U. Cincinnati

  8. C, P and T - E/M Reminder • Introduce vector and scalar potentials (A, f): • Substitute into Maxwell’s equations: • Auxiliary conditions: Brian Meadows, U. Cincinnati

  9. 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

  10. Charge Conjugation C • Operator C turns particle into anti-particle. • C |+> = |-> ; C |K+> = |K-> ; C |q> = |q> ; etc. • C can only be a good quantum number for neutrals • 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

  11. Charge Conjugation for Charged Particles • For charged particles, it is convenient to define a related operator G = C ei I2 • The exponential rotates a state about the I2 axis by  • This flips the sign of I3 making charge change sign • Like P, this introduces a factor (-1)I • Then G|+> = C ei I2|+> = C(-1)I|-> = -|+> • Therefore, the G-parity of the  is -1. • Like C, G-parity is conserved in strong and e/m interactions • Example: What is the G-parity of the  meson? • The  is known to decay strongly to  • Therefore its G-parity is +1 Brian Meadows, U. Cincinnati

  12. Effect of C Brian Meadows, U. Cincinnati

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

  14. Effect of T Brian Meadows, U. Cincinnati

  15. 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

  16. 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

  17. CP and the K0 Particle • The K0 is a pseudo-scalar particle, 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

  18. 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 Brian Meadows, U. Cincinnati

  19. CP and K0-K0 Mixing • Rather than assigning KL to K1 and Ks to K2we define: • Fitch and Cronin’s (and subsequent) experiments lead to Brian Meadows, U. Cincinnati

  20. 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 q Brian Meadows, U. Cincinnati

  21. K0-K0 Mixing • The Ks and KL wave-functions have time-dependence related to their complex masses M - ½ i ~ / |Ks.L> ´s,L(t)=K,L (0) exp [- i (Ms,L /~ - ½ i/s,L) t ] • IF we make the simplification that KL = K1 and Ks = K2 K0 = [L(t) + s(t)] / \/2 • At time t the intensity of a beam of K0 will be IK0(t) = |K0|2 = ¼ [e -t /s + e -t /L + 2 e- ½ (1/ s +1/ L)t cos (m t / ~)] where  = +1 for K0 and -1 for K0(m ´ML -Ms) • This is referred to as “K0-K0 mixing” Brian Meadows, U. Cincinnati

  22. 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

  23. 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 See a recent review: http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.4534v2.pdf Brian Meadows, U. Cincinnati

More Related