Lecture 7: Symmetries II

1. Lecture 7:Symmetries II • Charge Conjugation • Time Reversal • CPT Theorem • Baryon & Lepton Number • Strangeness • Applying Conservation Laws Useful Sections in Martin & Shaw: Section 4.6, Section 2.2

2. a state characterized by a particle x and a wave function C∣ x, > = ∣ x,  > where∣ x,  > whereCyis a ''phase factor" of 1 (like for parity) used to determine C-conservation in interactions C∣ y,  > = Cy∣ y,  > C-Parity (charge conjugation) C-Parity Changes particle to anti-particle(without affecting linear or angular momentum) Electromagnetism is obviously symmetric with respect to C-parity (flip the signs of all charges and who would know?). It turns out that the strong force is as well. But, again, not the weak force (otherwise there would be left-handed anti-neutrinos that couple in a similar way to left-handed neutrinos... there aren’t !) For particles with distinct anti-particles(x = e, p, , n, ...) For particles which nonothave distinct anti-particles(y = , , ...)

3. C-Parity Assignment C∣ x, x > = ∣ x, x > = | x, x > depending on whether the system is symmetric or antisymmetric under the operation For multi-particle systems Example: consider a  pair in a state of definite orbital angular momentum L C∣ ,L > = (-1)L∣ ,L > since interchangingandreverses their relative position vector in the spatial wave function

4. C-Parity Conservation C Ah! So we can never get 0 if C-parity is conserved !! Example: Experimentally, 0but never 0 How is this reconciled with the concept of C-parity ?? using a similar argument as for parity,C = 1 C∣  > = C∣  > C∣  > = CCC (-1)L∣  > C∣  > = CC(-1)L∣  > But 0 spin is zero, so L=0 = C2C∣  > = C2∣  > = C∣  > = ∣  > = ∣  >

5. Time Reversal However, note that if we start with  t H (x,t) = i ℏ(x,t) Now apply t t H (x,t) = i ℏ(x,t)  t Wigner, 1931 T [H (x,t)] =  T i ℏ(x,t) ]  t  t H *(x,t) = i ℏ*(x,t) Both (x,t) and *(x,-t) satisfy the same equation. Thus we define TS(x,t) = S*(x, t) Time Reversal In analogy with parity, we could try t t But we want the Schrodinger equation to be invariant! This can be patched up by taking the complex conjugate

6. Non-Hermitian Operators Note that if x,tx,t TTx,tTx,t *{ Tx,t}*{ Tx,t} ≠{ Tx,t}{ Tx,t} ''anti-linear" So the operator is not Hermitian and the eigenvalues are not real! Hermitian** so n*Gn)  gnn* n gn  (Gn)* n gn* n* n gn*  gn gn* so gn must be real So, unlike other symmetries, time-reversal does not give rise to real conserved quantities (i.e. no conservation laws per se)

7. Test of T-Reversal Invariance the spin-averaged rates of these should thus be the same under TPsymmetry ''Principle of Detailed Balance"(confirmed in various EM & strong interactions) In practice this is difficult to test (how do you ''reverse time" in an experiment?) It’s more useful to consider the combination TP : PL=P(x  m dx/dt) = xm dx/dt TPL = (xm dx/dt) = L Pp = p TPp = p and Note that (assume this holds for spin) a(pa, sa) + b(pb, sb)  a(pa, sa) + b(pb, sb) apply TP a(pa, sa) + b(pb, sb)  a(pa, sa) + b(pb, sb)

8. Charge Conjugation -X +X

9. Parity Charge Conjugation -X +X -X +X Flip orientation in time

10. Charge Conjugation Parity -X +X -X +X +X -X Flip orientation in time Switch coordinate definitions

11. Charge Conjugation Parity -X +X -X +X +X -X Flip orientation in time Switch coordinate definitions

12. Charge Conjugation Parity -X +X -X +X -X +X Flip orientation in time Switch coordinate definitions

13. Charge Conjugation Parity -X +X -X +X -X +X Flip orientation in time Switch coordinate definitions

14. Time Reversal Charge Conjugation Parity -X +X -X +X -X +X Flip orientation in time Switch coordinate definitions

15. Charge Conjugation Time Reversal Parity -X +X -X +X -X +X -X +X Flip orientation in time Switch coordinate definitions Run movie backwards in time

16. CPT

17. CPT Theorem States that if a quantum field theory is invariant under Lorentz transformation, then CPT is an exact symmetry !! CPT Theorem (independently discovered by Pauli, Luders and Bell and Schwinger) • 1) Integer spin particles obey Bose-Einstein statistics (bosons) • 2) 1/2 - spin particles obey Fermi-Dirac statistics (fermions) • 3) Particles and antiparticles must have identical masses & lifetimes • 4) All internal quantum numbers of antiparticles are opposite to those of the corresponding particles ( Note that if, for example, CP is violated, then T must be violated )

18. NO Experimental evidence that this is the case!!! } e e  L } } } HOT OFF THE PRESS! These can be violated by ''neutrino oscillations" Le L L Baryon and Lepton Number It is an empirical observation that the number of baryons (fermions with masses  the proton mass) minus the number of antibaryons is conserved in all reaction thus far observed. Thus we define the ''baryon number" B  (# baryons) - (# antibaryons)as a conserved quantity Baryon and Lepton Number The same has been assumed to be true for Leptons. However, there is also a form of the rule that seems to operate which relates to individual lepton ''families" or ''generations" :

19. Proton Decay Among other things, conservation of B and L means that protons and electrons don’t decay (so matter is stable) and baryons don’t mix with leptons. GUT models therefore predict these laws to break down at some point

20. Symmetry Summary Symmetry Summary: TransformationSymmetry TypeConserved Quantity translation global, continuous linear momentum rotation global, continuous angular momentum time global, continuous energy (Lorentz global, continuous CM velocity) (''space-time") rotation in ''isospin-space" global, continuous isospin additive (''internal") electromagnetic scalar/vector potential local, continuous, gauge charge  (global, continuous, gauge) baryon number  (global, continuous, gauge) lepton number parity global, discreet ''P-value" charge conjugation global, discreet ''C-value" time reversal global, discreet none! multiplicative In Addition: Local, Lorentz-Invariant, Quantum Field TheoriesCPT

21.   K0   p  Strangeness Strangeness

22. Discovery of Strangeness It can be shown that this is about what you’d expect for aweakdecay: Originally found in cosmic ray cloud chambers in 1947, Then in bubble chambers at the Brookhaven Cosmotron in 1953 The new particles were always produced in pairs (''associated production"), suggesting a new conserved quantity  ''strangeness" so defineS1 and SK = +1 The cross-section for production indicates a strong interaction, however the decay timescale is much longer than expected for a strong decay: tS~ 1 fm / c = (1015m) / (3x108m/s) = 1023s but the observed decays took 1010s (practically forever!)

23. Timescale for Weak Decays since the distributions for e and e are unconstrained in a 3-body decay and, assuming the proton gets basically no kinetic energy, the energy, E , available for the e & e is determined. E 0 ~ Ee2 (EEe)2 dEe E5  First consider: n  p + e + e from Fermi Golden Rule: 1/t (density of final states) dNe~ pe2 dpe dN~ p2 dp  dN = dNe dN~ pe2 dpe p2 dp If E  Ee + E p≃ E  Ee then, for a given value of Ee, dp≃ dE Thus, d = dN/dE = dN/dp ~ pe2 (EEe)2 dpe and, in the limit Ee≫ me ,

24. Strangeness Violation in Weak Interactions so take an average value of E~750 MeV Thus, we’d expect a weak decay timescale of order tW = 1000 s (1.3/750)5 = 1011s Interpretation: Sis conserved by the strong interaction, which is why these particles are produced in pairs and why the individual particles cannot undergo strong decay to non-strange products. For “1st-order" Weak Interactions: S = 0, 1 For -decay, E ≃ mn mp~ 1.3 MeV and the neutron lifetime is ~ 1000 s In case of the strange particles: m = 1116 MeV mK = 498 MeV ( which is certainly alot closer to what is actually observed! ) However, Sisnotconserved by the weak interaction, which eventually does allow the  and K0 to decay !

25. Applying Conservation Laws Applying Conservation Laws:

26. Conservation Examples In the following pairs of proposed reactions, determine which ones are allowed and the relevant force at work Example:  + p 0 + 0  + p 0 + K0  + n + p strong weak Interaction: charge: lepton number: baryon number: strangeness: Isospin (I3) : 1 + 1 = 0 + 0 0 + 0 = 0 + 0 0 + 1 = 1 + 0 1 + 1 = 0 + 0 0 + 0 = 0 + 0 0 + 1 = 1 + 0 0 + 0 =  1 +1 1 + 1/2 = 0  1/2 1 = 1 + 0 0 = 0 + 0 1 = 0 + 1 1 = 1 + 1 0 = 0 + 0 1 = 0 + 1 0 + 0 = 1 + 0 1 + 1/2 = 0 + 0

27.   X L = 0 Lepton number: 0 = 0 + ? B = 0 Baryon number: 0 = 0 + ?  J = 0 Spin: 0 = 0 + ? Candidates:  X =  Example: Identify the neutral particle  + X But, mX < mm (X0)

28.   L = 0 Lepton #: ? = 0 + 0 X  B = 0 Baryon #: ? = 0 + 0  J = 0 Spin: ? = 0 + 0  Y (as before) Candidates:  p L = 0 Lepton #: ? = 0 + 0 X = Kor K B = 1 Baryon #: ? = 0 + 1  J = 1/2 Spin: ? = 0 + 1/2 Candidates: n X = K Y =  X +  But,mX > m + m (X) &  < 1018 s (c < 0.3 nm) (X) Y + p strong interaction  +p X +  But,mY > m + mp (Yn) S = +1 Strangeness: 0 + 0 = ? +   +  (Y) (Y) S=0,1 (weak decay)