1. Particle Physics Particle Physics Chris Parkes Symmetries,Invariances and Conservation laws (Or how to decide whether to shake hands with an alien!) Conserved quantities in QM Parity Scalars,Vectors and pseudo-S,axial-V CP,T 4 th Handout http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html

2. Symmetries and Conservation Laws Quantum numbers are associated with the conserved observables Some are universal laws of nature (p,E,L,CPT…), others are valid only in approximations (e.g. parity - valid for strong/EM force but not weak) In classical physics there are a number of quantities which are conserved – momentum, energy, angular momentum Conservation theorems also occur in QM In classical physics the conservation laws tend to be the starting points ( there are also more sophisticated way of deriving them) In QM however the conservation laws are deeply related to the principle of superposition of amplitudes and the symmetry of the system. we will deal with both continuous (e.g. displacement in space/time) and discrete symmetries (e.g. mirror like) Emmy Noether Noether’s theorem – Symmetries (invariances) naturally lead to conserved quantities

3. C P CP Parity Inversion Spatial mirror Charge Inversion Particle-antiparticle mirror

4. Conserved quantities in QM - Revision Any operator, Â, which is time independent (e.g. p) and commutes with the Hamiltonian is associated with a conserved quantity. Expectation value Of operator Conservation requires (e.g. momentum) Thus if A is indep.of time the expectation value is constant, as long as A,H commute Hamiltonian Minus sign from complex conjugate

5. Translational Invariance  linear momentum conservation M&S 4.1 i.e. wish to show p operator independent of time Invariance: All positions in space are physically indistinguishable Consider moving a particle a small distance Depends on derivatives not on position (natural units) define operator D that performs this translation Higher order terms Since in natural units Consider wavefunction 1) 2) Comparing 1)&2), 1, i  x just numbers so =0 Hence, linear momentum is conserved and is a good quantum number

6. Rotational Invariance  Angular Momentum Conservation M&S 4.2 All directions in space are physically indistinguishable Rotating a system of particles around its CM to a new orientation leaves its physical properties unchanged  Proof is very similar to translation, see lectures This proof considers only L, in general must also consider spin Get Translations in time  Energy Conservation Define an operator for time evolution, A But wait, we already have this operator, it is the hamiltonian TISE And H commutes with itself So time translation is the symmetry, H is operator, E is the conserved quantity N.B. time translation invariance is different from time reversal operator T– discussed later Laws of physics independent of time

7. Other conserved quantities Electric Charge, Colour Charge, Baryon number, lepton number, strangeness.. First three always conserved (strong,EM,weak) Last one not conserved in weak e.g. Other Discrete Symmetry operators Parity (P)– spatial Inversion Charge conjugation (C)– particles  anti-particles reverses:charge magnetic moments baryon number strangeness Only particles that are their own anti-particles are eigenstates of C Time (T)- Time reversal Discuss P,C,CP/T, particularly for the weak force CPT – combined is a fundamental symmetry of QFTs, arising from very basic assumptions like Lorentz invariance Q) Is there a difference in behaviour between matter and anti-matter ?

8. Parity - Spatial Inversion P operator acts on a state |  (r, t)> as Hence for eigenstates P=±1  (r, t)>= cos x has P=+1, even  (r, t)>= sin x has P=-1, odd  (r, t)>= cos x + sin x, no eigenvalue e.g. hydrogen atom wavefn  (r, ,  )>=  (r)Y l m ( ,  ) Y l m ( ,  )= Y l m (  - ,  +  ) =(-1) l Y l m ( ,  ) So atomic s,d +ve, p,f –ve P Hence, Electric dipole transition  l=1  P  =- 1 Discrete symmetry Parity conserved when Hamiltonian invariant under Parity transformation (strong,EM)

9. Parity Examples Conventions – quarks and leptons have +ve parity Anti-quarks and anti-leptons have –ve parity Parity multiplicative For a meson made from q qbar pair with orbital angular momentum l |  > =  a  b, P=P a P b (-1) l For ground state (l=0) P meson =-1, expect –ve parity for light mesons  -,  o,K -,K o all P=-1 For baryons: So, expect+ve parity for low lying states For anti-baryons: expect-ve parity for low lying states l 12 l3l3 q1q1 q3q3 q2q2

10. Scalars,Vectors,Pseudo-Scalars,Axial Vectors Scalar – unaffected by parity (+ve parity) Vector – reverses (-ve parity) Can also form quantities from ‘.’ and ‘X’ products of vectors. How do the resultant scalars/vectors behave ? Axial vector: consider cross-product of two vectors Both reverse under parity, so L unaltered p r -r-r -p Pseudo-scalar: consider dot product of two vectors In a parity conserving theory you can’t add an axial vector to a vector Acts like a scalar Now, consider dot product of vector, axial vector Changes sign, a pseudo-scalar This leads to parity violation in weak interactions

11. Weak Force Parity Violation Discovery “  -  ” problem  Same mass, same lifetime, BUT   +    , (21%) P  =+1   +    +  -, (6%) P  =-1 Actually K + Postulated Yang& Lee, 1956 C.S. Wu et. al., Phys. Rev. 105, 1413 (1957) B field e- (E,p) Co 60 Nuclei spin aligned Beta decay to Ni* 60 e- (E,-p) Parity   Spin axial vector -> maximal violation V-A theory, neutrino handedness Experimental discovery Revision

12. Operating with P on this reverses p, not spin, produces a right-handed neutrino. Do not observe: Helicity and the neutrino In angular momentum we choose the axis of quantisation to be the z axis. Lets choose this axis along the particle momentum direction. Helicity is the component of the spin along the momentum direction. A spin ½ particle can thus have helicity +1 (m s =+ ½) or –1 (m s =- ½ ) Not so interesting for a massive particle, as not Lorentz invaraint, but consider the neutrino. p s +1 p s Right-handed Left-handed 1)Only left-handed neutrinos exist and right-handed anti-  2)Helicity is a pseudo-scalar Operating with C on this produces a left-handed anti-neutrino. Do not observe: Operating with C and P on this produces a right-handed anti-neutrino. Do observe! Weak force violates Parity, but CP OK?

13. Measuring Helicity of the Neutrino Goldhaber et. al. 1958 Electron capture K shell, l=0 photon emission Consider the following decay: Eu at rest Select photons in Sm* dir n Neutrino, Sm In opposite dirns e-e- Momenta, p spin OR  S=+ ½ S=- ½ Left-handed S=+ 1 S=- 1 right-handed Left-handed right-handed Helicities of forward photon and neutrino same Measure photon helicity, find neutrino helicity

14. Neutrino Helicity Experiment  Tricky bit: identify forward γ  Use resonant scattering!  Measure γ polarisation with different B-field orientations magnetic field Pb NaI PMT 152 Sm 152 Eu γ γ Fe Similar experiment with Hg carried out for anti-neutrinos Vary magnetic field to vary photon absorbtion. Photons absorbed by e- in iron only if spins of photon and electron opposite. Forward photons, (opposite p to neutrino), Have slightly higher p than backward and cause resonant scattering Only left-handed neutrinos exist

15. CP Violation Parity is violated by weak force But neutrino analysis shows CP looks OK. History repeats itself, just as we expected parity to be conserved, we then expected CP to be conserved. Actually violated by a tiny amount – currently a hot research topic CPT is conserved so CP violation is equivalent to T violation QM + relativity: Gave us matter/anti-matter symmetry So why is our world full of protons,neutrons,electrons and not anti-protons, anti-neutrons, positrons? Historical accident that our corner of universe has more matter than anti-matter ? No, astronomical evidence tells us that observable universe is all made of matter. CP/T violation is the key….

16. C P CP Parity Inversion Spatial mirror Charge Inversion Particle-antiparticle mirror

17. Time reversal – T Leaving all position vectors x unchanged but p,J reverse Detailed balance Compare reaction With time reversed counter-part Where m are spin Z-components Conserved for Strong,EM We wish to test this for the weak force Inverse experiments are difficult to do with the weak force, need to avoid strong/EM contamination e.g. reversed would be Would be dominated by strong interaction of proton,pion. Neutrino expt. would be possible, but difficult and looking for a small effect Particle are eigenstates of P, neutral particles can be of C, but cannot be identical to itself going backwards in time

18. Let us have a quick look at nature.... Neutral kaon system flavour eigenstatesCP conjugated mass eigenstates KSKS KLKL Short lifetime Long lifetime are a mixture of flavour eigenstates Time dilation -  factor needed for actual flight distance in lab Mainly Decays to three pions (34%) 3 x m(pion) ~ m(Kaon) Mainly Decays to two pions (99.9%)

19. CPLEAR- some parameters Beam – 10 6 anti-protons /s into Hydrogen target Fast online trigger selection of events ~ 10 3 /s Ability to separate charged pions / kaons using Cherenkov, dE/dx, Time of flight discriminate in momentum range 350-700 MeV/c Can detect and reconstruct K s vertex to ~ 60 lifetimes c  ~2.6 cm Observe events over ~ 4  Magnetic field (0.4T) and tracking leads to particle momentum determination (~5% accuracy) Kaon Oscillation Rate difference K o  K o  K o  K o is T violation Particle can turn into anti-particle. So say at t=0, pure K o, later a superposition of states d s u, c, t WW WW _ s d _ d s WW WW _ s d _ _ _ _ K0K0 K0K0 

20. 1) Identify K o / K o at production: produced in association with K + /K - 2) Identify K o / K o at decay: observe leptonic decay CPLEAR T invariance test Initial state at t = 0 S = 0 Get positron: Or electron:

21. Experiment at LEAR ring at CERN 1990-1996 Pions from kaon decay

22. Discovery of T violation  Currently the only direct observation of T violation Measure asymmetry in rates CPLEAR,1998 Number of lifetimes T, or equivalently CP, violated by this tiny amount

23. CP Violation: Why is it interesting ?  Fundamental: The Alien test C violation does not distinguish between matter/anti-matter Left-handed / right-handed are simply conventions We cannot define what we mean by Co 60 e - emission asymmetry unless we can define difference from anti-Co 60 (or charge) CP violaton says preferred decay K L  e + v e  - Never shake hands with an alien whose `electrons’ are the preferred decay state of the long lived kaon! Hence, it allows us to unambiguously distinguish between matter and anti- matter.  Least Understood: CP Violation is ‘add-on’ in SM Parity violation naturally imbedded from coupling structure Left-handed and right-handed couplings There is a matrix (CKM matrix) that tells us how likely transitions are from one quark generation to another e.g. b quark to decay to a c quark or a u quark. CP violation can be accommodated in this matrix by adding a complex phase. It is an ‘add-on’ justified only by the observation of CP violation.

24. CP: Why ?  Powerful: delicately broken symmetry Very sensitive to New Physics models Historical: Predicted 3 rd generation !  Baryogenesis: there is more matter !  N(antibaryon) << N(baryon) << N(photons) Fortunately! 1:10 9  Sakharov (1968) Conditions for matter dominated universe Baryon number violation CP violation Not in thermal equilibrium Assuming not initial conditions, but dynamic. Cannot allow all inverse reactions to have happened CP Violation key dates  1964 CP Violation discovery in Kaons  1973 KM predict 3 or more families  …..  …..erm…not…much…  ….  1999 Direct CP Violation NA48/KTeV  2001 BaBar/Belle CP Violation in B mesons  200? LHCb physics beyond the SM? Are we just the left over matter after CP violating matter/ anti-matter annihilation processes?