Role of symmetries in physics –e.g. translational -> momentum conservation – rotational -> angular momentum conservation –Time -> energy conservation Fundamental Symmetries we will study –Parity (P) – spatial inversion –Charge Conjugation (C) – particle/ anti- particle –CP –CPT Emmy Noether
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
Parity cont. Conserved in strong & emag. Interactions Parity multiplicative | > = a b, P=P a P b Proton –Convention P p =+1 QFT –Parity fermion -> opposite parity anti-fermion –Parity boson -> same parity anti-particle Angular momentum –Use intrisnic parity with GROUND STATES –Also multiply spatial config. Term (-1) l scalar, pseudo-scalar, Vector, axial(pseudo)-vector, J p = 0 +, 0 -, 1 -, 1 + -, o,K -,K o all 0 -, photon 1 -
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
Charge Conjugation C operator acts on a state | (x, t)> as Particle to anti-particle Only a particle that is its own anti-particle can be eigenstate of C, e.g. C | o > = ±1 | o > o + A = J , hence C =-1 Thus, C | o > =(-1) 2 | o > = +1 | o > G, isospin rotation I 3 ->-I 3, e.g. + -> -
Discovery of T violation direct observation of T violation –Detailed balance expts difficult due to strong/em. effects
Electric Dipole Moments Energy shift due to say, neutron, being in weak electric field – e.d.m. (measured in e cm) –TdT -1 = d, but only available direction is J so –d=const.J –TJT -1 = -J, hence d=0 Also for electron, and (less obviously) atomic nuclei –(linear term in E not present) Spin precession fequency of ultracold neutrons in a weak magnetic field. d(n) 6.3x10 -26 ecm, also d(e) 1.6x10 -27 ecm (sussex)
CPT Invariance Particle->anti-particle, reverse time, invert space. CPT | (r,t)> = | (-r,-t)> Lagrangian invariant under CPT –Lorentz invariant –Unique ground state –Spin-statistics (Fermi/Bose)…. No appealing theory of CPT violation exists
CPT Consequences(1) Particle/anti-particle mass equality
CP Violation Introduction: Why is it interesting ? Fundamental: The Martian test –C violation does not distinguish between matter/anti-matter. LH /RH are conventions –CP says preferred decay K L e + v e - Least Understood: CP Violation is ‘add-on’ in SM –Parity violation naturally imbedded from V-A coupling structure –CP requires a complex phase in 3 generation CKM matrix, allowed but not natural
CP: Why ? cont. 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 –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’s 200? LHCb physics beyond the SM?
CP Violation in SM: CKM matrix SM weak charged current –V-A form LH states L V ij U i ( ) D j W V ij is the quark mixing matrix, the CKM matrix for 3 famillies this is a 3x3 matrix U,D are up/down type quark vectors U = uctuct D = dsbdsb V ud V us V ub V cd V cs V cb V td V ts V tb e.g. W-W- c d Coupling V cd
CKM continued Cabibbo (1963) and Kobayashi & Maskawa (1973) Realised mass and flavour eigenstates –need not be the same Weak interaction generations Related to physical quark states by CKM matrix d’ s’ b’ dsbdsb = V CKM u d’ c s’ t b’ Values of elements a purely experimental matter
Number of Parameters in CKM n x n complex matrix, –2n 2 parameters Unitarity n 2 constraints –n 2 parameters Phases of quark fields can be rotated freely –(n-1) 2 parameters Real parameters, rotation (Euler) angles –n(n-1)/2 real Phases –(n-1)(n-2)/2 phases n=2, 1 real, 0 phase n=3, 3 real, 1 phase
K&M Predict 3 famillies (Prog. Theor. Phys. 49, 652(1973) ) Only 3 quarks discovered –Charm predicted by GIM mechanism –CP violation discovered Phase e i(wt+ ) T e i(-wt+ ) –i.e. Violates T/CP Hence predict three (or more) famillies! Now parameterize 3x3 CKM in 4 parameters
PDG, 3 angles + phase C 12 S 12 0 -S 12 C 12 0 0 0 1 10 0 0 C 23 S 23 0 -S 23 C 23 3 angles 12, 23, 13 phase C ij = cos ij S ij =sin ij V CKM = R 23 x R 13 x R 12 R 12 =R 23 = R 13 = C 13 0 S 13 e -i 0 1 0 -S 13 e -i 0 C 13
Wolfenstein’s parameters A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ??? V CKM = S 12, A=S 23 /S 2 12, =S 13 cos / S 13 S 23, =S 13 sin / S 12 S 23 V CKM (3) terms in up to 3 CKM terms in 4, 5
Unitarity conditions hence 6 triangles in complex plane j=1,3 No phase info. j,k =1,3 j k db: sb: ds: ut: ct: uc:
More triangles Area of all the triangles is the same ( 6 A 2 ) Two triangles (db) and (ut) have sides of similar size Easier to measure, (db) is often called the unitarity triangle Bottom side A 3 normalised to 1 (,)(,) ’= , = - ’, = - ’ = -arg(Vts) = arg(Vts)
CP in SM summary Study of CP violation is the analysis of the CKM matrix to verify if it is consistent with the standard model. If not New Physics! Will CP lead to SM ?
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