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:The mirror did not seem to be operating properly:
A guide to CP violation
Chris Parkes
12/01/2006
:Section 1:
Symmetries
Symmetries• 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 InversionP operator acts on a state |(r, t)> as
),(),(
),(),(
2 ttP
ttP P
rr
rr
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)Ylm(,)
Ylm(,)= Yl
m(-,+)
=(-1)l Ylm(,)
So atomic s,d +ve, p,f –ve P
Hence, Electric dipole transition l=1P=- 1
Parity cont.• Conserved in strong & emag. Interactions• Parity multiplicative |> = a b, P=PaPb
• Proton– Convention Pp=+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,
Jp = 0+ , 0-, 1-, 1+ -,o,K-,Ko 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)
Co60Nuclei
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),(),(
),(),(
2 ttC
ttC C
rr
rr
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 I3 ->-I3, e.g. + -> -
Neutrino helicity
left-handed
right-handed
Parity ->
left-handed
right-handed
Charge & Parity ->
•Massless approximation
•Goldhaber et al. Phys Rev 109 1015 (1958)
Time
Let us have a quick look at nature....
Neutral kaon system sdK0 sdK0 flavour eigenstates CP conjugated
110SL
8
LL
10
SS
s100014.05301.0=
s1004.017.51
s100008.08934.01
mmm
mass eigenstatesKS
KL
Three pion decay, very little phase
space
CPLEAR T invariance testInitial state at t = 0
KK
KKpp
0
0
S = 0 S = 0)su(K
)su(K
Rate difference Ko Ko Ko Ko is T violation
Experiment at LEAR ring
at CERN 1990-1996
Discovery of T violation• direct observation of T violation
– Detailed balance expts difficult due to strong/em. effects
3106.16.6 TA
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)
ii
ie rd
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
m
PHHP
PCPTHHCPTP
PCPTHHCPTP
PCPTCPTHHCPTCPTP
PHHPm
ems
ems
ems
ems
ems
))(()(
))(()(
)())()(()(
†
†
††
CPT Consequences (2)• Particle/anti-particle width equality
:Section 2:
Introducing CP in SM
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 KLe+ve-
• 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 3rd generation !
• Baryogenesis: there is more matter !• N(antibaryon) << N(baryon) << N(photons)
– Fortunately! 1 : 109
• 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 Vij Ui () Dj W
• Vij 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 =
uct
D =
dsb
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
e.g.
W-
c d
Coupling Vcd
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’
dsb
= VCKM
ud’
c s’
t b’
Values of elements a
purely experimental
matter
Number of Parameters in CKM
• n x n complex matrix, – 2n2 parameters
• Unitarity n2 constraints– n2 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
ikjkj
ijVV *
n=2, 1 real, 0 phasen=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 ei(wt+) Tei(-wt+)
– i.e. Violates T/CP
• Hence predict three (or more) famillies!
• Now parameterize 3x3 CKM in 4 parameters
PDG, 3 angles + phase
C12 S12 0
-S12 C12 0
0 0 1
1 0 00 C23 S23
0 -S23 C23
3 angles 12, 23, 13 phase
Cij= cos ij
Sij=sin ij
VCKM = R23 x R13 x R12
R12 = R23 =
R13 =
C13 0 S13 e-i
0 1 0
-S13 e-i 0 C13
Wolfenstein’s parameters
1ˆˆ12
1
21
423
22
52
32
iAAiA
AiA
iA
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
21ˆ ,
21ˆ
22
VCKM
= S12, A=S23/S212, =S13cos/ S13S23, =S13sin/ S12S23
VCKM(3) terms in up to 3
CKM terms in 4,5
Unitarity conditions
ikjkj
ijVV *
hence 6 triangles in complex plane
123
1
i
ijV j=1,3
No phase info.
3
1
* 0i
ikijVV j,k =1,3 jk
0
0
0
0
0
0
***
***
***
***
***
***
cbubcsuscdud
tbcbtscstdcd
tbubtsustdud
tstdcscdusud
tbtdcbcsubus
tbtdcbcdubud
VVVVVV
VVVVVV
VVVVVV
VVVVVV
VVVVVV
VVVVVVdb:
sb:
ds:
ut:
ct:
uc:
More triangles•Area of all the triangles is the same (6A2)
•Two triangles (db) and (ut) have sides of similar size
•Easier to measure, (db) is often called the unitarity triangle
*ubudVV
*tbtdVV
*cbcdVV
Bottom side A3 normalised to 1
(,)
*ubtbVV
*udtdVV
*ustsVV
’=, =-’, =-’ = -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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