: section 3: mixing and cp violation in (mostly) neutral mesons
TRANSCRIPT
:Section 3:
Mixing and CP violation in (mostly) neutral mesons
State evolution in neutral mesons• Neutral meson states Po, Po
– Could be Ko, Do, Bo
• With internal quantum number F– Such that F=0 emag/strong, F0 for Hweak
• Obeys time dependent Schrödinger eqn
M, Hermitian 2x2 matrices, mass matrix and decay matrix,
H11=H22 from CPT (mass/decay particle = anti-particle)
oo PtbPtat )()()(
b
ai
b
aH
b
a
dt
di W )
2( ΓM
• Solve Schrödinger for Eigenstates of Hw
• From characteristic equation
• If E1=M1-i1/2, E2= M2-i2/2
and M=M2-M1, = 2- 1
oo
oo
PqPpP
PqPpP
2
1 Complex coeff. p,q obey |q|2+|p|2=1
0 EIHW)
2)(
2()
2(, *
1212*
12122
iM
iME
iMhence
)2
(2
)2
(
)2
(2
)2
(
2
1
immE
immE
• Eigenvector Eqn
• Eigenstates have time evolution:
• then
0)(
q
pEIHW
12*
12*
1212
2
2
iM
iM
q
p
ti
Mi
ti
Mi
ePtP
ePtP
)2
(
22
)2
(
11
22
11
)(
)(
000
000
)()()(
)()()(
Ptfq
pPtftP
Ptfp
qPtftP
with
ti
Miti
Mieetf
)2
()2
( 2211
2
1)(
Finally probabilities…
• With
• The oscillations depend upon the parameter x the speed of oscillations c.f. lifetime
22
22
P(
P(
(t)fp
q(t)PP;t)PP
(t)f(t)PP;t)PP
oooo
oooo
)cos(24
1)( )(2
21 mteetf tt Where = (1+ 2)/2
Interference term
Mx
Mixing probabilities
x in Bs
not yet measure
d
>19
Probability
of finding
Po from initial pure Po beam
• CP in decay
• CP in mixing
• CP in interference between mixing and decay
Types of CP violation
Pff P
ff P
Pff P
P P P
ff PP P P+ +
1) CP Violation in mixing Indirect CP Violation
• Mass eigenstates being different from CP eigenstates• Mixing rate PoPo is different to PoPo
• If conserved
oo
oo
PqPpP
PqPpP
2
1
2
1qp CP|P1> = +1 |P1>
CP|P2> = -1 |P2> with
1212
*1212
*2
2
2
iM
iM
p
q
•If violated Such asymmetries usually small
Need to calculate M,, involve hadronic uncertainties
Hence, tricky to relate to CKM parameters
2) CP Violation in decaydirect CP Violation
• Two types of phase– Weak phase: due to weak interactions (complex CKM elements)– Strong phase: contribution from intermediate states, CP conserving (same sign in both)
fP
fP
PHfA
PHfA
f
ff is final state common to both decays
1
i
iii
i
iii
f
f
ii
ii
eeA
eeA
A
A
occurs for both charged and neutral states P
3)CP violation in the interference of mixing and
decay• Choose state* f, Pof, Pof• Two possible decay chains, with or w/o mixing
• CP can be conserved in mixing and in decay and still be violated !
*Not necessary to be CP eigenstate
1p
q 1f
f
A
Af
f
A
A
p
q
•Interference term depends on
Can put and get 1 but 1Im
:Section 4:
Neutral Kaon system
Ko Ko system• Kaon mesons in two isospin doublets
K+ = us
Ko = ds
K- = us
Ko = ds
S=+1
S=-1
Part of pseudo-scalar JP=0- meson octet with ,
I3=+1/2
I3=-1/2
• Kaon production
Ko : - + p o + Ko
But from baryon number conservation:
Ko : + + p K+ + Ko + p
Or
Ko : - + p o + Ko + n +n
Requires higher energy
Much higher
S 0 0 -1 +1
S 0 0 +1 -1 0
S 0 0 +1 -1 0 0
Kaon oscillations
• So say at t=0, pure Ko, – later a superposition of states
d
su, c, t W
W_
s
d_u, c, t
ds
u, c, tW W
_ sd_
u, c, t___
K0K0
Ko Decay
• In that case
00
2
001
2
12
1
KKKK
KKKK
L
s
CP=+1
CP=-1
2
1qp
Ksoo
Ks+-
KL+-o
KLooo
CP=+1
CP=-1
95.0
s100014.05301.0=
s1004.017.51
s100008.08934.01
110SL
8
LL
10
SS
Mx
mmm
KS branching fractions: 69%, 31%KL branching fractions: 21%, 13%, 66%
mass eigenstatesKS
KL
show
Assume CP
Time dependent probabilities for the neutral kaon case.
t (1/)
Ko Regeneration
• Start with Pure Ko beam– After time all Ks component decayed
• Introduce slab of material in beam– reactions
pKnK
nKpK
0
0
1) Elastic scatttering
2) Charge exchange
3) Hyperon production
00 pK
•Hence Ko absorbed more strongly
sL
emerge
KffKff
KfKfK
2
1
2
12
1 00
i.e. Ks regenerated
Discovery of CP ViolationK1oo
K1+-
K2+-o
K2ooo
CP=+1
CP=-1
So if KL =K1 CP eigenstate, Observe no two pion component
But if broken get:
212
122
||1
1
||1
1
KKK
KKK
s
L
Where quantifies degree of CP violation
BUT Can one find KL decaying into +-?
KL X
KL
p = p + p
= angle between pKL and p
If X = 0, p = pKL: cos = 1If X 0, p pKL: cos 1
cos
KL
m () < mK
J.H. Christenson et al., PRL 13,138 (1964)
Discovery of CP violation
m () = mK
m () > mK
•Mass and angular spectrum
So CP symmetry is violated in the neutral kaon system.
Mass eigenstates (KS and KL) CP eigenstates
Both KS and KL could decay into +--.
•Experimentally well known:The majority of KS decays into - and KL into --.
015.0
0
N
N Small but with profound implications
In KL
Decay final state at time t
- Spin() = 0L- = 0
Initial state at t = 0
KK
KKpp
0
0
S = 0 S = 0)su(K
)su(K
CP(-
i.e. CP eigenstate
K0 at t = 0 decays into
vsK0 at t = 0 decays into
any difference = CP violation
CPLEAR revisited
•Tag Ko/Ko from charged pion/kaon
K0
K0_
CPLEARR+-(t)andR+-(t)_
CP violation
CPLEARCP asymmetry
A(t) = R+-(t) R+-(t)
R+-(t) R+-(t)_
_
large difference!
•CP violation in mixing
•The two mass eigenstates are not CP eigenstates
Kaons: CP violation in DecayCP violation first through existence of certain decay
modes
SW
LW
SW
LW
KH
KH
KH
KH00
00
00,
~2.3x10-3
If CP violation is only in mixing, i.e. independent of decay
00
So, put channel independent term and channel dependent ’ '2,' 00
S
L2
K
K
N
NHence, by measuring only rates:
00S
00L2
00 K
K
N
N
Re612
2
00get
So, two expts in the 80’s did it:
• NA31 (CERN)• E731 (Fermilab)
• Ambiguous result!
So, two expts did it again…….
3
3
1059.074.0'
Re
1065.03.2'
Re
NA48
KTeV
Measure
and at the same time: L
00LS
00S , NNNN
NA31, NA48
KS is regenerated from KL: SL
00S
00L , rNNrNN
E731, KTeV
No normalization is required,
but efficiencies, acceptances etc. have to be corrected…
Normalisation constants
Effort over30 years!
Note; 3 Re = |/ |i.e.Re 0
Not easy to
compare with SM
theory
CKM parameters with CP conserving parameters
• |Vud| : nuclear Beta decay 0.97340.0008
• |Vus| : semileptonic Kaon and hyperon decay (CCFR) 0.21960.0026
• |Vcd| : Neutrino and anti-neutrino production of charm off valence d quarks0.224 0.016
• |Vcs| : W decays (LEP/me!)0.9960.013
• |Vcb| : semileptonic inclusive and exclusive B decays (LEP/CLEO)0.04120.0020
• |Vub| end point spectrum in semileptonic B decays(LEP/CLEO)0.0036 0.0007
• Bo mixing xd, + lattice gauge inputs |Vtb*Vtd| 0.0079 0.0015• Can use Unitarity constraints