Oscillations in sector of
neutral mesons
Oscillation phenomena occur in the system of neutral
K, D and B mesons:
K 0  K 0 , D0  D 0 , B 0  B 0
Strangeness, Charm and Beauty are not conserved
 S  2 , C  2 ,  B  2
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Oscillations in B-meson sector
B-meson oscillations: Bd0   db  : Bd0  Bd0 and Bs0   sb  : Bs0  Bs0
Vqd
Vqb*
SM:
no QCD corr.
Vqb*
Vqd
0
0
Essence of oscillation: B , B are produced in strong interactions,
but they are not eigenstates of full Hamiltonian (mass operator).
Convention for CP transformation: CP B0   B0
CP eigenstates:

1
B1 
B0  B0
2
CP B1  B1
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

1
B2 
B0  B0
2
CP B2   B2
S. Tokar, Neutral meson oscillations

2
Evolution of B0 state
B0  B0 time evolution:
B0 ( t )  B0
B (t )  B
0
0
produced at t=0
0

B
(t )
d
i  0
dt  B ( t )
0
 

B
(t )

   M  i  0
 
 B (t )
2







Hermiticity of mass and decay matrix: M=M and =
*
Off-diagonal elements: M12  M 21
and 12  *21
correspond to
B0 mixing  M12 and 12 stem from box diagram.
12- from the real final states – decays of both B0 and B0
Main contribution: CKM favored tree u- and c-decay  not
sensitive to new physics
M12- induced by short-distance physics – top quark gives main
contribution  new physics affects the2 mixing phase  M  arg M12
*
SM prediction (for BS):  M  arg VtbVts 
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Mass eigenstates
Eigenstates are linear combinations of B0 and anti-B0:
BL  p B0  q B 0 ,
BH  p B0  q B 0 ,
Egenvalues and eigenstates are obtained from:
det H   I  0 and
Masses
 p
H


I

   0
 q 
2
2
CPT: H11=H22
M 12  i 12 / 2 
 M  i / 2
H  *
*
M  i  / 2 
 M 12  i 12 / 2
 M12  i12 / 2   M12*  i12* / 2 
M H , L  Re     M  Re  Q  ,  H , L  2Im      2Im  Q 
M  M H  M L  2 Re  Q  ,
   H   L  4Im  Q 
  M  i / 2  Q ,
Q
Eigenvectors
*
*
M12
 i12
/2
q
Q
 

p M12  i 12 / 2 M12  i12 / 2
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p  q 1
12

BH , L
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CP-eigenstates on =1
1

B0   B0
12


4
Evolution od B-system
Evolution of physical states:
BH , L ( t )  BH , L e
  iM H ,L  H ,L 2 t
1
B
 B0
0
1
Using the relation between physical and strong interaction eigenstates 
2
B (t )  f (t ) B   f  (t ) B ,
0
where
f (t )  e
0
0
B (t ) 
0

 iM H t  H t 
1   iM Lt  2L t
2
f (t )   e
e

2


 iMt  t
2
  
cos 
t ,
 2 
f  (t )  e

 iMt  t
2
f (t )

B0  f (t ) B0
can be transformed into:
  
sin 
t  ,   M  i  2
 2 
For B-mesons =H-L  0 and = M = MH-ML
For B-oscillation we need: M, , =2Q ( Q 
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 M12  i12 / 2  M12*  i12* / 2
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)
5

Box diagram implications
M12 and 12 values: from box diagram
12 : box diagram cut  decays of B0 and B0 to common u-,c-final states

GF2
12 
M Bq ( Bˆ Bq f B2q )mb2 VtbVtq*
8
0
B  2
0
M12 -effective Hamiltonian : M12  Bq H eff Bq
H
B  2
eff

2
long distance QCD corr
GF2

MW2 1c2 S0  xc   2t2 S0  xt   23c t S0  xc , xt   Cl QCD  O B  2 (  )  h.c.
2
16
Short distance QCD corr
Vib
Vid
Box diagram w-o ext. legs
i  VibViq* , xi  mi2 MW2 , S0 ( x )  x




x 0
O B  2  b   1   5 q  b   1   5 q

id
V
Vib
OB=2
1
1
 2
2
2
MW  q
MW

GF2
M 12  
 M Bq ( Bˆ Bq f B2q ) MW2 S0 ( xt ) VtbVtq*
2
12
q  d , s for Bd0 , Bs0
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
2
Box diagram implications
Comparing M12 and 12 using leading terms, ignoring QCD corrections:
 12
mb2
3
1

M12
2 S0  mt2 MW2  MW2
Ratio 12/M12 for leading terms is a real
(*)
quantity though 12,M12 are complex.
 SM expected: CP violation in B0-mixing is very small:
 2
mt2 1 
2  3
M 12 ~  t  mt  M B  ln 2  
 4 MB 3 

2

mt2
2
  2t c mc ln 2
MB

 12

  12 e i at SM sin   0.01
M 12
M 12
 From *  |12|<<|M12| as
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,  12 ~ t2 mb2  t c mc2 
3
For SM no CP-violation (=0) is a
good approximation
S0  mt MW  ~ mt MW 
2
2
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2
2
 12
M 12
3 mb2

2 mt2
7
B-mixing frequency: theo vs exp
Using 12  M12

GF2
M=2|M12|: M  2  M Bq  Bˆ Bq f Bq  MW2 S0 ( xt ) VtbVtq*


6
Mixing frequency for Bd and Bs mesons
2
, q  d, s
used approximation:
M Bd
 f B BB   m

d
d
t


 0.50 ps


 200 MeV   170 GeV 
M Bs
2
1.52
 f B BB   m


Vts


s
t
 
 15.1 ps 1  s
  0.040  0.55
240
MeV
170
GeV

 
 

1
1.52

2
2
 Vtd  


0.0088

 0.55
 mt

S0 ( xt )  2.4 

170
GeV


1.52
2
Lattice QCD: f Bd BBd  244  11  24 MeV and f Bs BBs
f Bd BBd  1.21  0.040.04
0.01
Perturb. QCD:  = 0.55 ± 0.01 Tevatron: mt = 172.8 ± 1.8 GeV/c2
3
3
CKM unitarity: Vtd   8.140.32
and Vts   41.610.12
0.64   10
0.78   10
measurement
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M Bd  0.507  0.004 ps 1
M Bs  17.77  0.10( stat )  0.07( syst ) ps 1
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Mixing parameters
Oscillation case characteristics: variables x and y
x

M
1

 decay ,  M  2 M12

1  M  mix
y
 2  12



Taking imaginary part from Q2 where
Q  12   M  i  2  
We get:
 M 
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 Re  M12  12*   M12  12 
M
*
*

i

M

i



12
12
12
12 
  2  12
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Experimental view on mixing
• Make a Bs meson and look for its flavor at decay
• Identify the flavor at the production(tagging)
• Look at oscillation probability vs proper decay
time
ct  Lxy
m BS
pT
Fourier component
of asymmetry:
Mixing amplitude
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Present CDF results
CDF’s previous result:
• p-prob=0.2
• Assuming that it is signal:
31 ( stat )  0.07 ( syst )
 m S  17.31 00..18
Bs oscillation observed with a
significance of 5.4 (p=810-8):
 m S  17.77  0.10( stat )  0.07 ( syst ) ps-1
Vts
Vtd
 0.2060  0.0007 (exp) 0.0081 ( theo )
0.0060
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Charm Mixing: D0 -D0
Or mixing:
D*++D0+K+Doubly Cabibbo Supressed
“wrong sign” (WS)
u
W+
D0
s
c
d
u
u
K
-
D*++D0+K-+
Cabibbo favored
“right sign” (RS)
W+
+
“tagging” 
d
c
D0
s
u
u
m  m K  m K  m
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u
S. Tokar, Neutral meson oscillations
K
R( t ) 
P  D0  K   ; t 
P  D0  K   ; t 
12
Charm Mixing: rate of DCS/CF
• Comparison of DCS (D0K ) to CF (D0K+-) decay rate as a
function of time : R(t)
• Assuming CP conservation and small values of x=M/ and y=/:
x  2  y 2 2
R( t )  Rd  y Rd t 
t
4
2
A( D0  K  )
A( D0  K   )
charm mixing var. x’, y’ related to m and  of DL0, DH0 mass
eigenstates:
Rd 


x  x cos   y sin  and y   x sin   y cos 
Strong interaction phase difference between the DCS and CF amplitude
No mixing: (x’,y’) =(0,0)  R(t)=Rd
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Charm Mixing: results
 Performed a binned fit to ratio of
WS/RS D0 decay as function of
time: R(t)
 Probability of no mixing (x’,y’)=(0,0)
is 0.013%
 Equivalent to 3.8 significance
Rd = 3.04±0.05 x 10-3
y’ = 8.5±7.6 x 10-3
x’2 = -0.12±0.35 x 10-3
Evidence for charm mixing!
(significance competitive to Belle & BaBar)
Decay time in D0 lifetimes
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Conclusions
 Neutral mesons oscillation phenomena are a big challenge
for experimental particle physics.
 It is an important window for a new physics.
 Real chance to see D0-oscillation at Tevatron
Comparison of mixing cases
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Backup slides
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Box diagram implications – decay width
Decay width 
• experiment: from the lifetime measurement
• theory: main contribution from spectator diagram
(b  qW ) ~ mb5 Vqb
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2
Backup slides
box diagrams with QCD corrections
Box diagram expressions
without QCD corrections
c.f. Sachrajda ‘98
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