Theory of D0 - D0 mixing
Alexey A Petrov
Wayne State University
Table of Contents:
• Introduction
• Experimental constraints
• Theoretical expectations
• Conclusions and outlook
Based on works with S. Bergmann,
A.F. Falk, Y. Grossman, Z. Ligeti, Y. Nir
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Introduction: why do we care?
B-mixing:
DQ=2: only at one loop in the Standard Model
rate  m12  m22
GIM mechanism:
sensitive to ultra-heavy particles in the loop
Expectation: rate is “large” in B system
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Introduction: why do we care?
B-mixing:
DQ=2: only at one loop in the Standard Model
rate  m12  m22
GIM mechanism:
sensitive to ultra-heavy particles in the loop
 a(t ) 
0
0
B(t )  
 b(t ) 
  a(t ) B  b(t ) B


Time-dependence: coupled Schrödinger equations
A

i 

i
B(t )   M    B(t )   2
t
2 

q
Diagonalize: mass eigenstates
p2 
 B(t )
A
 flavor eigenstates
B1, 2  p B 0  q B 0
x
Alexey Petrov
(Wayne State Univ.)
M 2  M1
  1
, y 2

2
FPCP 2002 May 16-18
Introduction: why do we care?
D-mixing:
DQ=2: only at one loop in the Standard Model
rate  m12  m22
GIM mechanism:
sensitive to ultra-heavy particles in the loop
 a(t ) 
0
0
D(t )  
 b(t ) 
  a(t ) D  b(t ) D


Time-dependence: coupled Schrodinger equations
A

i 

i
D(t )   M    D(t )   2
t
2 

q
Diagonalize: mass eigenstates
p2 
 D(t )
A
 flavor eigenstates
D1, 2  p D 0  q D 0
x
Alexey Petrov
(Wayne State Univ.)
M 2  M1
  1
, y 2

2
FPCP 2002 May 16-18
Introduction: why do we care?
B-mixing:
DQ=2: only at one loop in the Standard Model
rate  m12  m22
GIM mechanism:
sensitive to ultra-heavy particles in the loop
D-mixing:
the only probe of down-type quark dynamics
GIM mechanism: no ultra-heavy quarks in the loop
b-quark contribution  Vub mb2 can be neglected
rate  f (ms )  f (md )  0 (SU(3)F limit)
very sensitive to long-distance QCD, as mc ~ 1 GeV
Clean probe of New Physics?
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
How would new physics affect mixing?
Local operator  possible
New Physics!
D-D mass matrix:
i 
1

(0)
Di0 H WDC  2 D 0j
 M     mD  ij 
2  ij
2 mD

1

2 mD
Real intermediate states, affect
both x and y  Standard Model

I
Di0 H WDC 1 I
I H WDC 1 D 0j
mD2  mI2  i
1. x  y : signal for New Physics?
x  y : Standard Model?
2. CP violation in mixing/decay
With b-quark contribution neglected:
only 2 generations contribute
 real 2x2 Cabibbo matrix
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Experimental constraints
1. Time-dependent D 0 (t )  K   analysis


 D 0 (t )  K    e t AK  
2

Rm2 2
2
  R  R Rm  y ' cos   x' sin   t 
y  x 2 t  
4



R
AK  
2
q
Rm e 
p
i
A K  

if D 0  D 0 :
Rm  Rm1
x'   x'
2. Lifetime difference analysis
yCP
Am
 D    K  


1

y
cos


x
sin

 D  K  K  
2
3. Semileptonic analysis
Picture courtesy of S. McGee
Alexey Petrov
(Wayne State Univ.)
rate  x 2  y 2
Quadratic in x,y: not so sensitive
FPCP 2002 May 16-18
Experimental constraints
Several groups have measured yCP
Experiment
FOCUS (2000)
Value
(3.42±1.39±0.74)%
E791(2001)
(0.8±2.9±1.0)%
CLEO (2002)
(-1.2±2.5±1.4)%
Belle (2002)
(-0.5±1.0
 0 .7
 0 .8 )%
BaBar (2002)
(1.4±1.0
 0 .6
)%
 0 .7
World average: (1.0±0.7)%
G. Raz
What are the expectations for x and y?
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Theoretical estimates
•
•
•
•
Alexey Petrov
(Wayne State Univ.)
Theoretical predictions are all over
the board
Can y ~ 1% be convincingly
accommodated?
Is it possible to have y >> x?
Does it still mean that y ~ x?
FPCP 2002 May 16-18
Theoretical estimates I
A. Short distance gives a tiny contribution, consider y as an example mc is quite large !!!
y
1
D0 Τ D0
mD 
… as can be seen form the straightforward computation…
y sd

NC  1
ms2  md2

XD
2 NC 
mc2

2
22 N C  1 BD'
M D2 C22

1  NC
BD mc  mu 2
with
D
0

u  c u c D
0

ms2  md2
2
2
C

2
C
C

C
NC
2
1
2
1
2
mc
2


C
1
1   N C 2  2 C1


C2
C2


1  N C 4 FD2 mD2

BD , etc.
NC
2 mD
 2  NC 



 2 NC 1 
4 unknown matrix elements
similar for x (trust me!)
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Theoretical estimates I
A. Short distance + “subleading corrections” (in 1/mc expansion):
y
(6)
sd
(6)
xsd
m

 md2
mc2
2
s
m

2
s
 md2
mc2

2

ms2  md2  2
 had  ms6 6
2
mc
2
4 unknown matrix elements
2
 had
 ms4  4
…subleading effects?
(9)
ysd
 ms3 3
15 unknown matrix elements
(9)
xsd
 ms3 3
(12)
y sd
  0  2S   ms2 2
x
(12)
sd

 S   m 
2
s
2
Leading contribution!!!
Alexey Petrov
(Wayne State Univ.)
Georgi, …
Bigi, Uraltsev
Twenty-something unknown
matrix elements
Guestimate:
x ~ y ~ 10-3 ?
FPCP 2002 May 16-18
Resume: model-independent computation
with model-dependent result
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Theoretical estimates II
B. Long distance might give a large result? Let’s see…
y
1
 n
2 n

mc is NOT large !!!
D 0 HWDC 1 n n HWDC 1 D 0  D 0 HWDC 1 n n HWDC 1 D 0

… with n being all states to which D0 and D0 can decay. Consider , K, KK
intermediate states as an example…
y2  Br D 0  K  K    Br D 0     
 2 cos 
cancellation
expected!
Br D 0  K    Br D 0    K  
If every Br is known up to O(1%)
the result is expected to be O(1%)!
The result here is a series of large numbers with alternating signs, SU(3) forces 0
x = ? Extremely hard…
Alexey Petrov
(Wayne State Univ.)
need to restructure the calculation…
FPCP 2002 May 16-18
Resume: model-dependent computation
with model-dependent result
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Questions:
1. Can any model-independent statements be
made for x or y ?
What is the order of SU(3) breaking?
n
i.e. if x, y  ms what is n?
2. Can one claim that y ~ 1% is natural?
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Theoretical expectations
At which order in SU(3)F breaking does the effect occur? Group theory?
D 0 HW HW D 0  0 D HW HW D 0
is a singlet with D  Di that belongs to 3 of SU(3)F (one light quark)
The DC=1 part of HW is
q cq q , i.e. 3  3  3  15  6  3  3  H
i
j
k
ij
k
O15  ( sd )(ud )  (uc )( sd )  s1 ( d c )(ud )  s1 (uc )( d d )
 s1 ( sc )(us )  s1 (uc )( ss )  s12 ( d c )(us )  s12 (uc )( d s )
O6  ( sd )(ud )  (uc )( sd )  s1 ( d c )(ud )  s1 (uc )( d d )
 s1 ( sc )(us )  s1 (uc )( ss )  s12 ( d c )(us )  s12 (uc )( d s )
i
Introduce SU(3) breaking via the quark mass operator M j  diag (mu , md , ms )
All nonzero matrix elements built of Di , H kij , M ij must be SU(3) singlets
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Theoretical expectations
note that DiDj is symmetric

belongs to 6 of SU(3)F
D 0 HW HW D 0  0 D HW HW D 0
Explicitly,
DD  D6
H W H W  O60  O42  O15'
1. No 6 in the decomposition of HW HW  no SU(3) singlet can be formed
D mixing is prohibited by SU(3) symmetry
2. Consider a single insertion of M ij  D6 M transforms as 6  8  24  15  6  3 
still no SU(3) singlet can be formed
NO D mixing at first order in SU(3) breaking
3. Consider double insertion of M  DDM : 6  (8  8) S  (60  42  24  15  15  6)
 (24  15  6  3)  6
D mixing occurs only at the second order in SU(3) breaking
Alexey Petrov
(Wayne State Univ.)
A.F., Y.G., Z.L., and A.A.P.
FPCP 2002 May 16-18
Theoretical expectations
• Does it always work? SU(3) breaking must enter perturbatively…
Ai  ASU ( 3)   i
• Known counter-examples:
1. Very narrow light quark resonance with mR~mD
2
2
g DR
g DR
x, y ~ 2
~ 2
2
mD  mR mD  m02  2m0 R
Most probably don’t exists…
see E.Golowich and A.A.P.
2. Part of the multiplet is kinematically forbidden
0
0
Example: both D  4 and D  4 K are from the same multiplet, but the
latter is kinematically forbidden
see A.F., Y.G., Z.L., and A.A.P.
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Theoretical expectations
• Two major sources of SU(3) breaking
1.
phase space
mK  m  m ...
2a. matrix elements (absolute value)
f K  f ...
2b. matrix elements (phases aka FSI)
Im

A D 0  K  
0
0
A( D  K   )
Take into account only the first source (computable)!
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
SU(3) and phase space
• “Repackage” the analysis: look at the complete multiplet contribution


y   yF , R Br D 0  FR ~  yF , R
FR
FR

1
0

D
n

 nFR
y for each SU(3) multiplet

Each is 0 in SU(3)
• Does it help? If only phase space is taken into account: no (mild) model dependence
yF ,R 

D 0 HW n  n n HW D 0

D 0 HW n  n n HW D 0

D 0 HW n  n n HW D 0
nFR
nFR
if CP is conserved

nFR
 ( D
nFR
Alexey Petrov
(Wayne State Univ.)
0
 n)
Can consistently compute !
FPCP 2002 May 16-18
Results
• Product is naturally O(1%)
• No (symmetry-enforced) cancellations
• Does NOT occur for x
naturally implies that y~1% and x < y !
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18
Conclusions
• x,y=0 in the SU(3) limit (as Vub is very small)
• it is a second order effect
• it is quite possible that y ~ 1% with x<y
if true, search for New Physics
is complicated
• expect new data from BaBar/Belle/CLEO/CLEOc/CDF(?)
• currently: x  (2.8  2.5)%, y  (0.9  3.6)% (allowing NP)
• CP-violation in mixing is a “smoking gun” signal for New Physics
Alexey Petrov
(Wayne State Univ.)
FPCP 2002 May 16-18