theory of d 0 - d 0 mixing

Alexey Petrov (Wayne State Univ.)

FPCP 2002 May 16-18

Theory of D0 - D0 mixing

Alexey A PetrovWayne 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


FPCP 2002 May 16-18

Introduction: why do we care?

B-mixing: Q=2: only at one loop in the Standard Model GIM mechanism: sensitive to ultra-heavy particles in the loop

22

21 mmrate

Expectation: rate is “large” in B system


FPCP 2002 May 16-18



22

21 mmrate

00 )()()()(

)( BtbBtatbta

tB

)()(2

)(2

2

tBAqpA

tBiMtBt

i

Time-dependence: coupled Schrödinger equations

002,1 BqBpB

Diagonalize: mass eigenstates flavor eigenstates

2

, 1212 yMMx


FPCP 2002 May 16-18


D-mixing: Q=2: only at one loop in the Standard Model GIM mechanism: sensitive to ultra-heavy particles in the loop

22

21 mmrate

00 )()()()(

)( DtbDtatbta

tD

)()(2

)(2

2

tDAqpA

tDiMtDt

i

Time-dependence: coupled Schrodinger equations

002,1 DqDpD

Diagonalize: mass eigenstates flavor eigenstates

2

, 1212 yMMx


FPCP 2002 May 16-18



22

21 mmrate

D-mixing: the only probe of down-type quark dynamics GIM mechanism: no ultra-heavy quarks in the loop b-quark contribution can be neglected

(SU(3)F limit)

very sensitive to long-distance QCD, as

Clean probe of New Physics?

0)()( ds mfmfrate

2bub mV

GeVmc 1~


FPCP 2002 May 16-18

How would new physics affect mixing?

D-D mass matrix:

I ID

jC

WC

Wi

D

jC

WiD

ijDij

imm

DHIIHD

m

DHDm

miM

22

0110

020)0(

21

21

2

Real intermediate states, affect both x and y Standard Model

Local operator possible New Physics!

1. : signal for New Physics? : Standard Model?

2. CP violation in mixing/decay

yx yx

With b-quark contribution neglected: only 2 generations contribute real 2x2 Cabibbo matrix


FPCP 2002 May 16-18

Experimental constraints

2222

20

4sin'cos'

)(

txyRtxyRRR

AeKtD

mm

Kt

2

sincos1 mCP

AxyKKDKDy

1. Time-dependent analysis KtD )(0

pqeR i

m 2

K

K

AA

R

2. Lifetime difference analysis

3. Semileptonic analysis22 yxrate

''

:1

00

xxRR

DDif

mm

Picture courtesy of S. McGeeQuadratic in x,y: not so sensitive


FPCP 2002 May 16-18

Experimental constraintsSeveral groups have measured yCP

Experiment Value

FOCUS (2000) (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 )%

BaBar (2002) (1.4±1.0 )%

7.08.0

6.07.0

World average: (1.0±0.7)%G. Raz

What are the expectations for x and y?


FPCP 2002 May 16-18

Theoretical estimates

• 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 IA. Short distance gives a tiny contribution, consider y as an example

12221

1122

22

1

2

122

21

2

22

2'

2121

222

22

2

222

C

CC

uc

D

D

D

C

C

Cc

ds

c

dsD

C

Csd

NN

CC

CCN

mmCM

BB

NN

NCCCCm

mmm

mmXN

Ny

001 DΤDm

yD

… as can be seen form the straightforward computation…

with .,2

41 2200 etcB

mmF

NNDcucuD D

D

DD

C

C

4 unknown matrix elements

similar for x (trust me!)

mc is quite large !!!


FPCP 2002 May 16-18

Theoretical estimates IA. Short distance + “subleading corrections” (in 1/mc expansion):

4422

222)6(

6622

22

2

222)6(

shadc

dssd

shadc

ds

c

dssd

mm

mmx

mm

mmm

mmy

…subleading effects?

4 unknown matrix elements

33)9(

33)9(

ssd

ssd

mx

my15 unknown matrix elements

22)12(

2220

)12(

sSsd

ssd

mx

myS

Twenty-something unknown matrix elements

Guestimate: x ~ y ~ 10-3 ?Leading contribution!!!

Georgi, …Bigi, Uraltsev


FPCP 2002 May 16-18

Resume: model-independent computation with model-dependent result


FPCP 2002 May 16-18

Theoretical estimates IIB. Long distance might give a large result? Let’s see…

KDBrKDBr

DBrKKDBry

00

002

cos2

If every Br is known up to O(1%) the result is expected to be O(1%)!

mc is NOT large !!!

… with n being all states to which D0 and D0 can decay. Consider K, KK intermediate states as an example…

01100110

21 DHnnHDDHnnHDy C

WC

WC

WC

Wn

n

cancellation expected! The result here is a series of large numbers with alternating signs, SU(3) forces 0

need to restructure the calculation… x = ? Extremely hard…


FPCP 2002 May 16-18

Resume: model-dependent computation with model-dependent result


FPCP 2002 May 16-18

Questions:

1. Can any model-independent statements be made for x or y ?

2. Can one claim that y ~ 1% is natural?

What is the order of SU(3) breaking? i.e. if what is n? n

smyx ,


FPCP 2002 May 16-18

Theoretical expectationsAt which order in SU(3)F breaking does the effect occur? Group theory?

))(())(())(())((

))(())(())(())((

))(())(())(())((

))(())(())(())((

21

2111

116

21

2111

1115

sdcussucdssscussucss

ddcusducdsdscududsO

sdcussucdssscussucss

ddcusducdsdscududsO

0000 DHHDDHHD WWWW

is a singlet with that belongs to 3 of SU(3)F (one light quark)

Introduce SU(3) breaking via the quark mass operator

ijkkji Heiqqcq 33615333..,

All nonzero matrix elements built of must be SU(3) singlets

iDD

The C=1 part of HW is

),,( sduij mmmdiagM

ij

ijki MHD ,,


FPCP 2002 May 16-18

Theoretical expectations

0000 DHHDDHHD WWWW

note that DiDj is symmetric belongs to 6 of SU(3)F

36152486

D mixing is prohibited by SU(3) symmetry

Explicitly,

WW HH6

'154260

6

OOOHHDDD

WW

1. No in the decomposition of no SU(3) singlet can be formed

2. Consider a single insertion of transforms as still no SU(3) singlet can be formed

MDM ij 6

NO D mixing at first order in SU(3) breaking

3. Consider double insertion of 6)361524(

)61515244260()88(6:

SDDMM

D mixing occurs only at the second order in SU(3) breaking

A.F., Y.G., Z.L., and A.A.P.


FPCP 2002 May 16-18


iSUi AA )3(

KDandD 44 00

Most probably don’t exists…

• Does it always work? SU(3) breaking must enter perturbatively…

• Known counter-examples:

1. Very narrow light quark resonance with mR~mD

RD

DR

RD

DR

mmmg

mmgyx

020

2

2

22

2

2~~,

2. Part of the multiplet is kinematically forbidden

see E.Golowich and A.A.P.

Example: both are from the same multiplet, but the latter is kinematically forbidden

see A.F., Y.G., Z.L., and A.A.P.


FPCP 2002 May 16-18


... mmmK

• Two major sources of SU(3) breaking

1. phase space

2a. matrix elements (absolute value)

2b. matrix elements (phases aka FSI)

...ffK

0)(

Im 0

0

KDA

KDA

Take into account only the first source (computable)!


FPCP 2002 May 16-18

SU(3) and phase space

RRR FnF

RFRF

RF nDyFDBryy 0,

0,

1~

• “Repackage” the analysis: look at the complete multiplet contribution

• Does it help? If only phase space is taken into account: no (mild) model dependence

Each is 0 in SU(3)y for each SU(3) multiplet

R

R

R

R

Fn

FnWnW

FnWnW

FnWnW

RF

nD

DHnnHD

DHnnHD

DHnnHDy

)( 0

00

00

00

,

if CP is conserved

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 !


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

• expect new data from BaBar/Belle/CLEO/CLEOc/CDF(?) • currently: (allowing NP)• CP-violation in mixing is a “smoking gun” signal for New Physics

if true, search for New Physics is complicated

)%6.39.0()%,5.28.2( yx

theory of d 0 - d 0 mixing

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