Note!!!

Note!!!Note!!!

2s bsince both the perturbative and the non-perturbative 1 m corrections a

The FCNC process b s is a th

re known to be small.

Howeve

eoretically very clean mode in S

r, it might be extremely difficul t

M

t

o measure precisely the inclusive mode

because it requires to reconstruct all (together with two neutrinos).

(very small Brs)

s

s

B X

X

( )Experimentally it could be much easier to measure the exclusive modes !!!B K

' ' 'FCNC

2cosZ Z

qp qR pRW

gU d d Z

L

' ' 'FCNC

2cos R

Z ZR R

W

gQ Z

L

quark quark

neutrino neutrino

We can obtain the effective HamiltonianWe can obtain the effective Hamiltonian

2 2 2' '

using Feynman g

auge

Z Z

ig ig

k M M

2 5 5' '

2 2'

' ' 5 52 2 2

'

1 1 1 4cos 2 2

1 1 1

4sin cos

R

R

Z Zeff qp

W Z

Z Zqp

W W Z

gH i U Q q b

M

i U Q q bM

We can calculate branching ratio with the effective HamiltonianWe can calculate branching ratio with the effective Hamiltonian

1 2 1 1 2 1 ( , ) ( , ) ( , ) ( , ) : B PB p m P p m k m k m 1 2 1 1 2 1 ( , ) ( , ) ( , ) ( , ) : B VB p m V p m k m k m

The determination of the elements of the CKM matrix is one of The determination of the elements of the CKM matrix is one of the most important issues of quark flavor physicsthe most important issues of quark flavor physics

In SM In SM

The process B M is decribed by the b q transition at quark level,

and dominated by the same Z-penguin and box diagrams involving top quark exchanges.

The effective Hamiltonian The effective Hamiltonian

10 5 5 1 12 2

Feff tb tq

GH C V V q b

where : Fermi constant,

: fine structure constant (at the Z mass scale),

: elements of the CKM matrix.

F

ij

G

V

Note!!!Note!!!

10described by only Wilson coefficienone t C, ! !!

( C.S.Kim, T.M. Aliev PRD58,013003(1998) )( C.S.Kim, T.M. Aliev PRD58,013003(1998) )Theory Theory

Since hadrons are involved in all the decaysSince hadrons are involved in all the decays

QCD effects are unavoidable and must be quantitatively understood.QCD effects are unavoidable and must be quantitatively understood.

⇒ ⇒ OPEOPE (Wilson and Zimmermann,1972) (Wilson and Zimmermann,1972)

⇒ ⇒ RG RG ( ‘t Hooft, 1973 ; Weinberg, 1973 ) ( ‘t Hooft, 1973 ; Weinberg, 1973 )

OPE (Operator Product Expansion )OPE (Operator Product Expansion )

12 1 1 2 1

The amplitude can be extracted from the Green's function

( ; , , () ( ) (( ) )) 0m mxG x y y y y

121 2 12 where ( ) are c-number function( ) (0) ( ) (

For small s

0)

eparation,

nnn

n

C xx C x

Wilson(1969) proposed that the effects of the operator product could be computed by Wilson(1969) proposed that the effects of the operator product could be computed by replacing the product of operators with a linear combination of local operators replacing the product of operators with a linear combination of local operators (OPE)(OPE)

1 2 ,

.

This OPE will depend only on the operators and their separation,

and will be independent of the identity and location of the other fields appearing in the Green;s function

QFT

OPE in Weak decayOPE in Weak decay

Weak decays of hadrons are mediated through weak interactions of quarks, whose strong interactions,

binding the quarks into hadrons, are charaterized by typical hadronic energy scale of (1GeV), m

W,Z

uch

lower than the scale of weak interactions : (M )

Therefore, derive an effective low energy theory describing the weak interactions of quarks

!!!

2

2 2

2

2

tree-level W-exchange amplitude

A ( ) ( )2

( ) ( )2

WFcs ud V A V A

W

Fcs ud V A V A

W

MGi V V sc uc

k M

G ki V V sc uc

M

Now this term may obviously also be obtained from an effective Hamiltonian

( ) ( ) high D Operators,2F

eff cs ud V A V A

GH V V sc uc

OPE (Operator Product Expansion )OPE (Operator Product Expansion )

i

OPE separates the full problem into two distinct parts,

The physics contributions described by the Wilson coefficients ( )

T

he contribu

short

tions

-dist

colong-dis ntained

ance

in tan the operator

C

ce

, WM

matrix eleme ( )nts ( )iQ

i

i

A = , ( ) 2

, : Wilson coefficients correspondi

Q

ng coupling constants

( ) : local operators effective vertices

C

iiFCKM Wff i

i

W

e

GB F F B VH BC M F

M

Q

iC , WM

s

s

Due to the asymptotic freedom of QCD,

the short-distance strong interaction are calculable in perturbation theory in the strong coupling( ( )).

However, the presence of ln( ) in the calculatioWM

in of the coefficients C ( , ) spoils

the validity of the usual perturbation series

disappear through Renormalization Group (RG) improved perturbation expansion

(RG : the group of transform

WM

ations between different choices of the renormalization scale )

Rev.Mod.Phys.68,1125(1996)Rev.Mod.Phys.68,1125(1996)


10 Wilson coefficient, C

2 2210 where sin ( )

t t WWt x m mC X x

0 2

3 2 4 3 2 4

1 2 2

Inami-Lim function(1981)

The QCD correction (G.Buchalla and A.J.Buras, NPB400,225(1

2 3 6( ) ln( )

8 1 1

4 5 23 11( ) ln( )

3 1 1

9

3

9 3))

t t tt t

t t

t t t t t t t t tt t

t t

X

X

x x xx xx x

x x x x x x x x xx xx x

2 2

3 22

3

30

22 where

4 8ln ( )

2 1

4 ( ) Li (1 ) 8 ln( )

1W

t tt

t

t t tt t

tt

x mX

x x xx

x x xx x xxx

2 1

ln( )Spence function Li (1 ) :

1tx

t

tx dt

t

0 14( ) ( ) ( ) ( )st t tX x X x X x

the scale dependence : (10%) (2%)O to O

one-loop contribution

two-loop contribution


10 5 5 1 12 2

Feff tb tq

GH C V V q b

eff has been investigated through different approaches !

chiral perturbation thoery

three point QCD sum rules

heavy quark effective theory

The matrix ele

(HQET

ment M H B

light-cone QC

)

D sum

rules (LCSRs)

Summarized the physics contributions to the amplitude A(B→F) from the scales lowSummarized the physics contributions to the amplitude A(B→F) from the scales lower than er than μμ ( long distance contribution) ( long distance contribution)

Non-perturbative !!!Non-perturbative !!!

Constitute the most important source of theoretical uncertainty Constitute the most important source of theoretical uncertainty

LCSRs (Light-Cone Sum Rules) LCSRs (Light-Cone Sum Rules)

QCD sum rules on the light-cone allow the calculation of form factors in a QCD sum rules on the light-cone allow the calculation of form factors in a kinetic regime where the final-state meson has large energy in the rest-kinetic regime where the final-state meson has large energy in the rest-system of the decaying B (large momentum transfer) system of the decaying B (large momentum transfer)

LCSRs treat both hard and soft-gluon contribution on the same footingLCSRs treat both hard and soft-gluon contribution on the same footing

Extension of the original method of QCD sum rules devised by SExtension of the original method of QCD sum rules devised by Shifman, Vainshtein and Zakharov (SVZ)hifman, Vainshtein and Zakharov (SVZ)

QCD sum rules QCD sum rules combine the concepts of correlation functions and quark-hadron duality ingenuous way that allow the calculation of the properties of non-excited hadron states with a very reasonable theoretical uncertainty

Two different parton configuration Two different parton configuration

1. hard-gluon exchange ( factorizable )1. hard-gluon exchange ( factorizable )

: all quarks have large momenta and the momentum transfer happen via the exchange of a hard gluon: all quarks have large momenta and the momentum transfer happen via the exchange of a hard gluon

2. soft-gluon exchange ( non-factorizable )2. soft-gluon exchange ( non-factorizable )

: one quark is soft and interact with the other partons only via soft-gluon exchange: one quark is soft and interact with the other partons only via soft-gluon exchange

LCSRs rely on the factorization of the underlying correlation function into genuinely nonperturbative LCSRs rely on the factorization of the underlying correlation function into genuinely nonperturbative and universal hadron distribution amplitudes (DAs) and universal hadron distribution amplitudes (DAs) ΦΦ which are convoluted with process-dependent which are convoluted with process-dependent amplitudes T.amplitudes T.

12 ( ) 2 2 ( )

0n

( , ) ( , , , ) ( , )n nB B IR IRq p du T u q p u

2 2 2 22 2

5 02 2( ) | (1 ) | ( ) ( ) ( ) ( )

where

P PB P B PB B

B

m m m mP p q b B p p p q f q q f q

q q

q p p

eff The matrix element M H B


2 20 ( , ) transition : , ( ) ( ) P PB P f fK q q

25 1

22

2 23 02

( , ) | (1 ) | ( ) ( ) ( )

( ) ( ) ( )

2 ( ) ( ) ( )

B B V

BB V

V

V p q b B p i m m A q

A qi p p q

m m

miq q A q A q

q

22 ( ) B

B V

V qp p

m m

2 2 2 20 1 2 ( , ) transition : ( ) ( ) ( ) V( ) , , , A qB V A q AK q q

2 2 2 2 23 1 2 0 3

1with ( ) ( ) ( ) ( ) ( ) and ( 0) ( 0)

2 B V B VV

A q m m A q m m A q A q A qm


1 2 1 2 ( , ) ( , ) ( ,0) ( ,0) : B PB p m P p m k k

We haveWe have

2 22 2 23/ 2 3 2

102 8 5( ) (1, , ) ( )

2pF

tq tb p B

GdB P V V r s m C f q

dq

2 2

22 2 2

(1, , ) 1 2 2 2

with ,

M M M M

M M B B

r s r s r s r s

r m m s q m

1 2 1 2 ( , ) ( , ) ( ,0) ( ,0) : B VB p m V p m k k

We haveWe have

1

2 22 21/ 2 3

102 10 5

222 2 2

V 2 V 1 22 2V

V V

( ) (1, , )2

8 s 1 1 r 12 2 1 r Re

r1+ r 1 r

Ftq tb v B

V

GdB V V V r s m C

dq

V r s A A s A A


( , ) B P K

2 22 2 23/ 2 3 2

102 8 5( ) (1, , ) ( )

2pF

tq tb p B

GdB P V V r s m C f q

dq

Numeric Numeric

( P.Ball, R.Zwichy PRD71,014015(2005 )( P.Ball, R.Zwichy PRD71,014015(2005 )( C.S.Kim, T.M. Aliev PRD58,013003(1998) )( C.S.Kim, T.M. Aliev PRD58,013003(1998) )

6

6

( ) 5.263 10

( ) 0.144 10

B K

B

B

B

0

2 20 2 2

2 1 22 2 2 2

1

2 1 222 2 2 2

1 1

For :

( ) 1

For :

( ) + 1 1

For K :

( ) + 1 1

fit

fit

f

rf q

q m

r rf q

q m q m

r rf q

q m q m

2

2 12 2 2( )

1

1 22 2 2 2

1

( )1

+ 1 1

B Pm m

fit

rf q ds

q m s q

r r

q m q m

2

+

D.Becirevic and A.B.Kaidalov(PLB478,417)(2000)

suggest to write the form factor f as a dispersion relation in q with a lowest-lying pole

plus a contribution from multiparticle states, which in turn is to be replaced by an effective pole at higher mass

5.279

0.4937

0.1396

B

K

m

m

m

( , )B V K

Numeric Numeric

( P.Ball, R.Zwichy PRD71,014029(2005 )( P.Ball, R.Zwichy PRD71,014029(2005 )( C.S.Kim, T.M. Aliev PRD58,013003(1998) )( C.S.Kim, T.M. Aliev PRD58,013003(1998) )

5

5

( ) 1.105 10

( ) 0.032 10

B K

B

B

B

0

2 1 22 2 2 2

1

2 22 2

2

2 1 222 2 2 2

For V,A :

( ) +1 1

For A :

( ) 1

For A :

( ) + 1 1

R fit

fit

fit fit

r rF q

q m q m

rF q

q m

r rF q

q m q m

1

2 22 21/ 2 3

102 10 5

222 2 2

V 2 V 1 22 2V

V V

( ) (1, , )2

8 s 1 1 r 12 2 1 r Re

r1+ r 1 r

Ftq tb v B

V

GdB V V V r s m C

dq

V r s A A s A A

5.279

0.4937

0.1396

B

K

m

m

m

B K B

Documents

Note!!!