review of qcd introduction to hqet applications conclusion

Review of QCD

Introduction to HQET

Applications

ConclusionPaper:M.Neubert PRPL 245,256(1994)

Yoon yeowoong( 윤여웅 ) Yonsei Univ. 2004.04.30

•Introduction to HQET - Review of QCD

Bjorken scaling : structure function only depend on . (1969)

→ Point-like structure inside proton, Asymtotic freedom

→ Non-Abelian gauge field theory. Yang, Mills

→ Asymtotic freedom in Non-Abelian gauge field theory. t’Hooft(1973)

→ Gell-Mann propose extra symmetry of non-Abelian color symmetry(1972) QCD was born

→ Quark confinement( Only colorless states are physically observable) is explained in QCD by infrared divergences due to the massless gluons

22 2

12( )

(33 2 ) log( / )sf QCD

Qn Q

2Q

v

High Energy probe

Asymtotic freedom

Colo

r ch

arg

e

Con

fin

em

en

t

1 fermi

1S

Barr

ier

Distance from the bare quark color chage

•Introduction to HQET - Review of QCD

Summary of Non Abelian Gauge theory SU(3)

[ , ]a b abc cT T if T

i ij jU exp( ) , 1,2,..,8a aU iT a

( )L i D m a aD igT G

a a a abc b cvG G G gf G G [ , ] a a

vD D igT G

1a abc b c aG f Gg

1( )

4a aL i D m G G

1( ( ) )

4

qNk k a ai ij ij j

k

L i D m G G

•Introduction to HQET - physical picture

Heavy Quark : m Q > ΛQCD

Heavy Quark limit : mQ →∞

Heavy Quark + light quark system

Qq

q

Q

“Brown muck”

light quark q cannot see the quantum numbers of Heavy Quark

Comptom wavelength of Q : λQ~1

Qm

To resolve the quantum number of Heavy quark,need a hard probe with 2 2

QQ m

•Introduction to HQET - physical picture

The configuration light Degree of freedoms with different heavy quark flavor, spin system of hadron does not change if the velocity of heave quark is same.

Heavy Quark velocity ≒ Meson velocityMomentum transfer ~ ΛQCD ⇒ velocity change ~ ΛQCD /mQ ~ 0

We can regard heavy quark velocity as conserved quantity

vv

Therefore this picture gives spin – flavor symmetry in QCD under mQ →∞ limit. Nh heavy quark flavor → SU(2Nh) spin-flavor symmetry group

It provide the relations between the properties of hadrons with different flavor and spin of heavy quark.Such as B, D, B*,D*, Λb Λc

•Introduction to HQET - details with elementary field theory

Heavy quark momentum almost on-shell Q QP m v k 2 1v

Divide quark field by large and small component respectivelyQ

( ) ( )

( ) ( )

Q

Q

im v x

v

im v x

v

h x e P Q x

H x e PQ x

,v vh H

(1 )

2

vP

( ) [ ( ) ( )]Qim v x

v vQ x e h x H x

v v

v v

v h h

vH H

QCD Lagrangian

( )

( )( )( )

( 2 )

( 2 )

Q

v v Q Q v v

v v v Q v v v v v

v v v Q v v v v v

L Q i D m Q

h H m v m iD h H

h iDh H iD m H h iDH H iDh

h iv Dh H iv D m H h iDH H iDh

a aD igT G

1 1

2 2

1 1 1,

2 2 2

1 1

2 2

1 1

2 2

v v v v

v v

v v

v v

v vh iDh h iD h

vand iD ivD iD

iv D iDv iD

h iDh

v vh iv D iD h

h iv Dh

where

,D D v v D then v D


On a classical level, DOF of H v can be eliminated by EOM of QCD

( 2 )Q v v v Q v v v v vL h iv Dh H iv D m H h iDH H iDh

( 2 ) ,Q v viv D m H iD h 1

( 2 )v vQ

H iD hiv D m

1

( 2 )eff v v v vQ

L h iv Dh h iD iD hiv D m

0

1

2 2

n

eff v v v vnQ Q

iv DL h iv Dh h iD iD h

m m

21 1

1x x

a x a a a

Variation of Lagrangian with respect to vH

Considering order of 1/mQ (n=0)

And using the relation

2( )2

giD iD iD G

2

2

2

1 1{ , } [ , ]

2 21

( ) [ , ] ( , )2

12 2( ) [ , ] ( , )

21

[ , ] ( , )2

1( ) [ , ][ , ]

4

iD iD iD iD

iD iD iD iD

iD iD iD Commute

iD iD iD iD iD Commute

iD iD Commute

iD iD iD iD iD


Inserting gluon field strength tensor [ , ] a aiD iD igT G igG

It can be shown by

and, 1

[ , ]2

Then the effective Lagrangian of order 1/mQ is

2 21(1/ )

2 4eff v v v v v v QQ Q

gL h iv Dh h iD h h G h O m

m m

Kinetic termFrom residual mome

ntum kPQ=mQv+k

hv=eimQv·xP+Qv

Chromo-magnetic momentum interaction

(Halzen Ex6.2)

v c vQ

gh S B h

m

1

2i ijk jkcB G


Now we consider heavy quark limit mQ →∞

eff v vL h iv Dh

1. It has spin symmetry

Associated group is SU(2) symmetry group under which Leff is invariant

An infinitesimal SU(2) transformation

05

01 1

2 20

ii i

iS

5

1

2i iS ve 0iv e

[ , ]i j ijk kS S i S [ , ] 0iv S

(1 )v vh i S h

[ , ] 0eff vL h iv D i S h

On-Shell condition satisfied

5

5

5

1

21

22

1

2

i i

i i

i i

vS v ve

v v e e v

ve v S v

(1 ) (1 ) (1 )v v vv i S h i S vh i S h


2. It has flavor symmetry

When there are Nh heavy quark flavor

hNi i

eff v vi i

L h iv Dh

Because this Lagrangian do not contain heavy quark mass, It is invariant under rotations in flavor space

Combined with spin symmetry the effective Lagrangian belong to SU(2Nh) symmetry group.


Now consider Feynman rules

Feynman propagator, and vertex factor can be derived by effective Lagrangian

eff v vL h iv Dh

Propagator ,v k

i j1

2 ji

i v

v k

Vertex ( )a jiig T v

It can be also derived by taking the appropriate limit of the QCD Feynman rules

2 2 2 2 2

( ) ( ) ( ) 1

( ) 2 2Q Q Q Q Q Q

Q Q Q Q Q

i p m i m v m k i m v m k v i

p m i m v k m i m v k k i v k i

i j

,a

v

For the heavy quark gluon vertex


aigT

Using the relation P P P v P

21 1 1

2 2 2 2

1 1 1

2 2 2

v vv v vP P

v v vv v v P

v P P P v P

Therefore vertex factor in Heavy quark limit become

aigT v

•Application - Spectroscopy

Strong Interaction dynamics is independent of the spin and mass of the heavy quark by heavy quark symmetry.

Therefore hadronic states can be classified by the quantum number of the light DOF such as flavor, spin, parity, etc.

Spin-flavor symmetry in HQET predict some relations of properties of hadron states, typically mass spectrum of different Hadrons states

Meson Constituent Quarks J P

D c, (u or d) 0 -

D* c, (u or d) 1 -

D1 c, (u or d) 1 +

D2* c, (u or d) 2 +

Ds c, s 0 -Ds* c,s 1 -

Meson Constituent Quarks J P

B b, (u or d) 0 -

B* b, (u or d) 1 -

B1 b, (u or d) ? ?

B2* b, (u or d) ? ?

Bs b, s 0 -Bs* b,s 1 -


1. Ground state mesons

1

2l lj s 1

2lJ j 0 1J or J

*

*

*

46MeV

142MeV

142MeVSS

BB

DD

DD

m m

m m

m m

Experimentally

degenerate states

Need a hyperfine correction of order 1/mQ

*

1~MM

Q

m mm

Quite small as expected

So we can expect * *

2 2 2 2 .B DB Dm m m m const

*

*

2 2 2

2 2 2

0.49GeV

0.55GeV

BB

DD

m m

m m


2. Excited state mesons

1 3,

2 2l ls j 1*2

1: (2420)

2 : (2460)

J D

J D

degenerate states

*12

35MeVDDm m

It is small mass splitting supporting our assertion

One can expect also

* *1 12 2

2 2 2 2 20.17 GeVB DB Dm m m m

3. Excitation energy

1 1

* *2 2

100MeV

557 MeV

593MeV

S SB B D D

B B D D

B DB D

m m m m

m m m m

m m m m

•Application - Weak decay form factors

Physical picture of weak decay

b

d

c

d

0B D

e

eW

5(1 )2 2

gi

5(1 )cbV

(*)5( ) | (1 ) | ( )D p c b B p Hadronic matrix element

parameterized by several form factors.

5( ) (1 ) ( )2 e

Fcb e

GM V u p v p

(*)

5( ) | (1 ) | ( )D p c b B p

2 2'0 ( )M Mq m m


Q

q

Kinematical picture

Q’

qv v

0t t 0t t

w v v 2 2 2

2 2' '

( ) ( )

2

Q Q

Q Q Q Q

q P P m v m v

m m m m v v

2 2 2

2M M

M M

m m qv v

m m

Maximum q2=(mM’-mM)2 ; minimum w=1 Zero recoil

Q

q

Q’

Qq

Minimum q2=0 ; maximum w

Q’

M M e

e

e


0

3 32' | ( ) (2 ) ( )

M

pM p M p p p

m

0 3 3' | ( ) 2 (2 ) ( )M p M p p p p

1( ) ( )

M

M v M pm

2 2 2( ) 2 ( 1)light QCD QCD QCDq v v v v 1 1.6w

*

*

2

2 2 2 22 2

1 02 2

* * 2

** * 2

1

( ') | | ( ) ( )( )

( ') | | ( ) ( ) ( ) ( ) ( ) ( )

2( ') | | ( ) ( )

( ') | | ( ) ( ) ( )

el

cb B D B D

cb

B D

cbB D

B

B p b b B p F q p p

m m m mD p V B p F q p p p p F q p p

q q

iD p V B p p p V q

m m

qD p A B p m m A q

m m

*

*

*

*2 2

2 32

*2

02

( ) ( ) 2 ( ) ( )

2 ( )

DD

D

qp p A q m p p A q

q

qm q A q

q

Typical hadronic matrix element M.Wirbel ZPHY C29,637(1985)

5,cb cbV c b A c b 0 1(0) (0)F F

* *

* *

2 2 23 1 2, ( ) ( ) ( )

2 2B BD D

D D

m m m mA q A q A q

m m

Now in HQET


| | ( ) ( )( )v vP v h h P v v v v v Why not ( )( )v v v v

' | | ( ) ( )( )v vP v h h P v v v v v

Using flavor symmetry

( )v v

Is called Isgur-Wise function 2

| | ( ) ( )( ) ( )( )

| ( ') | ( ) 0 ( )( )

0 2 ( )(1 2 )

v v

v v

P v h h P v v v v v v v v v

P v h v v h P v v v v v

v v v v

Normalized at zero recoil as (1) 1

( )v v

For equal velocityv v v vJ h h h v h is conserved current

3 0 3( ) ( ) ( )Q Q v vN d xJ x d x h x h x

( ) ( )QQN P v P v ( ) ( )Q QN P v P v

explained by following

0 3 3

3 0 2 0 3 3 0 3 3

'( ) ( ) ( ) | ( ) 2 (2 ) (0)

'( ) ( ) ( )2 (2 ) (0) 2 (1)(2 ) (0)

Q Q

v v

P v N P v P v P v v

P v d x h h P v v v v


Using spin symmetry

3 3( , ) 2SQV v P v

3 3

3

( , ) ( ) ( ) | 2[S , ] | ( )

( ) | (2 ) | ( )

v v Q v v

v v

V v h h P v P v h h P v

P v h S h P v

( )

*( )

3 ( ) *( )

10

21

12

1

2

Q

Q l Q l

Q

Q l Q l

Q QQ

P J

P J

S P P

3

3 0 35

(1,0,0,0)

(0,0,0,1)

1

2

v

S

3 0 0 3 3

3 3 3 0 0

3 1 1 2 2

3 2 2 1 1

2[S , ]

2[S , ]

2[S , ] ( )

2[S , ] ( )

Q

Q

Q

Q

V A A V

V A A V

V A i A V

V A i A V

In the rest frame of the final state meson

3 *

3 * *5

( , ) ( ) ( )

( , ) ( ) ( ) ( 1)

v v

v v

V v h h P v v v i v v

V v h h P v v v v v v v

HQET

Typical


Summarize parameterization

*

*

2

2 2 2 22 2

1 02 2

* * 2

** * 2

1

( ') | | ( ) ( )( )

( ') | | ( ) ( ) ( ) ( ) ( ) ( )

2( ', ) | | ( ) ( )

( ', ) | | ( ) ( ) ( )

el

cb B D B D

cb

B D

cbB D

B p b b B p F q p p

m m m mD p V B p F q p p p p F q p p

q q

iD p V B p p p V q

m m

qD p A B p m m A q

*

*

*

*2 2

2 32

*2

02

( ) ( ) 2 ( ) ( )

2 ( )

DB D

D

qp p A q m p p A q

m m q

qm q A q

q

3 *

3 * *5

( , ) ( ) ( )

( , ) ( ) ( ) ( 1)

v v

v v

V v h h P v v v i v v

V v h h P v v v v v v v

' | | ( ) ( )( )v vP v h h P v v v v v

| | ( ) ( )( )v vP v h h P v v v v v

Bp m v

(*)Dp m v


2( ) ( )elv v F q

Relations between form factors and Isgur-Wise function.

21( ) ( )v v RF q

122

02( ) 1 ( )

( )B D

qv v R F q

m m

* 2( ) ( )v v R V q

* 20( ) ( )v v R A q

* 22( ) ( )v v R A q

*

12

* 212

( ) 1 ( )( )B D

qv v R A q

m m

2 22 (1 ) 0Bq m v v

2 2 2 2 0B D B Dq m m m m v v

20.88B D

B D

m mR

m m

* *

2 2 2 2 0B BD Dq m m m m v v

*

*

20.89

B D

B D

m mR

m m

Renormalization group equation

Study hard !

Model independent Vcb

Inclusive decay with HQET

Conclusion - more study

review of qcd introduction to hqet applications conclusion

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