Model independent analysis of B meson light-cone wavefunction

BNM08 in AtamiJan.24-26 2008

Hiroyuki Kawamura (RIKEN)

Work in collaboration with K.Tanaka (Juntendo Univ.)

Hiroyuki Kawamura (RIKEN) BMN2008

BBNS vs. data

QCD Factorization for Exclusive B decays: B →ππ, πlν, γlν…

→ systematic framework to calculate QCD effects in the mb →∞ limit

Beneke et al. (’99)Bauer, Pirjol, Stewart (’01)Keum, Li, Sanda (’01)

1 2B M M

• successful to describe the hadronic B decays.

• In some cases, the ignorance of B meson wavefunction gives a major theoretical uncertainty. ex. hard spectator interactions in B→π0π0

( )Br B PP

Beneke (Beauty06)

Introduction

etc.

Hiroyuki Kawamura (RIKEN) BMN2008

• B meson light-cone wavefunction in HQET

=(1,0,0, 1) nm - light-cone vector:

In momentum space

0

tn

k v momentum of light quark

• UV structure (radiative tail)

( )lo(

g) sB iFaf

m

ww

w-:

0( ) ( 0)j

Bd jww f w¥

=- ¥ ³ò

B meson LCWF

HQET field:

Radiative corrections generate a hard negative tail. ↔ pion LCWF

2 ( =1)Bp m v vm m=

Hiroyuki Kawamura (RIKEN) BMN2008

— Contributions from higher dim. Operators in the IR region.

Kodaira, Tanaka, Qiao, HK (’01)

HQ symmetry + E.O.M.

( ( )) ( ) ( )( ) BWWBB

gf f wwf w = +

( )

2( ) (2 )

2

0 (0)( ) ( )

WWB

nv

iF

q v D h B v

wf w q w= L -

L

« ×

“Wandzura-Wilczek” part

( ) 0 ( )gB vq hG B vf :

— “twist = dimension - spin” is not a good quantum number

v vvh h 0vq D v Dh � ��������������

IR structure

→

Hiroyuki Kawamura (RIKEN) BMN2008

• non-analytic at .

• cusp singularity → radiative tail

Lange & Neubert (’03), Braun et al. (03),Li & Liao (’04), Lee & Neubert (’05)

2log ( )it

log( ) it

0(0) exp ( ) ( )v vh P ig dsv A sv h v

MSlog( ), EL it egm m m= =

• UV & IR structures are different!

Radiative corrections

1-loo

21

5

p

2

50 2

( , )

5(1 ) 0 ( )

2 24 1

( ) (0)

( ) (

1

0

11

20 ) (

2

1

IR

IR

UV UVUV

B

sv

v

F LL

t

Cd B vq tn n h

q tnL h Bn

e ee

f m

a px g

x g

p xe

e

xx d x

+

éì üæ ö æ öé ùï ï÷ ÷ï ïê ç ç ê ú÷ ÷= - + + + - + -ç çí ý÷ ÷ê ç ç ê ú÷÷ï ç ïç -è øè ø ë ûêï ïî þëæ ö÷ç ÷- +ç ÷ç ÷çè ø

ò

%

5

2

) 2 1 0 ( )

( ) ( ,

1

IR po

( ) (0)

higher-dim ople .s )s

IRvqv t B v

O t

L Dn hvtnx x ge

æ ö÷ç ÷- -+ ×-ç ÷ç ÷çè ø

ù+ úû

su

RG evolution

CUSP0t

Hiroyuki Kawamura (RIKEN) BMN2008

• gives a natural scale to separate UV & IR behaviors ⇒ OPE

Operator product expansion

• B meson LCWF in terms of HQET parameters

RG evolution for LCWF2

1 1,

UV UVe e

1

IRe

OPE at

( , ) ( , (() 0 ))B ii

it C t B vOf m m m=å% %

expansion in

F bm

t

1/ t

UV

IR

µ( , ) ( , ) ( , )B F F Bt U tf m m m f m= Ä% %

with higher dim. ops.

1

t

1/ t

2log ( ), log( )it it 1/ t correlated

Hiroyuki Kawamura (RIKEN) BMN2008

• Lee & Neubert PRD72(’05)094028

OPE at NLO

vq hG vqD hG

UV UV( , , ) ( , , 0 ( ))) (B ii

it C t O B vf m m mL L=å% %

• This work

( , ) ( , (() 0 ))B ii

it C t B vOf m m m=å% %

MS-bar scheme, up to dim-5 ops. vq hG vqD hG

{ }, v vDq hD qG hG G+

cut-off scheme, up to dim-4 ops.

Hiroyuki Kawamura (RIKEN) BMN2008

Calculation

• 1-loop matching of the non-local operator for B meson LCDA with non-local ops. up to dim.5

many dim. 5 ops.

→ calculation in x-space keeping gauge invariance explicitly.

(C)

1(C) (C)

0 (

(

) (

0

)

)x A x

A x duux G uxrm r

m

m

m

Þ =

=

ò

{ }50

( )P exp ( ) (0) , , , t

v v v v vD Dq tn ig d n A n n h q h q h q h q hD Gmml l g

æ ö÷ç « G G G G÷ç ÷è øò

→ decouple from Wilson line

• Fock-Schwinger gauge forclA

cl qA A A

• Background field method

Hiroyuki Kawamura (RIKEN) BMN2008

dim.3

dim.4

dim.5

OPE up to dim.5 (NLO in MS-bar scheme)

MSEe= gm mlog( )L i tm=

Hiroyuki Kawamura (RIKEN) BMN2008

Matrix elements

dim.3

dim.4

dim.5 (covariant tensor formalism)

“Chromo-electronic”

“Chromo-magnetic”

← decay constant

Hiroyuki Kawamura (RIKEN) BMN2008

B meosn LCWF from OPE

• Dim.3&4 terms reproduce the results in cut-off scheme by Lee & Neubert (’05)

dim.3

dim.4

dim.5

• Expressed in terms of 3 HQET parameters:

• in on-shell scheme suffers IR renormalon ambiguities.

→ “distribution amplitude” scheme

Hiroyuki Kawamura (RIKEN) BMN2008

real part imaginary part

• Contributions from dim. 5 operators are important.• Effects from are very small. • Evolution effects must be included.

LCWF from OPE at 1GeV

Inputs:Grozin,Neubert (’97)

Lee,Neubert (’05)

, E H

(QCD sum rule)

Hiroyuki Kawamura (RIKEN) BMN2008

• expressed in terms of 3 HQET parameters at

Summary

• Knowledge of B meson LCWF is important to reduce the theory error for exclusive B decays.

• Radiative correction to B meson LCWF

• OPE for B meson LCWF — up to dim.5 — NLO corrections to Wilson coefficient

1/ t

— different UV & IR structures

, , E H

• Model-independent study including RG evolution is underway.

• Similar analysis for shape function is possible.

— 2log ( ), log(it )it →

( , )B t

( , )B t 1GeV

Hiroyuki Kawamura (RIKEN) BMN2008

LCWF with analytic continuation t i