H. Kawamura (Univ. of Liverpool)

(1)

OPE for B-meson

distribution amplitude and dimension-5 HQET operators

Kazuhiro Tanaka (Juntendo U)

arXiv:0810.5628 [hep-ph]

(2)

, , ,

B QCD factorization for Exclusive B decays → ππ ργ π ν l

Beneke, Buchalla, Neubert, Sachrajda (’99) Bauer, Pirjol, Stewart (’01)

B meson’s LCDA in HQET

(

²

)

=(1, 0, 0, 1) 0

n

^µ

− n =

(

²

1 )

p

^µ

= m v

B ^µ

v =

tn

_µ

0 B

1 2 1

0 S q (0) D D

^ν ^ν

D

^ν^j

n

5

h

_v

(0) B v ( ) γ

µ

⎡

−

⎤

⎢ ⎥

⎣ ⎦

∼

t ⇔ µ

twist = dimension - spin

( )

+ 1 ,

( )

^{im v x}^b _v

( ) m

_b

v v ( ) v ( )

b x = e

⁻ ^⋅

h x Ο / h x = h x

m b → ∞ µ

ⁱ

= m

^b

Λ

^QCD

(3)

IR structure

Kawamura, Kodaira, Qiao, Tanaka, PLB523 (’01) 111

heavy quark symmetry:

( ) ( )

( ) B WW B

B

φ ω φ g ω

φ ω = +

( )

( ) 2 (2 )

2

WW

B iF ω

φ ω = θ Λ− ω

Λ

B b

m m

Λ = −

( ) g 0 v ( )

B q G h B v

φ ∼

radiative corrections from hard loops

( )

( lo g

) s

B iF α

φ µ

ω ω ω

−

∼

Lange, Neubert,PRL523 (’03) 102001

cusp singularity

5

Pexp

0

( ) ( )

_v

(0 )

t

ig d n A

q n t n γ ⎛ ⎜ ⎜ ⎝ ∫ λ

_µ ^µ

λ n ⎞⎟ ⎟ ⎟ ⎠ h

0

Pexp ig d sv A

_µ ^µ

( ) sv h

_v

( v )

−∞

⎛ ⎞⎟

⎜ ⎟

⎜ ⎟⎟

⎜ ⎠ −∞

⎝ ∫

UV structure

v

0 q D = ⋅ v Dh =

constraints from HQET eqs. of motion:

v v

vh / = h

“radiation tail”

0^∞

d ωω φ ω

^j _B

( ) = −∞

∫

(4)

Radiative corrections from hard and soft/collinear loops

4

4 2

1 1 1

UV IR

d q

q

ε

ε ε

−

∫ ∼

4 2 d = − ε

・ UV ～ IR “scaleless’’

・ analytic at t=0

( ) ( )

1 2 1-loop

0

2 2

5

log log 5

( , ) (1 )

2 2

log

l

( ) (

og

1

1 0)

4

0 ( )

1

2 1

1

1 1

2

_UV

IR IR

IR

V s

U B

v U

F

V

t C d

B

it it

it

i

v q tn n

t

h

t

α π

φ µ ξ δ

ξ γ

µ

ε ε µ ξ

µ µ

ε ε

π

ε ξ

ξ

+

⎡⎧ ⎛ ⎪ ⎪ ⎜ ⎞ ⎟

⎢ ⎟

= ⎢ ⎢⎪⎩ ⎣ ⎨ ⎪ ⎜⎝ − ⎜ ⎜ + + + ⎟ ⎟ ⎠ −

⎛ ⎞ ⎟ ⎡ ⎤ ⎛ ⎞ ⎟⎪ ⎫ ⎪

⎜ ⎟ ⎢ ⎥ ⎜ ⎟

+ ⎜ ⎜ ⎜ ⎝ − ⎟ ⎟ ⎠⎣ ⎢ − ⎥ ⎦ − ⎜ ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ ⎬ ⎪ ⎪⎭

⎛ ⎞⎟

− ⎜ ⎜⎜⎜⎝ + − − ⎠

∫

(

5

0

0 q ξ tn ) v D n γ h

_v

( ) B ( ) v ⎤

⎟ ⎥ +

⎟⎟ ⋅ ⎦

MS

e

E

= γ

µ µ

Brodsky-Lepage pot.

1-loop 1

0 0

( , )

5

( , ) 1 1 0 ( ) ( ) (

2

_UV _IR

)

s

C

F

q tn n q tn

t d d K p

ξ π

φ µ α ξ η π

π η ξ γ η

ε ξ

ε

⎛ ⎞⎟

⎜ ⎟

= ⎜ ⎜ ⎜⎝ − ⎟ ⎟ ⎠ ∫ ∫

double log

(5)

・ nonanalytic at t=0 log

²

( ) it µ , log ( ) it µ : nontrivial dependence on t µ

・ UV IR

2

1 1

,

UV UV

ε ε

1 with many higher dim. operators ε

IR

1/t

We have to use to separate UV and IR behaviors, in contrast to light -meson light-cone A

OPE

D ! qq

1

µ 1 t

( )

( , ) ( , ) 0 ( )

B i

i

t C t O i B v

φ µ = ∑ µ µ

( )

log

2

itµ

∼ local op.

QCD

i

m

b

µ = Λ

( , ) ˆ ( , ) ( , )

B t µ i U µ i B t

φ = µ ⊗ φ µ

Sudakov-type [Lange, Neubert (’03)]

(6)

UV

) 0

(

( UV , ) 0

( , ) i N i ( ) ( )

N

i

B B v

d ωω φ ω µ µ O µ

Λ = ∑ Λ

∫ C

OPE for cut-moments:

0,1

( )

: up to ( ) : up to dim.4

N

i

i O s

O

= α C

qD Γ h v

Lee, Neubert,

q h Γ v

PRD72 (’05) 094028

OPE in coordinate space (MS scheme) : ( )

( , ) ( , ) 0 ( )

B i

i

t C t O i B v

φ µ = ∑ µ µ

dim.

: up to ( ) : u p t o 5

i s

O i

C O α this work

{ q D D Γ h v , qG Γ h v }

q h Γ v qD Γ h v

arXiv:0810.5628 [hep-ph]

(7)

background field method

( C ) ( C ) ( C ) ( C ) ( C ) ( C ) ( C )

(

( Q ) ( C ) ( Q ) ( C ) ( Q ) ( C )

0 , 0 ,

C )

, , ,

v

a a

v

v v

i v D h

h h h q q q A A

i D q D G t q t q

A

µ ρ ρ

µ

µ µ µ

γ

⋅ = = =

→ + → + → +

Fock-Schwinger gauge:

^{( C} ^{( C )} ¹ ⁽^{C )}

0

)

( ) x 0 A ( ) x d u u x G ( x )

A u

x

^µ _µ

= ⇒

_µ

= ∫

^ρ _{ρ µ}

( Q ) ( Q ) ( 1) ( C )

0

( ) ( 0 ) ( ) ( ) 1 1 )

2 (

d v

x v

v

h x h = θ v x ⋅ δ

⁻

x

_⊥

+ / ⎛ ⎜ + ⎜ ⎝ ig ∫

^⋅

d λ v A

^µ _µ

λ v + ⎞⎟ ⎟ ⎟ ⎠

x 0 + +

( )

/2 1 /2 2

2 2

/2 /2

(C) (C)

/ 2 1 / 2

(0) 2 (0)

8 8

d d

gG y x

i d g i d

x y i x y

gG i

η ρ

ηρ µν

µν

π ε

⁻

π ε

⁻

− Γ − − Γ −

= +

⎡ − − + ⎤ ⎡ − − + ⎤

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎦

+

y ⎣

x

y

x

( ) ( )

( )

( ) ( )

( )

/ 2 / 2 1

2 2

/ 2

(C)

(0)

/ 2

/ 2 1

(C)

(

4 16

d d

0)

d d

d x y i d x y

x y i x y i

gG

^µρ

y x

_{µ ρ}

gG

^µρ _µρ

π π ε

σ

ε

⁻

−Γ − Γ − −

= + +

⎡ − − + ⎤ ⎡ − − + ⎤

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

0 5

( )P exp ( ) (0)

t

q tn ⎛ ⎜ ⎜ ⎝ ig ∫ d n A λ

_µ ^µ

λ n ⎞⎟ ⎟ ⎟ ⎠ n γ h

v

(8)

dim.3 dim.4

dim.5

MS

2 n n v

_µ ^µ

+

^µ

≡

2 2

0

n = n =

(9)

Matrix elements and HQET parameters

dim.3 : decay constant

dim.4

B b

m m Λ = −

dim.5

“Chromo-electronic”

“Chromo-magnetic”

(10)

LCDA from OPE MS scheme

dim.3 dim.4 dim.5

： dim.3&4 terms reproduce the results by Lee & Neubert (’05)

{ } { }

with up to (

Complete OPE re sult C

_i

O α

_s

) and O

_i

up to dim.5

2 2

completely represented by HQET p aram eters Λ , λ E , λ H

double logs due to cusp singularity

UV

0

N

( ) d ω ω φ ω

Λ

∫

+

DA

(1 GeV) 0.52 GeV Λ

( ) ( )

2 2 2 2

1 GeV 0.11 0.06 GeV , 1 GeV 0.18 0.07 GeV

λ

_E

= ± λ

_H

= ±

For quantitative estimate:

[Grozin, Neubert (’97)]

[Neubert (’05)]

DA

7 ( ) 9 ( )

( ) 1

16 8

F s F s

C α µ C α µ

µ µ

π π

⎛ ⎞

Λ = Λ ⎜ ⎝ + ⎟ ⎠ − from , B → X

_s

γ B → X

_u

ν

from QCD sum rules

(11)

dim.3 dim.4 dim.5

“3”

“ ３＋ 4”

“ ３＋４＋ 5”

LO

GeV -1

τ ⎡ ⎣ ⎤ ⎦

•

NLO perturbative corrections are very large for τ→０ and 10-30% level for moderate τ

• Nonperturbative corrections from dim. 5 as well as dim. 4 operators are important (20-30% level)

• Effects from are significant in dim. 5 contributions.

λ λ

_E

,

_H

L-N

(12)

Lee-Neubert’s two-component ansatz (’05)

0 D

2 0

/

1 4

A

( , ) ( ) ln 2 ln

2 3

ω

α

ω ω ω

φ ω µ θ ω

πω µ ω µ

ω ω

^F ^s

LN

N

t

C

+

e

−

⎡ ⎛ ⎞ Λ ⎛ ⎞ ⎤

= + − ⎢ ⎜ − ⎟ + ⎜ − ⎟ ⎥

⎝ ⎠ ⎝ ⎠

⎣ ⎦

( )

2.33 GeV, 0.963, 0 0.438 GeV 1 GeV

t N µ

ω = = ω = =

the first term produces also particular contributions associated with the operators of dimension d > 4

UV

)

0

(

( UV , ) 0

( , ) N ( ) ( )

N

i i

i

O

d ωω φ + ω µ µ µ B v

Λ = ∑ Λ

∫ C

OPE for cut-moments:

0,

( 1 )

: up to ( )

N

i = O α s

C

qD Γ h v

q h Γ v

: up to dim.4

O i

(13)

OPE up to

dim. 5 ops. ( )

/ 0

2 0 2

0 0

1

d e

^ωτ

N ω e

^{ω ω}

N

ω ω τω

∞ − −

=

∫ +

τ c

continuity at τ τ =

_c

2

2 2 2 2

0

9

2

1 1

4 2

E H

c DA

DA DA DA

N τ λ λ

ω

⎧ ⎡ ⎤ ⎫

⎪ ⎪

= Λ ⎨ ⎪ ⎩ + Λ ⎢ ⎣ Λ + Λ − ⎥ ⎦ ⎬ ⎪ ⎭ +

( i )

φ τ

+

τ

∂ −

∂

( i ), φ

₊

− τ

1

0

( , )

B

( ) d φ ω µ

λ µ ω

ω

−

= ∫

∞ +

0^c

( , ) ( , )

c

d i d i

τ

τφ

₊

τ µ

τ^∞

τφ

₊

τ µ

= ∫ − + ∫ −

L-N ansatz our OPE

1 1

0.6 GeV

⁻

1 1 τ

_c

1 GeV

⁻

“stable” behavior for

( 1 GeV) 0.37 GeV λ µ

B

=

( 1 GeV) 0.48 GeV

LN

λ

B

µ =

(14)

Summary

1 / t µ ⇔

OPE of the bilocal operator for B-meson LCDA up to dim.5 local operators

NLO corrections for Wilson coefficients

terms from cusp singularity

( )

log 2 i t µ

∼

B-meson LCDA for exclusive B decays novel UV & IR structures in the HQET LC bilocal operator

Model-independent behavior of B-meson LCDA from the OPE large NLO pert. effects

significant nonpert. effects completely expressed by three HQET parameters Λ , , λ λ

_E² _H²

1 t µ 1

1

0

( , )

B

( ) d φ ω µ

λ µ ω

ω

−

= ∫

∞ + ₀^c

( , ) ( , )

c

d i d i

τ

τφ

₊

τ µ

τ^∞

τφ

₊

τ µ

= ∫ − + ∫ −

Connecting smoothly to an ansatz for the long-distance behavior:

( =1 GeV ) 0.37 GeV

λ µ

B

decreases for increasing values of λ λ

_E²

,

_H²

( 0.2 ∼ 0.5 GeV )

precise nonperturbative estimate of λ λ

_E²

,

_H²

RG evolution to & Sudakov resummation functional form of long-distance behavior

Need µ

^F

= m

^b

Λ

^QCD

(15)

( 1 Ge V ) ( 0 τφ ( τ , 1 GeV ) )

λ µ

^B

= = ∫

^∞

d

₊

− i µ =

0.6 GeV

0.46 0.11 GeV 0.48 0.06 GeV 0.

0.37 G

35 0.15 Ge V

V e

±

QCD SR (LO)

QCD SR (NLO + power corr.)

OPE up to dim.4 ops., combined with model ansatz

Ball, Kou (’03)

Braun, Ivanov, Korchemsky (’04)

Lee, Neubert (’05)

this work

input for QCD factorization formula

Beneke, Buchalla, Neubert, Sachrajda (’99) Beneke, Neubert (’03), Beneke, Jager (’07)

QCD

1 ( )

φ + µ λ µ −

= Λ

⇐ ⊗ ∝

F m b B F

A H

1 + O ( α s )

＝

(16)

OPE up to

dim. 5 ops. ( )

/ 0

2 0 2

0 0

1

d e

^ωτ

N ω e

^{ω ω}

N

ω ω τω

∞ − −

=

∫ +

τ c

continuity at τ τ =

_c

2

2 2 2 2

0

9

2

1 1

4 2

E H

c DA

DA DA DA

N τ λ λ

ω

⎧ ⎡ ⎤ ⎫

⎪ ⎪

= Λ ⎨ ⎪ ⎩ + Λ ⎢ ⎣ Λ + Λ − ⎥ ⎦ ⎬ ⎪ ⎭ +

( i )

φ τ

+

τ

∂ −

∂

( i ), φ

₊

− τ

1

0

( , )

B

( ) d φ ω µ

λ µ ω

ω

−

= ∫

∞ +

0^c

( , ) ( , )

c

d i d i

τ

τφ

₊

τ µ

τ^∞

τφ

₊

τ µ

= ∫ − + ∫ −

with μ =1 GeV

L-N ansatz

our OPE