H. Kawamura, “QCD for B physics”

QCD for B Physics

KEK Theory meeting “Toward the New Era of Particle Physics ”

Dec.12 2007

Hiroyuki Kawamura (RIKEN)

H. Kawamura, “QCD for B physics”

B Physics

• NP search by (over)constraining “unitarity triangle”

• Flavor mixing + CP violation via weak interaction

HFAG: LP07

H. Kawamura, “QCD for B physics”

ex. (semi-leptonic decay)

QCD for B Physics• Extraction of |VCKM| and φCP from hadronic weak decays requires a good understanding of QCD effects.

• QCD effects are even necessary for “direct CP violation” — phase is detected through quantum interference

+

ex. (hadronic decay)

or B B

T (tree) P (penguin)

(*)B D l

B

H. Kawamura, “QCD for B physics”

Theoretical tools

• Light flavor symmetry: isospin, SU(3) symmetry, …

• Heavy quark effective theory (HQET)

• Λ/mb expansion ↔ separation of scales (factorization)

ex. isospin analysis for B → ππ etc.

• QCD factorization for inclusive semi-leptonic decay

• Lattice simulation

(Observable) ( , ( ), ) ( , ) ( )

p

iji W NP b i bi j b

C M M m H m O Om

• Light-cone sum rule (quark hadron duality, spectral function,OPE,…)

This talkMw

mb

Λ

μ

,cB X l

, ,B D • Soft-Collinear effective theory

• QCD factorization for exclusive hadronic decay

MNP

perturbative

H. Kawamura, “QCD for B physics”

HQET

• Can be described by an effective field theory which includes only soft modes of “large component “of Q + soft quarks + soft gluon

Heavy-light meson system

HQET field:

• Perturbative matching with full QCD (αs(mb) small) :

Wilson coeff.

Leading term : SU(2Nf) Spin-Flavor symmetry

mb

Λ

Mw

μ

QCD

HQET

H. Kawamura, “QCD for B physics”

Meson spectrum

•

•

H. Kawamura, “QCD for B physics”

Exclusive semi-leptonic decays

B (*)D

l

(*)B D l

• heavy-heavy form factorIsgur-Wise function

At the “zero-recoil limit” :

H. Kawamura, “QCD for B physics”

Experimental result

LP07

⇓

Extrapolation of the data to ω=1

2007 WA

H. Kawamura, “QCD for B physics”

Inclusive semi-leptonic decays

cB X l• total rate

OPE (short distance expansion)

⇒ ( ( ))i s bi

C m

iO

propagator1

↔

H. Kawamura, “QCD for B physics”

Moments vs. |Vcb| & HQET parameters

Moment → Vcb & HQET parameters

(kinetic energy)

⇓

Buchmuller & Flacher (‘05)

H. Kawamura, “QCD for B physics”

• differential rateuB X l

B→ Xulν

kinematical cuts to avoid Xc background. ex.

propagator

→ outgoing jet has low virtuality (sensitive to soft physics)

scale of outgoing jet :

non-local for b-quark residual momentum: shape function

Factorization Korchemsky, Sterman (’94)

H. Kawamura, “QCD for B physics”

• Shape function: new non-perturbative object

• Strategies for extracting |Vub| from B → Xulν

— ME of Light-cone non-local operator (similar to parton distribution)

1. Fit S(ω) from B → Xsγ, use it for B → Xulν

2. Use “shape-function free ” relation between B → Xsγand B → πlν

Lange, Neubert ,Paz (’05)

ex. Lange, Neubert ,Paz (’05)

: weight function (calculated at 2-loop)

: residual hadronic power corrections

Leibovich,Low,Rothstein (‘99)

B→ Xulν

H. Kawamura, “QCD for B physics”

LLR: Leibovich,Low,Rothstein (’99) ↔ 1-loop, without rhc LNP: Lange, Neubert ,Paz (’05)

|Vub| from BaBar data • ”SF free” analysis by Golubev, Skovpen, Luth : hep-ph/0702072v2

error: exp.(Xulν) +exp. (Xsγ) + th.

H. Kawamura, “QCD for B physics”

Exclusive hadronic decays

B D Effective Hamiltonian

• Naïve factorization

4-fermi operators

• QCD factorization holds for decays in which the spectator quark is absorbed into the final heavy meson.

from “color transparency” argument, but no μ dependence

B D

πT0,8

Фπ

FB→π

Beneke, Buchalla, Neubert, Sachrajda (’00)

H. Kawamura, “QCD for B physics”

B→Dπ• What must be shown?

— gluon exchange between (B,D) and π• Soft div. cancel among diagrams• Collinear div. absorbed into universal pion wave function

— shown up to 2-loop by BBNS (’00) All-order proof was given using SCET : Bauer et al. (‘01)

• SCET (Soft-Collinear effective theory)

HQET (soft) + collinear modes of quark & gluon + …

— systematic expansion in

— soft mode decouple with col. modes from power counting

→ Factorization proved at the leading power in operator language

light-like vector

H. Kawamura, “QCD for B physics”

, , B K • Key point: B → π form factor B

— end-point singularity ↔ soft nature of spectator quarks— “hard-collinear scale”

asymptotic form

• 3 formalisms have been developed in recent years

(1) QCDF (BBNS:’99 - )

• end-point singularity included in the form factor.• proof given by SCET.• power corrections partly included (parameterized)• NLO calculation completed

B → M1 M21

2 1 2

B M I IIM M M BF T T

B→ππ, Kπ

H. Kawamura, “QCD for B physics”

(2) SCET (BPSR ’04-) Bauer, Pirjol, Stewart, Rothstein

• end-point singularity included in the form factor• hard-collinear scale distinguished. → different formula from (1)• power correction neglected• modest → large number of fitted input

1

2 1 2

B M I IIM M M BF T T J

(3) PQCD (’01-) Li, Kuem, Sanda, Kurimoto, Mishima, Nagashima, , ,

• kT factorization + Sudakov →end-point suppressed• power counting different from (1) & (2).• more predictive power• NLO calculation has started

1 2

( ) ( , ) ( , ) ( , )S bM M BH e x b y b k b

b:impact parameter

B→ππ, Kπ

H. Kawamura, “QCD for B physics”

Theory (QCDF) vs. Data Beneke (Beauty06)

( )Br B PP ( )CPA B PP

• Good agreement with B → PP, PV data except “πK puzzle” and large direct CP of π+π- (input set S4: Beneke &Neubert (’03))

• BPRS, PQCD are also good.

• How can different formalism can give similar prediction?

H. Kawamura, “QCD for B physics”

CKM phase from data

123 8

0.58 0.09

65

S

63 6

0.03 0.09

69

S

123 8

0.13 0.19

65

S

Average:

3 (68 4)

3 (61 5)

UT fit

Beneke (Beauty06)

H. Kawamura, “QCD for B physics”

B meson LCWF

• Operator definition

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

momentum rep.

0

tn

k v momentum of light quark (at tree level)

Complicated object which contains soft + “hard” dynamics

• In SCET soft fieldscol. fields

• Hard radiative tail

( )lo(

g) sB iFaf

m

ww

w-:

0( )j

Bdwwf w¥

=- ¥ò

H. Kawamura, “QCD for B physics”

Operator relations Kodaira, Tanaka, Qiao, HK (’01)

HQ symmetry + EOM

→

3-body ops.

( ( )) ( ) ( )( ) BWWBB

gf f wwf w = +“Solution”

( )2

( ) (2 )2

WWB iF

wf w q w= L -

L“Wandzura-Wilczek approx.”

( ) 0 ( )vg

B q hG B vf :

— “Twist = Dimension - Spin” is not a good quantum number

B bm mL = -

— Higher dim. operators appear in IR region (at large tΛ)

v vvh h 0vq D v Dh �

“enhanced power correction” Lee,Neubert,Paz (’06) Shape Function

H. Kawamura, “QCD for B physics”

Radiative corrections

( )( )

( )

( )

211-loop

0

22

5

loglog

5( , ) (1 )

2 2

log

l

( ) (

4

0 ( )1

1

1

o

2

12

1

1

01

g

)

2 UVUV

IR IR

s

R

F

I

B

Uv

V

Ct d

B

itit

it

i

vq tn n

t

h

t

a pf m x d

x g

m

ex

pm

m

m

e e

x

e

e x

e

x

+

éì æ öï ÷ï çê ÷= - + + + -çí ÷ê ç ÷ï çè øêïîëüæ ö æ öé ù ï÷ ÷ïç çê ú÷ ÷+ - - +ç ç ý÷ ÷ç çê ú ÷÷ çç ï- è øè øë û ïþ

æ ö÷ç- - -çççè ø+

ò%

52( ) (00 ( ) ( ) (IR pole, higher-dim ps )) o .v B v Oq vt Dn tn hx g ù÷ +ú÷ ×

÷ û

su

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

• Cusp singularity2log ( )it

• non-analytic at t=0 1

( )t

→ hard radiative tail

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

1-loop

MSEe= gm m

• UV & IR structure different!

H. Kawamura, “QCD for B physics”

Evolution equation

Consistent with Lange & Neubert (’03)

H. Kawamura, “QCD for B physics”

Operator product expansion

• Separation of UV & IR behavior ⇒ OPE• Interpolate B meson LCWF to HQET parameters

RG evolution for LCWF

2

1 1,

UV UVe e

1

IRe

RG evolution for local ops.

( , ) ( , ) ( )( )0Bi

iiOt C t B vf m m m+ =å% %

many higher-dim. ops

expansion parameter

1-loop matching

H. Kawamura, “QCD for B physics”

• Lee & Neubert PRD72(’05)094028 : Cut-off scheme up to dim-4 ops.

Operator product expansion (cont’d)

vq hG vqD hGUV UV( , , ) ( , , 0 ( ))) (B ii

it C t O B vf m m mL L=å% %

• Our calculation (HK & K.Tanaka)

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

it C t B vOf m m m=å% %

→ OPE + exponential ansatz “hybrid model”

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

{ }, v vDq hD qG hG G+

H. Kawamura, “QCD for B physics”

Calculation

• 2-point + 3-point functions with non-local operators

• Operator identification calculation in x-space

• Keep gauge invariance explicitly background method + Fock-Schwinger gauge

(C)1

(C) (C)

0 (( ) (0 ))x A x A x duux G uxr

m rm mm Þ == ò

50

( )P exp ( ) (0)t

vq tn ig d n A n n hmml l g

æ ö÷ç ÷ç ÷è øò

→ decouple from Wilson line

H. Kawamura, “QCD for B physics”

dim.3

dim.4

dim.5

ResultMS

Ee= gm m

H. Kawamura, “QCD for B physics”

Matrix elements

dim.3

dim.4

B bm mL = -

dim.5 (covariant tensor formalism)

“Chromo-electronic”

“Chromo-magnetic”

: decay constant

H. Kawamura, “QCD for B physics”

LCWF from OPE

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

dim.3

dim.4

dim.5

• Represented by HQET parameter: ↔ Lattice, QCD sum rule

Evolution & phenomenological studies underway!

H. Kawamura, “QCD for B physics”

Summary

• Understanding of QCD effects in B physics has been largely improved in recent years.

• B meson wave function for exclusive B decay is quite different from pion wave function.

• OPE for bilocal operator for B meson LCWF to dim.5 → expressed by 3 HQET parameters

1/ t different UV & IR structures

, , E H

• Model-independent study of B meson LCWF is underway.

• Similar analysis for shape function is ongoing.