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Revisiting the B → πρ, πω Decays in the Perturbative QCD Approach beyond the leading order

By Zhou Rui(周锐 )

Collaborator: Cai-Dian Lu, Xiang-dong Gao

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Outline

MotivationTheoretical framework Perturbative QCD

ApproachNumerical ResultsSummary

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Motivation

The B → πρ decays are useful to determine the CKM angle .

the CP asymmetries are sensitive to high order contributions.

It is necessary to calculate the NLO corrections to those channels in order to improve the reliability of the theoretical predictions.

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Theoretical framework

Effective Hamiltonian is the basic tool to study B physics

are Wilson coefficients are Effective operators

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tree operators

QCD-penguin and QED-penguin

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Magnetic Penguins

bGqmg

Q

bFqme

Q

bs

g

b

)1(8

,)1(8

528

527

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The amplitude of is

The key is to tackle :

Naïve factorizationGeneralized FactorizationQCD factorization (QCDF)Soft-collinear effective theory (SCET)Perturbative QCD approach (PQCD)…

32MMB

BOMMCVG

BHfΑ iii

iCKM

Feff )()(

232

BOMM i )(32

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Picture of PQCD Approach--kT 因子化

Six quark interaction inside the dotted line

4-quark operator

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The End-point singularity (x→0,1). Introducing the transverse momentum of the light quark can remove the end-point singularities of hard kernel.

])(][)1[(

1

)1(

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312

3123

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4331 TTBTBB kkMxxkMxMxxx

the Sudakov form factor Large double logarithms KT resummation –-Sudakov form factor

Suppressed the long distance contributions Improve the applicability of PQCD

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)(log2 Ts k

large double logarithms

summed by the threshold resummation,and they lead to St(x) which smears the the end-point singularities on x ,we parameteried this term as below:

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3.0,)]1([)1(

)23(2)(

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cxxc

cxS c

c

t

)(log2 xs

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In pQCD approach ,the end-point divergence was removed effectively . the non-perturbative contributions were absorbed into the meson wave functions ,and the perturbative contributions can be calculated in the hard kernel .the calculation is reliable. In this frame ,the amplitude can be written as[PPNP51,85]

Ф： universal H ： process dependent

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Feynman diagrams which may contribute at leading order to B → πρ, πω decays

Calculate in leading order (LO)

Calculate in next-to-leading order (NLO)

we add two sorts of subleading corrections which include:

1. the NLO Wilson coefficient , the NLO Sudakov factor.

2. the NLO hard kernel contains the vertex corrections; the quark-loop and the chromo-magnetic penguin contributions.

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)()()( 2)1()0()0(sss HHH

)1(H

Feynman diagrams for NLO contributions: the

vertex corrections (a-f); the quark-loops(g-h) and the chromo-magnetic penguin contributions (i-j).

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Numerical Results

♫CP averaging branching ratios

♫Direct CPV

♫Mixing induced CPV

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The CP-averaged branching ratio (10^-6)

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The scale dependence of in LO and NLO pQCD

)/( 00 BBBr

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The pQCD predictions for the direct CP-violating asymmetries (in units of %)

||*

*

T

P

VV

VVz

udub

tdtb )arg(*

*

udub

tdtb

VV

VV )arg(

T

P

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The pQCD predictions for the CP-violating parameters S_f and the total CP-violating parameters A_CP of B0 → π0ρ0, π0ω (in units of %)

)cos()sin(

))(())((

))(())(()(

00

00

mtAmtS

ftBftB

ftBftBtA

dirCPf

CP

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The LO and NLO pQCD predictions for the CP-violating parameters Cf , Sf ,∆C and ∆S (f = π−ρ+) of B→ π±ρ∓ (in units of %)

Summary We calculate the branching ratios and CP-saymmetries

of the B → πρ, πω decays in the perturbative QCD factorization approach up to the NLO contributions。

NLO correction have significant effects on some of the decay channels, most our NLO predictions agree well with the measured values。

The NLO corrections play an important role in modifying direct CP asymmetries。

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Thank you for your attention !

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C=0.3 comes from the best fit to the next-to-leading-logarithm threshold resummation in moment space.（由mellin变换决定的）

KT regularization scheme The vertex corrections can be absorbed into the

redefinition of the Wilson coefficients by adding a vertex-function to them

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)(ia

)(MVi

The contribution from the so-called “quark-loops” is a kind of penguin correction with the four quark operators insertion . For the b → d transition ,the effective Hamiltonian can be written as (PRD72 114005)

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)'')()1()(,(22

52*

,, '

qTqbTdlCVVG

H aaqsqdqb

tcuq q

Fqleff

The magnetic penguin is another kind penguin correction induced by the insertion of the operator O8g The corresponding weak effective Hamiltonian contains the b → dg transition can be written as

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jaa

ijibs

g

geffgtdtb

Fmpeff

bGTdmg

O

OCVVG

H

)1(

8

2

528

88*

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The NLO decay amplitude

the NLO contributions can be included in a simple way:

the vertex corrections have been absorbed into the redefinition of the Wilson coefficient )(ia

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Wilson coefficients and Effective operators

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the hard-scattering form factor ζJ is relatively large and comparable with the soft form factor ζ. Besides, this term has a large Wilson coefficient.

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