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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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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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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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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(
12
312
3123
23
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:
-12-
3.0,)]1([)1(
)23(2)(
21
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).
-16-
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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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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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