Introduction
Results
Summary and Outlook
B → D (∗) Form Factors from QCD Sum Rules
with B-Meson Distribution Amplitudes
Sven Faller
(with A. Khodjamirian, C. Klein and Th. Mannel)
Theoretical Physics 1
University of Siegen
EF Orsay, 14. November 07
Sven Faller
B → D (∗) form factors
Introduction
Results
Summary and Outlook
Motivation
|Vcb | from exclusive B → D (∗) decay
G2 |Vcb |2 mB5 q 2
dΓ(B̄ → Deν̄e )
= F
(w − 1)3 r 3 (1 + r )2 FD (w)2 ,
dw
48π 3
G2 |Vcb |2 mB5 p
dΓ(B̄ → D ∗ eν̄e )
= F
(w − 1)(w + 1)r ∗3 (1 − r ∗ )2
dw
48π 3
»
–
4w 1 − 2wr ∗ + r ∗2
· 1+
FD ∗ (w)2 .
∗
2
w +1
(1 − r )
r = mD /mB , r ∗ = mD ∗ /mB
FD and FD ∗ combinations of form factors h+ (w), h− (w)
and hV (w)
v µ = (p + q)µ /mB and v 0µ = pµ /mD (∗) four-velocities
2
2
w = (v · v 0 ) = [mB2 + mD
(∗) − q ]/[2mB mD (∗) ]
1/mc , 1/mb -corrections to form factors, need full QCD
Sven Faller
B → D (∗) form factors
Introduction
Results
Summary and Outlook
new method: LCSR with B-meson distribution amplitudes (DA0 s)
form factors B → π, K and B → ρ, K ∗ [A. Khodjamirian, Th. Mannel
and N. Offen (2007)]
s
B
b
=⇒
B
b D, D *
p, r, K *
apply this technique for B → D (∗) form factors
p
=
c
B
B
()
b
D*
q
Sven Faller
B → D (∗) form factors
Introduction
Results
Summary and Outlook
Form Factors
standard definition
+
±
hD(p)|c̄γµ b|B̄(p + q)i = 2pµ fBD
(q 2 ) + qµ fBD
∗
hD ∗ (p)|V µ |B̄(p + q)i =
2V BD
µναβ ∗ν qα pβ
mB + mD ∗
form factors adjusted to Isgur-Wise limit
[Isgur and Wise (1989, 1990), Neubert (1994)]
hD(p)|V µ |B̄(p + q)i
= h+ (w)(v + v 0 )µ + h− (w)(v − v 0 )µ
√
mB mD
hD ∗ (v 0 , )|V µ |B̄(v )i
p
= hV (w)µναβ ∗ν vα0 vβ
∗
mB mD
Sven Faller
B → D (∗) form factors
Introduction
Results
Summary and Outlook
B → D (∗) correlation functions, contributions of two- and
three-particle B-meson DA0 s
p
c
B
h0|q̄α (x)[x, 0]hv β (0)|B̄v i =
»
Z ∞
ifB mB
−iωv ·x
−
dωe
(1 + /
v)
4
0

ff –
B
B
φ+ (ω) − φ− (ω)
B
· φ+ (ω) −
x γ5
/
2v · x
βα
B
b
q
finite c-quark mass, c-quark virtual
B-meson DA0 s defined in heavy-quark effective theory (HQET)
two particle DA0 s φB+ (ω) and φB− (ω) [Grozin and Neubert (1997), Braun
et al. (2004)]
B→D
current
iγ5
+
±
fBD
, fBD
form factor
h+ , h−
B → D∗
γµ
VBD ∗
Sven Faller
hV
model
φ+ =
ω
ω2
0
e−(ω/ω0 ) = ωω φB
−
0
ω0 = λB = 460 ± 110 MeV
B → D (∗) form factors
Introduction
Results
Summary and Outlook
Isgur-Wise Limit
f+ (q 2 ), f− (q 2 ) and V (q 2 ) → h+ (w), h− (w) and hV (w),
heavy quark limit
mB = mb + Λ̄, mD = mc + Λ̄
Borel-parameter M 2 = 2mc τ
(∗)
threshold s0D = mc2 + mc β0
√
fB,D(∗) → f̃B,D(∗) / mB,D(∗)
Isgur Wise limit
mc
mb,c → ∞, but m
= const. = κ2
b
h+ (w) = hV (w) = ξ(w) and h− (w) = 0
zero recoil point ξ(1) = 1
kinematic range for w
1≤w ≤1+
Sven Faller
[mB − mD (∗) ]2
2mB mD (∗)
B → D (∗) form factors
Introduction
Results
Summary and Outlook
Results for B → D ∗
sum rule for the form factor V BD
Isgur-Wise function: hV (w) =
∗
[Khodjamirian et al. (2005)]
√
2 mB mD ∗
BD ∗ (w)
mB +mD ∗ V
Isgur-Wise limit
hV (w) =
f̃B
β0 /w
Z
f̃D ∗
where κ =
0
q
 „ 2
«
ff »
„
«
–
Λ̄
ω κ
1 B
1
dω exp
−w +
·
φ− (ω) + 1 −
φB
(ω)
+
τ
2
τ
2w
2w
mc
mb
and ω0 =
β0
w.
three-particle contribution suppressed
Sven Faller
B → D (∗) form factors
Introduction
Results
Summary and Outlook
Isgur-Wise limit: hV (w)
1.4
hV HwL
HmB - mD* L2
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ +1
2 mB mD*
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
1.2
1.4
1.6
w
2
2
3 < M ≤ 6 GeV
Sven Faller
B → D (∗) form factors
1.8
Introduction
Results
Summary and Outlook
Results for B → D
Isgur-Wise h+ (w)
BD
h+ (w) =
fB mB mc
2f
2κmD
D
Z σ(s ,q 2 )
0
0
„
+

2 ff »
mc mB [mc + mB (κ2 − σ)] B
−s(σ, q 2 (w) + mD
·
exp
φ− (ω)
2 + m2 − q 2
2
M
σ̄ 2 mB
c
mc + mB (κ2 − σ)
σ̄
−
mB mc [mc + mB (κ2 − σ)]
2 + m2 − q 2
σ̄ 2 mB
c
«
B
φ+ (ω)
«
–
„
2
1
mc mB
2mB
σ̄[mc + mB (κ2 − σ)]
B
BD
−
−
+
Φ± (ω) + ∆h+
2
2
2
2
2
2
2
2
2
σ̄
σ̄ mB + mc − q
(σ̄ mB + mc − q )
Isgur-Wise limit
h+ (w) =
f̃B
f̃D
β0 /w
Z
dω exp
0
 „ 2
«
ff »
„
«
–
ω κ
Λ̄
1 B
1
−w +
·
φ− (ω) + 1 −
φB
(ω)
+
τ
2
τ
2w
2w
BD → 0
Isgur-Wise limit: three particle contribution ∆h+
Sven Faller
B → D (∗) form factors
Introduction
Results
Summary and Outlook
Isgur-Wise limit: h+ (w)
1.4
h+ HwL
HmB - mD L2
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ +1
2 mB mD
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
1.2
1.4
1.6
w
Sven Faller
B → D (∗) form factors
1.8
Introduction
Results
Summary and Outlook
Results for B → D
Isgur-Wise function h− (w)
BD
h− (w) =
fB mB mc
Z σ(s ,q 2 )
0
2f
2κmD
D

exp
2
−s(σ, q 2 (w) + mD
M2
0
»
·
mc mB [mc − mB (κ2 + σ)] B
φ− (ω)
2 + m2 − q 2
σ̄ 2 mB
c
„
+
mc − mB (κ2 + σ)
σ̄
„
−
ff »
mc mB [mc + mB (κ2 − σ)] B
·
φ− (ω)
2 + m2 − q 2
σ̄ 2 mB
c
1
σ̄
−
+
mB mc [mB (κ2 + σ) − mc ]
mc mB
2
σ̄ 2 mB
+
mc2
−
q2
2 + m2 − q 2
σ̄ 2 mB
c
−
2
(σ̄ 2 mB
+
mc2
−
BD
and ∆h−
→0
Sven Faller
B
φ+ (ω)
2
2σ̄mB
mc [mB (κ2 + σ) − mc
Isgur-Wise limit
h− (w) ≡ 0
«
B → D (∗) form factors
q 2 )2
–
«
B
BD
Φ̄± (ω) + ∆h−
Introduction
Results
Summary and Outlook
Summary and Outlook
Summary
new light-cone sum rules for B → D (∗) in terms of B-meson
DA0 s
obey Isgur-Wise limit for the form factors h+ , h− and hV
Outlook
1/mc and partial 1/mb -expansion of the Isgur-Wise
functions
BD
BD
ABD
1 , A2 and A3 form factors for the differential decay
rates dΓ(B̄ → D (∗) eν̄e ) → |Vcb |
dynamics at the zero-recoil point (w = 1)
perturbative αs -corrections
Sven Faller
B → D (∗) form factors