B → ππ semileptonic form factors
from QCD Light-Cone Sum Rules
Alexander Khodjamirian
UNIVERSITÄT
SIEGEN
Theoretische Physik 1
RESEARCH UNIT
f
qe
t
Mini-workshop on multiparticle final states in B decays
February 21, 2017, Siegen
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
1 / 26
B → ππ form factors:
•
•
used in B → ππ`ν` for |Vub | determination
determine the rich set of observables in B → ππ`ν` ,
S. Faller, T. Feldmann, A. Khodjamirian, T. Mannel and D. van Dyk, 1310.6660 [hep-ph]
•
provide factorizable parts of B → 3π nonleptonic amplitudes
Krä nkl, T. Mannel and J. Virto, 1505.04111 [hep-ph]
not yet accessible in the lattice QCD
QCDF
SCET
HQET
2
MB
=
w
Heavy meson ChPT ⊕ dispersion relations
k2
•
0.
7
in QCD-based effective theories:
0.
5
X. W. Kang, B. Kubis, C. Hanhart and U. G. Meißner,
w
=
0.
3
QCD factorization:
w
•
=
1312.1193 [hep-ph];
P. Böer, T. Feldmann and D. van Dyk, 1608.07127 [hep-ph].
4Mπ2
0
q2
2
qmax
we calculate B → 2π from QCD Light-Cone Sum Rules
accessible kinematical region: dipion with a small invariant mass and large recoil:
k 2 . 1 GeV2 , 0 ≤ q 2 ≤ 12-14 GeV2 .
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
2 / 26
Outline
the form factors for B̄ 0 → π + π 0 `− ν` ,
dipion with isospin 1, L = 1, 3, , ..
Two different LCSR methods:
•
•
with dipion distribution amplitudes (2π-DAs)
Ch. Hambrock, AK, 1511.02509 [hep-ph]
with B-meson distribution amplitudes (B-DAs)
[S.Cheng, AK, J.Virto, 1701.01633 [hep-ph] ]
Questions to address:
•
•
how important are L > 1 partial waves of 2π states?
dominance of ρ -resonance, “nonresonance” background
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
3 / 26
LCSRs with dipion distribution amplitudes
Ch. Hambrock, AK, 1511.02509 [hep-ph]
The correlation function:
k = k1 + k2 , k = k1 − k2 , p = (k + q)
Πµ (q, k1 , k2 ) =
R
= i d 4 x eiqx hπ + (k1 )π 0 (k2 )|T {ū(x)γµ (1−γ5 )b(x), imb b̄(0)γ5 d(0)}|0i
= iµαβρ q α k1β k2ρ Π(V ) + qµ Π(A,q) + kµ Π(A,k ) + k µ Π(A,k ) ,
the invariant amplitudes Π(V ),(A,q),..) (p2 , q 2 , k 2 , q · k̄ ),
near light-cone dominance x 2 ∼ 0
at q 2 mb2 (b-quark virtual), k 2 mb2
LO diagram:
hb(x)b̄(0)i → Sb (x, 0)
dipion DA’s
vacuum → hπ + π 0 |
hadronic matrix elements
of bilocal ū(x)d(0)
operators
with ρ-resonances "embedded"
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
4 / 26
Dipion light-cone DAs
introduced for γ ∗ γ → 2π processes
M. Diehl, T. Gousset, B. Pire and O. Teryaev, (1998);
D. Müller, D. Robaschik, B. Geyer, F.-M. Dittes and J. Horejsi, (1994); M. V. Polyakov, (1999).
twist-2 DAs:
Z1
√
2
hπ + (k1 )π 0 (k2 )|ū(x)γµ [x, 0]d(0)|0i = − 2kµ du eiu(k ·x) ΦI=1
k (u, ζ, k ) ,
0
√
k1µ k2ν − k2µ k1ν
hπ (k1 )π (k2 )|ū(x)σµν [x, 0]d(0)|0i = 2 2i
2ζ − 1
0
+
Z1
2
du eiu(k ·x) ΦI=1
⊥ (u, ζ, k ) ,
0
•
the “angular” variable: ζ = k1+ /k + , 1−ζ = k2+ /k + , ζ(1 − ζ) ≥
q · k̄ =
•
1
(2ζ − 1)λ1/2 (p2 , q 2 , k 2 ) ,
2
.
2
(2ζ − 1) = (1 − 4mπ
/k 2 )1/2 cosθπ ,
normalization conditions → pion timelike form factors ,
Z1
du
0
•
•
in dipion c.m.
2
mπ
k2
n F em (k 2 )
n ΦI=1 (u, ζ, k 2 )
π
k
= (2ζ − 1)
I=1
2
Fπt (k 2 )
Φ⊥ (u, ζ, k )
pion e.m. form factor
pion “tensor” form factor
2 contain Im part
Φ⊥,k (u, ζ, k 2 ) at k 2 > 4mπ
Fπem (0) = 1,
Alexander Khodjamirian
•
⊥
“tensor” charge of the pion Fπt (0) = 1/f2π
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
5 / 26
Result for the correlation function
at LO, twist-2 accuracy:
Z1
√
Πµ (q, k1 , k2 ) = i 2mb
0
nh
du
(q · k )kµ −
2
(q + uk )2 − mb
+iµαβρ q α k1β k2ρ
„
«
(q · k ) + uk 2 k µ
i Φ (u, ζ, k 2 )
o
⊥
− mb kµ Φk (u, ζ, k 2 ) .
2ζ − 1
read off invariant amplitudes: Π(V ) , Π(A,k ) , Π(A,k ) , and Π(A,q) = 0
transform to a form of dispersion integral in the variable p2 :
m2 −q 2 ū+k 2 u ū
s(u) = b
, e.g.,
u
Π
(V )
2
2
2
Z∞
(p , q , k , ζ) =
ds
s − p2
mb2
„
|
du
ds
«
Φ⊥ (u, ζ, k 2 )
.
2ζ − 1
{z
}
ρOPE(V ) (s, q 2 , k 2 , ζ)
similar for Π(A,k ) , but not for Π(A,k ) ,
due to kinematical singularity in the Källen function λ1/2 (p2 , q 2 , k 2 )
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
6 / 26
Hadronic dispersion relation
the hadronic sum in B-meson channel (unitarity condition):
(V )
Πµ (q, k1 , k2 ) =
hπ + π 0 |ūγµ b|B̄ 0 (p)ifB mB2
mB2
−
p2
+
X hπ + π 0 |ūγµ b|Bh ihBh |imb b̄γ5 d|0i
mB2 − p2
Bh
|
{z
}
R∞
quark-hadron duality approximation ⇒
sB
0
OPE,(V )
ρ
ds µ
(s)
s−p2
B → ππ matrix element vs form factors:
+
0
µ
0
2
2
ihπ (k1 )π (k2 )|ūγ (1 − γ5 )b|B̄ (p)i = −F⊥ (q , k , ζ) q
4
k 2 λB
µαβγ
i
qα k1β k2γ
1 “ µ
4(q · k )(q · k ) µ
4k 2 (q · k ) µ ”
2
2
+Fk (q , k , ζ) √
k +
q
,
k −
λB
λB
k2
q
2 q2 “ µ
qµ
k · q µ”
2
2
2
2
+Ft (q , k , ζ) q
+ F0 (q , k , ζ) p
k −
q
λ
q2
2
B
q
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
7 / 26
LCSRs for the form factors
applying quark-hadron duality and Borel transformation:
F⊥ (q 2 , k 2 , ζ)
mb
√ √
=√
2
2
2f
m
k λB
B B (1 − 2ζ)
Fk (q 2 , k 2 , ζ)
mb
√
=√
2fB mB2 (1 − 2ζ)
k2
Z1
2
2
2
m −q ū+k
m
B− b
du
uM 2
Φ⊥ (u, ζ, k 2 ) e M 2
u
2 uū
u0 (s0B )
Z1
2
2
2
m −q ū+k
m
B− b
du “ 2 2 2 2 ”
uM 2
mb −q +k u Φ⊥ (u, ζ, k 2 ) e M 2
2
u
2 uū
u0 (s0B )
in both sum rules only the chiral-odd twist-2 DA contributes
an additional relation between the axial-current form factors:
√
1
√ (mB2 −q 2 −k 2 )F0 (q 2 , k 2 , ζ) = Ft (q 2 , k 2 , ζ)+2
λB
Alexander Khodjamirian
k2
p
q 2 (2ζ − 1)
√
Fk (q 2 , k 2 , ζ) .
λB
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
8 / 26
The sum rule for the form factor Ft
S.Cheng, AK, J.Virto, [paper in preparation]
using a different correlation function:
Z
i d 4 x eiqx hπ + (k1 )π 0 (k2 )|T {ū(x)imb γ5 b(x), imb b̄(0)γ5 d(0)}|0i .
= Π(5) (p2 , q 2 , k 2 , q · k̄ )
B → ππ matrix element of the pseudoscalar current
relating to the divergence of the axial-vector current:
p
0
hπ + (k1 )π 0 (k2 )|ū(x)imb γ5 b(x)|B i = Ft (q 2 , k 2 , ζ) q 2
following the same method,
LCSR with twist-2 accuracy, only the chiral-even DA enters:
m2
Ft (q 2 , k 2 , ζ)
p
=√ b
2fB mB2
q2
Alexander Khodjamirian
Z1
u0
2
2
2
m −q ū+k
m
”
B− b
du “ 2
uM 2
mb − q 2 + k 2 u 2 Φk (u, ζ, k 2 ) e M 2
2
u
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
2 uū
,
9 / 26
The structure of dipion DAs
[M. V. Polyakov, Nucl. Phys. B 555 (1999) 231.]
double expansion in Legendre and Gegenbauer polynomials:
6u(1−u)
⊥
f2π
Φ⊥ (u, ζ, k 2 ) =
Φk (u, ζ, k 2 ) =
∞
P
n+1
P
3/2
n=0,2,.. `=1,3,..
6u(1 − u)
∞
P
⊥ (k 2 )C
Bn`
n
n+1
P
n=0,2,.. `=1,3,..
k
(0)
(2u − 1)βπ P`
3/2
Bn` (k 2 )Cn
“
(0)
(2u − 1)βπ P`
2ζ−1
βπ
“
”
,
2ζ−1
βπ
”
,
Gegenbauer moments, (multiplicativey renormalizable)
•
•
⊥,k
2
Bn` (k 2 ) - complex functions at k 2 > 4mπ
k
B01 (k 2 ) = Fπ (q 2 ) , pion timelike form factor
instanton vacuum model for the coefficients,
2
n = 0, 2, 4, valid at small k 2 ∼ 4mπ
[M. V. Polyakov and C. Weiss, (1999)]
!
!
7
k2
k2
⊥ 2
=1+
=
1−
, B23 (k ) =
1+
,
36
30M02
36
30M02
!
!
!
5k 2
77
k2
11
k2
⊥ 2
⊥ 2
1−
, B43 (k ) =
1−
, B45 (k ) =
1+
.
168M02
675
630M02
135
56M02
⊥ 2
B01 (k )
⊥
2
B41 (k ) =
11
225
k2
⊥ 2
, B21 (k )
12M02
7
⊥
2
f2π
= 4π 2 fπ
/3M0 ' 650 MeV, where fπ = 132 MeV is the pion decay constant.
2 < k 2 . 1 GeV2 , the other FFs at k 2 ∼ 4m2
only Ft can be predicted at 4mπ
π
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
10 / 26
Sum rules for partial waves
The form factors expanded in partial waves:
F⊥,k (q 2 , k 2 , ζ) =
∞ √
P
`=1
(1)
(`)
2` + 1F⊥,k (q 2 , k 2 )
(m)
ζ ∼ cos θ, Pl
P` (cos θπ )
sin θπ
,
-the (associated) Legendre polynomials
sum rules for separate partial waves
√
•
•
•
√
n+1
X
n=0,2,.. `0 =1,3,..
k2
mb3
X
n+1
X
⊥
2f2π
mB2 fB
n=0,2,4,..
`0 =1,3,..
⊥
2f2π
√
(`)
Fk (q 2 , k 2 ) = √
k2
λB mb m2 /M 2
e B
mB2 fB
(`)
F⊥ (q 2 , k 2 ) = √
2
emB /M
2
X
⊥
2 ⊥ 2
2
2 B
I``0 Bn`
0 (k )Jn (q , k , M , s0 ) ,
k
⊥
2
2
2
2 B
I``0 Bn`
0 (k )Jn (q , k , M , s0 ) ,
I``0 - integrals over Legendre polynomials,
⊥,k
Jn
3/2
- the Borel-weighted integrals over Cn
(2u − 1)
in the limit of asymptotic DA, (B01 6= 0, Bn>0,` = 0),
only P-wave form factors are 6= 0
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
11 / 26
Numerical results
P-wave form factors: (only twist-2)
(`=1)
F⊥
4.0
(`=1)
2
(q 2 , kmin
)
3.5
3.5
3.0
3.0
2.5
2.5
2.0
0
2
4
Fk
4.0
6
8
10
12
2.0
0
2
(q 2 , kmin
)
2
4
6
8
q 2 [GeV 2 ]
10
12
q 2 [GeV 2 ]
P-wave dominance: ratios of F - and P-wave form factors
(`=3)
R⊥
0.03
(`=3)
2
(q 2 , kmin
)
0.02
0.02
0.01
0.01
0.00
0.00
-0.01
-0.01
-0.02
-0.03
Rk
0.03
2
(q 2 , kmin
)
-0.02
0
2
4
6
8
10
q 2 [GeV 2 ]
12
-0.03
0
2
4
6
8
10
q 2 [GeV 2 ]
12
- - - - uncertainties from the variation of M 2 .
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
12 / 26
How much B → ρ contributes to the B → 2π?
resonance model for the B → ππ P-wave (` = 1) form factors:
√ (`=1) 2 2
3F⊥
(q , k )
gρππ
V B→ρ (q 2 )
√ √
= 2
+ ...
mρ − k 2 − imρ Γρ (k 2 ) mB + mρ
k 2 λB
and
√
(`=1)
3Fk
√
(q 2 , k 2 )
k2
=
gρππ
(mB + mρ )A1B→ρ (q 2 ) + ...
mρ2 − k 2 − imρ Γρ (k 2 )
Γρ (k 2 ) =
2
mρ
k2
2
k 2 −4mπ
2 −4m2
mρ
π
!3/2
2
θ(k 2 − 4mπ
)Γtot
ρ ,
using the definition of B → ρ FFs:
∗(ρ) β γ
hρ+ (k )|ūγµ (1 − γ5 )b|B̄ 0 (p)i = µαβγ α
∗(ρ)
−iµ
p k
2V B→ρ (q 2 )
mB + mρ
(mB + mρ )AB→ρ
(q 2 ) + ...
1
using the LCSRs for V and A1 with ρ -meson DA ("quenched")
P. Ball and V. M. Braun,(1997)
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
13 / 26
Numerical estimates
[F
(`=1) 2 2
(q ,k
)](ρ)
[F
min
⊥
(`=1) 2 2
(q ,k
)](LCSR)
⊥
min
(`=1) 2 2
(q ,k
)](ρ)
min
k
(`=1) 2 2
(q ,k
)](LCSR)
min
k
[F
[F
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0
2
4
6
8
10
12
0.6
q 2 [GeV 2 ]
0
2
4
6
8
10
12
q 2 [GeV 2 ]
Relative contribution of ρ-meson to the B → π + π 0 P-wave form factors
(`=1)
F⊥
(`=1)
2
(q 2 , kmin
) (left panel) and Fk
2
(q 2 , kmin
) (right panel) from LCSRs.
Dashed lines - the uncertainty due to the variation of the Borel parameter.
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
14 / 26
Results for form factors Ft and F0
PREIMINARY
250
50
q 2 Ftl =1 (s)
200
60
q 2 Ftl =1 q 2 
150
40
100
30
20
50
10
0
0
2
4
6
8
10
0
0.2
0.4
0.6
q 2 (GeV 2 )
0.8
1.0
1.2
1.4
k 2 (GeV 2 )
additional relation between form factors yields F0
60
50
q 2 F 0l =1 q 2 
40
30
20
10
0
0
2
4
6
8
10
q 2 (GeV 2 )
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
15 / 26
LCSRs with B-meson distribution amplitudes
[S.Cheng, AK, J.Virto, 1701.01633 [hep-ph] ]
LCSRs with B-meson DA and ūγµ d interpolating current
originally introduced to calculate B → P, V form factors,
A.K., N. Offen, Th. Mannel (2005),(2007); "SCET sum rules", F. De Fazio, Th. Feldmann, T.Hurth (2006);
NLO corr.to B → π ,Y-M. Wang, Y-L.Shen (2015)
d
k
B̄ 0
PB
B̄ 0
u
q
b
q
PB
(a)
.
The correlation function:
Z
Fµν (k , q) = i
k
(b)
d 4 xeik ·x h0|T{d̄(x)γµ u(x), ū(0)γν (1 − γ5 )b(0)}|B̄ 0 (q + k )i,
= εµνρσ q ρ k σ F(ε) (k 2 , q 2 ) + igµν F(g) (k 2 , q 2 ) + iqµ kν F(qk ) (k 2 , q 2 ) + ...
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
16 / 26
B-meson DAs
defined in HQET:
h0|d̄α (x)[x, 0]hv β (0)|B̄v0 i = −
ifB mB
4
Z∞
˘
dωe−iωv ·x (1 + /
v ) φB
+ (ω)
0
+
h0|q¯2 α (x)Gλρ (ux)hv β (0)|B̄ 0 (v )i =
fB mB
4
Z∞
Z∞
dω
0
"
B
φB
− (ω) − φ+ (ω)
2v · x
!
x γ5
/
¯
βα
,
dξ e−i(ω+uξ)v ·x
0
(
“
”
× (1 + /
v ) (vλ γρ − vρ γλ ) ΨA (ω, ξ) − ΨV (ω, ξ) − ....
#
,
βα
key parameter: inverse moment
1
=
λB
Alexander Khodjamirian
Z∞
dω
φB
+ (ω)
ω
.
0
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
17 / 26
Accessing B → ππ form factors
OPE diagrams ⇒ invariant amplitudes ⇒ dispersion form in k 2 :
OPE 2
F(ε)
(k , q 2 ) = fB mB
φB
+ (σmB )
∞
Z
dσ
σ̄(s − k 2 )
0
+ {3 − particle DAs}
2
s = s(σ, q 2 ) = σmB
− σq 2 /σ̄ ,
σ̄ ≡ 1 − σ
hadronic dispersion.relation and unitarity:
F(ε) (k 2 , q 2 ) =
Z∞
1
π
ds
ImF(ε) (s, q 2 )
s − k2
.
2
4mπ
Z
2 ImFµν (k , q) =
dτ2π h0|d̄γµ u |π + π 0 i hπ + π 0 |ūγν (1 − γ5 )b|B̄ 0 (q + k )i + · · · ,
{z
}
{z
}|
|
B → 2π (` = 1) form factors
Fπ (s)
k1
k
d
d
π+
k
d
u
π0
u
B̄ 0
PB
Alexander Khodjamirian
b
B̄ 0
q
PB
u
b
k2
+···
q
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
18 / 26
Resulting sum rules
(`=1)
e,g,. for the form factor F⊥
Z
s02π
ds e−s/M
2
4mπ
of the vector current
√
s [βπ (s)]3 ?
(`=1)
√ Fπ (s) F⊥
√
(s, q 2 )
4 6π 2 λ
#
" Z 2π
B
σ0
2
2 φ+ (σmB )
+ mB ∆V BV (q 2 , σ02π , M 2 ) ,
= fB mB
dσ e−s(σ,q )/M
σ̄
0
2
2
σ02π - the solution of σmB
− σq 2 /σ̄ = s02π , three-particle DA contribution ∆V BV cumbersome
similar sum rules for all other P-wave B → 2π form factors
not a direct calculation, given the shape of the B → 2π form factors,
these sum rules can provide normalization
the complex phases of B → 2π FFs and Fπ (s) equal at low s:
a usual Watson theorem
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
19 / 26
Probing ρ-resonance models
ansatz for the B → ππ FF:
inspired by experimental fit of Fπ (s)
√
+
0
gρ0 ππ
mρ2 0
−
k2
Model 1:
− imρ0 Γρ
•
(`=1)
3F⊥
(q 2 , k 2 )
V B→ρ (q 2 )
gρππ
√ √
= 2
2 − im Γ (k 2 ) m + m
2
m
−
k
ρ ρ
ρ
k λB
B
ρ
0 (k 2 )
00
gρ00 ππ
V B→ρ (q 2 )
V B→ρ (q 2 )
+ 2
2
2
mB + mρ0
mρ00 − k − imρ00 Γρ00 (k ) mB + mρ00
V B→ρ (q 2 ) from LCSR with ρ-meson DAs (in which Γρ = 0)
taken from A.Bharucha, D.Straub and R.Zwicky,1503.05534
•
neglect
ρ00
and substitute in LCSR
⇒ an apppreciable contribution of ρ0 is consistent with the fit,
Model 2 :
•
Alexander Khodjamirian
•
all three resonances taken into account
their proportion taken as in Fπ (s) Belle fit
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
20 / 26
B → 2π (` = 1) FFs: dipion mass dependence
50
50
---•-
40
30
30
20
20
10
10
0
0.0
0.2
0.4
0.6
0.8
1.0
50
1.2
1.4
0
0.0
30
20
20
10
10
Alexander Khodjamirian
0.2
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
1.2
1.4
0
0.0
1.2
1.4
1.2
1.4
---•-
40
30
0
0.0
0.2
50
---•-
40
---•-
40
0.2
0.4
0.6
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
0.8
1.0
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B → 2π (` = 1) FFs: q 2 -dependence at small k 2
---•___
8
6
---•___
8
6
4
4
2
2
0
0
0
5
2
4
6
8
10
12
---•-
0
4
6
8
10
12
2
4
6
8
10
12
---•-
8
4
2
6
3
4
2
2
1
0
0
0
2
Alexander Khodjamirian
4
6
8
10
12
0
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
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Conclusion and outlook
B → ππ form factors are calculable from LCSRs with dipion DAs
at small dipion mass and large recoil
sum rules provide estimates of higher partial waves,
allow to go beyond the ρ-resonance contribution
LCSRs with B meson DAs yield additional independent
constraints on the B → ππ form factors
more accurate correlation functions:
• twist 3-4 dipion DAs
•
model-independent Bnl (k 2 ) at k 2 .1.0-1.5 GeV2
•
NLO gluon radiative corrections
⊥,k
(dispersion relations, Omnes representation)
extending these methods to B → π + π − , π 0 π 0 , K π, K K̄ FFs,
including dimesons in S, D, ..-waves (` = 0, 2, ..)
LCSR for S-wave K π state in B → K π, U. G. Meißner, W. Wang, (2014)
the results can be used to assess models used for experiments
B → ππ`ν` and B → K π``
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
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Backup
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
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LCSRs for B → ρ form factors
e.g., [ P. Ball and V. M. Braun, Phys. Rev. D 55 (1997) 5561]
LCSRs for B → ρ form factors (Γρ = 0)
in terms of the ρ-meson DAs in the twist-2 approximation:
2
V B→ρ (q 2 ) =
AB→ρ
(q 2 ) =
1
(mB + mρ )mb ⊥ mB2
fρ e M
2mB2 fB
mb3
2(mB + mρ )mB2 fB
m2
B
fρ⊥ e M 2
Z1
u0
Z1
u0
du (ρ)
−
φ (u) e
u ⊥
2 uū
m2 −q 2 ū+mρ
b
uM 2
,
„
« mb2 −q 2 ū+mρ2 uū
q 2 − mρ2 u 2
du (ρ)
−
uM 2
φ
(u)
1−
e
.
⊥
u2
mb2
both sum rules determined by the chiral-odd ρ-meson DA:
` ∗(ρ)
∗(ρ) ´
hρ+ (k )|ū(x)σµν [x, 0]d(0)|0i = −ifρ⊥ µ kν − kµ ν
Z1
(ρ)
dueiuk ·x φ⊥ (u) ,
0
the Gegenbauer polynomial expansion:
0
(ρ)
φ⊥ (u)
= 6u(1 − u) @1 +
1
X
(ρ)⊥ 3/2
an Cn (2u
− 1)A ,
n=2,4,,..
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
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Input for the pion vector form factor
Measurement of τ → π − π 0 ντ decay [ Belle Collab. (2008)];
Fπ (s) fitted to the combination of three ρ-resonances,
Fπ (s) =
iφγ BW GS (s)
BWρGS (s) + |β|eiφβ BWρGS
0 (s) + |γ|e
ρ00
1 + |β|eiφβ + |γ|eiφγ
,
where the Gounaris-Sakurai model was adopted
GS
BWR (s) =
2
mR
+ mR ΓR d
,
√
s ΓR (s)
2 − s + f (s) − i
mR
Resonance
mR (MeV)
ΓR (MeV)
weight factor
ρ
774.6 ± 0.2 ± 0.5
148.1 ± 0.4 ± 1.7
1.0
ρ0
1446 ± 7 ± 28
434 ± 16 ± 60
|β| = 0.15 ± 0.05+0.15
−0.04
φβ = (202 ± 4+41
)o
−8
ρ00
1728 ± 17 ± 89
164 ± 21+80
−26
|γ| = 0.037 ± 0.006+0.065
−0.009
φγ = (24 ± 9+118
)o
−28
Alexander Khodjamirian
B → ππ semileptonic form factors from QCD Light-Cone Sum Rules
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