B → ππ form factors from QCD Light-Cone
Sum Rules
Alexander Khodjamirian
UNIVERSITÄT
SIEGEN
Theoretische Physik 1
RESEARCH UNIT
f
qe
t
"Continuous Advances in QCD" (CAQCD 2016), FTPI, Minneapolis, May 13, 2016
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
1 / 20
Quark flavour physics: current tensions between SM and experiment
b → u transition , |Vub | determination:
exclusive B → π`ν̄` vs inclusive B → Xu `ν̄` decays
figure taken from [J. A. Bailey et al. [Fermilab Lattice and MILC Collaborations], arXiv:1503.07839 [hep-lat].]
B → π form factors
from lattice QCD and
Light-Cone Sum Rules
need other exclusive b → u channels, e.g., B → ρ`ν̄` ,
"anomalies" in the observables of B → K ∗ `+ `− FCNC decays
need B → ρ, K ∗ form factors, calculated in "quenched" approximation
LCSRs see e.g., [ A.Bharucha, D.Straub and R.Zwicky,1503.05534]
for more precision need general B → ππ or B → K π form factors
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
2 / 20
B → 2π form factors:
to be used for |Vub | determination B → ππ`ν` ,
determine a rich set of observables in B → ππ`ν` ,
[S. Faller, T. Feldmann, A. Khodjamirian, T. Mannel and D. van Dyk, (2013)]
enter factorizable parts of B → 3π, K ππ nonleptonic amplitudes
[ see also the talk of Thomas Mannel ]
can we apply QCD Sum Rule methods to B → 2π FFs?
QCDF
SCET
HQET
2
MB
=
w
w
=
0.
3
w
=
0.
5
GeV2 .
k2
dipion with a small invariant mass
and large recoil:
k 2 . 1 GeV2 , 0 ≤ q 2 ≤ 12-14
0.
7
Dalitz plot:
4Mπ2
0
Alexander Khodjamirian
q2
B → ππ form factors from QCD Light-Cone Sum Rules
2
qmax
3 / 20
An exploratory study
[Ch. Hambrock, AK, Nucl. Phys. B (2016); 1511.02509 [hep-ph] ]
the method: similar to the Light Cone Sum Rules (LCSR) for
B → π form factors,
nonperturbative input: dipion distribution amplitudes (DAs)
we consider only B̄ 0 → π + π 0 `− ν` , isospin 1, L = 1, 3, , ..
only LO, twist-2 approximation for dipion DAs available
questions to be addressed:
•
•
how important are L > 1 partial waves of 2π state in B → ππ?
B → ρ dominance in the P-wave?
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
4 / 20
The method of LCSRs
[ I.I.Balitsky, V.M.Braun, A.V. Kolesnichenko (1986); V.L.Chernyak, I.Zhitnisky (1990)
The correlation function:
]
k = k1 + k2 , k = k1 − k2
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 ), p = (k + q)
OPE valid at q 2 mb2 (b-quark virtual)
k 2 mb2 (2-pion system produced near the LC)
LO diagram:
hb(x)b̄(0)i → Sb (x, 0)
vacuum → on-shell dipion
hadronic matrix elements
of nonlocal ū(x)d(0)
operators
with ρ-meson "embedded"
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
5 / 20
Dipion light-cone DAs
introduced and developed 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 ) ,
1
0
√ k1µ k2ν − k2µ k1ν Z
2
du eiu(k ·x) ΦI=1
hπ + (k1 )π 0 (k2 )|ū(x)σµν [x, 0]d(0)|0i = 2 2i
⊥ (u, ζ, k ) ,
2ζ − 1
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 ΦI=1 (u, ζ, k 2 )
n F em (k 2 )
π
k
= (2ζ − 1)
2)
Fπt (k 2 )
ΦI=1
(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 form factors from QCD Light-Cone Sum Rules
6 / 20
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
Alexander Khodjamirian
„
«
(q · k ) + uk 2 k µ
o
i Φ (u, ζ, k 2 )
⊥
− mb kµ Φk (u, ζ, k 2 ) .
2ζ − 1
B → ππ form factors from QCD Light-Cone Sum Rules
7 / 20
Hadronic dispersion relation
the ground B-meson state contribution:
Πµ (q, k1 , k2 ) =
hπ + (k1 )π 0 (k2 )|ūγµ (1 − γ5 )b|B̄ 0 (p)ifB mB2
mB2 − p2
+ .... ,
expansion of B → ππ matrix element in form factors:
4
ihπ + (k1 )π 0 (k2 )|ūγ µ (1 − γ5 )b|B̄ 0 (p)i = −F⊥ (q 2 , k 2 , ζ) p
iµαβγ qα k1β k2γ
k 2 λB
p
qµ
2 q2 “ µ
k ·q ”
+Ft (q 2 , k 2 , ζ) p + F0 (q 2 , k 2 , ζ) √
k − 2 qµ
q
λB
q2
“
2
1
4(q · k )(q · k ) µ
4k (q · k ) µ ”
µ
k −
k +
q ,
+Fk (q 2 , k 2 , ζ) √
2
λ
λB
k
B
quark-hadron duality in the B-channel, ⇒ effective threshold s0 ,
Borel transformation , p2 → M 2
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
8 / 20
Final results: LCSRs for the form factors at twist-2, LO
in both sum rules only the chiral-odd twist-2 DA contributes:
mb
F⊥ (q 2 , k 2 , ζ)
√ √
=√
2fB mB2 (1 − 2ζ)
k 2 λB
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 (s0 )
Z1
2
2
2
m −q ū+k
m
”
B− b
du “ 2
uM 2
mb − q 2 + k 2 u 2 Φ⊥ (u, ζ, k 2 ) e M 2
2
u
u0 (s0 )
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
k2
p
i
q 2 (2ζ − 1)
√
Fk (q 2 , k 2 , ζ) .
λB
can we obtain a sum rule also for Ft ?
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
9 / 20
2
What do we know about LCDAs
[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 ) =
∞
P
n+1
P
3/2
n=0,2,.. `=1,3,..
⊥ (k 2 )C
Bn`
n
(0)
(2u − 1)βπ P`
“
2ζ−1
βπ
”
,
⊥
Gegenbauer moments, Bn`
(k 2 ) - complex functions at k 2 > 4mπ2
instanton vacuum model for the coefficients,
2
n = 0, 2, 4, valid at small k 2 ∼ 4mπ
⊥
2
B01 (k ) = 1 +
⊥
2
B41 (k ) =
11
225
1−
5k
2
168M02
k2
12M02
!
⊥
[M. V. Polyakov and C. Weiss, (1999)]
⊥
2
7
2
, B21 (k ) =
, B43 (k ) =
36
77
675
1−
1−
k2
!
k
2
630M02
⊥
2
, B23 (k ) =
30M02
!
⊥
2
, B45 (k ) =
7
36
11
135
1+
1+
k2
!
,
30M02
k
2
56M02
!
.
⊥
2
/3M0 ' 650 MeV, where fπ = 132 MeV is the pion decay constant.
f2π
= 4π 2 fπ
2 ' 4m2 for an exploratory numerical analysis
we confined ourselves by k 2 ∼ kmin
π
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
10 / 20
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 form factors from QCD Light-Cone Sum Rules
11 / 20
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 form factors from QCD Light-Cone Sum Rules
12 / 20
How much B → ρ contributes to the B → 2π?
dispersion relations for the B → ππ P-wave (` = 1) form factors:
√ (`=1) 2 2
3F⊥
(q , k )
gρππ
V B→ρ (q 2 )
√ √
= 2
+ ...
2
2
2
mρ − k − imρ Γρ (k ) mB + mρ
k 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
Alexander Khodjamirian
p k
2V B→ρ (q 2 )
mB + mρ
(mB + mρ )AB→ρ
(q 2 ) + ...
1
B → ππ form factors from QCD Light-Cone Sum Rules
13 / 20
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 form factors from QCD Light-Cone Sum Rules
14 / 20
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 form factors from QCD Light-Cone Sum Rules
15 / 20
What is there in B → ππ beyond the ρ -contribution ?
model-dependent question
− 0
the timelike pion form factor in e+ e− → π + π − or on τ →
√ π π ντ
is fitted ( in a model-dependent way !) in the region of s ≤ 1.5
GeV to a combination of ρ and ρ0 (1450), ρ00 (1750) resonances
analogous dispersion relation for the vector FF looks like :
√
(`=1)
3F⊥
(q 2 , k 2 )
gρππ
V B→ρ (q 2 )
√ √
= 2
2
2
mρ − k − imρ Γρ (k ) mB + mρ
k 2 λB
+
+
Alexander Khodjamirian
0
gρ0 ππ
mρ2 0
−
k2
− imρ0 Γρ
0 (k 2 )
00
gρ00 ππ
mρ2 00
−
k2
− imρ00 Γρ
V B→ρ (q 2 )
+
mB + mρ0
00 (k 2 )
V B→ρ (q 2 )
+ ...
mB + mρ00
B → ππ form factors from QCD Light-Cone Sum Rules
16 / 20
The sum rule for the form factor Ft
[AK, work in progress]
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
twist-2 accuracy, only the chiral-even DA enters:
m2
Ft (q 2 , k 2 , ζ)
p
=√ b
2fB mB2
q2
Z1
u0
2
2
2
m −q ū+k
m
”
B− b
du “ 2
2
2 2
2
M2
uM 2
m
−
q
+
k
u
Φ
(u,
ζ,
k
)
e
k
b
u2
2 uū
,
main input B01 (k 2 ) = Fπ (k 2 )
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
17 / 20
Further development and applications
⊥,k
ansatz for Gegenbauer functions Bnl (k 2 ) at k 2 .1 GeV2
(disp.relation, Omnes representation)
including twist-3,4 and q̄qG components of OPE,
NLO gluon radiative corrections
B → π + π − , π 0 π 0 channels, including dipions in S, D, ..-waves
(` = 0, 2, ..), scalar f0 and tensor f2 dominance?,...
B → K π(K ∗ ) form factors,
SU(3)-violating asymmetry of Gegenbauer moments
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
18 / 20
Alternative method to access B → ππ FFs
[S.Cheng, AK, J.Virto, work in progress]
LCSRs with B-meson DA and ūγµ d interpolating current
the method introduced to calculate B → P, V form factors,
[A.K., N. Offen, Th. Mannel (2006)] ,
"SCET sum rules", [F. De Fazio, Th. Feldmann, T.Hurth (2006)]
NLO corrections to B → π ,[Y-M. Wang, Y-L.Shen (2015)
p1
p
u
b
B−
q
PB
p
d
π
u
PB
u
u
d
B−
u
π0
d
−
p2
b
+···
q
insert a dispersion.relation for B → 2π form factors
and a (dispersion rel. ⊕ experiment) parametrization for Fπ
not a direct calculation, given the shape of the B → 2π form factors, these
sum rules can provide normalization
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
19 / 20
Conclusions
B → ππ form factors are calculable from LCSRs with dipion DAs
at small dipion mass and large recoil
first exploratory study: all B → π + π 0 form factors at LO, twist-2
provide quantitative estimates for P-wave dominance,
ρ-meson dominance in P-wave, etc.
more information /dedicated studies on dipion DAs needed
LCSR can provide "building blocks" for nonleptonic B → 3P
will help to build viable models for dimeson spectra measured in
B → ππ`ν` and B → K π``
Alexander Khodjamirian
B → ππ form factors from QCD Light-Cone Sum Rules
20 / 20