Consistent Analysis of the $B\toπ$ Transition Form Factor in the Whole Physical Region

0.29 MB·English

by  Tao Huang

Consistent Analysis of the B π Transition Form Factor in the
→
Whole Physical Region
Tao, Huang1,2 and Xing-Gang, Wu2
∗ †
1CCAST(World Laboratory), P.O.Box 8730, Beijing 100080, P.R.China
2Institute of High Energy Physics, Chinese Academy of Sciences,
P.O.Box 918(4), Beijing 100049, P.R. China
‡
5
0
Abstract
0
2
In the paper, we show that the B π transition form factor can be calculated by using the
n →
a
different approach in the different q2 regions and they are consistent with each other in the whole
J

1
physical region. For the B π transition form factor in the large recoil regions, one can apply the
3
→

2 PQCD approach, where the transverse momentum dependence for both the hard scattering part
v
7 and the non-perturbative wavefunction, the Sudakov effects and the threshold effects are included
1
4 to regulate the endpoint singularity and to derive a more reliable PQCD result. Pionic twist-3
2
1 contributions are carefully studied with a better endpoint behavior wavefunction for Ψ and we
p
4
0
find that its contribution is less than the leading twist contribution. Both the two wavefunctions
/
h
p Ψ and Ψ¯ of the B meson can give sizable contributions to the B π transition form factor
B B
- →
p
and should be kept for a better understanding of the B decays. The present obtained PQCD
e
h
: results can match with both the QCD light-cone sum rule results and the extrapolated lattice
v
i
X QCD results in the large recoil regions.
r
a
PACS numbers: 12.38.Aw, 12.38.Bx, 13.20.He
∗ email: [email protected]
† email: [email protected]
‡ Mailing address
1
I. INTRODUCTION
There are various approaches to calculate the B π transition form factor, such as the
→
lattice QCD technique[1, 2, 3], the QCD light-cone sum rules (LCSRs)[4, 5, 6, 7] and the
perturbative QCD (PQCD) approach[8, 9, 10, 11, 12, 13]. The PQCD calculation is reliable
only when the involved energy scale is hard enough, i.e. in the large recoil regions. Due to
the restriction to the π energies smaller than the inverse lattice spacing, the lattice QCD
calculation becomes more difficult in the large recoil regions and at the present, the lattice
QCD results of the B π transition form factor are available only for soft regions, i.e.
→
q2 > 15GeV2. The lattice QCD results can be extrapolated to small q2 regions, and the
different extrapolation methods might cause uncertainties about 5%[2]. While, the QCD
LCSRs can involve both the hard and the soft contributions below q2 < 18GeV2[4] and can
beextrapolatedtohigherq2 regions[5,6,7]. Therefore, theresultsfromthePQCDapproach,
the lattice QCD approach and the QCD LCSRs are complementary to each other, and by
combining the results from these three methods, one may obtain a full understanding of the
B π transition form factor in its physical region, 0 q2 (M M )2 25GeV2.
B π
→ ≤ ≤ − ≃
Certain exclusive process involving hadrons can be described by PQCD if the momentum
transferissufficiently large. Theamplitudecanbefactorizedintotheconvolutionofthenon-
perturbative wavefunction for each of the hadrons with a PQCD calculable hard-scattering
amplitude. The PQCD factorization theorem has been worked out in Refs.[14, 15] based
on the earlier works on the applications of PQCD to hard exclusive processes [16]. In the
present paper, we shall use the PQCD approach to calculate the B π transition form
→
factor in the large recoil regions.
In the PQCD approach based on collinear factorization theorem, a direct calculation of
the one-gluon-exchange diagram for the B meson transition form factor suffers singularities
from the endpoint region of a momentum fraction x 0. Because of these singularities,
→
it was claimed that B π transition form factor is dominated by soft dynamics and not
→
calculable in PQCD[17]. In fact, in the endpoint region the parton transverse momenta k
⊥
are not negligible. After including the parton transverse momenta, large double logarithmic
corrections α ln2k appear in higher order radiative corrections and must be summed to
s
⊥
all orders. In addition, there are also large logarithms α ln2x which should also be summed
s
(threshold resummation[18]). The relevant Sudakov form factors from both k and the
⊥
2
threshold resummation can cure the endpoint singularity which makes the calculation of the
hard amplitudes infrared safe, and then the main contribution comes from the perturbative
regions.
An important issue for calculating the B π transition form factor is whether we need
→
to take both the two wavefunctions Ψ and Ψ¯ into consideration or simply Ψ is enough?
B B B
Inliterature, many authors(see Refs.[9, 10, 11]) didthephenomenological analysis with only
Ψ , setting Ψ¯ = 0 (or strictly speaking, ignoring the contributions from Ψ¯ ). However, As
B B B
has been argued in Refs.[19, 20], one may observe that the distribution amplitudes (DAs)
of those two wavefunctions have a quite different endpoint behavior, such difference may be
strongly enhanced by the hard scattering kernel. Even though Ψ¯ (with the definition in
B
Ref.[13]) is of subleading order contribution, there is no convincing motivation for setting
Ψ¯ = 0. In the present paper, we shall keep both the two wavefunctions Ψ and Ψ¯ to
B B B
do our calculations and show to what extent the Ψ¯ can affect the final results. Another
B
issue we need to be more careful is about the pionic twist-3 contributions. Based on the
asymptotic behavior of the twist-3 DAs, especially φas(x) 1, most of the people pointed
p ≡
out a large twist-3 contribution[12, 21] to the B π transition form factor, i.e. bigger
→
than that of the leading twist in almost all of the energy regions. In Ref.[22], the authors
have made a detailed analysis on the model dependence of the twist-3 contributions to the
pion electro-magnetic form factor, and have raised a new twist-3 wavefunction with a better
endpoint behavior for Ψ , which is derived from the QCD sum rule moment calculation[23].
p
And their results show that with such new form for Ψ , the twist-3 contributions to the
p
pion electro-magnetic form factor are power suppressed in comparison to the leading twist
contributions. According to the power counting rules in Ref.[21], the pionic twist-2 and
twist-3 contributions should be of the same order for the case of the B meson decays. With
the new form for Ψ [22], we show that for the case of the B π transition form factor, even
p
→
though the twist-3 contributions are of the same order of the leading twist contributions, its
values are less than the leading twist contribution.
The purpose of the paper is to examine the B π transition form factor in the PQCD
→
approach,andtoshowhowthePQCDresultscanmatchwiththeQCDLCSRresultsandthe
extrapolated lattice QCD results. In the PQCD approach, the full transverse momentum
dependence (k -dependence) for both the hard scattering part and the non-perturbative
T
wavefunction, the Sudakov effects and thethreshold effects areincluded to cure theendpoint
3
singularity. In section II, based on the k factorization formulism, we give the PQCD
T
formulae for the B π transition form factor in the large recoil regions. In section III, we
→
give our numerical results and carefully study the contributions from Ψ and Ψ¯ , and those
B B
from the different pionic twist structures. The slope of the obtained form factors FBπ(q2) in
+,0
the large recoil regions can match with those obtained from other approaches. Conclusion
and a brief summary are presented in the final section.
II. B π TRANSITION FORM FACTOR IN THE LARGE RECOIL REGIONS
→
First, we give our convention on the kinematics. For convenience, all the momenta are
described in terms of the light cone (LC) variables. In the LC coordinate, the momentum
is described in the form, k = (k+, k−,k ), with k = k0 k3 and k = (k1,k2). The scalar
√2 √2 ⊥ ± ± ⊥
product of two arbitrary vectors A and B is, A B = A+B−+A−B+ A B . The pion mass
· 2 − ⊥· ⊥
is neglected and its momentum is chosen to be in the minus direction. Under the above
convention, we have P = MB(1,1,0 ), P = MB(0,η,0 ) and P¯ = MB(η,0,0 ), with
B √2 ⊥ π √2 ⊥ π √2 ⊥
η = 1 q2 and q = P P .
− MB2 B − π
The two B π transition form factors FBπ(q2) and FBπ(q2) are defined as follows:
→ + 0
M2 m2 M2 m2
π(P ) u¯γ b B¯(P ) = (P +P ) B − πq FBπ(q2)+ B − πq FBπ(q2), (1)
h π | µ | B i B π µ − q2 µ! + q2 µ 0
where FBπ(0) should be equal to FBπ(0) so as to cancel the poles at q2 = 0.
+ 0
TheamplitudefortheB π transitionformfactorcanbefactorizedinto theconvolution
→
of the wavefunctions for the respective hadrons with the hard-scattering amplitude. The
wavefunctions arenon-perturbative anduniversal. The momentum projectionfor thematrix
element of the pion has the following form,
if Ψ (x,k ) Ψ (x,k ) ∂
Mαπβ = 4π(p/γ5Ψπ(x,k⊥)−mp0γ5 Ψp(x,k⊥)−iσµν nµn¯ν ′σ 6 ⊥ −pµ σ 6 ⊥ ∂k ν!!)αβ,
⊥
(2)
where f is the pion decay constant and mp is the parameter that can be determined
π 0
by QCD sum rules[23]. Ψ (x,k ) is the leading twist (twist-2) wave function, Ψ (x,k )
π p
⊥ ⊥
and Ψ (x,k ) are sub-leading twist (twist-3) wave functions, respectively. Ψ (x,k ) =
σ ′σ
⊥ ⊥
∂Ψ (x,k )/∂x, n = (√2,0,0 ) and n¯ = (0,√2,0 ) are two null vectors that point to
σ
⊥ ⊥ ⊥
the plus and the minus directions, respectively. The momentum projection for the matrix
4
FIG. 1: Lowest order hard-scattering kernel for B π form factor, where the cross denotes an
→
appropriate gamma matrix.
element of the B meson can be written as [12, 24]:
if p/ +M ∂
MB = B B B n/Ψ+(ξ,l )+n¯/Ψ (ξ,l ) ∆(ξ,l )γµ γ , (3)
αβ − 4 ( 2 " B ⊥ −B ⊥ − ⊥ ∂lµ # 5)
αβ
⊥
where ξ = l+ is the momentum fraction for the light spectator quark in the B meson
MB
and ∆(ξ,l ) = M ξdξ (Ψ (ξ ,l ) Ψ+(ξ ,l )). Note the four-component lµ in Eq.(3) is
⊥ B 0 ′ −B ′ ⊥ − B ′ ⊥ ⊥
defined through, lµ R= lµ (l+nµ+l−n¯µ) with l = (l+ , l−,l ).
⊥ − 2 √2 √2 ⊥
In the large recoil regions, the B π transition form factor is dominated by a single
→
gluon exchange in the lowest order as depicted in Fig.(1). In the hard scattering kernel, the
transverse momentum in the denominators are retained to regulate the endpoint singularity.
¯
The masses of the light quarks and the mass difference (Λ) between the b quark and the
B meson are neglected. The terms proportional to k2 or l2 in the numerator are dropped,
⊥ ⊥
which are power suppressed compared to other (M2) terms. Under these treatment, the
O B
Sudakov form factor from k resummation can be introduced into the PQCD factorization
T
theorem without breaking the gauge invariance[21]. In the transverse configuration b-space
and by including the Sudakov form factors and the threshold resummation effects, we obtain
the formulae for FBπ(q2) and FBπ(q2) as following,
+ 0
πC
FBπ(q2) = Ff f M2 dξdx b db b db α (t) exp( S(x,ξ,b ,b ;t))
+ N π B B B B π π s × − π B
c Z Z
¯
S (x)S (ξ) Ψ (x,b ) (xη +1)Ψ (ξ,b )+(xη 1)Ψ (ξ,b )
t t π π B B B B
× (" −
(cid:16) (cid:17)
mp 2 mp Ψ (x,b )
+ 0 Ψ (x,b ) (1 2x)Ψ (ξ,b )+ 1 Ψ¯ (ξ,b ) 0 ′σ π
p π B B B B
MB · − η − ! !− MB 6 ·
5
2 mp Ψ (x,b )
1+2x Ψ (ξ,b ) Ψ¯ (ξ,b ) +6 0 σ π Ψ (ξ,b ) h (x,ξ,b ,b )
B B B B B B 1 π B
− η! − ! MB 6 #
mp Ψ (x,b )
(1+η +xη) 0 σ π [M ∆(ξ,b )]h (x,ξ,b ,b )
B B 2 π B
− M 6
B
∆(ξ,b ) mp
+ Ψ (x,b ) ξη¯[Ψ (ξ,b )+Ψ¯ (ξ,b )]+ B +2 0 Ψ (x,b )
π π B B B B p π
" − MB ! MB ·
2ξ ∆(ξ,b )
(1 ξ)Ψ (ξ,b )+(1+ξ )Ψ¯ (ξ,b )+2 B h (ξ,x,b ,b ) , (4)
B B B B 1 B π
− − η MB !# )
and
πC
FBπ(q2) = Ff f M2 dξdx b db b db α (t) exp( S(x,ξ,b ,b ;t))
0 N π B B B B π π s × − π B
c Z Z
S (x)S (ξ) Ψ (x,b )η (xη +1)Ψ (ξ,b )+(xη 1)Ψ¯ (ξ,b )
t t π π B B B B
× (" −
(cid:16) (cid:17)
mp
+ 0 Ψ (x,b )((2 η 2xη)Ψ (ξ,b )+ηΨ¯ (ξ,b ))
p π B B B B
M − −
B
mp Ψ (x,b )
0 ′σ π (η(2x 1)Ψ (ξ,b ) (2 η)Ψ¯ (ξ,b ))
B B B B
− M 6 · − − −
B
mp Ψ (x,b )
+ 6 0 η σ π Ψ (ξ,b ) h (x,ξ,b ,b )
B B 1 π B
MB 6 #
mp Ψ (x,b )
[3 η xη] 0 σ π [M ∆(ξ,b )]h (x,ξ,b ,b )
B B 2 π B
− − − M 6
B
∆(ξ,b )
+ Ψ (x,b )η ξη¯(Ψ (ξ,b )+Ψ¯ (ξ,b ))+ B
π π B B B B
" MB !
mp
+ 2 0 Ψ (x,b ) (η(1+ξ) 2ξ)Ψ (ξ,b )+η(1 ξ)Ψ¯ (ξ,b )
p π B B B B
M · − −
B
(cid:16)
∆(ξ,b )
B
+ 2(2 η) h (ξ,x,b ,b ) , (5)
1 B π
− MB # )
(cid:17)
where
h (x,ξ,b ,b ) = K ( ξxη M b ) θ(b b )I (√xη M b )K (√xη M b )
1 π B 0 B B B π 0 B π 0 B B
" −
q
+θ(b b )I (√xη M b )K (√xη M b ) , (6)
π B 0 B B 0 B π
− #
b
B
h (x,ξ,b ,b ) = K ( ξxη M b ) θ(b b )I (√xη M b )K (√xη M b )
2 π B 2√ξxyMB 1 q B B " B − π 0 B π 0 B B
+θ(b b )I (√xη M b )K (√xη M b ) , (7)
π B 0 B B 0 B π
− #
and we have set,
Ψ+ +Ψ Ψ+ Ψ
Ψ = B −B , Ψ¯ = B − −B . (8)
B B
2 2
6
The functions I (K ) are the modified Bessel functions of the first (second) kind with the
i i
i-th order. The angular integrations inthe transverse plane have been performed. The factor
exp( S(x,ξ,b ,b ;t))containstheSudakov logarithmiccorrectionsandtherenormalization
π B
−
group evolution effects of both the wave functions and the hard scattering amplitude,
1 tˆ 1 tˆ
S(x,ξ,b ,b ;t) = s(x,b ,M )+s(x¯,b ,M )+s(ξ,b ,M ) ln ln , (9)
π B " π b π b B b − β1 ˆbπ − β1 ˆbB#
wheretˆ= ln(t/Λ ),ˆb = ln(1/b Λ ),ˆb = ln(1/b Λ )ands(x,b,Q) istheSudakov
QCD B B QCD π π QCD
exponent factor, whose explicit form up to next-to-leading log approximation can be found
in Ref.[15]. S (x) and S (ξ) come from the threshold resummation effects and here we take
t t
a simple parametrization proposed in Refs.[21, 25],
21+2cΓ(3/2+c)
S (x) = [x(1 x)]c , (10)
t
√πΓ(1+c) −
where the parameter c is determined around 0.3 for the present case.
The hard scale t in α (t) and the Sudakov form factor might be varied for the different
s
hard scattering parts and here we need two t [13, 21], whose values are chose as the largest
i
scale of the virtualities of internal particles, i.e.
t = MAX(√xηM ,1/b ,1/b ), t = MAX ξηM ,1/b ,1/b . (11)
1 B π B 2 B π B
(cid:18)q (cid:19)
The Fourier transformation for the transverse part of the wave function is defined as
Ψ(x,b) = d2k exp( ik b)Ψ(x,k ), (12)
Z|k|<1/b ⊥ − ⊥ · ⊥
where Ψ stands for Ψ , Ψ , Ψ , Ψ , Ψ¯ and ∆, respectively. The upper edge of the
π p σ B B
integration k < 1/b is necessary to ensure that the wave function is soft enough[26].
| ⊥|
In summary, we compare the results in Eqs.(4,5) with those in Refs.[12, 13, 20, 21]. In
Ref.[20], only leading twist (Ψ ) of the pion is discussed. Setting the twist-3 terms to zero,
π
the two formulae in Eqs.(4,5) and Ref.[20] are in agreement. In Ref.[21], the single B meson
wave function Ψ is assumed and the terms of Ψ¯ and ∆ are neglected. And in Ref.[13],
B B
with a new definition for Ψ and Ψ¯ , i.e.
B B
Ψ = Ψ+ , Ψ¯ = (Ψ+ Ψ ), (13)
B B B B − −B
both contributions from Ψ and Ψ¯ are taken into consideration, with only the terms of ∆
B B
are neglected. The momentum projector used in [13, 21] for the pion is different from the
7
present projector in Eq.(2), i.e. there is no term proportional to Ψ in Refs.[13, 21]. Except
σ
for these differences1, the formulae in [13, 21] are consistent with ours. Our results agree
with Ref.[12], except for several minus errors that should be corrected there.
III. NUMERICAL CALCULATIONS
In the numerical calculations, we use
Λ(nf=4) = 250MeV, f = 131MeV, f = 190MeV, mp = 1.30GeV. (14)
π B 0
MS
The wavefunctions in the compact parameter b-space, ΨB(ξ,b ), ΨB(ξ,b ), Ψ (x,b ),
+ B B π π
−
Ψ (x,b ) and Ψ (x,b ) can be found in the appendix. The k -dependence has been
p π σ π T
kept in both the B meson and the pion wavefunctions. As has been argued in several
papers[22, 27, 28, 29], the intrinsic k -dependence of the wave function is important and the
T
results will be overestimated without including this effect, so it is necessary to include the
transverse momentum dependence into the wave functions not only for the B meson but also
for the pion. As has been argued in Ref.[22], we take mp = 1.30GeV for latter discussions,
0
which is a little below the value given by the chiral perturbation theory[30].
The two wavefunctions Ψ and Ψ¯ of the B meson shown in the appendix depend only
B B
on the effective mass (Λ¯ = M m ) of the B meson. An estimate of Λ¯ using QCD sum
B b
−
rule approach gives Λ¯ = 0.57 0.07GeV[31]. In Fig.(2), we show the B π transition form
± →
factor with different value of Λ¯, where the shaded band is drawn with a broader range for
Λ¯, i.e. Λ¯ (0.4GeV,0.7GeV). And for comparison, we show the QCD LCSR result [5] in
∈
solid line and its theoretical error ( 10%) by a fuscous shaded band in Fig.(2). The results
±
show that the B π transition form factor will decrease with the increment of Λ¯. When
→
Λ¯ (0.5GeV,0.6GeV), one may observe that the present results agree well with the QCD
∈
LCSR results[4, 5]uptoq2 14GeV2. InFig.(2), forsimplicity, onlytheQCDLCSR results
∼
of Ref.[5] are shown. The LCSR results in Refs.[4, 5] are in agreement with each other even
though they have taken different ways to improve the QCD LCSR calculation precision, i.e.
in Ref.[4], an alternative way to do the QCD LCSR calculation is adopted in which the
pionic twist-3 contributions are avoided by calculating the correlator with a proper chiral
1
According to the power counting rules in Ref.[21], the terms that do not existent in Ref.[21] are defined
as sub-leading terms in 1/MB and are neglected accordingly. And here, we keep all the terms with care.
8
1.5 1.0
0.8
1.0
)
) 2 q 0.6
2 q (
( B 0
B F
F+ 0.4
0.5
0.2
0.0 0.0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
2 2 2 2
q (GeV ) q (GeV )
FIG. 2: PQCD results for the B π transition form factors FBπ(q2) (Left) and FBπ(q2) (Right)
→ + 0
with different values for Λ¯. The dashed line stands for Λ¯ = 0.5GeV, the dotted line stands for
Λ¯ = 0.6GeV, the upper edge of the shaded band corresponds to Λ¯ = 0.40GeV and the lower edge
of the band corresponds to Λ¯ = 0.70GeV. For comparison, the solid line comes from the QCD
LCSR[4, 5] and the fuscous shaded band shows its theoretical error 10%.
±
current and then the leading twist contributions are calculated up to next-to-leading order;
while in Ref.[5], the usual QCD LCSR approach is adopted and both the twist-2 and twist-3
contributions are calculated up to next-to-leading order. In Ref.[13], Λ¯ is treated as a free
parameter and a bigger value is adopted there, i.e. Λ¯ = (0.70 0.05)GeV. The main reason
±
is that in the present paper, we have used an improved form (with better endpoint behavior
than that of the asymptotic one) for the pionic twist-3 wavefunction Ψ , while in Ref.[13],
p
they took φ in Ref.[7] (with an endpoint behavior even worse than the asymptotic one)
p
other than Ψ to do the calculations, so the value of Λ¯ in Ref.[13] must be big enough to
p
suppress the endpoint singularity coming from the hard kernel. For clarity, if not specially
stated, we shall fix Λ¯ to be 0.5GeV in the following discussions.
Second, to get a deep understanding of the B π transition form factor, we discuss the
→
contributions from different parts of the B meson wavefunction or the pion wave function,
correspondingly. Here we take FBπ(q2) to do our discussions and the case of FBπ(q2) can
+ 0
be done in a similar way. In Fig.(3a), we show the contributions from the different twist
structures of the pion wave function, i.e. Ψ , Ψ and Ψ (the contributions from the terms
π p σ
involving Ψ are included in Ψ ), respectively. From Fig.(3a), one may observe that the
′σ σ
9
4
1.0
All 3 All
B
0.8
p 2
2 q) 0.6 2 q) B
( (
B B 1
+ +
F 0.4 F
0
0.2
-1
0.0
-2
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
2 2
q(GeV) 2 2
q (GeV )
(a)
(b)
FIG. 3: PQCD results for the B π transition form factor FBπ(q2) with fixed Λ¯ = 0.5GeV. The
→ +
left diagram is for the different pion twist structures, Ψ , Ψ and Ψ . The right diagram is for the
π p σ
different B meson structures, Ψ , Ψ¯ and ∆, where Ψ and Ψ¯ are defined in Eq.(8).
B B B B
contribution from Ψ is the biggest, then comes that of Ψ and Ψ . And the ratio between
π p σ
all the twist-3 contributions and the leading twist contribution is 70% in the large recoil
∼
regions. This behavior is quite different fromthe conclusion that has been drawn in Refs.[12,
21], in which they concluded that the twist-3 contribution is bigger than that of twist-2
contribution, especially in Ref.[12], it claimed that the twist-3 contribution is about three
times bigger than that of twist-2 at q2 = 0. Such kind of big twist-3 contributions are due to
the fact that they only took the pion distribution amplitudes into consideration (or simply
adding a harmonic transverse momentum dependence for the pion wavefunctions), and then
the endpoint singularity coming from the hard kernel can not be effectively suppressed,
especially for Ψ whose DA’s asymptotic behavior is φ 1. In Ref.[22], the authors have
p p
≡
made a detailed analysis on the model dependence of the twist-3 contributions to the pion
electro-magnetic form factor, and have raised a new twist-3 wavefunction (as is shown in
the appendix) with a better endpoint behavior for Ψ , which is inspired from QCD sum rule
p
moment calculation. With this model wave function for Ψ , Ref.[22] shows that the twist-3
p
contributions of the pion electro-magnetic form factor agree well with the power counting
rule, i.e. the twist-3 contribution dropsfast andit becomes less thanthetwist-2 contribution
at Q2 10GeV2. For the present B meson case, according to the power counting rules in
∼
Ref.[21], the twist-3 contribution andthe twist-2 contribution are of the same order, however
10