$B_{(s)}\to S$ transitions in the light cone sum rules with the chiral current

0.51 MB·English

by  Yan-Jun Sun

B S transitions in the light cone sum rules with the chiral
(s)
→
current
Yan-Jun Sun1,2 , Zuo-Hong Li3 and Tao Huang2
∗ † ‡
1Department of Modern Physics, University of Science
and Technology of China, Hefei 230026, P.R. China
1 2Institute of High Energy Physics and Theoretical Physics Enter for Science Facilities,
1
0 Chinese Academy of Sciences, Beijing 100049, P.R. China
2

n 3Department of Physics, Yantai University, Yantai 264005,China
a
J
Abstract
7
1
We make a QCD light-cone sum rule (LCSR) assessment of B semi-leptonic decays to a light
(s)

]
h scalar meson, B Slν¯,Sl¯l (l = e,µ,τ). Chiral current correlators are used and calculations
(s) l
p →
- are performed at leading order in α . Having little knowledge of ingredients of the scalar mesons,
p s
e
h we confine ourself to the two quark picture for them and work with the two possible scenarios.
[

The resulting sum rules for the form factors receive no contributions from the twist-3 distribution
2
v
amplitudes (DA’s), in comparison with the calculation of the conventional LCSR approach where
1
0
9 the twist-3 parts play usually an important role. We specify the range of the squared momentum
3
. transfer q2, in which the operator product expansion (OPE) for the correlators remains valid
1
1
approximately. It is found that the form factors satisfy a relation consistent with the prediction of
0
1
: soft collinear effective theory (SCET). In the effective range we investigate behaviors of the form
v
i
X factors and differential decay widthes and compare our calculations with the observations from
r
a other approaches.
PACS numbers: numbers: 13.25.Hw, 13.60.Le, 12.38.Lg
∗ Email: [email protected]
† Corresponding author Email: [email protected]
‡ Corresponding author Email: [email protected]
1
I. INTRODUCTION
With numerous scalar meson states being discovered experimentally, most of efforts have
been devoted to a study about their inner structure and how they are classified. How-
ever, much controversy persists regarding the underlying components of them. Currently
one of our main concerns is that these scalar particles can whether or not be described
consistently in a quark picture. Recently, from a survey of the accumulated experimen-
tal data the two possible scenarios are suggested [1, 2], where the scalar mesons below
and above 1 GeV are assumed to enter their respective nonets in two different ways. In
scenario 1, there are the two scalar nonets formed by the two quark bound states. One
contains, as the lowest lying scalar states, the isoscalars σ(600) and f (980), isodoublets
0
(κ+(800),κ0(800)) and (κ¯0(800),κ (800)) and isovector (a+(980),a0(980),a (980)). The
− 0 0 −0
other is made up of the corresponding first excited states: the isoscalars f(1370) and
f (1500), isodoublets (K +(1430),K 0(1430)) and (K¯ 0(1430),K¯ (1430)) and isovector
0 0∗ 0∗ 0∗ 0∗−
(a+(1450),a0(1450),a (1450)). In scenario 2, those scalar states below 1 GeV are taken
0 0 −0
to be the members of a four-quark nonet, while f(1370), f (1500), a (1450) and K (1430)
0 0 0∗
are treated as the lowest lying two-quark resonances and arranged into another nonet, with
the corresponding first excited states between 2.0 2.3 GeV.
∼
Althoughnowwearenotabletodiscriminateamongalltheexistingschemes forthescalar
mesons, the above two are intriguing in that they can provide us with a ground to make a
systematic study on the scalar mesons. In such assignment scenarios, an investigation has
been madeinto therelated decay constants andlight-conedistribution amplitudes (DA’s)[1].
Moreimportantly, togaininsight into thescalar mesons someoftheBdecays involving them
have been explored in the same context. In Ref.[1, 2], the hadronic decays with a scalar final
state are discussed in detail in the framework of QCD factorization, important implications
being drawn for the properties of the scalar particles. More attentions are paid to the
¯
semileptonic decays with a potential interest B Slν¯,Sll. Especially, one shows a great
(s) l
→
interest in the knowledge of their differential rates, since it is critical, as confronted with
the coming experimental observations, for acquiring valuable information on ingredients of
the scalar particles. Unfortunately, among the existing approaches no one can afford the
task to understand the underlying form factors in the whole regions of q2, with q being the
momentum transfers. An effective range of q2, in which the calculations are believable, has
2
even not been specified in literature, the computations being carried out in just a small or
intermediate kinematical region arbitrarily selected. So the results are less persuasive.
Superior tothe three-point QCD sum rules inevaluating heavy tolight meson transitions,
the LCSR approach, which starts with a two-point correlation function, adopts the operator
product expansion (OPE) near the light cone x2 = 0 in terms of nonlocal operators, whose
matrix elements are parameterized as the hadronic DA’s of increasing twist. Such that the
resulting LCSR for form factors, in addition to having an estimable effective region of q2,
can embody as many long-distance effects as possible involved in the decaying processes.
However, a better understanding of these DA’s is critical to have the calculation more
reliable. Together with the leading twist-2 DA, in general, the twist-3 ones enter and play
an important role in a LCSR calculation on the form factors. In the case of the scalar
mesons, the probe into the twist-2 and -3 DA’s has been conducted in the framework of
QCD sum rules and a DA model, in an expansion form in the Gegenbauer polynomials, has
been formulated, but with a sizable error in some of the model parameters. To try our best
to reduce uncertainty in LCSR calculation from the long distance parameters, a practical
improvement scenario has been worked out with its validity examined and confirmed, in
which a chiral correlator is so chosen that the twist-3 DA’s make no contribution [3]. In the
present work, we intend to apply the same trick to revaluate the semileptonic transitions
¯
B Slν¯,Sll, in the two quark picture for the scalar mesons. We will work in the effective
(s) l
→
regions required by the OPE validity and with the two different scenarios aforementioned,
and calculation is to be performed at leading order in α .
s
The paper is organized as follows: In the following section, we present the correlation
functions with a chiral current and use them to derive the LCSR for the form factors for the
B S transitions. The discussion and comment are made on the important inputs-the
(s)
→
DA’s and decay constants of the scalar mesons, in Sec.III. Sec.VI is devoted to a detailed
¯
numerical discussion about the form factors and differential widths for B Slν¯,Sll,
(s) l
→
including a numerical comparison with the estimates of some other approaches. The final
section is reserved for a summary.
3
II. THE LCSR FOR THE B S FORM FACTORS
(s)
→
¯
In the standard model (SM), the semileptonic decays B Slν¯,Sll are induced by the
(s) l
→
following effective Hamiltonian:
G
= FV u¯γ (1 γ )bℓ¯γµ(1 γ )ν
eff ub µ 5 5 ℓ
H √2 − −
G αV V
+ F t∗b ts Ceffs¯γ (1 γ )b ℓ¯γµℓ
√2π 9 µ − 5
+ C s¯γ (1 hγ )bℓ¯γµγ ℓ
10 µ 5 5
−
2m Ceff(m )
b 7 b s¯iσ qν(1+γ )b ℓ¯γµℓ . (1)
− q2 µν 5
i
(eff)
HereV aretheCabibbo-Kobayashi-Maskawa (CKM)matrixelements, andC theWilson
ij i
coefficients, among which Ceff and C are scale-independent for thecorresponding operators
9 10
have a vanishing anomalous dimension. Ceff and Ceff are expressed as
7 9
Ceff(µ) = C (µ)+C (µ), (2)
7 7 b sγ
→
Ceff = C (µ)+Y (s)+Y (s), (3)
9 9 pert ′ LD ′
where C (µ) stems from the absorptive part of b scc¯ sγ rescattering which will be
b sγ
→ → →
neglected here, Y and Y stand for, respectively, the short-and long-distance contribu-
pert LD
tions from the four quark operators [4], with
1
Y (s) = h(z,s)C h(1,s)(4C +4C +3C +C )
pert ′ ′ 0 ′ 3 4 5 6
− 2
1 2
h(0,s)(C +3C )+ (3C +C +3C +C ), (4)
′ 3 4 3 4 5 6
−2 9
C = 3C +C +3C +C +3C +C , and
0 1 2 3 4 5 6
h(z,s) = 8lnz + 8 + 4x 2(2+x) 1 x 1/2 ln √√11−xx+11 −iπ for x ≡ 4z2/s′ < 1 ,
′ − −
−9 27 9 − 9 | − |  2a(cid:12)rctan (cid:12)1 for x 4z2/s > 1
 (cid:12)(cid:12) √x(cid:12)(cid:12)−1 ≡ ′
8 8 m 4 4
b
h(0,s′) = ln lns′ + iπ ,  (5)
27 − 9 µ − 9 9
TABLE I: The values of Wilson coefficients C (m ) in the leading logarithmic approximation in
i b
Standard Model, with m = 80.4GeV, m = 173.8GeV, m = 4.8GeV[5].
W t b
C C C C C C C C C
1 2 3 4 5 6 7 9 10
1.119 0.270 0.013 0.027 0.009 0.033 0.322 4.344 4.669
− − − − −
4
where z = m /m and s = q2/m2. The Wilson coefficients C (m ), listed in Tab.I, are given
c b ′ b i b
in the leading logarithmic accuracy.
¯
Aiming at an evaluation of the semileptonic decays B Slν¯,Sll, we need to confront
(s) l
→
the hadronic matrix elements S(p) q¯γ γ b B (p+ q) and S(p) q¯σ γ qνb B (p+ q) .
2 µ 5 (s) 2 µν 5 (s)
h | | i h | | i
They can be parameterized, in terms of the form factors f (q2), f (q2) and f (q2), as
+ T
−
S(p) q¯ γ γ b B(p+q) = 2ip f (q2) i[f (q2)+f (q2)]q , (6)
2 µ 5 µ + + µ
h | | i − − −
f (q2)
S(p) q¯ σ γ qνb B(p+q) = [2p q2 2q (q p)] − T , (7)
2 µν 5 µ µ
h | | i − · m +m
B S
where the B mesons are signified by B for short. The relative form factors could be
(s)
calculated in the LCSR. Instead of the correlation functions used in Ref.[6], we would like
to consider the following two correlators, with the T product of chiral current operators
sandwiched between the vacuum and one on-shell scalar meson state [7]:
Π (p,q) = i d4xeiqx S(p) T q¯(x)γ (1 γ )b(x),¯b(0)i(1 γ )q (0) 0 , (8)
µ 2 µ 5 5 1
h | { − − }| i
Z
Π (p,q) = i d4xeiqx S(p) T q¯(x)σ (1+γ )qνb(x),¯b(0)i(1 γ )q (0) 0 , (9)
µ 2 µν 5 5 1
h | { − }| i
Z
e
where q ,q denotes the light quark field.
1 2
The hadronic representations for them are easy to achieve, by inserting between the
currents a complete set of resonance states with the same quantum numbers as the operator
¯
b(0)i(1 γ )q (0). On thedesired pole contributions due to the lowest pseudoscalar B-meson
5 1
−
are insolated, we obtain the hadronic representations:
¯
S(p) q¯ γ γ b B(p+q) B(p+q) biγ q 0
Πh(p,q) = h | 2 µ 5 | ih | 5 1| i
µ m2 (p+q)2
B −
S(p) q¯ γ (1 γ )b Bh(p+q) Bh(p+q)¯bi(1 γ )q 0
2 µ 5 5 1
+ h | − | ih | − | i, (10)
m2 (p+q)2
h B −
X
S(p) q¯ σ (1+γ )qνb B(p+q) B(p+q)¯biγ q 0
Πh(p,q) = h | 2 µν 5 | ih | 5 1| i
µ − m2 (p+q)2
B −
S(p) q¯ σ (1+γ )qνb Bh(p+q) Bh(p+q)¯bi(1 γ )q 0
e 2 µν 5 5 1
+ h | | ih | − | i. (11)
m2 (p+q)2
h B −
X
It should be stressed that the correlation functions receive contributions from the scalar
resonances included in the intermediate statesBh[7], in addition to the higher pseudoscalar
ones, and the ground-state scalar meson is a bit lighter than the pseudoscalar resonance
lying in the first excited state.
5
With the definitions of B-meson decay constant B ¯biγ q 0 = m2BfB and Eqs.(6) and
h | 5 1| i mq1+mb
(7), the phenomenological representations of the correlation functions read
i m2 f
Πh(p,q) = − B B [2f (q2)p +(f (q2)+f (q2))q ]
µ m2B −(p+q)2mq1 +mb + µ + − µ
1 2ρh(s)p +(ρh(s)+ρh(s))q
∞ds + µ + µ, (12)
−
−π s (p+q)2
Zs0 −
1 m2 f f
Πh(p,q) = B B T [2p q2 2q (q p)]
µ m2 (p+q)2m +m m +m µ − µ ·
B − q1 b B S
1 ρh(s)[2p q2 2q (q p)]
e ∞ds T µ − µ · . (13)
−π s (p+q)2
Zs0 −
Here we have replaced the summations in (10) and (11) with the dispersion integrations
starting with the threshold s near the squared mass of the lowest scalar B-meson[7]. The
0
spectral densities can be approximated as, by invoking the quark-hadron duality ansatz
ρh (s) = ρQCD (s)θ(s s ). (14)
+, ,T +, ,T − 0
− −
The QCD spectral densities ρQCD (s) can be derived by calculating the correctors in QCD
+, ,T
−
theory. To this end, we work in the large space-like momentum regions (p + q)2 << m2
b
for the bq¯ channel and a larger recoil region of the decaying B-meson as given later, which
1
correspond to the small light-cone distance x2 0 and are required by the validity of the
≈
OPE [10]. Considering the effect of the background gluon field, we can write down a full
b-quark propagator
¯
0 Tb(x)b(0) 0 (15)
h | | i
d4k k +m 1
= iS (x,0) ig e ikx dv[ 6 b Gµν(vx)σ + vx Gµν(vx)γ ].
0 − s (2π)4 − (m2 k2)2 µν m2 k2 µ ν
Z Z b − b −
Here G is the gluonic field strength, g denotes the strong coupling constant and S (x,0)
µν s 0
expresses a free b-quark propagator
d4k k +m
iS (x,0) = i e ikx 6 b . (16)
0 − (2π)4 − m2 k2
Z b −
The large virtuality of the underlying heavy quarks makes it sound to neglect the contribu-
tions of soft gluon emission from the heavy quarks, which, in fact, is just a twist-4 effect.
In this accuracy and leading order in α , we find that as contrasted with the results of
s
the traditional LCSR [6], only the nonlocal matrix element S(p) q¯ (x)γ q (0) 0 remains,
2 µ 1
h | | i
while those concerning the nonlocal operators q¯ (x)q (0) and q¯ (x)σ q (0) cancel out. As
2 1 2 µν 1
6
usual, applying the light-cone OPE to the matrix element S(p) q¯ (x)γ q (0) 0 , we could
2 µ 1
h | | i
be led to the leading twist-2 DA’s of the scalar mesons Φ (u,µ) as defined in [1]. We are
S
going to return to this point in the following section. Now the light-cone OPE forms for the
correlators can be written as follows:
1 Φ (u)
ΠQCD(p,q) = 2ip m du S , (17)
µ µ b m2 (q +up)2
Z0 b −
1 Φ (u)
ΠQCD(p,q) = 2(p q2 q (q p)) du S . (18)
µ − µ − µ · m2 (q +up)2
Z0 b −
We would likeeto convert them into a form of dispersion integration in order to facilitate
the ensuing subtraction of the effect of the higher resonances and continuum states in the
phenomenological representations (12) and (13). To this end, invoking the relation m2
b −
(q+up)2 = u(s (p+q)2) we make a replacement of u with s. Matching both the forms of
−
the correlators, subtracting continuum contributions and making Borel transformation [11]
with respect to the variable (p+q)2,
1 1 m2B
BM2m2 (q +p)2 = M2e−M2,
B −
BM2m2 (q1+up)2 = uM1 2eu−M12[m2b+u(1−u)p2−(1−u)q2], (19)
b −
with M2 being the Borel parameter and m the scalar meson mass, we get the sum rules
S
for the form factors:
m +m 1 Φ (u)
f (q2) = q1 bm du S eΛ, (20)
+ − m2 f b u
B B Z∆
m +m 1 Φ (u)
f (q2) = q1 bm du S eΛ, (21)
− m2BfB bZ∆ u
m +m 1 Φ (u)
f (q2) = q1 b(m +m ) du S eΛ, (22)
T − m2 f B S u
B B Z∆
where
1
∆ = (s m2 q2)2 +4(m2 q2)m2 (s m2 q2) ,
2m2 0 − S − b − S − 0 − S −
S (cid:20)q (cid:21)
1 m2
Λ = m2 +u(1 u)m2 (1 u)q2 + B. (23)
−uM2 b − S − − M2
(cid:2) (cid:3)
We find, as a by-product, that the form factors in question respect the following LCSR
relations:
f (q2) = f (q2), (24)
+
− −
(m +m )
f (q2) = B S f (q2). (25)
T +
m
b
7
Actually, apart from that the same is observed in the LCSR involving a pseudoscalar
meson, a simple relation is obtained also for the form factors in the vector meson case [13].
All these observations, upto thehard-exchange corrections, areconsistent withthe results of
soft collinear effective theory (SCET) [12]. Having these relations at hand, in the numerical
discussion we will put our focus on the form factor f (q2).
+
III. DECAY CONSTANTS AND DISTRIBUTION AMPLITUDES OF SCALAR
MESONS
In this section, we give a brief review and discussion on the decay constants and DA’s of
the related scalar mesons, which are the basic inputs for the LCSR calculation.
For a light scalar meson in the two quark picture, it could couple to the corresponding
vector and scalar quark current operators thus we can define its decay constants as [1],
S(p) q¯(0)γ q (0) 0 = p f , (26)
2 µ 1 µ S
h | | i
¯
S(p) q¯(0)q (0) 0 = m f . (27)
2 1 S S
h | | i
¯
It is observed readily that the decay constants f and f are scale independent and de-
S S
pendent, respectively. The neutral scalar mesons like a0 and f ( if considered purely a ss¯
0 0
bound state) cannot couple with a vector current operator owing to the charge conjugation
invariance or conservation of the vector current and thus we have
f = f = 0. (28)
f0 a00
¯
For the other scalar mesons, the decay constants f and f are connected by equation of
S S
motion
¯
f = µ f , (29)
S S S
where
m
S
µ = , (30)
S
m (µ) m (µ)
2 1
−
the running quark masses m (µ) respect the renormalization group equation (RGE):
i
4/b
αs(µ0) −
m (µ) = m (µ ) , (31)
i i 0
α (µ)
(cid:18) s (cid:19)
8
with b = (33 2n )/3, n being the number of active quark flavors. The decay constants
f f
−
f hence are either zero or small of order m m .
S 2 1
−
Similar to the case of pseudoscalar mesons, the twist-2 DA Φ (u,µ) of the scalar meson
S
is defined as [1]
1
S(p) q¯(x)γ q (y) 0 = p dueiupx+u¯pyΦ (u,µ), (32)
2 µ 1 µ · · S
h | | i
Z0
with u being the fraction of the light-cone momentum of the scalar meson carried by q and
2
u¯ = 1 u, and obeys the normalization
−
1
duΦ (u,µ) = f . (33)
S S
Z0
With reference to the DA’s of scalar mesons, a few words should be given. From the
definition of Φ (u,µ), the corresponding scalar mesons have to carry a large light-cone
S
momentum p + p . Along with the requirement of the OPE validity, such a constrain
0 3
condition demands that we work in a region assigned as,
0 q2 < (m m )2 2(m m )Λ , (34)
b S b S QCD
≤ − − −
which, to be specific, is 0 q2 < 11GeV2 for a scalar meson below 1 GeV and 0 q2 <
≤ ≤
8GeV2 for one above 1 GeV. Also, it is important to realize that the DA’s of scalar meson,
strictly speaking, become meaningful just at a scale µ m , since the constituent quark of
S
≥
the scalar meson is in essence off-shell and in particular, it is far from its mass shell by the
virtuality of m2 as carrying the total momentum of the scalar meson. Considering the DA’s
S
at a scale below m means that we are dealing with the situation that these off-shell modes
S
are in part or in full integrated out, however, which is meaningless.
Based on the conformal symmetry hidden in the QCD Lagrangian, Φ (u,µ) can be ex-
S
3/2
panded in a series of Gegenbauer polynomials C (x) with increasing conformal spin as
m
Φ (u,µ) = f¯ (µ)6uu¯ B (µ)+ B (µ)C3/2(2u 1) ,
S S 0 m m −
" #
m=1
X
where Gegenbauer moments B (µ), which are scale dependent, are given as
m
1 2(2m+1) 1
B (µ) = C3/2(2u 1)Φ (u,µ)du. (35)
m f¯ 3(m+1)(m+2) m − S
s Z0
9
The scale evolutions of Φ (u,µ) are determined using the following RGE:
S
4/b
α (µ )
¯ ¯ s 0
f (µ) = f (µ ) ,
S S 0
α (µ)
s !
αs(µ0) −(γ(m)+4)/b
B (µ) = B (µ ) , (36)
m m 0
α (µ)
(cid:18) s (cid:19)
where the one-loop anomalous dimensions is [14]
m+1
2 1
γ = C 1 +4 ,
(m) F
− (m+1)(m+2) j
!
j=2
X
with C = 4/3. The conservation of charge parity demands an antisymmetric Φ (u,µ)
F S
under the interchange u 1 u, namely, Φ (u,µ) = Φ (1 u,µ), for the neutral scalar
S S
↔ − − −
mesons of a qq¯ content. Accordingly, for the scalar mesons a0 and f we could write down
0 0
their leading twist DA’s as
¯ 3/2
Φ (u,µ) = f (µ)6uu¯ B (µ)C (2u 1). (37)
S S 2m+1 2m+1 −
m=0
X
In the two quark picture, it is concluded that the twist-2 DA’s of all the light scalar mesons
are antisymmetric under the interchange u 1 u in the flavor SU(3) limit, thus the odd
↔ −
Gegenbauer moments dominateintheDA’s, formingastriking contrast tothecorresponding
situations of the pseudoscalar mesons where the leading DA of the pion, for instance, covers
no odd Gegenbauer moments and so is symmetric. Indeed the zeroth Gegenbauer moment
B , which is equal to µ 1, vanishes in the SU(3) limit. In the following, we will neglect the
0 −S
contributions of the even Gegenbauer moments and take only into account the first two odd
moments.
To proceed, we must add that the LCSR for the form factor f (q2) would have a distinct
+
scale dependence, due to the absence of the QCD radiative corrections. In such a case, it
should be in order that we work at the scale µ = m2 m2, which denotes the typical
b Bs − b
virtuality of the underlying b quark. At this scale, tqhe related parameters can be evaluated
making use of the RGE (36) with an initial scale µ m . As the initial conditions we
0 S
≥
prefer using the QCD sum rule estimates at µ = 1 GeV [1], which though is a bit inadequate
¯
for the situation involving the scalar mesons above 1GeV. The numerical results for f (µ)
S
and B are collected in Tab.II and III, and the shapes of the DA’s in the two scenarios are
1,3
illustrated in Fig.1.
10