Table Of ContentEXCITED HEAVY MESONS DECAY FORMFACTORS IN LIGHT CONE QCD
T. M. ALIEV ∗, A. O¨ZPI˙NECI˙ † and M. SAVCI ‡
Physics Department, Middle East Technical University
7
9 06531 Ankara, Turkey
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1
February 1, 2008
n
a
J
2
2 Abstract
v
6
The formfactors of B π and D π transitions, where B (D ) is the 1+
5 1 1 1 1
→ →
2 P-wave Q¯γ γ q meson state, are calculated in the framework of the light cone QCD.
0 ν 5
1
Furthermore these formfactors are compared with the pole dominance model predic-
6
9
/ tion using the values of the strong coupling constants gB1B∗π and gD1D∗π, and a good
h
p agreement between these two different descriptions is observed.
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p
e
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v PACS numbers: 13.20.He, 13.25.Hw, 13.30.Eg
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∗e-mail:[email protected]
†e-mail:[email protected]
‡e-mail:[email protected]
1 Introduction
Understanding the formfactors of the hadronic currents is one of the main problems in
particle physics. As is well known, the knowledge of the formfactors allows one the de-
termination of the quark mixing parameters, in a unique way [1]. Moreover, it gives us a
direct information about the dynamics of the hadronic processes at large distance, where
perturbation theory of QCD is invalid. At present there exists no theoretical method for
calculating the formfactors, starting from a fundamental QCD Lagrangian. Therefore, in
estimating the formfactors, usually phenomenological or semiphenomenological methods
are used.
Along these lines, there exists two well known methods for calculating the hadronic form
factors. First of these is the pole model description based on the vector dominance idea,
that suggests a momentum dependence dominated by the nearest pole, and the other one
is the QCD sum rules method. The pole model approach does not have its roots on the
basic principles of the theory and is purely phenomenological. Generally, it is assumed that
the vector dominance approximation is valid at zero recoil, that is at p2 m2, where p2 is
→
the square of the momentum transfer, that depends on the formfactor, and m is the mass
of the nearest vector meson. However, there are no reliable arguments in favour of the pole
model, for it to be also valid at small p2, that are interesting for practical applications. The
effective heavy quark theory predicts somewhat larger region of validity, characterized by
(m2 p2)/m (1 GeV), where m is the heavy quark mass (for more detail see [2, 3] and
Q Q
− ∼
the references cited therein). It is shown in [4, 5] that the pole behaviour of the formfactors
is consistent with the p2 dependence at p2 0 predicted by the sum rule. This result has
→
been confirmed independently by the calculations carried within the framework of the light
cone sum rules [2, 6, 7].
In contraryto the polemodel, theQCD sum rules method isbased onthe first principles
and on the fundamental QCD Lagrangian. In this work we employ an alternative version
of the QCD sum rules, namely light cone sum rules method for the calculation of the
formfactors. We note that using this approach, the formfactors of the πA0γ∗ transition
[8], the pion formfactors at the intermediate momentum transfer [9], semileptonic B and D
decays [10], radiative B K∗γ decay formfactor [11] and B∗ Bπ, D∗ Dπ transition
→ → →
formfactors [3] are calculated. The results of all works that have been devoted to the
calculationoftheformfactorsconfirmthat, thepredictions ofthepolemodelarecompatible
1
with that of the QCD sum rules method. Note that, in all these works the transition
formfactors between the ground state mesons are considered.
In connection with these observations, there follows an immediate question to whether
the same agreement holds for the prediction of the formfactors resulting from the decays
of the excited meson states. This article is devoted to find an answer to the question
cited above, more precisely, we calculate the formfactors for the excited 1+ P wave decays
P P∗π, where P (B ,D ) is the excited 1+ P wave meson state, and P∗ is the 1− state.
1 1 1 1
→
This article is organized as follows: In sect.2, we derive the sum rules for the P P∗π
1
→
transition formfactors. Sect.3 is devoted to the numerical analysis and discussions.
2 Sum Rules for the P P π transition formfactors
1 ∗
→
For calculation of the sum rules formfactors for the P P∗π transition, we consider the
1
→
following correlator,
Π (q ,q) = i d4x eiq2x π−(q) d¯(x)γ Q(x)Q¯(0)γ γ u(0) 0 , (1)
µν 2 µ ν 5
h | | i
Z
where, d¯γ Q and Q¯γ γ u are the interpolating currents of 1− and 1+, respectively, Q is
µ ν 5
the heavy quark, q is the momentum of the 1− meson and q is the pion’s momentum.
2
When the pion is on the mass shell, i.e., q2 = m2, the correlator depends on two invariant
π
variables q2 and (q +q)2. In further calculations, we will set m = 0.
2 2 π
We start by considering the physical part of eq.(1). Before calculating the physical part
of the eq.(1), few words are in order. According to the value of angular momentum of the
light degrees of freedom (sP = 1+, 3+), the heavy quark effective theory [12, 13], predicts
l 2 2
the existence of two multiplets, each including the 1+ meson: the first one contains (0+,1+)
and the second (2+,1+) mesons. In terms of the conventional (2s+1)P states, the 1+ states
J
defined above (physical states), are given by the following linear combinations [12, 14]:
2 1
1+ = 1P + 3P
3 1 1
| 2i s3| i s3| i
1 2
1+ = 1P 3P .
1 1 1
| 2i s3| i−s3| i
Therefore, the physical part of the eq.(1) must contain the contributions from both of the
1+ mesons, and it can be written as
π−(q) d¯γ Q 1+ 1+ Q¯γ γ u 0
Π = h | µ | l ih l | ν 5 | i . (2)
µν (q +q)2 m2
l=X12,23 2 − 1l
2
In [15, 16], it was shown that in the heavy quark limit, m >> m ,
Q q
1+ Q¯γ γ u 0 = 0 .
3 ν 5
h 2| | i
The full theory can only bring small corrections to this result. For this reason the contri-
bution of the 1+ meson, in eq.(2), can safely be neglected. Therefore, in future analysis we
3
2
will consider the contribution of the 1+ meson only, and for simplicity we denote it as P .
1 1
2
The matrix elements entering in eq.(2) are defined in the standard manner:
P Q¯γ γ u 0 = if m ǫ (p) ,
h 1| ν 5 | i − P1 P1 ν
¯
π dγ Q P = F ǫ (p)+F (qǫ)(p+q) +F (qǫ)(p q) , (3)
ν 1 0 ν 1 ν 2 ν
h | | i −
where f , ǫ and m are the leptonic decay constant, vector polarization and mass of
P1 ν P1
the excited 1+ meson, respectively. F , F and F are the formfactors that describe the
0 1 2
P π transition. The contributions of the coefficients of F and F to the decay rate are
1 1 2
→
proportional to
m2π +(mP1 −mP∗)2 .
From this expression it is obvious that, their contributions are small in comparison to the
F ’s contribution. Our numerical analysis shows that the contributions of F and F to the
0 1 2
decay width constitute about 20% of F , and hence, we shall neglect such terms. Thus,
0
f m p p
Π = i P1 P1 F g + µ ν ,
µν − (q2 +q)2 −m2P1 0 − µν m2P1 !
and in further analysis we will pay our attention to the structure g and denote its coeffi-
µν
cient by Π. The physical part of the above relation takes the following form:
f m F (q2) ∞ ρh(q2,s)ds
Π q2,(q +q)2 = i P1 P1 0 2 +i 2 . (4)
2 2 − m2 (q +q)2 s (q +q)2
(cid:16) (cid:17) P1 − 2 Zs0 − 2
The first term on the right hand side is the pole term due to the ground state in the heavy
channel, while the continuum and the higher order states are taken into account by the
dispersion integral above the threshold s .
0
Let us now consider the theoretical part of the correlator (1). This correlator function
can be calculated in QCD at large Euclidean momenta (q +q)2. In this case the virtuality
2
of the heavy quark is of the order m2 (q +q)2, and this is a large quantity. Therefore, we
Q− 2
can expand the heavy quark propagator in powers of the slowly varying fields residing in
3
the pion, that acts as the external field on the propagating heavy quark. The expansion in
powers of the external fields is also the expansion of the propagator in powers of a deviation
from the light cone at x2 = 0. The leading contribution is obtained by using the free heavy
quark propagator in eq.(1)
d4x (k +m )
S0(x) = i e−ikx 6 Q .
Q (2π)4 k2 m2
Z − Q
Hence,
d4xd4k ei(q2−k)x
Π (q ,q) = π(q) q¯(x)γ (k +m )γ γ u(0) 0 . (5)
µν 2 (2π)4 m2 k2 h | µ 6 Q ν 5 | i
Z Q −
(cid:16) (cid:17)
We point out that, in eq.(5) and throughout the course of the present analysis, the path-
ordered factor P exp ig 1x Aµ(ux)du has been omitted, since in the Fock-Schwinger
s 0 µ
gauge x Aµ = 0, this(cid:16)factRor is trivial. It (cid:17)follows from eq.(5) that, the answer is expressed
µ
via the pion matrix element of the gauge invariant, nonlocal operator with a light cone
separation x2 0. Following [17], the two particle pion wave functions are defined as:
≃
1
π(q) d¯(x)γ γ u(0) 0 = iq f du eiqux ϕ (u)+x2g (u) +
µ 5 µ π π 1
h | | i −
Z0 h i
x2q 1
+q f x µ du eiquxg (u) , (6)
µ π µ 2
− qx !Z0
f m2 1
π(q) d¯(x)γ u(0) 0 = π π du eiquxϕ (u) , (7)
5 P
h | | i m +m
u d Z0
where ϕ (u) is the leading twist τ = 2, ϕ (u) is τ = 3 and g (u), g (u) are the τ = 4 two
π P 1 2
particle pion wave function. Using eq.(6) and eq.(7) after integrating over x and k, we get
for the structure g :
µν
m2 1 ϕ (u) 1 g (u)
Π(q ,q) = i π m du P +2m2 du 2
2 − (mu +md QZ0 ∆ QZ0 ∆2 −
1 (g (u)+G (u)) 2m2 1 ϕ (u)
q q 4 du 1 2 1+ Q du π , (8)
2
− " Zo ∆2 ∆ !−Z0 ∆ #)
where,
∆ = m2 (q +qu)2 ,
Q − 2
u
G (u) = g (v)dv .
2 2
−
Z0
Since the higher order twist contributions are taken into account in eq.(8), the terms
up to τ = 4, which comes from the propagator expansion, must also be considered. The
4
complete expansion of the propagatoris given in[18], andit contains the contributions from
the nonlocal operators q¯Gq, q¯GGq , and q¯qq¯q. The only operator we consider here is q¯Gq,
because the contributions from the other two are very small, and hence they are neglected
(for more detail see [17, 18]). Under these approximations, the heavy quark propagator is
given by the following expressions [3, 17]:
d4k 1 1 k +m
S (x) = S0(x) ig e−ikx du 6 Q Gµν(ux)σ +
Q Q − sZ (2π)4 Z0 " 2 m2 k2 2 µν
Q −
1 (cid:16) (cid:17)
+ ux Gµν(ux)γ . (9)
m2 k2 µ ν#
Q −
Substituting second term in eq.(9) into eq.(1), we get,
d4x d4k du ei(q2−k)x
Π (q ,q) = π(q) d¯(x)γ ux Gρλ(ux)γ +
µν 2 Z (2π)4 m2Q −k2 2 h | µh ρ λ
k +m (cid:16)1 (cid:17)
+ 6 Q Gρλ(ux)σ γ γ u(0) 0 . (10)
m2 k2 2 2 ρλ ν 5 | i
Q − i
(cid:16) (cid:17)
Using the following identities,
γ γ γ = g γ +g γ g γ iǫ γτγ ,
µ λ ν µλ ν λν µ µν λ µλντ 5
− −
γ σ = i(g γ g γ )+ǫ γτγ ,
µ ρλ µρ λ µλ ρ µρλτ 5
−
and the definition of the three particle pion wave functions [17],
¯
π(q) d(x)γ γ g G (ux)u(0) 0 =
µ 5 s αβ
h | | i
x q x q
f q g α µ q g β µ α ϕ (α )eiqx(α1+uα3) +
π β αµ α βµ i ⊥ i
(" − qx !− qx !# D
Z
q
+ µ (q x q x ) α ϕ (α )eiqx(α1+uα3) , (11)
α β β α i k i
qx − D )
Z
π(q) d¯(x)γ g G˜ (ux)u(0) 0 =
µ s αβ
h | | i
x q x q
if q g α µ q g β µ α ϕ˜ (α )eiqx(α1+uα3) +
π β αµ α βµ i ⊥ i
(" − qx !− qx !# D
Z
q
+ µ (q x q x ) α ϕ˜ (α )eiqx(α1+uα3) , (12)
α β β α i k i
qx − D )
Z
we obtain from eq.(10), for the structure g
µν
1 (1 2u)ϕ (α )+ϕ˜ (α )
k i k i
Π = iq q du α − , (13)
2 Z0 Z D i m2Q −[q2 +q1(α1 +uα3)]2 2
h i
5
where, α dα dα dα δ(1 α α α ), and G˜ = 1ǫ Gµν, is the dual tensor to
D i ≡ 1 2 3 − 1 − 2 − 3 αβ 2 αβµν
Gµν, and ϕ’s are all twist four fuctions. Collecting all the terms (eqs.(8) and (9))for the
theoretical part of the of the correlator function (1), we get,
m2 1 ϕ (u) 1 g (u)
Πtheor = i π m du P +2m2 du 2 +
− (mu +md QZ0 ∆ QZ0 ∆2
1 ϕ (u) 1 (g (u)+G (u)) 2m2
+q q du π 4 du 1 2 1+ Q
2
"Z0 ∆ − Z0 ∆2 ∆ !#−
1 (1 2u)ϕ (α )+ϕ˜ (α )
k i k i
q q du α − , (14)
− 2 Z0 Z D i ∆21 )
where,
∆ = m2 [q +q(α +uα )]2 .
1 Q − 2 1 3
The sum rule for the formfactor F (q2) is obtained by equating the expressions (4) and (14)
0 2
for the invariant amplitude Π(q2,(q +q)2), which results as:
2 2
f m F (q2) ∞ ρh(q2,s)ds
P1 P1 0 2 + 2 = Πtheor.
m2 (q +q)2 s (q +q)2
P1 − 2 Zs0 − 2
Invoking the duality prescripton and performing the Borel transformation in the variable
(q +q)2, we get the following sum rules for the form factor F (q2) :
2 0 2
f 1 du m2 m2 q2u¯
F (q2) = π exp P1 Q − 2 Φ (u,M2,q2)+
0 2 fP1mP1(Zδ u "M2 − uM2 # 2
1 1 α
i
+ du D Θ(α +uα δ)
2 Z0 Z (α1 +uα3)2 1 3 − ×
m2 m2 q2u¯
exp P1 Q − 2 Φ (u,M2,q2) , (15)
× "M2 − uM2 # 3 )
where,
m2 q2
δ = Q − 2 ,
s q2
0 − 2
u¯ = 1 u ,
−
and,
m2 m2 q2
Φ = π m ϕ (u)+ Q − 2ϕ (u)+
2 Q P π
m +m u
u d
g (u) m2 q2 m2(m2 q2)
+2 1 1+ Q − 2 Q Q − 2 +
u " uM2 − u2M4 #
G (u) m2 q2
+2 2 1 Q − 2 ,
u " − uM2 #
m2 q2
Φ = (1 2u)ϕ (α )+ϕ˜ (α ) 1 Q − 2 .
3 − k i k i − (α1 +uα3)M2!
h i
6
3 Numerical analysis and discussion
Now we turn to the numerical calculations. In expression (15), we use the following set of
parameters: m = 5.738 GeV, m = 2.4 GeV, (s ) = 35 GeV2, (s ) = 8 GeV2. For
B1 D1 0 B 0 D
leptonic decay constants f and f , we use the results given in [19]: f 0.2 0.02 GeV
B1 D1 B1 ≃ ±
and f 0.3 0.03 GeV. The highest value of the Borel Parameter M2 is fixed by
D1 ≃ ±
imposing the condition that the continuum contribution is 30% of the resonance. With
the help of this restriction, we calculate the maximum value of the Borel parameter for
B(D) to be M2 20 GeV2( 8 GeV2). The minimal value of the same parameter is
max ≃ ≃
usually fixed by the condition that the terms proportional to the higher powers of 1 are
M2
negligible and it is found to have the value M2 8 GeV2( 2 GeV2) for B(D) mesons.
min ≃ ≃
Our calculations show that, the variationof M2 within the above-mentioned limits, changes
the result by less than 10%, which means that the dependence of the formfactor F (q2) on
0 2
the Borel paremeter M2 is quite weak. Note that, a similiar situation exists for the B π
→
and D π transitions too (see for example [2, 3]). Because of that, in our analysis we take
→
M2 = 15 GeV2 for the B π and M2 = 4 GeV2 for the D π transition formfactors.
1 1
→ →
The maximum momentum transfer q2 at which the sum rule (15) is applicable, is about
2
15 20 GeV2 for B meson and 1 GeV2 for the D meson cases, respectively(for more detail
−
see [2, 3]). The explicit form of the pion wave function, used in eq.(15) can be found in [17].
The momentum transfer q2 dependence of the formfactors FB(q2) and FD(q2) are plotted
2 0 2 0 2
in Fig.1 and Fig.2. From these figures we observe that,
FB(q2 = 0) = 1.2 ,
0 2
FD(q2 = 0) = 1.5 . (16)
0 2
Let us turn our attention to the pole model prediction for FB and FD formfactors. In
0 0
[3], it is shown that the coupling constants gB∗Bπ and gD∗Dπ fix the normalization of the
formfactors of the B π and D π transitions, respectively, within the context of pole
→ →
model(seealso,[4,5]). Usingthepolemodeldescriptioninasimilar manner, theformfactor
F0(q22) can be expressed via the gP1P∗π coupling constant, that is calculated using the same
correlator function (1) in [19]. Indeed, the formfactor F (q2) defined by the matrix element,
0 2
π(q) d¯γ Q P (p) = F (q2)ǫ (p)+F (q2)(ǫq)(p+q) +F (q2)(ǫq)(p q) , (17)
h | µ | 1 i 0 2 µ 1 2 µ 2 2 − µ
7
is predicted to be (for the structure g )
µν
F (q2) = gP1P∗πfP∗ . (18)
0 2 mP∗ 1− mq2P22∗
(cid:18) (cid:19)
In deriving eq.(18), we used the following definition,
hπP∗|P1i ≡ gP1P∗π(ǫǫ∗) , (19)
where ǫ and ǫ∗ are the 4-polarization vectors of the P and P∗ mesons, respectively. The
1
dependence of the formfactors FB(q2) and FD(q2) on q2 in the pole model (eq.(18)) are
0 2 0 2 2
plotted in Fig.1 and Fig.2, where we use gB1B∗π = 24±3 GeV and gD1D∗π = 10±2 GeV
[19]. From these figures we conclude that, in the overlap region both calculations agree,
approximately, with each other. Quantitatively, at q2 = 0, it follows from eq.(18) that,
2
FB(q2 = 0) = 0.72 ,
0 2
FD(q2 = 0) = 1.20 . (20)
0 2
If we compare eqs.(16) and (20), we see that, in the region for which m2 q2 > (1 GeV2),
Q− 2
(Q = b, c), the numerical agreement between the two approaches is better than 20% for D,
but only within 35% for the B meson case.
Finally we would like to point out that, these two approaches lead to absolutely different
asymptotic behaviours of the relevant formfactors. Using the HQFT results, we get
fP1√m = fˆP1 , fP∗√m = fˆP∗
and,
2m
gP1P∗π ≃ gˆ (see also [3]) .
From eq.(18) it follows that, at m we get,
b
→ ∞
FB(D) (m )−1/2 . (21)
0 ∼ B(D)
Then from eq.(15) we see that, in this limit
FB(D) (m )−3/2 . (see also [20]) (22)
0 ∼ B(D)
In conclusion, we calculated the formfactors of the excited state 1+ mesons, namely,
B π and D π transitions, in the framework of the light cone QCD sum rules
1 1
→ →
8
and compared our results with the pole dominance model predictions. The comparision
elaborates that, the agreement between the two descriptions is rather good. This justifica-
tion demonstrates that, the pole dominance model works quite good for the p-wave meson
transitions, as it does for the s-wave ones.
9
