Semileptonic (D_s^+ -> phi l nu) decay in QCD light cone sum rule

0.13 MB·

by  T. M. Aliev

¯
Semileptonic D+ φℓν decay in QCD light cone sum
s
→
rule
4 T. M. Alieva ∗, A. O¨zpinecib †, M. Savcıa ‡
0
0 a Physics Department, Middle East Technical University, 06531 Ankara, Turkey
2
b The Abdus Salam International Center for Theoretical Physics, I-34100, Trieste, Italy

n
a
J

3
2
Abstract

1
v Wecalculate theformfactors V,A1,A2 andA0 appearingintheD φtransition
→
1 in light cone QCD sum rule method. We compare our results on these form factors
8
with the current experimental results and existing theoretical calculations.
1
1
0 PACS numbers: 11.55.Hx, 13.20.Fc
4
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p
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∗e-mail: [email protected]
†e-mail: [email protected]
‡e-mail: [email protected]
1 Introduction
Semileptonic decays of mesons containing charm and beauty quarks constitute a very im-
portant class of decays for studying strong and weak interactions. These decay modes of
heavy flavored mesons are much more clear samples compared to that of the hadronic decay
modes, since leptons do not participate in the strong interaction.
Therefore, the study of these decays is one efficient way for determining the elements of
the Cabibbo–Kobayashi–Maskawa (CKM) matrix, as well as for understanding the origin
of CP violation which is related to the structure of the CKM matrix in Standard Model
(SM).
An accurate determination of CKM matrix elements, obviously, depends crucially on
the possibility of controlling the effects of the strong interactions. For exclusive decays,
where initial and final states of hadrons are known, the main job is to calculate various
transition form factors, which involve all the long distance QCD dynamics. So, some non–
perturbative approach for estimating the long distance effects is needed. Several methods
have been used to treat these effects, such as quark model, QCD sum rules, lattice theory,
chiral perturbation theory, etc. Among these approaches, QCD sum rules occupies a special
place, since it is based on the very first principles of QCD.
The method of QCD sum rules [1] has been successfully applied to wide variety of
problems of hadron physics (see [2, 3] and references therein). In this method, physical
observables of hadrons are related to QCD vacuum via a few condensates. The semileptonic
decay D K¯0eν¯ was firstly studied in QCD sum rules with 3–point correlation function
e
→
in [4]. This method, then, successfully extended to study other semileptonic decay decays
of D and B mesons, i.e., D+ K¯0e+ν , D+ K¯0∗e+ν [5], D πeν¯ , D ρeν¯ [6],
e e e e
→ → → →
B D(D∗)ℓν¯ [7] and B πℓν¯ [8].
ℓ ℓ
→ →
However, this method inherits some problems, the main one being that some of the form
factors have nasty behavior in the heavy quark limit m . In order to overcome the
Q
→ ∞
problems of the traditional QCD sum rules, an alternative method, namely light cone QCD
sum rules (LCQSR) was developed in [9] and is regarded as an efficient tool in studying
exclusive processes which involve the emission of a light particle.
The LCQSR is based on the operator product product expansion (OPE) near the light
cone x2 0, which is an expansion over the twist of the operators, rather than the dimen-
≈
sions as in the traditional QCD sum rules. All non–perturbative dynamics is parametrized
by the so–called light cone wave functions, instead of the vacuum condensates in the tradi-
tional sum rules, which represents the matrix elements of the nonlocal operators between
thevacuum andthecorresponding particle (moreaboutthismethodcanbefoundin[3,10])
The LCQSR has wide range of applications to numerous problems of hadron physics.
One of the promising ways for obtaining information about CKM matrix elements, as well
as about wave functions, is studying the semileptonic decays.
In this work we study D+ φℓ¯ν decay in LCQSR. This decay mode has been measured
s →
in experiments in [11]–[14]. Note that D φ transition form factors are calculated in the
→
framework of traditional 3–point QCD sum rules in [15, 16], but the results don’t confirm
each other. Therefore we decided to calculate D φ form factors using light cone sum
→
rules as an alternative approach to the traditional sum rules.
The paper is organized as follows. In section 2 we derive the sum rules for the transition
1
form factor. Section 3 is devoted to the numerical analysis and discussions and contains a
summary of results and conclusions.
2 Light cone sum rules for the D φ transition form
s
→
factors
We start by defining the form factors of D φ weak form factors in the following way
s
→
φ(P) s¯γ (1 γ )c D (p ) = iε∗(m +m )A (q2)
h | µ − 5 | s Ds i − µ Ds φ 1
A (q2) 2m
+ i(p +P) (ε∗q) 2 +iq (ε∗q) φ[A (q2) A (q2)]
Ds µ m +m µ q2 3 − 0
Ds φ
2V(q2)
+ ǫ ε∗αqβPγ , (1)
µαβγ
m +m
Ds φ
where q = p P is the momentum transfer, P and ε are four momentum vector polar-
Ds −
ization of the vector φ meson, respectively, and p is the four momentum of D meson.
Ds s
In this section we derive sum rules for these form factors. In order to calculate the form
factors of the semileptonic D φℓν decay, we consider the following correlator function
s
→
Π (P,q) = i d4xeiqx φ(P)T[s¯(x)γ (1 γ )c(x)c¯(0)(1 γ )s(0)] 0
µ µ 5 5
h − − | i
Z
ε∗q ε∗q
= Γ0ε∗ Γ+ (2P +q) Γ− q +iΓVε ε∗αqβPγ . (2)
µ − Pq µ − Pq µ µαβγ
The Lorentz invariant functions Γ0,±,V can be calculated in QCD for large Euclidean p2 ,
Ds
to put it more correctly, when m2 p2 0, the correlation function (1) is dominated by
c − Ds ≪
the region of small x2 and can be systematically expanded in powers of deviation from the
light cone x2 = 0.
The main reason for choosing the chiral current c¯(1 γ )s is that, in this case many
5
−
of the twist–3 wave functions which are poorly known and cause the main uncertainties
to the sum rules, can effectively be eliminated and provide results with less uncertainties.
The chiral current approach has been applied to studying B π [17, 18], B η [19] weak
→ →
form factors.
Let us discuss firstly the hadronic representation of the correlator. This can be done by
inserting the complete set of intermediate states with the same quantum numbers of the
current operator c¯(1 γ )s in the correlation function. By isolating the pole term of the
5
−
lowest pseudoscalar D meson, we get thefollowing representation of the correlator function
s
from hadron side
φ s¯γ (1 γ )c D D c¯(1 γ )s 0
µ 5 s s 5
Π (P,q) = h | − | ih | − | i
µ m2 (P +q)2
Ds −
φ s¯γ (1 γ )c h h c¯(1 γ )s 0
µ 5 5
+ h | − | ih | − | i . (3)
m2 (P +q)2
Xh h −
2
For the invariant amplitudes Γ0,±,V, one can write a general dispersion relation in the
p2 = (P +q)2 limit
Ds
ρi(s)
Γi(q2,(P +q)2) = ds +subtr. ,
s (P +q)2
Z −
where the spectral densities corresponding Eq. (2) can easily be calculated. As an illustra-
tion of this fact, we present the result for Γ0
f m2
ρ(0)(s) = Ds Ds (m +m )A (q2)δ(s m2 )+ρ(0)h(s) . (4)
m +m B V 1 − Ds
c s
The first term in Eq. (4) represents the contribution of the ground state D meson. In
s
deriving Eq. (2), we have used
f m2
D c¯(1 γ )s 0 = i Ds Ds .
s 5
h | − | i m +m
c s
The second term in Eq. (4) corresponds to the spectral density of the higher resonances
and continuum. The spectral density ρ(0)h(s) can be approximated by invoking the quark
hadron duality anzats
ρ(0)h(s) = ρ(0)QCD(s s ) .
0
−
So for the hadronic representation of the invariant amplitude Γ(0) we have
f m2 m +m ∞ ρ(0)QCD(s)
Γ(0) = Ds Ds B φ A (q2)+ ds +subtr. . (5)
m +m m2 (P +q)2 1 s (P +q)2
c s Ds − Zs0 −
Hadronic representations for other invariant amplitudes can be constructed in precisely the
same manner.
In order to obtain sum rules for the form factors A , A , A and V, we must calculate
1 2 0
the correlator from QCD side. This calculation can be performed by using the light cone
OPE. The contributions to OPE can be obtained by contracting the quark fields to a full
c–quark propagator, i.e.,
Π (P,q) = i d4xeiqx φ s¯γ (1 γ )S (x)(1 γ )s(0) 0
µ µ 5 c 5
h | − − | i
Z
i
= d4xeiqx Trγ (1 γ )S (x)(1 γ )Γ φ s¯Γis 0 , (6)
µ 5 c 5 i
4 − − h | | i
Z h i
where Γi is the full set of the Dirac matrices Γi = (I, γ , γ , γ γ , σ ), and
5 α α 5 αβ
d4k 1 1 k +m
iS (x) = iS(0)(x) ig e−ikx du 6 c G (ux)σµν
c c − sZ (2π)4 Z0 "2(m2c −k2)2 µν
1
+ ux Gµν(ux)γ . (7)
m2 k2 µ ν#
c −
3
Here, G is the gluonic field strength, g is the strong coupling constant and S(0) represents
µν s c
a free c–quark propagator
d4k k +m
S(0)(x) = e−ikx 6 c . (8)
c (2π)4 k2 m2
Z − c
From Eqs. (6)–(8) we see that, in order to calculate the theoretical part of the correlator,
the matrix elements of the nonlocal operators between vector φ meson and vacuum states
are needed. We see from Eq. (6) that, the contribution to the correlator comes only from
the wave functions that contain odd–number γ matrices.
Up to twist–4, the φ meson wave functions containing odd–number of γ matrices, and
appearing in our calculations are:
eλx 1 m2x2
φ(P,λ) s¯(x)γ s(0) 0 = f m P dueiuPx Φ (u,µ2)+ φ A(u,µ2)
µ φ φ µ k
h | | i " Px Z0 16 !
eλx 1
+ eλ P dueiuPxg(v)(u,µ2)
µ − µPx!Z0 ⊥
1 eλx 1
x m2 dueiuPxC(u,µ2) , (9)
− 2 µ(Px)2 φ
Z0
1 2fTm 1
φ(P,λ) s¯(x)γ γ s(0) 0 = f φ s m ǫ ναβeλP x dueiuPxg(a)(u,µ2) ,
h | µ 5 | i 4 φ − mφ ! φ µ ν α βZ0 ⊥
(10)
φ(P,λ) s¯(x)gG (ux)iγ s(0) 0 = f m p (p eλ p eλ ) (u,px)
h | µν α | i φ φ α ν ⊥µ − µ ⊥ν V
eλx
+ f m3 (p g⊥ p g⊥ )Φ(u,px)
φ φ px µ αν − ν αµ
eλx
+ f m3 p (p x p x )Ψ(u,px) , (11)
φ φ(px)2 α µ ν − ν µ
φ(P,λ) s¯(x)gG (ux)iγ γ s(0) 0 = f m p (p eλ p eλ ) (u,px)
h | µν α 5 | i φ φ α ν ⊥µ − µ ⊥ν V
eλx
e + f m3 (p g⊥ p g⊥ )eΦ(u,px)
φ φ px µ αν − ν αµ
eλx e
+ f m3 p (p x p x )Ψ(u,px) . (12)
φ φ px α µ ν − ν µ
e
In all expressions, we have used
1 m2
φ
p = P x ,
µ µ µ
− 2 px
eλx m2
eλ = p φ x +eλ ,
µ px µ − 2(px) µ! ⊥µ
1
g⊥ = g (p x +p x ) , (13)
µν µν − px µ ν ν µ
4
(v) (a)
where Φ is the leading twist–2 wave function, while g , g , are twist–3 and all the
k ⊥ ⊥ V
remaining ones are twist–4 wave functions. The notation used in Eqs. (11)–(14) is the
following
K(u,Px) = αeiPx(α1+uα3)K(α) , (14)
D
Z
where
α = dα dα dα δ(1 α α α ) .
1 2 3 1 2 3
D − − −
Inserting Eqs. (7) and (8) into Eq. (6) and using the definitions of φ meson wave functions,
the invariant structures Γ0,±,V take the following form
(v)
2f m m g (u)
Γ = du φ φ c ⊥ , (15)
0
∆
Z 1
f m m f m3m uC(u)
Γ+ = du φ φ c m2m2A(u)+4(Pq)∆ Φ(i)(u) φ φ c
Z ( ∆31 h c φ 1 k i− ∆21
f m3m
φ φ c
α 2Φ 2Φ+Ψ Ψ V + V , (16)
− Z D ∆22 − − − 2 2e!)
e e
f m m f m3m (2 u)C(u)
Γ− = du − φ φ c m2m2A(u)+4(Pq)∆ Φ(i)(u) φ φ c −
Z ( ∆31 h c φ 1 k i− ∆21
f m3m
+ α φ φ c 2Φ 2Φ+Ψ Ψ V + V , (17)
Z D ∆22 − − − 2 2e!)
e e
2fTm f m m
ΓV = du 1 φ s g(a) φ φ c , (18)
Z − fφmφ ! ⊥ ∆21
where
u
(i) (v)
Φ (u) = dv Φ (v) g (v) ,
k k − ⊥
Z0 h i
∆ = m2 (q +Pu)2 ,
1 c −
∆ = m2 [q +(α +uα )P]2 ,
2 c − 1 3
Equating expressions of the invariant structures Γ0,±,V coming from QCD and phe-
nomenological parts of the correlation function and making the Borel transformation with
respect to (P + q)2 in both parts, in order to suppress the contributions of higher states
and continuum and also to eliminate the subtraction terms in the dispersion integral, we
get the following sum rules for the D φ transition form factors:
→
(v)
m +m 1 1 g (u)
A (q2) = c s em2Ds/M2 2f m m du ⊥ e−s(u)/M2 , (19)
1 fDsm2Ds mDs +mφ ( φ φ cZδ u )
(m +m )(m +m ) 2
A (q2) = c s Ds φ em2Ds/M2
2 f m2 m2 m2 q2
Ds Ds Ds − φ −
5
1 1 1 1 1 1
f m m m2m2 A(u) e−s(u)/M2 du Φ(i)(u)e−s(u)/M2
× ( φ φ c"2 φ c Zδ u 2(M2u)2 −Zδ u2 k
(i)
1 1 Φ (u) 1 uC(i)(u)
+ du (m2 m2u2 q2) k e−s(u)/M2 m2 du e−s(u)/M2
Zδ u c − φ − u2M2 − φZδ u2M2 #
1 1
f m3m du αθ(s s(k)) 2Φ(α) 2Φ(α)+Ψ(α) Ψ(α)
− φ φ cZδ D 0 − k2M2" − −
e e
V + V e−s(k)/M2 . (20)
− 2 2# )
e
The form factor A (q2) can be obtained from the exact result
3
m +m m m
A (q2) = Ds φA (q2) Ds − φA (q2) , (21)
3 1 2
2m − 2m
φ φ
and A (q2) can be calculated from the following sum rule
0
m +m q2 1
A (q2) A (q2) = c s em2Ds/M2
3 − 0 f m2 2m m2 m2 q2
Ds Ds φ Ds − φ −
1 1 1 1 1 1
f m m m2m2 A(u) e−s(u)/M2 + du Φ(i)(u)e−s(u)/M2
× ( φ φ c"− 4 φ c Zδ u 2(M2u)2 Zδ u2 k
(i)
1 1 Φ (u) 1 (2 u)C(i)(u)
du (m2 m2u2 q2) k e−s(u)/M2 m2 du − e−s(u)/M2
− Zδ u c − φ − u2M2 − φZδ u2M2 #
1 1
+ f m3m du αθ(s s(k)) 2Φ(α) 2Φ(α)+Ψ(α) Ψ(α)
φ φ cZδ D 0 − k2M2" − −
e e
V + V e−s(k)/M2 , (22)
− 2 2# )
e
(m +m )(m +m )
V(q2) = Ds φ c s em2Ds/M2
2f m2
Ds Ds
2m fT 1 1
1 s φ f m m dug(a)(u) e−s(u)/M2 , (23)
× ( − fφmφ ! φ φ cZδ ⊥ u2M2 )
where M2 is the Borel parameter and
m2 q2t¯+m2tt¯
s(t) = c − φ ,
t
u, or ,
t =

 k = α1 +uα3 ,
t¯= 1−t ,
1
δ = (m2 +q2 s )+ (s m2 q2)2 4m2(q2 m2) .
2m2 φ − 0 0 − φ − − φ − c
φh q i
6
3 Numerical analysis
In this section we present our numerical calculation of the form factors A , A , A and V.
1 2 0
As can easily be seen from the expressions of these form factors, the main input parameters
are the φ meson wave functions, whose explicit forms are given in [20] and we use them
in our study. The values of the other input parameters appearing in sum rules for form
factors are: m = 1.9686GeV, m = 0.14 GeV, m = 1.3 GeV, f = (0.214 0.038)GeV
Ds s c Ds ±
[3], m = 1.02 GeV. The leptonic decay constant of φ meson, which is f = 0.234 GeV,
φ φ
is extracted from the experimental result of the φ ℓ+ℓ− decay [21]. The threshold
→
s = (6.5 0.5) GeV2 is determined from the analysis of the three point function sum rules
0
±
for f (see for example [3]).
Ds
With the above–mentioned input parameters, we now proceed by carrying out our nu-
merical analysis. The first step, according to sum rules philosophy, is to look a working
region of the auxiliary Borel parameter M2, where numerical results should be stable for a
given threshold s . The lower limit of M2 is determined by the condition that the terms
0
M−2n (n > 1) remains subdominant. The upper bound of M2 is determined by requiring
that the continuum and higher state contributions constitute maximum 30% of the total
result. Our numerical analysis shows that both requirements are satisfied in the region
3 GeV2 M2 4.5 GeV2. We should note that LCQSR for the form factors are reliable
≤ ≤
at the region q2 < 0.4 GeV2. Moreover, we analyse the M2 dependencies of the form factors
∼
A , A , A and V at q2 = 0 GeV2 and q2 = 0.2 GeV2, for three different values of the
1 2 0
continuum threshold, namely, s = 6.0, 6.5 and s = 7.0 GeV2. Our analysis shows that
0 0
the form factors are practically independent of the Borel mass when M2 varies between
3 GeV2 and 4 GeV2. Variation of the form factors in relation to the continuum threshold
is also very weak. The results for all form factors change about 5% at q2 = 0. Our final
results for the form factors at q2 = 0 and s = 6.5 GeV2, are
0
A (0) = 0.65 0.15 ,
1
±
A (0) = 0.85 0.20 ,
2
±
A (0) = A (0) = 0.56 0.1 ,
0 3
±
V(0) = 0.90 0.20 . (24)
±
It should be noted that in the region q2 0.4 GeV2 the applicability of the light
≥
cone QCD sum rule is questionable. In order to extend our results to the full physical
region, we look for a parametrization of the form factors in such a way that in the region
0 q2 0.4 GeV2, the above–mentioned parametrization coincides with the light cone
≤ ≤
QCD sum rules prediction. The most convenient parametrization of the q2 dependence of
the form factors is given in terms of three parameters in the following form
F (0)
F (q2) = i . (25)
i 1 a (q2/m2 )+b (q2/m2 )2
− Fi Ds Fi Ds
The values of the parameters F (0), a and b are listed in Table (1).
i Fi Fi
We proceed by discussing the uncertainties related to the input parameters and wave
functions. Wenote first that theradiative corrections to the leading twist–2 function, which
is calculated in [20], is about 10%. As has already been noted, the results depend weakly
∼
7
F(0) a b
F F
A 0.65 1.36 0.31
1
−
A 0.85 4.5 5.55
2
A 0.56 0.13 0.46
0
−
V 0.9 2.82 1.51
Table 1: Parameters of the form factors given in Eq. (25), for the D decay in a three–
s
parameter fit. We take the central values of the form factors for F(0).
on the continuum threshold s and Borel parameter M2, and the uncertainty due to these
0
parameters is about 5%–7% in the working region of M2. Moreover, the results are also
quite weakly dependent on the vector meson decay constant f and fT, which results in
φ φ
an uncertainty about 5%. Additional uncertainty coming from the Gegenbauer moments
are about 10%. Summing up all these above–mentioned errors, the overall uncertainty
∼
in the values of the form factors is of the order of 17%.
In the experiments, the ratios
V(0) A (0)
2
r = , and r = ,
1 2
A (0) A (0)
1 1
are measured. In the present work, within the framework of the light cone QCD sum rules,
we get r = 1.57 0.36 and r = 1.30 0.34. In Table (2), we present a comparison of our
1 2
± ±
results with the existing experimental data and 3–point sum rule (3PSR).
Using the parametrization of the D φ transition in terms of the form factors A ,
s 1
→
A , V, A A , the differential decay width as a function of q2, in terms of the helicity
2 3 0
−
amplitudes can be written as
dΓ G2 V 2
= F | cs| λ1/2(m2 ,m2,q2)q2 H2 +H2 +H2
dq2 192π3m3 Ds φ 0 + −
Ds h i
dΓ dΓ dΓ
L + −
+ + , (26)
≡ dq2 dq2 dq2
wheretheindicesindΓ /dq2 andH denotethepolarizationoftheφmeson,λ(m2 ,m2,q2) =
i i Ds φ
(m2 +m2 q2)2 4m2 m2, and
Ds φ − − Ds φ
λ1/2(m2 ,m2,q2)
H = (m +m )A (q2) Ds φ V(q2) , (27)
± Ds φ 1 ∓ m +m
Ds φ
1 λ(m2 ,m2,q2)
H = (m2 m2 q2)(m +m )A (q2) Ds φ A (q2) . (28)
0 2mφ√q2" Ds − φ − Ds φ 1 − mDs +mφ 2 #
The differential decay rate when the final state φ meson is transversally polarized is
determined to be
dΓ dΓ dΓ
T + −
= + . (29)
dq2 dq2 dq2
8
r r
1 2
E791 2.27 0.35 0.22 1.57 0.25 0.19
± ± ± ±
CLEO 0.9 0.6 0.3 1.4 0.5 0.3
± ± ± ±
E687 1.8 0.9 0.2 1.1 0.8 0.1
± ± ± ±
E653 2.3+1.1 0.4 2.1+0.6 0.2
−0.9 ± −0.5 ±
Average 1.92 0.32 1.60 0.24
± ±
3PSR [15] 2.20 0.85 1.16 0.46
± ±
3PSR [16] 2.16 0.38 1.08 0.17
± − ±
Our Results 1.38 0.44 1.31 0.43
± ±
Table 2: Comparison of our results for r and r with the experimental results and 3–point
1 2
sum rule.
Integrating the differential decay widths over q2 in the region from q2 = 0 to (m m )2,
Ds − φ
we obtain
Γ = (1.52−0.61) 10−14 GeV ,
L +0.77 ×
Γ = (2.21−0.90) 10−14 GeV ,
T +1.13 ×
and for their ratio, we get
Γ
L
= (0.69 0.44) ,
Γ ±
T
which is in good agreement with the existing experimental data
Γ
L
= 0.72 0.16 , [21]
ΓT!exp ±
Using the value of the total decay width Γ = 1.34 10−12 GeV [21] of the D meson,
Ds × ¯ s
we get the following result for the branching ratio of the D φℓν decay
s
→
(D φℓ¯ν) = (2.78−1.13)% ,
B s → +1.42
which is consistent with the experimental result
¯
(D φℓν) = (2.0 0.5)% .
s exp
B → ±
9