Diquarks and the Semi-Leptonic Decay of $Λ_{b}$ in the Hybrid Scheme

0.25 MB·English

by  Peng Guo

Diquarks and the Semi-Leptonic Decay of Λ in the Hybrid
b
Scheme
Peng Guo1, Hong-Wei Ke1, Xue-Qian Li1, Cai-Dian Lu¨2 and Yu-Ming Wang1,2
1. Department of Physics, Nankai University, Tianjin, 300071, P.R. China
7 2. CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China and
0
0 Institute of High Energy Physics, CAS, P.O. Box 918, Beijing 100049, P.R. China
2
(Dated: February 2, 2008)

n
a
J Inthisworkweusethe heavy-quark-light-diquarkpicturetostudy thesemileptonicdecay

4
Λb Λc+l+ν¯l inthe so-calledhybridscheme. Namely,we applythe heavyquarkeffective
2 →
theory (HQET) for larger q2 (corresponding to small recoil), which is the invariant mass

6
2
v squareofl+ν¯,whereasthe perturbativeQCDapproachforsmaller q to calculatethe form
8
5 factors. The turning point where we require the form factors derived in the two approaches
0
1 to be connected, is chosen near ρcut = 1.1. It is noted that the kinematic parameter ρ
0
which is usually adopted in the perturbative QCD approach, is in fact exactly the same as
5
0 the recoil factor ω = v v′ used in HQET where v, v′ are the four velocities of Λb and Λc
/ ·
h
respectively. We find that the final result is not much sensitive to the choice, so that it is
p
- relatively reliable. Moreover, we apply a proper numerical program within a small range
p
e aroundρcut tomakethe connectionsufficientlysmoothandweparameterizethe formfactor
h
: by fitting the curve gained in the hybrid scheme. The expression and involved parameters
v
i
X can be compared with the ones gained by fitting the experimental data. In this scheme the
r end-point singularities do not appear at all. The calculated value is satisfactorily consistent
a
withthedatawhichisrecentlymeasuredbytheDELPHIcollaborationwithintwostandard
deviations.
PACS numbers: 14.20.Mr, 14.20.Lq, 12.39.Hg, 12.38.Bx, 12.38.-t
I. INTRODUCTION
The general theory of QCD has been developed for more than 40 years, and at present, nobody
ever doubts its validity. However, on the other side there is still not a reliable way to deal with
the long-distance effects of QCD which are responsible for the quark confinement and hadronic
transition matrix elements, because their evaluations cannot be done in perturbative approach.
Thus, one needs to factorize the perturbative sub-processes and the non-perturbative parts which
2
correspond to different energy scales. The perturbative parts are in principle, calculable to any
order within the framework of quantum field theory, whereas the non-perturbative part must be
evaluated by either fitting data while its universality is assumed, or invoking concrete models.
The perturbative QCD method (PQCD) has been applied to study processes where transitions
from heavy mesons or baryons to light hadrons are concerned [1, 2, 3], namely the PQCD which
includes the Sudakov resummation, is proved to be successful for handling processes with small
4-momentum transfer q2. Indeed the processes involving heavy hadrons may provide us with an
opportunity to study strong interaction, because compared to Λ there exist natural energy
QCD
scales (heavy quark masses) which can beused to factorize the perturbativecontributions from the
non-perturbative effects. On the other hand, for the processes involving heavy hadrons, at small
recoil region, where v v′ is close to unity (v and v′ denote the four-velocities of the initial and
·
final hadrons), i.e. the momentum transfer q2 is sufficiently large, the heavy quark effective theory
(HQET) works well due to an extra symmetry SU (2) SU (2) [4]. Therefore the HQET and
f s
⊗
PQCD seem to apply at different regions of q2. For a two-body decay, the momentum transfer is
fixed by the kinematics, however, for a three-body decay, q2 would span the two different regions.
Amongall theprocesses, thesemi-leptonic decay of hadronsplays an importantrole for probing
the underlying principles and employed models because this process is relatively simple and less
dependent on the non-perturbative QCD effects. Namely leptons do not participate in strong in-
teraction, and thereis no contamination from thecrossed gluon-exchanges between quarks residing
in differenthadronswhich areproducedin theweak transitions, whereas sucheffects areimportant
for the non-leptonic decays. Thus one might gain more model-independent information, such as
extraction of the Cabibbo-Kobayashi-Maskawa matrix elements from data. In the semi-leptonic
decays of heavy hadrons it is expected to factorize the perturbative and non-perturbative parts
more naturally. Recently the DELPHI collaboration reported their measurement on the Λ decay
b
form factor in the semi-leptonic process Λ Λ +l+ν¯ and determined the parameter ρˆ2 in the
b c l
→
Isgur-Wise function ξ(v v′)= 1 ρˆ2(1 v v′) [5].
· − − ·
Thereisafloodofpaperstodiscussthesemi-leptonicdecays ofheavymesonsandtheconcerned
factorization. By contraries, the studies on heavy baryons are much behind [6, 7, 8], because
baryons consist of three constituent quarks and their inner structures are much more complicated
than mesons. In this work, we are going to employ the one-heavy-quark-one-light-diquark picture
fortheheavyΛ andΛ toevaluate theformfactorsofthissemi-leptonic transitionΛ Λ +l+ν¯.
b c b c
→
Even though the subject of diquark is still in dispute, it is commonly believed that the quark-
diquark picture may be a plausible description of baryons [9], especially for the heavy baryons
3
which possess one or two heavy quarks.
The kinematic region for the semileptonic decay Λ Λ +l+ν¯ can be characterized by the
b c
→
quantity ρ, which is defined as
p p′
ρ · , (1)
≡ M M
Λb Λc
where p, p′ are the four-momenta of Λ and Λ respectively. It is noted that this parameter
b c
ρ which is commonly adopted in the PQCD approach is exactly the same as the recoil factor
ω = v v′ used in the HQET. The momentum transfer q2 in the process is within the range of
·
(M2 +M2 )
(m +m )2 q2 (M M )2, equivalently, it is 1 ρ Λb Λc ρ . In the framework
l ν ≤ ≤ Λb− Λc ≤ ≤ MΛbMΛc ≡ max
of the HQET [4], this process was investigated by some authors [6, 7, 8, 10]. For larger q2, the
HQETworks well, whereas onecan expect that for smaller q2, thePQCDapproach applies. Inthis
work, following Ref.[2], we calculate the contribution from the region with small q2, i.e. ρ ρ ,
max
→
to the amplitude in PQCD. One believes that the PQCD makes better sense in this region. Ko¨rner
et al. discussed similar cases and suggested that the symmetry for smaller recoil is different from
thatforlarger recoil, sotheyusedtheIsgur-Wisefunctiontoobtaintheamplitudeinthekinematic
region of smaller recoil, but Brodsky-Lepage function for larger recoils [11]. Our strategy is similar
that we apply the PQCD for small q2 while apply HQET for large q2 where the PQCD is no longer
reliable, instead.
Concretely, whenintegratingtheamplitudesquarefromminimumtomaximumofq2 togainthe
decayrate,wedividethewholekinematicregionintotwoparts,smallandlargeq2 (ρ,equivalently).
We phenomenologically adopt a turning point at a certain ρ value, to derive the form factors
(defined below in the text) in terms of PQCD in the region from ρ to this point and then
max
beyond it we use the HQET instead [8]. We let the two parts connect at the turning point. From
Ref. [3], we notice that as ρ 1.1, the PQCD result is not reliable, so that we choose the turning
≤
point at vicinity of ρ= 1.1. To testify if the choice is reasonable, we slightly vary the values of the
turning point as choosing ρ = 1.05, 1.10 and 1.15 to see how sensitive the result is to the choice.
Moreover, itisnotedthatasρ = 1.1and1.15arechosen, thetwopartsconnectalmostsmoothly.
cut
Even though, to make more sense, we adopt a proper numerical program to make the connection
sufficiently smooth, namely we let not only the two parts connect, but also the derivatives from
two sides are exactly equal. In fact, small differences in the derivatives are easily smeared out by
the program. Later in the text, we will explicitly show that the final result is not much sensitive
to it, thus one can trust its validity. We name the scheme as the “hybrid” approach. We also
parameterize the form factor with respect to ρ based on our numerical results. In fact, when we
4
integrate over the whole kinematic range of ρ, we just use the parameterized expression.
Moreover, we not only reevaluate the form factors f and g of the exclusive process in the
1 1
diquarkpicture,butalsocalculatetheformfactorsf andg whichwereneglectedinpreviousworks
2 2
[3]. So far, the data on the Λ semi-leptonic decay are only provided by the DELPHI collaboration
b
[5]andnotrich enough tosingle outthecontributions fromf andg . More accurate measurement
2 2
in the future may offer information about them. Our treatment has another advantage. In the
pure PQCD approach, there is an end-point divergence at ρ 1, even though it is mild and the
→
decay rate which includes an integration over the phase space of final states, i.e. over ρ, is finite.
As calculating the contribution from the region with large q2 (ρ 1)1 to the form factors in terms
→
of the HQET, the end-point divergence does not exist at all.
Weorganizeourpaperasfollows,insectionII,wederivethefactorizationformulaforΛ Λ lν¯.
b c
→
OurnumericalresultsarepresentedinSectionIII.Finally,SectionIVisdevotedtosomediscussions
and our conclusion.
II. FORMULATIONS
The amplitude of Λ Λ lν¯decay process is written as:
b c
→
G
= FV lγµ(1 γ )ν < Λ (p′) cγ (1 γ )b Λ (p) >, (2)
cb 5 l c µ 5 b
M √2 − | − |
where p and p′ are the momenta of Λ and Λ respectively. According to its Lorentz structure, the
b c
hadronic transition matrix element can be parameterized as
< Λ (p′) cγ (1 γ )b Λ (p) > (3)
µ c µ 5 b
M ≡ | − |
qν q
= Λ (p′) γ f (q2)+γ g (q2) +σ f (q2)+γ g (q2) + µ (f (q2)+γ g (q2)) Λ (p),
c µ 1 5 1 µν 2 5 2 3 5 3 b
" MΛb MΛb #
(cid:16) (cid:17) (cid:16) (cid:17)
where q p p′ and σ [γ ,γ ]/2.
µν µ ν
≡ − ≡
For the convenience of comparing with the works in literature, we rewrite the above equation
in the following form according to Ref.[8]
p
= Λ (p′) γ F (q2)+γ G (q2) + µ (F (q2)+γ G (q2))
µ c µ 1 5 1 2 5 2
M " MΛb
(cid:16) (cid:17)
p′
+ µ (F (q2)+γ G (q2)) Λ (p). (4)
3 5 3 b
MΛc #
1 If ml is not zero, ρ cannot be exactly 1, thusthesuperficial singularity does not exist at all, but theform factors
are obviously proportional to1/ml which has the singular property.
5
For the case of massless leptons,
q lγµ(1 γ )ν = 0, (5)
µ 5 l
−
thus the form factors f and g result in null contributions. The contributions from f and g were
3 3 2 2
neglected in previous literature [3], nevertheless in our work, we will consider their contributions
to the matrix elements and calculate them in terms of the diquark picture and our hybrid scheme.
The kinematic variables are defined as follows. In the rest frame of Λ
b
M M
p (p+,p−,p )= Λb, Λb,0 , (6)
T T
≡ √2 √2
(cid:18) (cid:19)
and
ρ+ ρ2 1 ρ ρ2 1
p′ = − M , − − M ,0 , (7)
p√2 Λc p√2 Λc T!
the diquark momenta inside Λ and Λ are parameterized respectively as
b c
M M
k1 = 0, Λbx1,k1T , k2 = Λcξ1x2,0,k2T , (8)
√2 √2
(cid:18) (cid:19) (cid:18) (cid:19)
whereρ p·p′ ,ξ ρ+ ρ2 1,x k−/p− andx k+/p′+. Accordingtothefactorization
≡ MΛbMΛc 1 ≡ − 1 ≡ 1 2 ≡ 2
theorem [1, 2, 3, 12, 13], thephadronic matrix element is factorized in the b-space as
1 d2b d2b
= dx dx 1 2 Ψ (x ,b ,p′,µ)
Mµ 1 2 (2π)2 (2π)2 Λc 2 2
Z0 Z
H (x ,x ,b ,b ,r,M ,µ)Ψ (x ,b ,p,µ), (9)
× µ 1 2 1 2 Λb Λb 1 1
e
where r M /M . The renormalization group evolution of the hard amplitude H is shown as
≡ Λc Λb µ
follows [14]
e
ta(b) dµ
H (x ,x ,b ,b ,r,M ,µ) = exp[ 4 γ (α (µ))]H (x ,x ,b ,b ,r,M ,t ),(10)
µ 1 2 1 2 Λb − µ µ q s µ 1 2 1 2 Λb a(b)
Z
e e
where γ (α (µ)) is the anomalous dimension.
q s
The wave function of Λ which has the heavy-quark and light-diquark structure, is given as
b
[11, 15]
Ψ (x ,b ,p,µ) = fS ΦS (x ,b ,p,µ)χS Λ (p,λ), (11)
Λb 1 1 Λb Λb 1 1 Λb b
whereΛ (p,λ)isthebaryonspinor,andthesuperscriptSdenotesscalardiquark(spin=0,isospin=
b
0). fS is a constant introduced in literature. χS is the flavor component of the baryon, namely
Λb Λb
χS = b†S† 0 >, where b† and S† are the creation operator of b-quark and the scalar diquark
Λb [u,d]| [u,d]
of ud quarks.
−
6
b c b c
N N
ud ud ud ud
(a) (b)
FIG.1: Thelowestorderdiagramsofthehardpartsofthetransitionprocessesinthequark-diquarkpicture.
The Λ distribution amplitude bears a similar form,
c
Ψ (x ,b ,p′,µ) = fS ΦS (x ,b ,p′,µ)χS Λ (p′,λ ). (12)
Λc 2 2 Λc Λc 2 2 Λc c 2
Includingthe Sudakov evolution of hadronicwave functions, i.e. runningthe scale of wave function
from µ down to ω (ω ) [14]:
1 2
µ dµ
ΦSΛb(x1,b1,p,µ) = exp[−s(ω1,(1−xl)p−)−2 µ γq(αs(µ))]ΦSΛb(x1,b1),
Zω1
µ dµ
ΦSΛc(x2,b2,p′,µ) = exp[−s(ω2,(1−x2)p′+)−2 µ γq(αs(µ))]ΦSΛc(x2,b2), (13)
Zω2
where ω = 1/b (i=1,2).
i i
In our work, the effective gluon-diquark vertices are defined as [11, 15]
SgS : ig tα(p +p ) F (Q2),
s 1 2 µ S
Q2
F (Q2) = δ 0 ,
S sQ2+Q2
0
δ = α (Q2)/α (Q2) (if Q2 Q2); δ = 1 (if Q2 < Q2), (14)
s s s 0 ≥ 0 s 0
where Q2 (p p )2.
1 2
≡ − −
According to the factorization scheme which is depicted in Fig.1, it is straightforward to obtain
the analytic expressions of the form factors F , G , F , G , F and G , by comparing eqs. (4) with
1 1 2 2 3 3
(9). In the above derivation, the following transformation has been used.
k = Ap+Bp′, k = Cp+Dp′ (15)
1 2
and the explicit expressions of A,B,C and D are given in the appendix.
Thenwecanobtaintheanalyticalformoff andg (i=1,2,3)makinguseoftherelationsbetween
i i
them and F ,G (j=1,2,3) listed below
j j
1 F F 1 G G
2 3 2 3
f = F + ( + )(M +M ), g = G ( + )(M M ),
1 1 2 M M Λb Λc 1 1− 2 M M Λb − Λc
Λb Λc Λb Λc
1 F F 1 G G
f = ( 2 + 3 )M2 , g = ( 2 + 3 )M2 ,
2 −2 M M Λb 2 −2 M M Λb
Λb Λc Λb Λc
1 F F 1 G G
f = ( 2 3 )M2 , g = ( 2 3 )M2 . (16)
3 2 M − M Λb 3 2 M − M Λb
Λb Λc Λb Λc
7
We do not display the expressions of f and g for the reason given above. The form factors are
3 3
integrations which convolute over three parts [12]: the hard-part kernel function, the Sudakov
factor and the wave functions of the concerned hadrons as
1 ∞ 2π
f (g ) = 4πC f f dx dx b db b db dθ Φ (x ,b ) (17)
i i F Λb Λc 1 2 1 1 2 2 Λc 2 2
Z0 Z0 Z0
kernSgS Φ (x ,b )exp[ S(x ,x ,b ,b ,r,M )],
× fi(gi) Λb 1 1 − 1 2 1 2 Λb
where the explicit expression of the kernel functions kernSgS is given in the appendix for
fi(gi)
concision of the text. The explicit form of the Sudakov factor appearing in the above equations is
given in Ref.[12] as
S(x ,x ,b ,b ,r,M )
1 2 1 2 Λb
=s(ω ,(1 x )p−)+s(ω ,(1 x )p′+)+2 ta(b) dµγ (α (µ))+2 ta(b) dµγ (α (µ)), (18)
1 l 2 2 q s q s
− − µ µ
Zω1 Zω2
where C = 4/3 is the color factor.
F
The wave function Φ (x ,k ) is [13, 16]
Λb 1 1T
2N x6(1 x )3
Φ (x ,k ) = b 1 − 1 , (19)
Λb 1 1T π[(1 a x )2x2(1 x )+ε (1 x )2x2+k2 ]3
− b− 1 1 − 1 pb − 1 1 1T
and
Φ (x ,b ) = d2k Φ (x ,k )eik1T·b1
Λb 1 1 1T Λb 1 1T
Z
N x6(1 x )3b2K ( (1 a x )2x2(1 x )+ε (1 x )2x2b )
= b 1 − 1 1 2 − b− 1 1 − 1 pb − 1 1 1 , (20)
2[(1 a q x )2x2(1 x )+ε (1 x )2x2]
− b− 1 1 − 1 pb − 1 1
whereK isthemodifiedBesselfunctionofthesecondkind. Ifneglectingthetransversemomentum
2
k , i.e. set k 0, the wave function can be simplified as
1T 1T
∼
N x2(1 x )
Φ (x ) b 1 − 1 . (21)
Λb 1 ≃ [(1 a x )2+ε (1 x )]2
b 1 pb 1
− − −
The normalization conditions are set as [13]
1
Φ (x )dx = 1,
Λb 1 1
Z0
1 Λ¯
Φ (x )x dx = ,
Λb 1 1 1 M
Z0 Λb
1 Λ¯2 λ
Φ (x )x2dx = ( + 1 ). (22)
Λb 1 1 1 M2 3M2
Z0 Λb Λb
Thefirstformuladeterminesthenormalizationofthepartondistributionofthebaryon,whereasthe
second oneis related totheeffective mass ofthelight diquarkΛ¯ M m , andthethirdformula
∼ Λb− b
8
reflects connection between hadronic matrix element of the kinematic operator λ = 1 Λ
1 −2MΛbh b |
¯b (iD )2b Λ and hadronic distribution amplitude. To satisfy the above three normalization
v ⊥ v b
| i
conditions, the parameters would take the following values Λ¯ = 0.848 GeV, λ = 0.76 GeV2,
1
−
a = 0.916, ε = 0.0051 and N = 0.0219, and in the following numerical evaluation, we will use
b pb b
them as inputs.
Forthewave functionofΛ ,theexpressionsarethesameasthatforΛ , whilethecorresponding
c b
parameters are Λ¯ = 0.8849 GeV, λ = 1.87GeV2 , a = 1.48, ε = 0.080 and N = 12.14. Thus
1 c pc c
−
we can write the differential decay width as
dΓ G2 V 2
= F| cb| M5 r3 ρ2 1
dρ 24π3 Λb −
q
[ f 2(ρ 1)(3r2 4ρr+2r+3)+ g 2(ρ+1)(3r2 4ρr 2r+3)
1 1
| | − − | | − −
+6 f f (r+1)(ρ 1)(r2 2 rρ+1)+6 g g (r 1)(r2 2 rρ+1)
1 2 1 2
− − − −
+ f 2(ρ 1)(3r2 2ρr+4r+3)(r2 2ρr+1)
2
| | − − −
+ g 2(ρ+1)(3r2 2ρr 4r+3)(r2 2ρr+1) ], (23)
2
| | − − −
with
1 M M
1 ρ ρ = Λb + Λc . (24)
max
≤ ≤ 2 MΛc MΛb!
III. NUMERICAL RESULTS
A. The results of PQCD
In the one-heavy-quark-one-light-diquark picture, the (ud) diquark in Λ is considered as a
¯3 b
scalar of color anti-triplet. To calculate the form factors in the framework of PQCD, one can
adjust the product fS fS to fit the empirical formula f (ρ ) 1.32/ρ5.18 given by the authors
Λb Λc 1 max ∼ max
of Ref.[3]2. Then the corresponding parameters are obtained as
m M = 5.624 GeV, m M = 2.2849 GeV, V = 0.040,
b ≃ Λb c ≃ Λc cb
fS fS = 0.0096GeV2, Q2 = 3.22GeV2, Λ = 0.2GeV. (25)
Λb Λc 0 QCD
Using these values, we can continue to numerically estimate the form factors f and g . Fig.2(a)
1 1
shows that the form factor f is exactly equal to g in the heavy quark limit. The form factor f
1 1 2
| |
2 Whenfirstcalculatingtheformfactors, therewerenodataavailable,theauthorsof[3]usedareasonableestimate
of BR(Λb → Λclν) as about 2%. Nowadays, measurements have been done and with the data, we have made a
new fit.
9
1.8
and gi 11..46 a adcb PHPHQQQQCCEEDTDT((((ff f12f2a1))anndd | g|g1|1)|) 14 GeV) 56 PHQQCEDT
fi 1.2 - 10 4
1.0 (
c d 3
0.8 /
d
0.6 2
b
0.4 1
0.2
d 0
0.0
1.0 1.1 1.2 1.3 1.4 1.5 1.0 1.1 1.2 1.3 1.4 1.5
(a) (b)
FIG. 2: (a) Form factors f2, f1 and g1 (b) Differential decay width of Λb Λclν
| | →
and g are much smaller than f and g , thus they can in fact be safely neglected for the present
2 1 1
| |
experimental accuracy.
In Fig.2(b), we plot the dependenceof the differential width dΓ/dρ on ρ. Although both f and
1
g have an end-point divergence at ρ 1 in PQCD approach, the differential decay rate is finite.
1
→
If one extrapolates the PQCD calculation to the region with smaller ρ values, we obtain the
form factors f and g in that region where there are obvious end-point singularities at ρ 1.
1,2 1,2
→
Redo the computations with the extrapolation (in the original work [3], the authors extend the
tangent of the PQCD result at a small ρ value to ρ = 1, so the end-point singularity is avoided)
and obtain BR(Λ Λ lν) which is about 1.35% (slightly smaller than the value of 2% guessed
b c
→
by the authors of Ref.[3], because then no data were available. ).
Obviously, the calculation in PQCD depends on the factor fS fS , which regularly must be
Λb Λc
obtained by fitting the data of semileptonic decays, so that the theoretical predictions are less
meaningful. Instead, we will use our hybrid scheme where we do not need to obtain the factor
fS fS byfittingdata, sincetheconnectionrequirementsubstitutesthefittingprocedure(seebelow
Λb Λc
for details).
10
B. The results of HQET
The transition rate was evaluated in terms of HQET by the authors of Ref. [8]. According to
thedefinitions given in eq.(16), we re-calculate f , f , g , g whiledroppingoutf andg andalso
1 2 1 2 3 3
obtain similar conclusion that f and g are the same in amplitude, but opposite in sign, as shown
1 1
in Fig. 2(a), whereas f is very small and g is exactly zero in HQET. The theoretical prediction
2 2
on the rate of the semi-leptonic decay Λ Λ +l+ν¯ in the HQET is
b c
→
Γ(Λ Λ lν¯) = 1.54 10−14 GeV, withoutcontributions fromf and g ,
b c 2 2
→ ×
Γ(Λ Λ lν¯) = 1.56 10−14 GeV, with contributions fromf and g , (26)
b c 2 2
→ ×
and the branching ratio is 2.87%.
Thetransition rate of Λ Λ lν¯ has recently been measured by the DELPHI Collaboration [5],
b c
→
and the value of BR(Λ Λ lν) is 5.0+1.1(stat)+1.6(syst)%. It is noted that the result calculated
b → c −0.8 −1.2
in terms of HQET is only consistent with data within two standard deviations.
C. The hybrid scheme: Reconciling the two approaches
As widely discussed in literature, in the region with large q2 (small ρ values), the result of
−
HQET is reliable, whereas for the region with small q2 (larger ρ values) the PQCD is believed to
−
work well. Therefore, to reconcile the two approaches which work in different ρ regions, 1 ρ <
≤
ρ , we apply the HQET for small ρ, but use PQCD for larger ρ. Our strategy is that we let
max
the form factors f , g derived in the PQCD approach be equal to the value obtained in terms of
1 1
HQET at the point ρ . The numerical results of the form factor f = g in the hybrid scheme
cut 1 1
| |
are shown in Fig.3 for three different ρ values: 1.05, 1.10 and 1.15, respectively. It is noted
cut
−
that for ρ = 1.1 and 1.15, the left- and right-derivatives are very close and the connection is
cut
smooth, whereas, as ρ is chosen as 1.05, a difference between the derivatives at the two sides
cut
of ρ = 1.05 obviously manifests. We then adopt a proper numerical program to smoothen the
cut
curve, namely let the derivatives of the two sides meet each other for any ρ value near the cut
−
point. Fig.3 shows that such treatment in fact does not change the general form of the curve, but
makes it sufficiently smooth for all ρ values including the selected cut point ρ .
cut
Another advantage of adopting such a “hybrid” scheme is that there does not exist end-point
singularity for the form factors at ρ= 1. Since at the turningpoint, we let the form factors derived