Jul 25, 2012 - arXiv:1207.5922v1 [hep-ph] 25 Jul 2012. Semileptonic ... II. QCD SUM RULES FOR TRANSITION FORM FACTORS OF χb0 → Bcℓν .... p′2). + subtr...

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Department of Physics, Dogus University, Acıbadem-Kadık¨oy, 34722 Istanbul, Turkey 2

Department of Physics, Kocaeli University, 41380 Izmit, Turkey

arXiv:1207.5922v1 [hep-ph] 25 Jul 2012

† ‡

e-mail:[email protected]

e-mail:[email protected] ∗

e-mail:[email protected]

Taking into account the two-gluon condensate contributions, the transition form factors enrolled to the low energy effective Hamiltonian describing the semileptonic χb0 → Bc ℓν, (ℓ = (e, µ, τ )) decay channel are calculated within three-point QCD sum rules. The fit function of the form factors then are used to estimate the decay width of the decay mode under consideration. PACS numbers: 11.55.Hx, 14.40.Pq, 13.20.-v, 13.20.Gd

2 I.

INTRODUCTION

The quarkonia, especially the bottomonium b¯b states, are approximately non-relativistic systems since they do not contain intrinsically relativistic light quarks. Hence, these states are the best candidates to examine the hadronic dynamics and investigate both perturbative and non-perturbative characteristic of QCD. In the past, mainly theoretical calculations on the properties of these states had been made using potential model or its extensions like the Coulomb gauge model (see for instance [1–6] and references therein). In [7], both potential model and QCD sum rule approach have been applied to extract the groundstate decay constant of mesons containing heavy b quark. It is stated that the QCD sum rule technique gives more reliable and accurate determination of bound-state characteristics compared to the potential models by tunning the continuum threshold parameter. The QCD sum rule approach [8] is one of the most powerful and applicable tools to hadron physics. This model has been widely applied to investigate the spectroscopy of hadrons and their electromagnetic, weak and strong decays. The obtained results have very good consistencies with the experimental data to date within the typical (10-20)% error bars of the technique. The present work is dedicated to investigation of the semileptonic transition of scalar P wave bottomonium χb0 (1P ) meson with quantum numbers I G (J P C ) = 0+ (0++ ) into the pseudoscalar Bc meson. The χb0 (1P ) state has been observed first in radiative decay of the Υ(2S) [9] and recently has been confirmed by ATLAS Collaboration [10] together with the higher χb (2P ) and χb (3P ) states. In the latter, these quarkonia states have been produced √ in proton-proton collisions at the Large Hadron Collider (LHC) at s = 7 T eV and through their radiative decays to Υ(1S, 2S) with Υ → µ+ µ− . Our previous theoretical results [11]

on the mass of these states done both in vacuum and finite temperature QCD are in good agreement with the experimental results [10]. Note that, we also have applied the QCD sum rules approach both in vacuum and finite temperature to investigate the spectroscopy of the pseudoscalar, vector and tensor quarkonia in [12–14]. As we know the masses and decay constants of the quarkonia, it is possible to investigate their electromagnetic, weak (leptonicsemileptonic) and strong decays. Considering such decay channels can help us obtain more information about the nature of the scalar χb0 (1P ) meson as well as perturbative and nonperturbative aspects of QCD. The layout of this article is as follows. In the next section, we derive the QCD sum rules for the form factors appearing in the amplitude of the semileptonic decay channel under consideration. To do so, we take into account the two-gluon condensates as the non-perturbative contributions to the correlation function. Section III is devoted to our numerical analysis of the obtained form factors and their behavior in terms of the transferred momentum squared. In this section, we also numerically estimate the decay width of the

3 semileptonic χb0 → Bc ℓν decay mode. The last section encompasses our concluding remarks. II.

QCD SUM RULES FOR TRANSITION FORM FACTORS OF χb0 → Bc ℓν

The hadronic event under consideration can be described in terms of quark degrees of freedom by the process b → cl¯ ν at tree-level, whose effective Hamiltonian can be written as: GF Hef f = √ Vcb ν γµ (1 − γ5 )l c γµ (1 − γ5 )b, 2

(1)

where GF is the Fermi weak coupling constant and Vcb is an element of the CabibboKobayashi-Maskawa (CKM) mixing matrix. The transition amplitude is obtained via M = hBc (p′ ) | Hef f | χb0 (p)i,

(2)

or

GF (3) M = √ Vcb νγµ (1 − γ5 )lhBc (p′ ) | cγµ (1 − γ5 )b | χb0 (p)i. 2 To proceed, we need to know the transition matrix element hBc (p′ ) | cγµ (1 − γ5 )b | χb0 (p)i whose vector part do not contribute due to parity considerations, i.e., hBc (p′ )|cγµ b|χb0 (p)i = 0.

(4)

The axial-vector part of transition matrix element can be parameterized in terms of form factors as hBc (p′ ) | cγµ γ5 b | χb0 (p)i = f1 (q 2 )Pµ + f2 (q 2 )qµ ,

(5)

where f1 (q 2 ) and f2 (q 2 ) are transition form factors; and Pµ = (p + p′ )µ and qµ = (p − p′ )µ .

Our main goal in the present section is to calculate the transition form factors applying

the QCD sum rules technique. The starting point is to consider the following tree-point correlation function as the main ingredient of the model: Πµ = i2

Z

d4 x

Z

′

n

o

d4 ye−ipxeip y h0 | T JBc (y)JµA;V (0)Jχ† b0 (x) | 0i,

(6)

where T is the time ordering product, JBc (y) = cγ5 b and Jχb0 (x) = bUb are the interpolating

currents of the Bc and χb0 mesons, respectively; and JµV (0) = cγµ b and JµA (0) = cγµ γ5 b are the vector and axial-vector parts of the transition current. Following the general idea in the QCD sum rules technique, we calculate this correlation function once in terms of hadronic degrees of freedom called physical or phenomenological side and the second in terms of QCD degrees of freedom (quarks and gluons and their interaction with QCD vacuum) called the QCD side. The latter is done in the deep Euclidean region by the help of operator product expansion (OPE). These two representations are then matched together, using the

4 quark-hadron duality assumption, through a double dispersion relation to obtain the QCD sum rules for the form factors. As we deal with the ground states in this approach, we shall separate the ground state from the higher states and continuum. This is done by two mathematical operations called Borel transformation and continuum subtraction. Such transformations bring some auxiliary parameters namely two Borel mass parameters and two continuum thresholds for which we will find their working regions in the next section. The phenomenological side of the correlation function is obtained inserting two complete sets of intermediate states with the same quantum numbers as the interpolating currents JBc and Jχb0 . As a result, we obtain ΠPµ HY S =

h0 | JBc (0) | Bc (p′ )ihBc (p′ ) | JµA (0) | χb0 (p)ihχb0 (p) | Jχ† b0 (0) | 0i + ···, (p′2 − m2Bc )(p2 − m2χb0 )

(7)

where · · · represents the contributions coming from higher states and continuum. Besides the

transition matrix elements defined previously, the matrix elements of interpolating current

between the vacuum and hadronic states are parameterized in terms of the leptonic decay constants, i.e., h0 | JBc (0) | Bc (p′ )i = i

fBc m2Bc , mb + mc

hχb0 (p) | Jχ† b0 (0) | 0i = −imχb0 fχb0 .

(8)

Putting all expressions together, the final version of the phenomenological side of the correlation function is obtained as ΠPµ HY S (p2 , p′2 )

fBc m2Bc fχb0 mχb0 = ′2 f1 (q 2 )Pµ + f2 (q 2 )qµ + ..., (p − m2Bc )(p2 − m2χb0 ) mb + mc "

#

(9)

where we will choose the structures Pµ and qµ , to evaluate the form factors f1 and f2 , respectively. At QCD side, the correlation function is calculated in deep Euclidean region by the help of the OPE. For this aim, we write the coefficient of each structure in correlation function as a sum of a perturbative (diagram a in figure 1) and a non-perturbative (diagrams b, c, d, e, f and g in figure 1) parts as follows:

ΠQCD = (Πpert + Πnonpert )Pµ + (Πpert + Πnonpert )qµ 1 1 2 2 µ

(10)

where, the Πpert functions are written in terms of double dispersion integrals in the following i way: Πpert =− i

1 (2π)2

Z

ds

Z

ds′

ρi (s, s′ , q 2 ) + subtraction terms, (s − p2 )(s′ − p′2 )

(11)

5

ν¯

l

ν¯

l

W

W

W

γµ (1 − γ5 )

b

c

b

c

b

b γ5

U

W

γ5

U

(a)

b

c

b γ5

U

(b) l

γµ (1 − γ5 )

γµ (1 − γ5 )

γµ (1 − γ5 )

b

W

W γµ (1 − γ5 )

c

b

b γ5

b

ν¯

l

W γµ (1 − γ5 )

U

γ5

(d)

ν¯

l

γµ (1 − γ5 )

c

b γ5

U

b

c γ5

U

(f )

(e)

b c

U

(c)

ν¯

b

ν¯

l

ν¯

l

(g)

FIG. 1. Feynman diagrams contributing to the correlation function for the χb0 → Bc ℓν decay: (a) the bare loop and (b, c, d, e, f, g) two-gluon condensate diagrams.

where, ρi (s, s′ , q 2 ) are the spectral densities with i = 1 or 2. Applying the usual Feynman integral technique to the bare loop diagram, the spectral densities are calculated via Cutkosky rules, i.e., by replacing the quark propagators with Dirac delta functions: 1 p2 −m2

→ −2πδ(p2 − m2 ) implying that all quarks are real. After some calculations, the

spectral densities are obtained as follows: h

i

ρ1 (s, s′, q 2 ) = 2Nc I0 (s, s′, q 2 ) mb (mc − 3mb ) − A(h + s) − B(h + s′ ) , h

i

ρ2 (s, s′, q 2 ) = 2Nc I0 (s, s′, q 2 ) mb (mb + mc ) − A(h − s) + B(h − s′ ) ,

where I0 (s, s′ , q 2 ) =

1 4λ1/2 (s, s′ , q 2 )

(12)

,

i h 1 2 2 ′′ ′ , (m − m )u + us b c λ(s, s′ , q 2 ) i h 1 2 2 ′ , 2(m − m )s + su B= b c λ(s, s′ , q 2 ) h = 2mb (mb − mc ),

A=

(13)

here also λ(s, s′ , q 2) = s2 + s′2 + q 4 − 2ss′ − 2sq 2 − 2s′ q 2 , u = q 2 + s − s′ , u′ = q 2 − s + s′ ,

u′′ = q 2 − s − s′ and Nc = 3 is the number of colors. The integration region for the perturbative contribution in Eq. (11) (bare loop diagram) is determined requiring that the arguments of the three δ functions vanish, simultaneously. Therefore, the physical region in the s and s′ plane is described by the following inequality: ′′

2s[m2 − m2c + s′ + u2 ] − 1 ≤ f (s, s ) = 1/2 2b ≤ +1. λ (mb , s, m2b )λ1/2 (s, s′ , q 2 ) ′

(14)

6 For the non-perturbative part, we take into account the two-gluon condensate diagrams (b, c, d, e, f, g) in figure 1. Here we should mention that we deal with the heavy quarks in the present work and the heavy quarks’ condensates are suppressed by inverse of their masses, so we can ignore them safely. After lengthy calculations for the two-gluon condensates diagrams (b, c, d) correspond to the diagrams with two gluon lines coming out from different quark lines, we get Πnonpert i

1

Z

=

dx

0

1−x

Z

0

1 Θ2i Θ3i Θ4i Θ1i dyh0| αs G2 |0i , + + + π D5 D4 D3 D2 )

(

(15)

where D = m2c x − m2b r + p′2 xr + p2 yf + xy(p2 + p′2 − q 2 ),

(16)

and Θ11

(

"

1 = 9m5b mc y 2 v 2 (−1 + 2y) − 18m6b y 2 v 3 − 3m4b xv 3m2c r 2 x + 2rxy + 2wy 2 24 2

3

2

2

3

+ q y 3r x + 3r ty + ry (−14 + 15x) + y (−22 + 18x + 9y) +

3m3b mc xv

"

!#

m2c x 6 − 19x + 13x2 − 15y + 22xy + 12y 2 + q 2 y 13r 2 x2 − 6r 2 x 2

2

2

+ 3ry − 15ryx + 22ryx + y 15 − 22y − 26x + 12x + 12xy + 12y "

2

!#

+ mb mc q x yv m2c x 22 − 49x + 39x2 − 62y + 66xy + 36y 2 + q 2 y 39x4 − 88x3 2 2

3

+ 66x y + 3f y 3 − 7y + 6y 4 3 2

+ 3q x y

"

m2c

2

2

3

2

+ x 71 − 128y + 36y

2

+ xz 11 − 33y + 18y

2

!#

3

f + 6f y + x (−11 + 3x + 9y) + f x 10 − 17x + 18xy − 26y + 18y

2

!

+ q 2 y x3 31 + 3x2 − 14x + 9xy − 50y + 21y 2 + f x − 20 + 54y − 51y 2 + 15y 3 + 36x − 59xy + 27xy

2

2

2

+ f (−4 + 3yz )

+

3m2b q 2 x2 y

"

q 2 y 4r 3 − 2r 3 x + 16r 2 y

!#

− 8r 2xy + 3ry 2 8 − 7x + 2x2 + y 3 17 − 5y − 27x + 12x2 + 6xy 2

3

2

2 2

2

2

+ 5x − 2x − 5x y + 6x y + f 8y − 8xy + 12xy + 6y z − 1 (

"

!

!#)

+ m2c 6xy − 4x ,

1 Θ21 = 9m3b mc v rx2 (−7 + 13x) + 2x2 (−8 + 11x)y − 2y 2 3f + 5x − 6x2 + 12y 3(f + x) 96

−

9m4b v

"

2 2

2

2

2

3r x + 6rx y + y 7 − 14x + 15x − 16y + 18xy + 9y

2

#

+ 3m2b x

#

7 "

× 3m2c 6y 4 − xr 2 + 6y 2r 2 − 2xry + 12ry 3 + q 2 y 8r 2 x − 11r 2 x2 − 12ry + 54rxy 2

2

2

3

2

− 38ryx − y 43 + 153x − 48x + 54ty + 23y − 158x − 48xyt − 48xy "

2

!#

+ mb mc xv 9m2c x 4 + 13rx − 13y + 22xy + 12y 2 + q 2 y 312x4 + x3 (−535 + 528y)

+ x2 313 − 683y + 288y 2 + 3y − 9y + 53y − 88y 2 + 48y 3 + 6x − 9 + 40y − 58y 2 + 24y 3

!#

"

+ 3q 2 x2 y m2c r 2 3 − 29x + 24x2 − y 21 − x 151 − 206x + 72x2 2

+ 78y + 2xy(−127 + 72x) + 36y (−3 + 4x) + 48y

3

!

!

+ q 2 y 45x5 + x4 (−182 + 135y)

+ x3 306 − 584y + 315y 2 + x2 − 253 + 849y − 1004y 2 + 405y 3 + 3f 4 − 30y + 69y 2 3

− 59y + 15y (

4

"

2

3

+ x 96 − 500y + 965y − 792y + 225y

4

!#)

,

1 Θ31 = 9m2b − r 2 x2 − 2rx2 y + r 1 − 7xy 2 + 8x2 y 2 + 2y 3(1 + 8rx) + y 4(−1 + 8x) 96

+

"

9m2c x

2

2

2

r x(−3 + 4x) + 2rxy(−5 + 6x) + 2y 2 − 13x + 12x

"

3

+ 12ty + 8y

4

#

#

+ 4q yx r 2 x 17 − 64x + 45x2 + y 21 + x(−154 + 413x − 418x2 + 135x3 ) − 120y 2 2

2

2

3

2

+ y x(497 − 718x + 315x ) + 3y 76 − 198x + 135x "

4

2

2

3

4

+ 3y (−58 + 75x) + 45y 2

5

#

+ 3mb mc v 52x + 24xf y + 12f y (−1 + 2y) + x (−61 + 88y) + 3x 7 − 19y + 16y Θ41

(

2

#)

,

3 = 5x5 + y 2f 2 (−4 + 5y) + x4 (−14 + 15y) + f xy 2(−19 + 25y) + x3 13 − 28y + 35y 2 16 2

2

− x 4 − 13y + 48y − 45y

3

)

.

(17)

Here v = −1 + x + y, r = −1 + x, t = −1 + 2x, w = 1 + x, f = −1 + y and z = −2 + y.

The explicit expressions for Θ1,2,3,4 correspond to the structure qµ are too long, hence we do 2

not present them here. In a similar way, we calculate the contributions of the diagrams (e, f, g) correspond to two gluon lines coming out from the same quark line. Because of their very lengthy expressions, we do not also depict their explicit form here, but we will take into account their contributions in our numerical results. The next step is to equate the coefficients of selected structures from both sides in order to get sum rules for the form factors. After applying double Borel transformations with respect to the variables p2 and p′2 (p2 → M 2 , p′2 → M ′2 ) to suppress the contributions of

the higher states and continuum, the QCD sum rules for the form factors are obtained as:

8

2 2 1 1 (mb + mc ) ′2 2 emχb0 /M emBc /M − f1,2 (q ) = 2 fBc mBc fχb0 mχb0 (2π)2

2

′

2

2

′

−s/M 2 −s′ /M ′2

× ρ1,2 (s, s , q )θ[1 − f (s, s )]e

e

Z

s0

4m2b

ds

Z

ˆ nonpert , + BΠ 1,2

s′0

(mb +mc )2

ds′ (18)

where the operator Bˆ denotes double Borel transformation. Note that to subtract contributions of the higher states and continuum, we also apply the quark-hadron duality assumption, i.e., ρhigherstates (s, s′ ) = ρOP E (s, s′ )θ(s − s0 )θ(s′ − s′0 ).

(19)

We also perform the double Borel transformation as follows: • for the perturbative part, we use Bˆ

1 1 −m21 /M 2 −m22 /M ′2 1 1 m+n 1 → (−1) e e . 2 2 [p2 − m1 ]m [p′2 − m2 ]n Γ[m] Γ[n] (M 2 )m−1 (M ′2 )n−1 (20)

• For the non-perturbative part, first we make the transformation ′2

Bˆ

2

−f (p )/M 1 n e , = (−1) [p2 − f (p′ 2 )]n Γ[n](M)n−1

(21)

′

to write the terms containing p 2 in exponential form. Then we apply the following ′

rule to transform the (p 2 → M ′2 ): ′ ˆ −αp 2 = δ( 1 − α), Be M ′2

(22)

where α is a function of quarks’ masses as well as the parameters used in Feynman parametrization.

III.

NUMERICAL RESULTS

In this section, we numerically analyze the related form factors, obtain their fit function in terms of q 2 and estimate the decay width of the channel under consideration. In calculations, we use the input parameters as presented in table I. The sum rules for the form factors denote that they also depend on four auxiliary parameters, namely continuum thresholds s0 and s′0 and Borel mass parameters M 2 and M ′2 . The continuum thresholds are not completely arbitrary but they are in correlation with the energy of the excited state in initial and final channels. Considering this point and the

9 Parameters

Values

mc

(1.275 ± 0.015) GeV

mb

(4.7 ± 0.1) GeV

me

0.00051 GeV

mµ

0.1056 GeV

mτ

1.776 GeV

mχb0

(9859.44 ± 0.42 ± 0.31) M eV

mBc

(6.277 ± 0.006) GeV

fBc

(400 ± 40) M eV

fχb0

(175 ± 55) M eV

GF Vcb h0| π1 αs G2 |0i

1.17 × 10−5 GeV −2 (41.2 ± 1.1) × 10−3

(0.012 ± 0.004) GeV 4

TABLE I. Input parameters used in our calculations [9, 11, 15, 16].

fact that the result of the physical quantities (form factors) should weakly depend on these parameters, we choose the intervals s0 = (97.7 − 99.2) GeV 2 and s′0 = (40 − 41) GeV 2

slightly higher than the mass of pole squared of the initial and final mesonic channels for the continuum thresholds. The Borel parameters M 2 and M ′2 also are not physical quantities, hence the form factors should be independent of them. The reliable regions for the Borel parameters M 2 and M ′2 can be determined by requiring that not only the contributions of the higher states and continuum are effectively suppressed, but contribution of the operators with the higher dimensions are small, i.e., the sum rules for form factors converge. As a result of these requirements, the working regions for these parameters are determined to be 15 GeV 2 ≤ M 2 ≤ 30 GeV 2 and 10 GeV 2 ≤ M ′2 ≤ 20 GeV 2 . The dependence of form factors

f1 and f2 on Borel masses at q 2 = 1 GeV 2 are plotted in figure 2. From this figure, we see

good stability of the form factors with respect to the variations of the Borel mass parameters at their working regions. To see the convergence of the OPE, we compare both perturbative and non-perturbative contributions to the form factors in figure 3 at q 2 = 1 GeV 2 and the presented Borel windows. From this figure and our numerical calculations, it is found that the ratio of non-perturbative contribution to that of perturbative is 0.08 and 0.05 for f1 and f2 , respectively. Hence, the non-perturbative contribution constitutes only 7.5% and 4.8% of the total results respectively for the form factors f1 and f2 . This means that the series of sum rules for the form factors are convergent. In the presented Borel windows, the contributions of the excited and continuum states are exponentially suppressed. This

10

f1

0.012 0.011 0.01 0.009 15

20 f2 18 16 2 ¢2 14 M HGeV L

20 2

2

M HGeV L

12

25

-0.04 -0.0425 -0.045 -0.0475 -0.05 15

20 18 16 2 ¢2 14 M HGeV L

20 M2 HGeV2 L

25

12 3010

3010

FIG. 2. Dependence of the form factors f1 and f2 on Borel mass parameters M 2 and M ′2 at q 2 = 1 GeV 2 .

0 f1

0.01

20

0.005

17.5

0 15

15 20

M2 HGeV2 L

M¢2 HGeV2 L

f2

-0.02

20

-0.04

17.5 15 M¢2 HGeV2 L

15 20

12.5 M2 HGeV2 L

25 30 10

12.5 25 30 10

FIG. 3. Comparison between perturbative and non-perturbative contributions to the form factors at q 2 = 1 GeV 2 and chosen Borel windows. The upper (lower) plane belongs to the perturbative contribution in f1 (f2 ).

guarantees the reliability of the sum rules and isolation of the ground state from the excited states and continuum. Our calculations show that the form factors are truncated at q 2 ≃ 9 GeV 2 (see figure 4).

After this point up to the higher limit of the q 2 , the sum rules predictions are not reliable

(for details see for instance [17, 18]). However, we need their fit functions in the whole physical region, m2l ≤ q 2 ≤ (mχb0 − mBc )2 to estimate the decay width of the χb0 → Bc ℓν

transition. To extend our results to the full physical region, we search for parameterization of the form factors in such a way that in the region 0 ≤ q 2 ≤ 9 GeV 2 , predictions of this

11 parameterization coincide with the sum rules results. The following parametrization adjust well the q 2 dependence of the form factors: fi (q 2 ) =

b a + , 2 q2 (1 − m2 ) (1 − mq2 )2 f it

(23)

f it

where, the values of the parameters a, b and mf it obtained using M 2 = 25 GeV 2 and M

′2

= 15 GeV 2 for the χb0 → Bc ℓν channel are given in table II. a

b

m2f it (GeV 2 )

f1 (χb0 → Bc ℓν) -0.055 0.062

21.86

f2 (χb0 → Bc ℓν) 0.225 -0.254

19.79

TABLE II. Parameters appearing in the fit function of the form factors.

We depict the dependence of form factors f1 and f2 on q 2 obtained directly from the sum rules as well as the fit parametrization at whole physical region in figure 4. In the case of sum rules predictions, we present the perturbative, non-perturbative and total contributions in this figure. 0.0 Fit Total Perturbative Non-Perturbative

-0.2

0.10

-0.4 f2

f1

0.15

-0.6

Fit Total Perturbative Non-Perturbative

0.05 -0.8

0.00 0

2

4

6 2

8

10

12

-1.0 0

2

4

2

6 2

q HGeV L

8

10

12

2

q HGeV L

FIG. 4. Dependence of the form factors f1 and f2 on q 2 at M 2 = 25 GeV 2 and M

′2

= 15 GeV 2 .

Having obtained the behavior of the form factors in terms of q 2 at whole physical region, we would like to calculate the decay width of the process under consideration. Using the amplitude previously discussed, the differential decay width for χb0 → Bc ℓν is obtained in terms of form factors as:

G2F dΓ q 2 − m2ℓ 2 2 2 1/2 2 = , m , q ) |V | λ (m cb S Bc dq 2 192π 3m3Bc q2

!2 (

q2 |f1 (q 2 )|2 (2m2Bc + 2m2S − q 2 ) 2 "

12

+

2(m2Bc

+

2(m2Bc

−

m2S )Re[f1 (q 2 )f2∗ (q 2 )]

2

−

m2S )q 2 Re[f1 (q 2 )f2∗ (q 2 )]

2 2

+ |f2 (q )| q 2

#

(q 2 + m2ℓ ) |f1 (q 2 )|2 (m2Bc − m2S )2 − 2q 2

2 4

+ |f2 (q )| q

"

#)

.

(24)

Performing integration over q 2 in Eq. (24) in the interval m2l ≤ q 2 ≤ (mχb0 − mBc )2 , we

obtain the expression for the total decay width. The numerical values of the decay width at different lepton channels are presented in Table III. The errors in the values of the decay rates Γ(GeV ) χb0 → Bc eν e 1.46 × 10−14

χb0 → Bc µν µ 1.45 × 10−14 χb0 → Bc τ ν τ 0.91 × 10−14

TABLE III. Numerical results for decay rate at different lepton channels.

in table III are due to uncertainties in determination of the working regions for continuum thresholds and Borel mass parameters as well as errors of the other input parameters.

IV.

CONCLUSION

In the present work, we studied the semileptonic χb0 → Bc ℓν, (ℓ = (e, µ, τ )) decay channel

within the framework of the three-point QCD sum rules. In particular, taking into account the two-gluon diagrams as non-perturbative contributions, we obtained the QCD sum rules for the form factors entered the transition matrix elements. After obtaining the working regions for the auxiliary parameters, we found the behavior of the form factors in terms of q 2 in whole physical region. The fit function of the form factors were then used to estimate the decay rates at different lepton channels. Any measurement on the form factors as well as decay rate of the channel under consideration and comparison of the obtained results with theoretical predictions in the present study can give valuable information about the internal structures of the participating mesons specially nature of the scalar χb0 (1P ) state.

V.

ACKNOWLEDGMENT

This work is supported in part by Scientific and Technological Research Council of Turkey (TUBITAK) under project No: 110T284 and partly by Kocaeli University Scientific Research

13 Center (BAP) under project No: 2011/52.

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