Transition form factors of chi(b2)(1P) -> B-c(l)over-barv in QCD

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Research Article

Transition Form Factors of

𝜒

_𝑏2

(1𝑃) → 𝐵

_𝑐

𝑙] in QCD

K. Azizi,

1

H. Sundu,

2

J. Y. Süngü,

2

and N. Yinelek

2

1_{Department of Physics, Do˘gus¸ University, Acıbadem, Kadık¨oy, 34722 Istanbul, Turkey} 2_{Physics Department, Kocaeli University, 41380 ˙Izmit, Turkey}

Correspondence should be addressed to K. Azizi; kazizi@dogus.edu.tr Received 27 October 2015; Revised 13 January 2016; Accepted 28 January 2016 Academic Editor: Enrico Lunghi

Copyright © 2016 K. Azizi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The form factors of the semileptonic𝜒_𝑏2(1𝑃) → 𝐵_𝑐𝑙] decay are calculated by using the QCD sum rule technique. The results obtained are then used to estimate the decay widths of this transition in all lepton channels. The orders of decay rates indicate that this transition is accessible at LHC for all lepton channels.

1. Introduction

Quarkonia are flavorless bound states composed of combi-nations of quarks and their antiquarks. Since discovery of 𝐽/𝜓 meson in 1974, many new quarkonia states have been detected. The quarkonia systems consist of charmonium and bottomonium. Due to the large mass there are no toponium bound states and no light quark-antiquark states because of the mixture of the light quarks in experiments. The large mass difference between charm and bottom quarks prevents them from mixing. The heavy quark bound states may provide key tools for understanding the interactions between quarks, new hadronic production mechanisms and transitions, and the magnitude of the CKM matrix elements and also analyzing the results of heavy-ion experiments.

The𝐽/Ψ suppression in ultrarelativistic heavy-ion

colli-sions was first suggested as a signal of the formation of a quark-gluon plasma (QGP). However, most recently, atten-tions have shifted to the bottomonium states due to the fact that they are more massive than charmonium states. The bottom quarks and antiquarks are relatively rare within the plasma, so the probability for reproduction of the bottomo-nium states through recombination is much smaller than for charm quarks. Consequently, the bottomonium system is expected to be a cleaner probe of the QGP than the char-monium system. Hence investigations on the properties of bottomonium systems can help us get useful information

not only about the nature of the𝑏𝑏 systems, but also on the

existence of QGP.

Since the discovery of quarkonium states, QCD sum rule technique as one of the most powerful nonperturbative tools to hadron physics [1, 2] has played an important role in understanding the quarkonia spectrum. In order to find missing states one should know their physical properties to develop a successful search strategy. Clearly significant progress in understanding of quarkonium production cannot be reached without detailed measurements of the cross sections and fractions of the quarkonia. Thereby, form factors and decay widths of quarkonia become significant for com-pleting quarkonia spectrum. Comcom-pleting the bottomonium spectrum is a crucial validation of theoretical calculations and a test of our understanding of bottomonium states in the context of the quark model. Bottomonium states are considered as great laboratories to search for the properties od QCD at low energies.

In this connection, we investigate the decay properties of

tensor𝜒_𝑏2(1𝑃) meson as one of the important members of

the bottomonia to the heavy𝐵_𝑐 meson in the present work.

The𝐵_𝑐meson with𝐽𝑃 = 0− is the only meson consisting of

two heavy quarks with different flavors. Yet other possible𝐵_𝑐

states (the scalar, vector, axial-vector, and tensor) have not

been observed; however, many new𝐵_𝑐species are expected

to be produced at the Large Hadron Collider (LHC) in the near future.

Volume 2016, Article ID 4692341, 7 pages http://dx.doi.org/10.1155/2016/4692341

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Taking into account the two-gluon condensate cor-rections, the transition form factors of the semileptonic

𝜒_𝑏2(1𝑃) → 𝐵_𝑐𝑙] decay channel are calculated within the

three-point QCD sum rule. We use the values of transition form factors to estimate the decay rate of the transition under consideration at all lepton channels. The interpolating

current of𝜒_𝑏2(1𝑃) with quantum numbers 𝐼𝐺(𝐽𝑃𝐶) = 0+(2++)

contains derivatives with respect to space-time. So we start our calculations in the coordinate space; then we apply the Fourier transformation to go to the momentum space. To suppress the contributions of the higher states and contin-uum, we apply a double Borel transformation.

The paper is organized as follows. We derive the QCD sum rules for the transition form factors in Section 2. Last section is devoted to the numerical analysis of the obtained sum rules, estimation of the decay rates at all lepton channels, and concluding remarks.

2. QCD Sum Rules for

𝜒

_𝑏2

→ 𝐵

_𝑐

𝑙]

Transition Form Factors

The semileptonic𝜒_𝑏2 → 𝐵_𝑐𝑙] decay is based on 𝑏 → 𝑐𝑙]

transition at quark level whose effective Hamiltonian can be written as

Heff(𝑏 󳨀→ 𝑐𝑙]𝑙) =

𝐺𝐹

√2𝑉𝑐𝑏𝑐𝛾𝜇(1 − 𝛾5) 𝑏𝑙𝛾𝜇(1 − 𝛾5) ], (1)

where 𝐺_𝐹 is the Fermi weak coupling constant and 𝑉_𝑐𝑏 is

element of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. After sandwiching the effective Hamiltonian between the initial and final states, the amplitude of this transition is obtained in terms of transition matrix elements. These matrix elements will be parameterized in terms of transition form factors later.

In order to start our calculations, we consider the three-point correlation function

Π_𝜇𝛼𝛽(𝑝, 𝑝󸀠, 𝑞)

= 𝑖2∬ ⟨0 | T | 𝑗_𝐵_𝑐(𝑦) 𝑗tr,𝑉−𝐴_𝜇 (0) 𝑗†𝜒𝑏2

𝛼𝛽 (𝑥) | 0⟩

⋅ 𝑒−𝑖𝑝⋅𝑥𝑒𝑖𝑝󸀠⋅𝑦𝑑4𝑦 𝑑4𝑥,

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where T is the time-ordering operator and 𝑗_𝜇tr,𝑉−𝐴(0) =

𝑐(0)𝛾_𝜇(1−𝛾₅)𝑏(0) is the transition current. To proceed we also

need the interpolating currents of the initial and final mesons in terms of the quark fields, which are given as

𝑗𝜒𝑏2 𝛼𝛽(𝑥) = 𝑖 2[𝑏 (𝑥) 𝛾𝛼 ↔ D𝛽(𝑥) 𝑏 (𝑥) + 𝑏 (𝑥) 𝛾𝛽 ↔ D𝛼(𝑥) 𝑏 (𝑥)] , 𝑗_𝐵_𝑐(𝑦) = 𝑏 (𝑦) 𝛾₅𝑐 (𝑦) , (3)

where the covariant derivativeD↔𝛽(𝑥) denotes the four

deriva-tives with respect to𝑥 acting on two sides simultaneousely

and it is defined as ↔ D𝛽(𝑥) =1₂[ ⃗D𝛽(𝑥) − ⃖D𝛽(𝑥)] , (4) with ⃗ D_𝛽(𝑥) = ⃗𝜕_𝛽(𝑥) − 𝑖𝑔 2𝜆𝑎𝐴𝑎𝛽(𝑥) , ⃖ D_𝛽(𝑥) = ⃖𝜕_𝛽(𝑥) + 𝑖𝑔 2𝜆𝑎𝐴𝑎𝛽(𝑥) . (5)

Here,𝜆𝑎 are the Gell-Mann matrices and𝐴𝑎_𝛽(𝑥) denote the

external gluon fields.

According to the method used, the correlation function in (2) is calculated in two different ways. In physical or phenomenological side we obtain it in terms of hadronic parameters such as masses and decay constants. In QCD or theoretical side we evaluate it in terms of QCD degrees of freedom like quark masses as well as quark and gluon condensates via operator product expansion (OPE). The QCD sum rules for form factors are obtained by equating the above representations to each other. After applying a double Borel transformation, the contributions of the higher states and continuum are suppressed.

The hadronic side of the correlation function is obtained by inserting complete sets of intermediate states into (2).

After performing the four integrals over𝑥 and 𝑦, we get

ΠPHYS 𝜇𝛼𝛽 (𝑝, 𝑝󸀠, 𝑞) = ⟨0 | 𝑗_𝐵_𝑐(0) | 𝐵_𝑐(𝑝󸀠_{)⟩ ⟨𝐵} 𝑐(𝑝󸀠) | 𝑗tr,𝑉−𝐴𝜇 | 𝜒𝑏2(𝑝, 𝜀)⟩ ⟨𝜒𝑏2(𝑝, 𝜀) | 𝑗𝛼𝛽†𝜒𝑏2(0) | 0⟩ (𝑝󸀠2_{− 𝑚}2 𝐵𝑐) (𝑝2− 𝑚2𝜒𝑏2) + ⋅ ⋅ ⋅ , (6)

where “⋅ ⋅ ⋅ ” denotes the contribution of the higher states and continuum. To go further, we need to know the following matrix elements: ⟨𝐵_𝑐(𝑝󸀠_{) | 𝑗}tr,𝑉 𝜇 | 𝜒𝑏2(𝑝, 𝜀)⟩ = ℎ (𝑞2) 𝜖𝜇]𝜃𝜂𝜖]𝜆𝑃𝜆𝑃𝜃𝑞𝜂, ⟨𝐵_𝑐(𝑝󸀠_{) | 𝑗}tr,𝐴 𝜇 | 𝜒𝑏2(𝑝, 𝜀)⟩ = −𝑖 {𝐾 (𝑞2) 𝜖𝜇]𝑃] + 𝜖𝜃𝜂𝑃𝜃𝑃𝜂[𝑃𝜇𝑏+(𝑞2) + 𝑞𝜇𝑏−(𝑞2)]} , ⟨𝜒𝑏2(𝑝, 𝜀) | 𝑗𝛼𝛽†𝜒𝑏2 | 0⟩ = 𝑓𝜒𝑏2𝑚 3 𝜒𝑏2𝜀 ∗ 𝛼𝛽, ⟨0 | 𝑗_𝐵_𝑐 | 𝐵_𝑐(𝑝󸀠)⟩ = 𝑖 𝑓𝐵𝑐𝑚 2 𝐵𝑐 𝑚_𝑐+ 𝑚_𝑏, (7)

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where ℎ(𝑞2), 𝐾(𝑞2), 𝑏₊(𝑞2), and 𝑏₋(𝑞2) are transition form

factors;𝜖_𝛼𝛽is the polarization tensor associated with the𝜒_𝑏2

tensor meson;𝑓_𝜒_𝑏2and𝑓_𝐵_𝑐are leptonic decay constants of𝜒_𝑏2

and𝐵_𝑐mesons, respectively,𝑃_𝜇= (𝑝+𝑝󸀠)_𝜇and𝑞_𝜇= (𝑝−𝑝󸀠)_𝜇.

Putting all matrix elements given in (7) into (6), the final representation of the correlation function on physical side is obtained as ΠPHYS 𝜇𝛼𝛽 (𝑝, 𝑝󸀠, 𝑞) = 𝑓𝜒𝑏2𝑓𝐵𝑐𝑚𝜒𝑏2𝑚 2 𝐵𝑐 8 (𝑚_𝑏+ 𝑚_𝑐) (𝑝2_{− 𝑚}2 𝜒𝑏2) (𝑝 󸀠2_{− 𝑚}2 𝐵𝑐) {Δ𝐾 (𝑞2) ⋅ 𝑞_𝛼𝑔_𝛽𝜇−2₃[Δ󸀠𝑏₋(𝑞2) + Δ𝐾 (𝑞2)] 𝑞_𝜇𝑔_𝛼𝛽 −2 3[Δ󸀠𝑏+(𝑞2) + 𝐾 (𝑞2) (Δ + 4𝑚2𝜒𝑏2)] 𝑃𝜇𝑔𝛼𝛽 − 𝑖 (Δ − 4𝑚2_𝜒_𝑏2) ℎ (𝑞2) 𝜀𝜆𝜂𝛽𝜇𝑃𝜆𝑃𝜂𝑞𝛼 + other structures} + ⋅ ⋅ ⋅ , (8) where Δ = 𝑚2_𝐵_𝑐+ 3𝑚2_𝜒_𝑏2− 𝑞2, Δ󸀠= 𝑚4_𝐵_𝑐− 2𝑚2_𝐵_𝑐(𝑚2_𝜒_𝑏2+ 𝑞2) + (𝑚2_𝜒_𝑏2− 𝑞2)2. (9)

Note that we represented only the structures which we will use to find the corresponding form factors. Meanwhile, we used the following summation over the polarization tensors to obtain (8): ∑ 𝜆 𝜀𝜆_𝜇]𝜀_𝛼𝛽∗𝜆= 1 2𝜂𝜇𝛼𝜂]𝛽+ 1 2𝜂𝜇𝛽𝜂]𝛼− 1 3𝜂𝜇]𝜂𝛼𝛽, (10) where 𝜂_𝜇]= −𝑔_𝜇]+𝑝𝜇𝑝] 𝑚_𝜒2 𝑏2 . (11)

The next step is to calculate the QCD side of the correlation

function in deep Euclidean region, where𝑝2 → −∞ and

𝑝󸀠2 → −∞ via OPE. Placing the explicit expressions of the

interpolating currents into the correlation function and contracting out all quark pairs via Wick’s theorem, we obtain

ΠQCD 𝜇𝛼𝛽 (𝑝, 𝑝󸀠, 𝑞) = −𝑖 3 2 ∬ {Tr [𝑆𝑖𝑎𝑏 (𝑥 − 𝑦) 𝛾5𝑆𝑎𝑗𝑐 (𝑦) ⋅ 𝛾_𝜇(1 − 𝛾₅)D↔𝛽(𝑥) 𝑆𝑗𝑖𝑏 (−𝑥) 𝛾𝛼] + [𝛽 ←→ 𝛼]} ⋅ 𝑒−𝑖𝑝⋅𝑥𝑒𝑖𝑝󸀠⋅𝑦𝑑4𝑦 𝑑4𝑥, (12)

where𝑆 is the heavy quark propagator and it is given by [3]

𝑆𝑎𝑖_𝑄(𝑥) = 𝑖 (2𝜋)4∫ { { { 𝛿_𝑎𝑖 𝑘 − 𝑚𝑄 −𝑔𝑠𝐺 𝜓𝜑 𝑎𝑖 4 𝜎_𝜓𝜑_{(𝑘 + 𝑚}_𝑄_{) + (𝑘 + 𝑚}_𝑄) 𝜎_𝜓𝜑 (𝑘2_{− 𝑚}2 𝑄) 2 +𝜋2 3 ⟨ 𝛼_𝑠𝐺𝐺 𝜋 ⟩ 𝛿𝑎𝑖𝑚𝑄 𝑘2_{+ 𝑚} 𝑄𝑘 (𝑘2_{− 𝑚}2 𝑄) 4 + ⋅ ⋅ ⋅ } } } ⋅ 𝑒−𝑖𝑘⋅(𝑥)𝑑4𝑘, (13)

where𝑄 = 𝑏 or 𝑐 quark. Replacing the explicit expression

of the heavy quark propagators in (12) and applying integrals

over𝑥 and 𝑦, we find the QCD side as

ΠQCD 𝜇𝛼𝛽 (𝑝, 𝑝󸀠, 𝑞) = (Πpert 1 (𝑞2) + Πnonpert1 (𝑞2)) 𝑞𝛼𝑔𝛽𝜇 + (Πpert 2 (𝑞2) + Πnonpert2 (𝑞2)) 𝑞𝜇𝑔𝛽𝛼 + (Πpert 3 (𝑞2) + Πnonpert3 (𝑞2)) 𝑃𝜇𝑔𝛽𝛼 + (Πpert 4 (𝑞2) + Πnonpert4 (𝑞2)) 𝜀𝜆]𝛽𝜇𝑃𝜆𝑃𝛼𝑞] + other structures. (14)

HereΠpert_𝑖 (𝑞2) with 𝑖 = 1, 2, 3, 4 are the perturbative parts

which are expressed in terms of double dispersion integrals as Πpert 𝑖 (𝑞2) = ∬ 𝜌_𝑖(𝑠, 𝑠󸀠_{, 𝑞}2₎ (𝑠 − 𝑝2_{) (𝑠}󸀠_{− 𝑝}󸀠2₎𝑑𝑠󸀠𝑑𝑠 + subtracted terms, (15)

where the spectral densities are defined as 𝜌_𝑖(𝑠, 𝑠󸀠, 𝑞2) =

(1/𝜋)Im[Πpert_𝑖 ]. The spectral densities corresponding to four

different structures shown in (14) are obtained as

𝜌1(𝑠, 𝑠󸀠, 𝑞2) = ∫1 0 ∫ 1−𝑥 0 3 [𝑚𝑐(−3 + 4𝑥 + 2𝑦) + 𝑚𝑏(−5 + 8𝑥 + 4𝑦)] 16𝜋2 𝑑𝑦 𝑑𝑥, 𝜌2(𝑠, 𝑠󸀠, 𝑞2) = ∫1 0 ∫ 1−𝑥 0 3 [𝑚𝑐(3 − 4𝑥 − 2𝑦) + 𝑚𝑏(−1 + 4𝑥 + 2𝑦)] 8𝜋2 𝑑𝑦 𝑑𝑥, 𝜌3(𝑠, 𝑠󸀠, 𝑞2) = − ∫1 0 ∫ 1−𝑥 0 3 [𝑚𝑐(1 − 2𝑦) + 𝑚𝑏(1 + 2𝑦)] 8𝜋2 𝑑𝑦 𝑑𝑥, 𝜌4(𝑠, 𝑠󸀠, 𝑞2) = 0. (16)

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The functionΠnonpert_𝑖 (𝑞2) in nonperturbative parts is calcu-lated in a similar manner and by considering the two-gluon condensates contributes. Having calculated both the physical and OPE sides of the correlation function, now, we match them to find QCD sum rules for form factors. In the Borel scheme we get 𝐾 (𝑞2) = 8 (𝑚𝑏+ 𝑚𝑐) 𝑓_𝜒_𝑏2𝑓_𝐵_𝑐𝑚2 𝐵𝑐𝑚𝜒𝑏2Δ ⋅ 𝑒𝑚2𝜒𝑏2/𝑀2𝑒𝑚2𝐵𝑐/𝑀󸀠2{∫𝑠0 4𝑚2 𝑏 ∫𝑠 󸀠 0 (𝑚𝑏+𝑚𝑐)2 𝜌₁(𝑠, 𝑠󸀠, 𝑞2) ⋅ Θ [𝐿 (𝑠, 𝑠󸀠_{, 𝑞}2_{)] 𝑒}−𝑠/𝑀2 𝑒−𝑠󸀠_/𝑀󸀠2 𝑑𝑠󸀠_𝑑𝑠 + ̂B_𝑀2B̂_𝑀󸀠2Πnonpert₁ (𝑞2)} , 𝑏₋(𝑞2) = − 12 (𝑚𝑏+ 𝑚𝑐) 𝑓𝜒𝑏2𝑓𝐵𝑐𝑚2𝐵𝑐𝑚𝜒𝑏2Δ󸀠 ⋅ 𝑒𝑚2𝜒𝑏2/𝑀2𝑒𝑚2𝐵𝑐/𝑀󸀠2{∫𝑠0 4𝑚2 𝑏 ∫𝑠 󸀠 0 (𝑚𝑏+𝑚𝑐)2 𝜌₂(𝑠, 𝑠󸀠, 𝑞2) ⋅ Θ [𝐿 (𝑠, 𝑠󸀠, 𝑞2)] 𝑒−𝑠/𝑀2𝑒−𝑠󸀠/𝑀󸀠2𝑑𝑠󸀠𝑑𝑠 + ̂B_𝑀2B̂_𝑀󸀠2Πnonpert₂ (𝑞2) − Δ Δ󸀠𝐾 (𝑞2)} , 𝑏+(𝑞2) = −_𝑓12 (𝑚𝑏+ 𝑚𝑐) 𝜒𝑏2𝑓𝐵𝑐𝑚2𝐵𝑐𝑚𝜒𝑏2Δ󸀠 ⋅ 𝑒𝑚2𝜒𝑏2/𝑀2_𝑒𝑚2𝐵𝑐/𝑀󸀠2{∫ 𝑠0 4𝑚2 𝑏 ∫𝑠 󸀠 0 (𝑚𝑏+𝑚𝑐)2 𝜌3(𝑠, 𝑠󸀠, 𝑞2) ⋅ Θ [𝐿 (𝑠, 𝑠󸀠, 𝑞2)] 𝑒−𝑠/𝑀2𝑒−𝑠󸀠/𝑀󸀠2𝑑𝑠󸀠𝑑𝑠 + ̂B_𝑀2B̂_𝑀󸀠2Πnonpert₃ (𝑞2) − Δ + 4𝑚2 𝜒𝑏2 Δ󸀠 𝐾 (𝑞2)} , ℎ (𝑞2) = −𝑖 8 (𝑚𝑏+ 𝑚𝑐) 𝑓_𝜒_𝑏2𝑓_𝐵_𝑐𝑚2 𝐵𝑐𝑚𝜒𝑏2(4𝑚2𝜒𝑏2− Δ) ⋅ 𝑒𝑚2𝜒𝑏2/𝑀2𝑒𝑚2𝐵𝑐/𝑀󸀠2{∫𝑠0 4𝑚2 𝑏 ∫𝑠 󸀠 0 (𝑚𝑏+𝑚𝑐)2 𝜌₄(𝑠, 𝑠󸀠_{, 𝑞}2₎ ⋅ Θ [𝐿 (𝑠, 𝑠󸀠, 𝑞2)] 𝑒−𝑠/𝑀2𝑒−𝑠󸀠/𝑀󸀠2𝑑𝑠󸀠𝑑𝑠 + ̂B𝑀2B̂_𝑀󸀠2Πnonpert₄ (𝑞2)} , (17)

where𝑀2and𝑀󸀠2are Borel mass parameters;𝑠₀and𝑠󸀠₀are

continuum thresholds in the initial and final channels. Here,

Θ is the step function and 𝐿(𝑠, 𝑠󸀠_{, 𝑞}2_{) is given by}

𝐿 (𝑠, 𝑠󸀠, 𝑞2) = 𝑠󸀠𝑦 (1 − 𝑥 − 𝑦) − 𝑠𝑥𝑦 + 𝑚2_𝑐(𝑥 + 𝑦 − 1)

− 𝑚_𝑏2(𝑥 + 𝑦) + 𝑞2𝑥 (1 − 𝑥 − 𝑦) . (18)

The functions ̂B_𝑀2B̂_𝑀󸀠2Πnonpert_𝑖 (𝑞2) are written as

̂ B𝑀2B̂_𝑀󸀠2Πnonpert_𝑖 (𝑞2) = ⟨𝛼𝑠𝐺𝐺 𝜋 ⟩ ∫ 1 0𝑓𝑖(𝑞 2₎ ⋅ 𝑒(−𝑚2 𝑏(1+𝑀2𝑥/𝑀󸀠2)+(𝑀2𝑥/𝑀󸀠2)(𝑚𝑐2−𝑞2𝑥))/𝑀2𝑥(1+(−1+𝑀2/𝑀󸀠2)𝑥)𝑑𝑥, (19)

where𝑓_𝑖(𝑞2) are very lengthy functions and we do not present

their explicit expressions here.

3. Numerical Results

To numerically analyze the sum rules obtained for the form factors, we use the meson masses from PDG [4]. Considering the fact that the results of sum rules considerably depend on the quark masses, decay constants, and gluon condensate, we use the values of these parameters from different sources. For the quark masses we take into account all the pole values

and those obtained at𝑀𝑆 scheme from [5–15]. For 𝑓_𝐵_𝑐, we

consider all the values predicted using different methods in

[16–19]. In the case of𝑓_𝜒_𝑏2, we use the only value that exists

in the literature, that is,𝑓_𝜒_𝑏2 = (0.0122 ± 0.0072) [20]. For the

gluon condensate, we also use its value from different sources [1, 21–28] calculated via different approaches.

From the sum rules for the form factors it is also clear that they contain extra four auxiliary parameters, namely,

the Borel parameters 𝑀2 and 𝑀󸀠2 as well as continuum

thresholds𝑠₀and𝑠󸀠₀. The general criteria are that the physical

quantities like form factors should be independent of these parameters. Therefore, we need to determine their working regions such that the form factors weakly depend on these parameters. To find the Borel windows, we require that the higher states and continuum contributions are sufficiently suppressed and the perturbative parts exceed the nonpertur-bative contributions and the series of OPE converge. As a

result we get the windows:14 GeV2 ≤ 𝑀2 ≤ 20 GeV2 and

8 GeV2 _{≤ 𝑀}󸀠2 _{≤ 12 GeV}2_{. The continuum thresholds}_𝑠

0

and 𝑠󸀠₀ are not completely arbitrary but they are related to

the energy of the first excited states with the same quantum numbers as the interpolating currents of the initial and final channels. In this work the continuum thresholds are chosen

in the intervals104 GeV2 ≤ 𝑠₀ ≤ 108 GeV2and43 GeV2 ≤

𝑠󸀠

0≤ 45 GeV2.

The dependence of form factors𝐾 and 𝑏₊, as examples,

on Borel parameters 𝑀2 and𝑀󸀠2 at𝑞2 = 0 is plotted in

Figures 1 and 2. From these figures we see that form factors show overall weak dependence on the Borel mass parameters.

The behaviors of the form factors𝐾, 𝑏₊,𝑏₋, andℎ in terms

of 𝑞2 are shown in Figures 3–6. In these figures, the red

triangles show the QCD sum rule predictions, yellow-solid line denotes the prediction of fit function obtained using the central values of the input parameters, and the green band shows the uncertainty due to errors of input parameters. Note that to obtain the central values, we consider the average values of input parameters discussed above; however, to

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14 15 16 17 18 19 20

M2(GeV2)

s0= 104 GeV2, s󳰀0= 43 GeV2, M󳰀2= 10 GeV2

s0= 108 GeV2, s󳰀0= 45 GeV2, M󳰀2= 10 GeV2

s0= 106 GeV2, s󳰀0= 44 GeV2, M󳰀2= 10 GeV2

0 −1 −2 −3 −4 K( q 2=0 ) (a) 8 9 10 11 12 M󳰀2(GeV2)

s0= 104 GeV2, s󳰀0= 43 GeV2, M2= 17 GeV2

s0= 106 GeV2, s󳰀0= 44 GeV2, M2= 17 GeV2

s0= 108 GeV2, s󳰀0= 45 GeV2, M2= 17 GeV2

0 −1 −2 −3 −4 K( q 2=0 ) (b)

Figure 1: (a)𝐾(𝑞2 = 0) as a function of the Borel mass parameter 𝑀2at fixed values of𝑠₀,𝑠󸀠₀, and𝑀󸀠2. (b)𝐾(𝑞2 = 0) as a function of the Borel mass parameter𝑀󸀠2at fixed values of𝑠₀,𝑠󸀠₀, and𝑀2.

14 15 16 17 18 19 20

M2(GeV2)

s0= 104 GeV2, s0󳰀= 43 GeV2, M󳰀2= 10 GeV2

s0= 108 GeV2, s0󳰀= 45 GeV2, M󳰀2= 10 GeV2

s0= 106 GeV2, s0󳰀= 44 GeV2, M󳰀2= 10 GeV2

1.0 0.8 0.6 0.4 0.2 0.0 b+ (q 2 =0 ) (G eV −2 ) (a) 1.0 0.8 0.6 0.4 0.2 0.0 b+ (q 2=0 ) (G eV −2) 8 9 10 11 12 M󳰀2_(GeV2₎

s0= 104 GeV2, s󳰀0= 43 GeV2, M2= 17 GeV2

s0= 106 GeV2, s󳰀0= 44 GeV2, M2= 17 GeV2

s0= 108 GeV2, s󳰀0= 45 GeV2, M2= 17 GeV2

(b)

Figure 2: (a)𝑏₊(𝑞2 = 0) as a function of the Borel mass parameter 𝑀2at fixed values of𝑠₀,𝑠󸀠₀, and𝑀󸀠2. (b)𝑏₊(𝑞2 = 0) as a function of the Borel mass parameter𝑀󸀠2at fixed values of𝑠₀,𝑠󸀠₀, and𝑀2.

calculate the uncertainties we consider all errors of these parameters from different sources previously quoted. As it is seen from these figures the sum rules results are truncated at some points. Hence, to enlarge the region to whole physical region we need to find some fit functions such that their results coincide well with the QCD sum rules predictions at

reliable regions. For this reason we show the𝑞2dependence

of form factors including both the sum rules and fit results in Figures 3–6. Our numerical calculations reveal that the

following fit function well defines the form factors under consideration: 𝑓 (𝑞2) = 𝑓0exp [ [ 𝑎 (_𝑚𝑞₂2 𝜒𝑏2 ) + 𝑏 (_𝑚𝑞₂2 𝜒𝑏2 ) 2 ] ] , (20)

where the values of the parameters𝑓₀,𝑎, and 𝑏 obtained at

𝑀2_{= 17 GeV}2_and_𝑀󸀠2_{= 10 GeV}2_for_𝜒

𝑏2→ 𝐵𝑐𝑙] transition

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−2.0 −2.5 −1.5 −1.0 −0.5 0.0 K(q 2) 0 2 4 6 8 10 12 q2_(GeV2₎

Figure 3:𝐾(𝑞2) as a function of 𝑞2at𝑠₀= 106 GeV2,𝑠󸀠₀= 44 GeV2, 𝑀2 _{= 17 GeV}2_{, and}_𝑀󸀠2 _{= 10 GeV}2_{. The red triangles show the}

QCD sum rule predictions, yellow-solid line denotes the prediction of fit function obtained using the central values of the input parameters, and the green band shows the uncertainty due to errors of input parameters. −0.0002 −0.0004 −0.0006 −0.0008 h( q 2 ) (G eV −2 ) 0 2 4 6 8 10 12 q2_(GeV2₎

Figure 4: The same as Figure 3 but forℎ(𝑞2).

0 2 4 6 8 10 12 0 1 2 3 4 5 b+ (q 2) (G eV −2) q2(GeV2)

Figure 5: The same as Figure 3 but for𝑏₊(𝑞2).

0 2 4 6 8 10 12 −2.0 −1.5 −1.0 −0.5 0.0 b− (q 2 ) (G eV −2 ) q2(GeV2)

Figure 6: The same as Figure 3 but for𝑏₋(𝑞2).

Table 1: Parameters appearing in the fit function of the form factors.

𝑓₀ 𝑎 𝑏 𝐾(𝑞2₎ _{−0.871 ± 0.279} _{5.239 ± 1.677} _{−7.588 ± 2.428} 𝑏−(𝑞2) −0.134 ± 0.043 GeV−2 8.973 ± 2.871 56.462 ± 18.068 𝑏₊(𝑞2_{) 0.304 ± 0.097 GeV}−2 _{10.054 ± 3.217 53.922 ± 17.255} ℎ(𝑞2₎ (−2.594 ± 0.830) × 10−4_GeV−2 5.224 ± 1.672 3.891 ± 1.245 Our final purpose in this section is to obtain the decay

width of the𝜒_𝑏2→ 𝐵_𝑐𝑙] transition at all lepton channels. The

differential decay width for this transition is obtained as 𝑑Γ 𝑑𝑞2 = 𝐺2 𝐹𝑉𝑐𝑏2 210₃2_𝑚7 𝜒𝑏2𝜋3𝑞6 (𝑚2_𝑙 − 𝑞2)2Δ󸀠3/2_{{󵄨󵄨󵄨󵄨󵄨𝑏}₋(𝑞2_{)󵄨󵄨󵄨󵄨󵄨}2 ⋅ Δ󸀠𝑚2_𝑙𝑞4_{+ 󵄨󵄨󵄨󵄨󵄨𝑏}₊(𝑞2_{)󵄨󵄨󵄨󵄨󵄨}2Δ󸀠[(𝑚2_𝐵_𝑐− 𝑚2_𝜒_𝑏2)2𝑚2_𝑙 + (𝑚2_𝐵_𝑐− 𝑚2_𝜒_𝑏2)2𝑞2− 2 (𝑚2_𝐵_𝑐+ 𝑚2_𝜒_𝑏2) 𝑞4+ 𝑞6] + 2 ⋅ Re [𝐾 (𝑞2) 𝑏₊∗(𝑞2)] ⋅ Δ󸀠[−𝑞4+ 𝑚2_𝐵_𝑐(𝑚_𝑙2+ 𝑞2) − 𝑚2_𝜒_𝑏2(𝑚2_𝑙 + 𝑞2)] − 2 ⋅ Re [𝑏₋(𝑞2) 𝑏₊∗(𝑞2)] Δ󸀠𝑚2_𝑙𝑞2(𝑚2_𝐵_𝑐− 𝑚2_𝜒_𝑏2) + 󵄨󵄨󵄨󵄨󵄨𝐾(𝑞2)󵄨󵄨󵄨󵄨󵄨2[𝑚4_𝐵_𝑐(𝑚2_𝑙 + 𝑞2) + 𝑚4_𝜒_𝑏2(𝑚2_𝑙 + 𝑞2) + 𝑞4_(𝑚2 𝑙 + 𝑞2) − 2𝑚2𝐵𝑐(𝑚 2 𝜒𝑏2+ 𝑞 2_{) (𝑚}2 𝑙 + 𝑞2) + 𝑚2_𝜒_𝑏2𝑞2(𝑚2_𝑙 + 5𝑞2_{)] + 3 󵄨󵄨󵄨󵄨󵄨ℎ(𝑞}2_{)󵄨󵄨󵄨󵄨󵄨}2Δ󸀠𝑚2_𝜒_𝑏2𝑞2(𝑚2_𝑙 + 𝑞2) − 2 Re [𝐾 (𝑞2) 𝑏₋∗(𝑞2)] Δ󸀠𝑚2_𝑙𝑞2} . (21)

After performing integration over𝑞2 in (21) in the interval

𝑚2

𝑙 ≤ 𝑞2≤ (𝑚𝜒𝑏2− 𝑚𝐵𝑐)

2_{, we obtain the total decay widths as}

presented in Table 2 for different leptons. The errors belong to the uncertainties coming from the determination of the

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Table 2: Numerical results of decay widths at different lepton chan-nels. Γ (GeV) 𝜒𝑏2→ 𝐵𝑐𝑒]𝑒 (1.054 ± 0.506) × 10−13 𝜒_𝑏2→ 𝐵_𝑐𝜇]_𝜇 (1.041 ± 0.500) × 10−13 𝜒_𝑏2→ 𝐵_𝑐𝜏]_𝜏 (2.398 ± 1.175) × 10−14 working regions for auxiliary parameters as well as those of the other input parameters. The orders of decay rates at all lepton channels show that these transitions are accessible at LHC in near future.

In summary, we have calculated the transition form

fac-tors for the semileptonic𝜒_𝑏2 → 𝐵_𝑐𝑙] transition using QCD

sum rule technique. We took into account the two-gluon condensate contributions as nonperturbative effects. We used these form factors to estimate the order of decay widths at all lepton channels. The order of decay width reveals that these transitions can be seen at LHC in the near future. Any comparison of the experimental results with our predictions can provide us with essential knowledge on the nature of the

tensor𝜒_𝑏2(1𝑃) state.

Competing Interests

The authors declare that there is no competing interests regarding the publication of this paper.

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Transition form factors of chi(b2)(1P) -> B-c(l)over-barv in QCD

Research Article

Transition Form Factors of

𝜒

𝑏2

(1𝑃) → 𝐵

𝑐

𝑙] in QCD

K. Azizi,

H. Sundu,

J. Y. Süngü,

and N. Yinelek

1. Introduction

2. QCD Sum Rules for

𝜒

→ 𝐵

𝑙]

Transition Form Factors

3. Numerical Results

Competing Interests

References

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_𝑏2

_𝑐