Analysis of the 32± pentaquark states in the diquark–diquark–antiquark model with QCD sum rules

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Nuclear Physics B ••• (••••) •••–•••

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www.elsevier.com/locate/nuclphysb

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Analysis of the pentaquark states in the diquark–diquark–antiquark model with QCD sum rules

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Zhi-Gang Wang

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Department of Physics, North China Electric Power University, Baoding 071003, PR China

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Received 14 June 2016; received in revised form 3 August 2016; accepted 17 September 2016

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Editor: Hong-Jian He

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Abstract In this article, we construct the axialvector-diquark–scalar-diquark–antiquark type and the axialvector± diquark–axialvector-diquark–antiquark type interpolating currents, and study the J P = 32 hidden-charm pentaquark states with the strangeness S = 0, −1, −2, −3 systematically using the QCD sum rules. The  −  11 1  −   − 10 1 11 3 and Puud2 32 are compatible with predicted masses of the pentaquark states Puud2 32 , Puud2 32 the experimental value MPc (4380) = 4380 ± 8 ± 29 MeV from the LHCb collaboration, more experimental data are still needed to identify the Pc (4380) unambiguously. © 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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1. Introduction In 2015, the LHCb collaboration studied the 0b → J /ψK − p decays with the data sample corresponds to an integrated luminosity of 3 fb−1 acquired with the LHCb detector from 7 and 8 TeV pp collisions, and observed two pentaquark candidates Pc (4380) and Pc (4450) in the J /ψp invariant mass distributions with the significances of more than 9 standard deviations [1]. They performed the amplitude analysis on all relevant masses and decay angles of the sixdimensional data using the helicity formalism and Breit–Wigner amplitudes to describe all reso-

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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.nuclphysb.2016.09.009 0550-3213/© 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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nances. The Breit–Wigner masses and widths are MPc (4380) = 4380 ± 8 ± 29 MeV, MPc (4450) = 4449.8 ± 1.7 ± 2.5 MeV, Pc (4380) = 205 ± 18 ± 86 MeV, and Pc (4450) = 39 ± 5 ± 19 MeV, − respectively. The preferred quantum numbers of the Pc (4380) and Pc (4450) are J P = 32 and 5+ 2 ,

respectively. Recently, the LHCb collaboration inspected the 0b → J /ψK − p decays for the presence of J /ψp or J /ψK − contributions with minimal assumptions about K − p contributions (such as their number, their resonant or nonresonant nature, or their lineshapes), and observed that at more than 9 standard deviations the 0b → J /ψK − p decays cannot be described with the K − p contributions alone, and the J /ψp contributions play a dominant role in this incompatibility, and obtained model-independent support for the evidences of the Pc+ (4380/4500) → J /ψp decays [2]. Furthermore, the LHCb collaboration performed a full amplitude analysis of the 0b → J /ψπ − p decays allowing for previously observed conventional (pπ − ) and exotic (J /ψp and J /ψπ − ) resonances, and observed that a significantly better description of the data can be achieved by either including the two Pc+ (4380/4450) states or the Zc− (4200) state [3]. There have been several possible assignments since the observations of the Pc (4380) and Pc (4450), such as the molecule-like pentaquark states [4] (or not the molecular pentaquark states [5]), the diquark–triquark type pentaquark states [6], the diquark–diquark–antiquark type pentaquark states [7–11], re-scattering effects [12], etc. We can diagnose their resonant or nonresonant nature using photoproduction (or pionproduction) off a proton target [13]. The photoproduction process does not satisfy the so-called anomalous triangle singularity condition, their presence in the J /ψ photoproduction is crucial to conforming the resonant nature of such states as opposed to their being kinematical effects. The QCD sum rules is a powerful theoretical tool in studying the ground state heavy baryon states [14–17]. We usually take the diquarks as the basic constituents in constructing the baryon currents and pentaquark currents. The attractive interactions induced by one-gluon exchange favor formation of the diquark states in color antitriplet, flavor antitriplet and spin singlet εij k qjT Cγ5 qk or in color antitriplet, flavor sextet and spin triplet ε ij k qjT Cγμ qk , where the i, j and k are color indexes [18], the QCD sum rules also favor formation of the diquark states εij k qjT Cγ5 qk and ε ij k qjT Cγμ qk [19,20], so we can take the scalar or axialvector diquark states as the basic constituents in constructing currents to interpolate the lowest pentaquark states. In our previous works, we observed that the heavy-light scalar and axialvector diquark states have almost degenerate masses [19], the masses of the light axialvector diquark states lie (150–200) MeV above that of the corresponding light scalar diquark states [20] from the QCD sum rules. We expect that the ground state diquark–diquark–antiquark type hidden-charm pentaquark states consist of a light scalar (or axialvector) diquark, a charm scalar (or axialvector) diquark and an anti-charm-quark. In Refs. [9,10], we take the light scalar diquark states, heavy axialvector diquark states and heavy scalar diquark states as the basic constituents, construct the scalar-diquark–axialvectordiquark–antiquark type and scalar-diquark–scalar-diquark–antiquark type currents to study the − + ± J P = 32 , 52 and 12 hidden-charm pentaquark states using the QCD sum rules. The predicted masses MP ( 3 − ) = 4.38 ± 0.13 GeV and MP ( 5 + ) = 4.44 ± 0.14 GeV support assigning c 2

the Pc (4380) and Pc (4450) to be the

3− 2

and

5+ 2

c 2

hidden-charm pentaquark states, respectively  2 − (2MQ )2 [9]. In Ref. [9], we observe that the empirical energy scale formula, μ = MX/Y/Z with the effective heavy quark mass MQ , in determining the ideal energy scales of the QCD

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spectral densities in the QCD sum rules for the tetraquark states X, Y and Z [21], can be successfully applied to study the hidden-charm pentaquark states with a slight modification  2 μ = MP − (2Mc )2 . In Ref. [11], we take the light axialvector diquark states, heavy axialvector diquark states and heavy scalar diquark states as the basic constituents, construct the axialvectordiquark–axialvector-diquark–antiquark type and axialvector-diquark–scalar-diquark–antiquark ± type interpolating currents to study the J P = 12 hidden-charm pentaquark states with the QCD sum rules in a systematic way. In this article, we resort to the same routine as that in Refs. [9–11] to study the masses and pole ± residues of the J P = 32 hidden-charm pentaquark states with the QCD sum rules by constructing the axialvector-diquark–scalar-diquark–antiquark type and axialvector-diquark–axialvectordiquark–antiquark type interpolating currents, and revisit the assignment of the Pc (4380). The article is arranged as follows: we derive the QCD sum rules for the masses and pole ± residues of the 32 hidden-charm pentaquark states in Sect. 2; in Sect. 3, we present the numerical results and discussions; and Sect. 4 is reserved for our conclusion.

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2. QCD sum rules for the

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10 1

2 Juuu,μ (x) =

10 12 (x) = Juud,μ

JP

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10 12 Judd,μ (x) =

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10 12 Jddd,μ (x) =

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ε ila ε ij k ε lmn √ 3  T × uTj (x)Cγμ uk (x)dm (x)Cγ5 cn (x) + 2uTj (x)Cγμ dk (x)uTm (x)Cγ5 cn (x)

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ε ila ε ij k ε lmn √ 3  T × djT (x)Cγμ dk (x)uTm (x)Cγ5 cn (x) + 2djT (x)Cγμ uk (x)dm (x)Cγ5 cn (x)

ε ila ε ij k ε lmn T T dj (x)Cγμ dk (x)dm (x)Cγ5 cn (x)C c¯aT (x) , √ 3

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× C c¯aT (x) ,

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ε ila ε ij k ε lmn T uj (x)Cγμ uk (x)uTm (x)Cγ5 cn (x)C c¯aT (x) , √ 3

× C c¯aT (x) ,

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where the are currents to interpolate the = hidden-charm tetraquark states, the superscripts jL and jH are the spins of the light diquark state and the charm (or heavy) diquark state, respectively, j = jH + jc¯ , the jc¯ is the spin of the charm (or heavy) antiquark, the subscripts q1 , q2 , q3 are the light quark constituents u, d or s. Now we write down the j j j interpolating currents Jq1Lq2Hq3 ,μ (x) explicitly,

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j j j Jq1Lq2Hq3 ,μ (x)

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Firstly, let us write down the two-point correlation functions qL1 q2Hq3 ,μν (p) in the QCD sum rules,    j j j j j j j j j qL1 q2Hq3 ,μν (p) = i d 4 xeip·x 0|T Jq1Lq2Hq3 ,μ (x)J¯q1Lq2Hq3 ,ν (0) |0 , (1)

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hidden charm pentaquark states

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1 2

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2 Juus,μ (x) =

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2 Juds,μ (x) =

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2 Jdds,μ (x) =

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Jdss,μ (x) =

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2 Jsss,μ (x) =

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2 (x) = Juud,μ

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11 12 Judd,μ (x) =

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 T T (x)Cγ5 cn (x) + uTj (x)Cγμ sk (x)dm (x)Cγ5 cn (x) × uTj (x)Cγμ dk (x)sm + djT (x)Cγμ sk (x)uTm (x)Cγ5 cn (x) C c¯aT (x) ,

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ε ila ε ij k ε lmn √ 3  T T (x)Cγ5 cn (x) + 2djT (x)Cγμ sk (x)dm (x)Cγ5 cn (x) × djT (x)Cγμ dk (x)sm

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ε ila ε ij k ε lmn √ 3  T T (x)Cγ5 cn (x) × sj (x)Cγμ sk (x)uTm (x)Cγ5 cn (x) + 2sjT (x)Cγμ uk (x)sm

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× C c¯aT (x) ,

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T T (x)Cγ5 cn (x) + 2sjT (x)Cγμ dk (x)sm (x)Cγ5 cn (x) × sjT (x)Cγμ sk (x)dm 

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× C c¯aT (x) ,

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ε ila ε ij k ε lmn T T sj (x)Cγμ sk (x)sm (x)Cγ5 cn (x)C c¯aT (x) √ 3

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ε ila ε ij k ε lmn

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 T (x)Cγα cn (x) + 2uTj (x)Cγμ dk (x)uTm (x)Cγα cn (x) × uTj (x)Cγμ uk (x)dm

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√ 3

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× γ5 γ α C c¯aT (x) ,

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2 (x) = ε ila ε ij k ε lmn uTj (x)Cγμ uk (x)uTm (x)Cγα cn (x)γ5 γ α C c¯aT (x) , Juuu,μ

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× C c¯aT (x) , ε ila ε ij k ε lmn

× C c¯aT (x) ,

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ε ila ε ij k ε lmn √ 3  T (x)Cγ5 cn (x) + 2uTj (x)Cγμ sk (x)uTm (x)Cγ5 cn (x) × uTj (x)Cγμ uk (x)sm

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ε ila ε ij k ε lmn √ 3  T T (x)Cγα cn (x) × dj (x)Cγμ dk (x)uTm (x)Cγα cn (x) + 2djT (x)Cγμ uk (x)dm

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× γ5 γ

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C c¯aT (x) ,

T 2 (x) = ε ila ε ij k ε lmn djT (x)Cγμ dk (x)dm (x)Cγα cn (x)γ5 γ α C c¯aT (x) , Jddd,μ

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2 Juus,μ (x) =

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ε ila ε ij k ε lmn √ 3  T (x)Cγα cn (x) + 2uTj (x)Cγμ sk (x)uTm (x)Cγα cn (x) × uTj (x)Cγμ uk (x)sm

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× γ5 γ C c¯aT (x) , ε ila ε ij k ε lmn α

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2 Jdds,μ (x) =

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 T T (x)Cγα cn (x) + uTj (x)Cγμ sk (x)dm (x)Cγα cn (x) × uTj (x)Cγμ dk (x)sm + djT (x)Cγμ sk (x)uTm (x)Cγα cn (x) γ5 γ α C c¯aT (x) ,

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ε ila ε ij k ε lmn √ 3  T T (x)Cγα cn (x) + 2djT (x)Cγμ sk (x)dm (x)Cγα cn (x) × djT (x)Cγμ dk (x)sm

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× γ5 γ α C c¯aT (x) ,

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11 12 (x) = Juss,μ

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ε ila ε ij k ε lmn √ 3  T T (x)Cγα cn (x) × sj (x)Cγμ sk (x)uTm (x)Cγα cn (x) + 2sjT (x)Cγμ uk (x)sm

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× γ5 γ C c¯aT (x) , ε ila ε ij k ε lmn α

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2 Jdss,μ (x) =

√

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×

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T T (x)Cγα cn (x) + 2sjT (x)Cγμ dk (x)sm (x)Cγα cn (x) sjT (x)Cγμ sk (x)dm

× γ5 γ α C c¯aT (x) ,

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2 (x) = Juud,μ

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 T (x)Cγμ cn (x) + 2uTj (x)Cγα dk (x)uTm (x)Cγμ cn (x) × uTj (x)Cγα uk (x)dm

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√ 3

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× γ5 γ α C c¯aT (x) ,

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2 Juuu,μ (x) = ε ila ε ij k ε lmn uTj (x)Cγα uk (x)uTm (x)Cγμ cn (x)γ5 γ α C c¯aT (x) ,

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ε ila ε ij k ε lmn √ 3  T T (x)Cγμ cn (x) × dj (x)Cγα dk (x)uTm (x)Cγμ cn (x) + 2djT (x)Cγα uk (x)dm

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× γ5 γ

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C c¯aT (x) ,

T 2 (x) = ε ila ε ij k ε lmn djT (x)Cγα dk (x)dm (x)Cγμ cn (x)γ5 γ α C c¯aT (x) , Jddd,μ

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1 2

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2 Juds,μ (x) =

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× γ5 γ C c¯aT (x) , ε ila ε ij k ε lmn

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 T T × uTj (x)Cγα dk (x)sm (x)Cγμ cn (x) + uTj (x)Cγα sk (x)dm (x)Cγμ cn (x) + djT (x)Cγα sk (x)uTm (x)Cγμ cn (x) γ5 γ α C c¯aT (x) ,

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ε ila ε ij k ε lmn √ 3  T T T × dj (x)Cγα dk (x)sm (x)Cγμ cn (x) + 2djT (x)Cγα sk (x)dm (x)Cγμ cn (x)

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α

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ε ila ε ij k ε lmn √ 3  T × uTj (x)Cγα uk (x)sm (x)Cγμ cn (x) + 2uTj (x)Cγα sk (x)uTm (x)Cγμ cn (x)

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2 (x) = Jdds,μ

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√ 3

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× γ5 γ α C c¯aT (x) ,

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11 32 Juss,μ (x) ε ila ε ij k ε lmn

=

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T (x)Cγμ cn (x) sjT (x)Cγα sk (x)uTm (x)Cγμ cn (x) + 2sjT (x)Cγα uk (x)sm

√ 3 × γ5 γ α C c¯aT (x) ,

Jdss,μ (x) =

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T T sjT (x)Cγα sk (x)dm (x)Cγμ cn (x) + 2sjT (x)Cγα dk (x)sm (x)Cγμ cn (x)

√ 3 × γ5 γ α C c¯aT (x) ,

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where the i, j , k, l, m, n and a are color indices, the C is the charge conjugation matrix. We take the isospin limit by assuming the u and d quarks have the degenerate masses, and classify the above thirty currents couple to the hidden-charm pentaquark states with degenerate masses into the following twelve types, 10 12

10 12

10 12

10 12

10 12

10 12

10 12

1. Juuu,μ (x) , Juud,μ (x) , Judd,μ (x) , Jddd,μ (x) ;

3.

10 12 (x) , Juss,μ

4.

10 12 (x) ; Jsss,μ

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2. Juus,μ (x) , Juds,μ (x) , Jdds,μ (x) ;

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ε ila ε ij k ε lmn

11 32 T Jsss,μ (x) = ε ila ε ij k ε lmn sjT (x)Cγα sk (x)sm (x)Cγμ cn (x)γ5 γ α C c¯aT (x) ,

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10 12 Jdss,μ (x) ;

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2 2 2 2 5. Juuu,μ (x) , Juud,μ (x) , Judd,μ (x) , Jddd,μ (x) ; 2 2 2 (x) , Juds,μ (x) , Jdds,μ (x) ; 6. Juus,μ

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2 2 7. Juss,μ (x) , Jdss,μ (x) ;

8.

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1 2

11 12 Jsss,μ (x) ;

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2 2 2 2 (x) , Juud,μ (x) , Judd,μ (x) , Jddd,μ (x) ; 9. Juuu,μ

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2 2 2 10. Juus,μ (x) , Juds,μ (x) , Jdds,μ (x) ;

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11. Juss,μ (x) , Jdss,μ (x) ;

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11 32

12. Jsss,μ (x) .

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In calculations, we choose the first current in each type for simplicity. ± j j j The currents Jq1Lq2Hq3 ,μ (0) couple potentially to the J P = 32 and J P = j j j

±

j j j

±

j j j j j j 3 0|Jq1Lq2Hq3 ,μ (0)|Pq1Lq2Hq3 (

−

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1± 2

hidden-charm

pentaquark states Pq1Lq2Hq3 ( 32 ) and Pq1Lq2Hq3 ( 12 ), respectively,

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− )(p) = λ− P Uμ (p, s) ,

2 + 3 j j j j j j + 0|Jq1Lq2Hq3 ,μ (0)|Pq1Lq2Hq3 ( )(p) = λ+ P iγ5 Uμ (p, s) , 2 + j j j j j j 1 + λ+ 0|Jq1Lq2Hq3 ,μ (0)|Pq1Lq2Hq3 ( )(p) =

P U (p, s) pμ , 2 − j j j j j j 1 − λ− 0|Jq1Lq2Hq3 ,μ (0)|Pq1Lq2Hq3 ( )(p) =

P iγ5 U (p, s) pμ , 2

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λ± P

± where the and

λ± P are the pole residues or current-hadron coupling constants, the Uμ (p, s) ± ± and U (p, s) are the Dirac spinors [22–25]. The Dirac spinors Uμ (p, s) obey the Rarita– Schwinger equation (/ p − MP ,± )Uμ± (p) = 0, and the relations γ μ Uμ± (p, s) = 0, p μ Uμ± (p, s) =

P ,± )U ± (p) = 0. 0; while the Dirac spinors U ± (p, s) obey the Dirac equation (/ p−M At the phenomenological side, we can insert a complete set of intermediate hidden-charm j j j pentaquark states with the same quantum numbers as the current operators Jq1Lq2Hq3 ,μ (x) and j j j j j j iγ5 Jq1Lq2Hq3 ,μ (x) into the correlation functions qL1 q2Hq3 ,μν (p) to obtain the hadronic representation [14,15]. After isolating the pole terms of the lowest states of the hidden-charm pentaquark ± ± states with J P = 32 and J P = 12 respectively, we obtain the following results:   p μ pν / + MP ,− / − MP ,+ jL jH j −2 p +2 p q1 q2 q3 ,μν (p) = λP + λ + −g μν P p2 MP2 ,− − p 2 MP2 ,+ − p 2 

P ,+

P ,− p / +M p / −M + − 2 2 +

λP +

λP pμ p ν + · · ·

2 − p2

2 − p2 M M P ,+ P ,−

 pμ pν j j j

qjL1 qj2Hqj3 (p 2 ) pμ pν . = qL1 q2Hq3 (p 2 ) −gμν + (16) + p2 j j j

In this article, we choose the component qL1 q2Hq3 (p 2 ) associated with the tensor structure −gμν + ± pμ pν to study the hidden-charm pentaquark states with J P = 32 , there are no contaminations p2

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

8

1 2 3 4 5 6 7 8 9

±

qjL1 qj2Hqj3 (p 2 ) come from the hidden-charm pentaquark states with J P = 12 in the component  j j j associated with the tensor structure pμ pν . It is easy to project out the component qL1 q2Hq3 (p 2 ),

 pμ pν 1 jL jH j j j j 2 q1 q2 q3 (p ) = (17) −gμν + qL1 q2Hq3 ,μν (p) . 3 p2

π

10 11 12

j j j,1

14 15 16 17 18 19 20

4m2c

22

s0 ds

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

√

10 11 12

j j j,0

(18)

  s  MP2 ,+ jL jH j,1 jL jH j,0 +2 sρq1 q2 q3 ,H (s) − ρq1 q2 q3 ,H (s) exp − 2 = 2MP ,+ λP exp − 2 , (20) T T

and use the example

10 21 sss,μν (p)

to illustrate the routine. Firstly, we contract the s 10 21

and c quark fields in the correlation function sss,μν (p) with Wick theorem,           10 12 d 4 xeip·x CCaT a (−x)C sss,μν (p) = i ε ila ε ij k ε lmn ε i l a ε i j k ε l m n    T T 2T r γμ Skk  (x)γν CSjj  (x)C T r γ5 Cnn (x)γ5 CSmm (x)C   T T − 4T r γμ Skk  (x)γν CSmj ,  (x)Cγ5 Cnn (x)γ5 CSj m (x)C

44 45 46 47

14 15 16 17 18 19 20 21 22 23 24

26 27 28 29 30 31 32 33 34 35 36 37 38

(21)

39 40

where the Sij (x) and Cij (x) are the full s and c quark propagators, respectively,

42 43

13

25

where the T 2 are the Borel parameters and the s0 are the continuum threshold parameters. In Eqs. (19)–(20), we separate the contributions of the negative-parity pentaquark states from that of the positive-parity pentaquark states explicitly. They do not contaminate each other, and warrant robust predictions. Now we briefly outline the operator product expansion for the correlation functions

40 41

5

9

4m2c

j j j qL1 q2Hq3 ,μν (p)

4

8

where the subscript index H denotes the   hadron side (or phenomenological side). Then we introduce the weight function exp − Ts2 to obtain the phenomenological side of the QCD sum rules,  s0   s  MP2 ,− √ jL jH j,1 jL jH j,0 −2 ds sρq1 q2 q3 ,H (s) + ρq1 q2 q3 ,H (s) exp − 2 = 2MP ,− λP exp − 2 , (19) T T

21

3

7

    2 +2 2 2 δ s − M δ s − M =p / λ− + λ P ,− P ,+ P P      2 +2 2 2 + MP ,− λ− δ s − M λ δ s − M − M , P ,+ P ,− P ,+ P P 

=p / ρq1Lq2Hq3 ,H (s) + ρq1Lq2Hq3 ,H (s) ,

13

2

6

We can obtain the hadronic spectral densities through dispersion relation, j j j ImqL1 q2Hq3 (s)

1

Sij (x) =

x 2 ¯s g

iδij x/ δij ms δij ¯s s iδij x/ ms ¯s s δij s σ Gs − − + − 2 4 2 2 12 48 192 2π x 4π x a t a (/ αβ + σ αβ x 2 ig G x σ / ) s iδij x x/ ms ¯s gs σ Gs 1 αβ ij + − ¯sj σ μν si σμν + · · · , − 1152 8 32π 2 x 2 (22)

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–••• 1 2

9





gs Gnαβ tijn σ αβ (/k + mc ) + (/k + mc )σ αβ δij − k/ − mc 4 (k 2 − m2c )2  gs2 (t a t b )ij Gaαβ Gbμν (f αβμν + f αμβν + f αμνβ ) − + ··· , 4(k 2 − m2c )5

i Cij (x) = (2π)4

3 4 5

4

d ke

1

−ik·x

2 3 4 5

6 7 8 9 10 11 12 13 14 15

6

f αβμν = (/k + mc )γ α (/k + mc )γ β (/k + mc )γ μ (/k + mc )γ ν (/k + mc ) , and t n =

λn 2

, the λn is the Gell–Mann matrix [15], then we compute all the integrals in the 10 21 (p) sss,μν

at coordinate space and in the momentum space to obtain the correlation function the quark level. The calculations are straightforward but tedious. All the correlation functions j j j qL1 q2Hq3 ,μν (p) at the QCD side can be written as

j j j

j j j

qL1 q2Hq3 ,μν (p) = qL1 q2Hq3 ,QCD (p 2 ) −gμν +

16 17 18 19 20 21

23

π

26 27 28 29 30 31 32 33

(24)

36 37 38 39 40 41 42 43 44 45 46 47

8 9 10 11 12 13

15 16 17 18 19 20 21

j j j,1 j j j,0

q1Lq2Hq3 (s) . =p / ρq1Lq2Hq3 (s) + mc ρ j j j,1

(25) j j j,0

The lengthy expressions of the QCD spectral densities ρq1Lq2Hq3 (s) and ρ

q1Lq2Hq3 (s) are given explicitly in the Appendix. We carry out the operator product expansion to the vacuum condensates up to dimension 10, and assume vacuum saturation for the higher dimension vacuum condensates. The gluon condensates are accompanied with large numerical denominators, their contributions to the total QCD spectral densities are less (or much less) than the contributions of the dimension 10 vacuum condensates D10 [9,10], see Eq. (26), so we neglect the vacuum αs GG ¯ s s αs πGG , qq ¯ 2  αs πGG , condensates involving the gluon condensate  αs πGG , qq π , ¯ qq¯ ¯ s s αs πGG , ¯s s2  αs πGG , the predictive ability will not be impaired. In calculations, we take into account the contributions of the terms D0 , D3 , D5 , D6 , D8 , D9 and D10 , where

34 35

7

14

+ ··· ,

j j j

ImqL1 q2Hq3 ,QCD (s)

25

pμ pν p2



where we add the notation QCD to denote the QCD side. Once the analytical expressions of j j j the correlation functions qL1 q2Hq3 ,μν (p) are gotten, we can obtain the QCD spectral densities j j j,1 j j j,0

q1Lq2Hq3 (s) through dispersion relation, ρq1Lq2Hq3 (s) and ρ

22

24

(23)

22 23 24 25 26 27 28 29 30 31 32 33 34

D0 = perturbative

terms ,

35 36

¯ , ¯s s , D3 ∝ qq

37

¯ s σ Gq , ¯s gs σ Gs , D5 ∝ qg

38 39

D6 ∝ qq ¯ 2 , qq¯ ¯ s s , ¯s s2 ,

40

¯ qg ¯ s σ Gq , ¯s sqg ¯ s σ Gq , qq¯ ¯ s gs σ Gs , ¯s s¯s gs σ Gs , D8 ∝ qq

41 42

¯ 3 , qq¯ ¯ s s2 , qq ¯ 2 ¯s s , ¯s s3 , D9 ∝ qq

43

D10 ∝ qg ¯ s σ Gq , qg ¯ s σ Gq¯s gs σ Gs , ¯s gs σ Gs . 2

2

(26)

Now we can take the quark-hadron duality below the continuum thresholds s0 and obtain the following QCD sum rules:

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 1 2

2 2MP ,− λ− P exp

3 4 5 6

s0 = 4m2c



2 2MP ,+ λ+ P exp

9

s0

10 11

=

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

−

MP2 ,−

1

T2

2 3

√  s  j j j,1 j j j,0 ds sρq1Lq2Hq3 (s) + mc ρ

q1Lq2Hq3 (s) exp − 2 , T

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

10

MP2 ,+ − 2 T

4

(27)

5 6



7 8 9

√  s  j j j,1 j j j,0 ds sρq1Lq2Hq3 (s) − mc ρ

q1Lq2Hq3 (s) exp − 2 . T

10

(28)

11 12

4m2c

13

1 , T2

We differentiate Eqs. (27)–(28) with respect to then eliminate the pole residues obtain the QCD sum rules for the masses of the hidden-charm pentaquark states MP2 ,∓ ,  √   s0 j j j,1 j j j,0 d sρq1Lq2Hq3 (s) + mc ρ

q1Lq2Hq3 (s) exp − Ts2 2 ds d(−1/T 2 ) 4m c √   , MP2 ,− =  s0 jL jH j,1 jL jH j,0 s ds sρ (s) + m ρ

(s) exp − c q q q q q q 2 2 1 2 3 1 2 3 4mc T  √   s0 jL jH j,1 jL jH j,0 d s ds sρ (s) − m ρ

(s) exp − c q1 q2 q3 q1 q2 q3 4m2c d(−1/T 2 ) T2 √   MP2 ,+ = .  s0 jL jH j,1 jL jH j,0 s ds sρ (s) − m ρ

(s) exp − c q q q q q q 2 2 1 2 3 1 2 3 4m T

λ± P

and

15 16 17

(29)

20 21

(30)

We obtain the masses through fractions, see Eqs. (29)–(30), the contributions involving the gluon condensate in the numerator and denominator are canceled out with each other, the main effects of the gluon condensates can be safely absorbed into the pole residues λ± P and the remaining tiny effects on the masses can be safely neglected.

22 23 24 25 26 27 28 29

3. Numerical results and discussions

30

The input parameters are shown explicitly in Table 1. The quark condensates, mixed quark condensates and MS masses evolve with the renormalization group equation, we take into account the energy-scale dependence according to the following equations, 4

αs (Q) 9 , αs (μ)  4 αs (Q) 9 ¯s s(μ) = ¯s s(Q) , αs (μ)  2 αs (Q) 27 qg ¯ s σ Gq(μ) = qg ¯ s σ Gq(Q) , αs (μ)  2 αs (Q) 27 ¯s gs σ Gs(μ) = ¯s gs σ Gs(Q) , αs (μ)   12 αs (μ) 25 mc (μ) = mc (mc ) , αs (mc ) qq(μ) ¯ = qq(Q) ¯

18 19

c



14

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–••• 1

11

1

3

Table 1 The input parameters in the QCD sum rules, the values in the bracket denote the energy scales μ = 1 GeV, 2 GeV and mc , respectively.

4

Parameters

Values

4

5

qq ¯ (1 GeV) ¯s s (1 GeV) qg ¯ s σ Gq (1 GeV)

−(0.24 ± 0.01 GeV)3 [14–16] (0.8 ± 0.1)qq ¯ (1 GeV) [14–16] m20 qq ¯ (1 GeV) [14–16]

5

¯s gs σ Gs (1 GeV) m20 (1 GeV) mc (mc ) ms (2 GeV)

m20 ¯s s (1 GeV) [14–16] (0.8 ± 0.1) GeV2 [14–16] (1.275 ± 0.025) GeV [26] (0.095 ± 0.005) GeV [26]

2

6 7 8 9 10 11

2 3

6 7 8 9 10 11

12

12



13 14 15 16 17 18

4

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

13

9 αs (μ) ms (μ) = ms (2 GeV) , αs (2 GeV)   b12 (log2 t − log t − 1) + b0 b2 1 b1 log t αs (μ) = 1− 2 + , b0 t b0 t b04 t 2 2

33−2n

153−19n

14 15 16

(31)

2857− 5033 nf + 325 n2

μ f f 9 27 f where t = log  , b2 = ,  = 213 MeV, 2 , b0 = 12π , b1 = 24π 2 128π 3 296 MeV and 339 MeV for the flavors nf = 5, 4 and 3, respectively [26]. Furthermore, we set the mu = md = 0. In Refs. [25,27], we separate the contributions come from the positive parity and negative ± ± parity baryon states explicitly, and study the masses and pole residues of the J P = 12 and 32 heavy, doubly-heavy and triply-heavy baryon states with the QCD sum rules systematically. In √ calculations, we observe that the continuum threshold parameters s0 = Mgr + (0.6–0.8) GeV can reproduce the experimental values of the masses of the observed heavy baryon states [26], where the subscript gr denotes the ground state baryon states. The pentaquark states are another type baryon states considering the fractional spins, such as 12 , 32 , 52 . In Ref. [9], we take √ the continuum threshold parameters as s0 = MPc (4380/4450) + (0.6–0.8) GeV in the QCD sum rules for the scalar-diquark–axialvector-diquark–antiquark type hidden-charm pentaquark states, which can reproduce the experimental values of the masses MPc (4380/4450) . In the present QCD √ sum rules, we take the continuum threshold parameters as s0 = MP + (0.6 − 0.8) GeV. ¯ can be described by a The hidden-charm or hidden-bottom five-quark systems q1 q2 q3 QQ double-well potential in the heavy quark limit. The two light quarks q1 and q2 combine together ¯ to form a light diquark state or correlation Di (q1 q2 ) in color antitriplet, the heavy antiquark Q 3¯

serves as one static well potential and combines with the light diquark state D3i¯ (q1 q2 ) to form a j heavy antitriquark state or correlation εij k D3i¯ (q1 q2 )Q¯ 3¯ in color triplet, where the i, j and k are color indexes. On the other hand, the heavy quark Q serves as the other static well potential and combines with the light quark q3 to form a heavy diquark state or correlation D3k¯ (q3 Q) in color j antitriplet. The heavy diquark state Dk (q3 Q) and the heavy antitriquark state εij k Di (q1 q2 )Q¯ 3¯

3¯ 3¯ j εij k D3i¯ (q1 q2 )Q¯ 3¯ D3k¯ (q3 Q),

combine together to form a double heavy physical pentaquark state ¯ stabilize the five-quark systems or pentaquark states, just like the two heavy quarks Q and Q the double heavy four-quark systems or tetraquark states [21,28]. In this article, we construct the interpolating currents according to such routines, see Eqs. (2)–(13).

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

In the heavy quark limit, the masses of the hidden-charm pentaquark states can be written as

 1 MP = 2mc +  + O + ··· , (32) mc   where the  is a parameter of the order O m0c , and independent on the heavy flavor. The mc can be taken to be the pole mass according to the reparameterization invariance. The fitted value from the mass of the D-mesons in the heavy quark effective theory is about mc = 1.3 GeV [29], which is approximate to the MS mass mc (mc ) = (1.275 ± 0.025) GeV from the Particle Data Group [26]. The MS mass mc (mc ) relates with the pole mass mc through the relation [26],   4αs (mc ) mc = mc (mc ) 1 + + ··· . (33) 3π   Up to corrections of the order O αs3 , the MS mass mc (mc ) = (1.275 ± 0.025) GeV corresponds to the pole mass mc = (1.67 ± 0.07) GeV [26], which is quite different from the fitted value mc = 1.3 GeV [29]. The heavy quark mass mQ is just a parameter. Now we assume that there exists an effective heavy quark mass Mc or Mb , which is a free parameter, and is larger than the fitted value of the pole mass mc = 1.3 GeV or mb = 4.7   GeV [29] and differs from the MS mass mc (mc ) or mb (mb ), some effects of the order O m0c or  0 O mb are embodied in the Mc or Mb . Then the double heavy five-quark systems q1 q2 q3 QQ¯ are  characterized by the effective heavy quark masses MQ and the virtuality V = MP2 − (2MQ )2 , just like the double heavy four-quark systems q1 q¯2 QQ¯ [21]. The QCD sum rules for the hidden-charm or hidden-bottom pentaquark states q1 q2 q3 QQ¯ have three typical energy scales μ2 , T 2 , V 2 , we can set the energy scales to be μ2 = V 2 = O(T 2 ), and obtain an energy scale formula,  μ = MP2 − (2MQ )2 , (34)

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

to determine the ideal energy scales of the QCD spectral densities [9–11]. Once the pentaquark mass MP is specialized, the effective heavy quark mass MQ determines the energy scale μ, which determines the MS mass mQ (μ) therefore the pentaquark mass MP based on the QCD

sum rules in return. In this article, we take the energy scale formula μ = MP2 − (2Mc )2 as a powerful constraint to obey. In Ref. [9], we choose the value Mc = 1.8 GeV determined in the QCD sum rules for the hidden-charm tetraquark states [21] and obtain the ideal energy scales μ = 2.5 GeV and μ = 2.6 GeV for the pentaquark states Pc (4380) and Pc (4450), respectively. The 

empirical energy scale formula μ = − (2Mc works well for the hidden-charm tetraquark states and hidden-charm pentaquark states [9–11,21]. In the present QCD sum rules, we choose the Borel parameters T 2 and continuum threshold parameters s0 to obey the following four criteria: 1. Pole dominance at the phenomenological side; 2. Convergence of the operator product expansion; 3. Appearance of the Borel platforms; 4. Satisfying the energy scale formula. In the QCD sum rules for the three-quark baryon states qq  Q, qQQ , QQ Q [25,27], the predicted masses increase slowly with the increase of the Borel parameters, the Borel platforms do not appear at the minimum values and are not very flat, we determine the Borel windows by the 2 MX/Y/Z/P

)2

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

13

criteria 1 and 2, the two necessary constraints warrant that the extracted masses are reliable. The pentaquark states are another type baryon states considering the fractional spins, so their Borel platforms maybe not appear at the minimum values of the predicted masses. In this article, we 2 − T 2 = 0.4 GeV2 , just like in the case of the hidden-charm choose small Borel windows Tmax min tetraquark states [21] and hidden-charm pentaquark states [9–11], and obtain the Borel platforms P by requiring the uncertainties δM MP induced by the Borel parameters are about 1%, the criterion 3 is modified slightly and satisfied automatically. We search for the ideal Borel parameters T 2 and continuum threshold parameters s0 according to the four criteria 1, 2, 3 and 4 using try and error. The resulting Borel parameters or Borel windows T 2 , continuum threshold parameters s0 , ideal energy scales of the QCD spectral densities, pole contributions of the ground state pentaquark states, and contributions of the vacuum condensates of dimension 9 and 10 in the operator product expansion are shown explicitly in Table 2. From the table, we can see that the first two criteria of the QCD sum rules are satisfied, so we expect to make reasonable predictions. We take into account all uncertainties of the input parameters, and obtain the masses and pole ± residues of the J P = 32 hidden-charm pentaquark states, which are shown explicitly in Table 3. From Table 2 and Table 3, we can see that the criterion 4 is also satisfied. Now the four criteria of the QCD sum rules are all satisfied, and we expect to make reliable predictions. The present predictions of the hidden-charm pentaquark masses can be confronted to the experimental data in the future. In the isospin limit,

−

−

−

− 10 12 3 10 12 3 10 12 3 10 12 3 MP = 4.39 ± 0.13 GeV for Puuu , Puud , Pudd , Pddd , 2 2 2 2







 − − − − 11 1 3 11 1 3 11 1 3 11 1 3 MP = 4.39 ± 0.14 GeV for Puuu2 , Puud2 , Pudd2 , Pddd2 , 2 2 2 2







 − − − − 11 3 3 11 3 3 11 3 3 11 3 3 MP = 4.39 ± 0.14 GeV for Puuu2 , Puud2 , Pudd2 , Pddd2 , 2 2 2 2 (35)

1

which are all compatible with the experimental value MPc (4380) = 4380 ± 8 ± 29 MeV from the LHCb collaboration [1]. The Pc (4380) can be assigned to be the pentaquark state  −  −  − 10 1 11 1 11 3 Puud2 32 , Puud2 32 or Puud2 32 , or the Pc (4380) has some pentaquark components  −  −  − 10 1 11 1 11 3 Puud2 32 , Puud2 32 and Puud2 32 . From Eq. (35), we can see that the pentaquark  −  11 1  −   − 10 1 11 3 states Puud2 32 , Puud2 32 and Puud2 32 have almost degenerate masses. All the pen      10 12 3 − 11 12 3 − 11 32 3 − taquark states Puud 2 , Puud 2 and Puud 2 have the light axialvector diquark

31

state ε ij k uTj Cγμ uk or ε ij k uTj Cγμ dk , moreover, the heavy-light scalar and axialvector diquark states have almost degenerate masses, so the axialvector-diquark–scalar-diquark–antiquark type hidden-charm pentaquark states and the axialvector-diquark–axialvector-diquark–antiquark type hidden-charm pentaquark states maybe have degenerate masses. The mass alone cannot identify the Pc (4380) unambiguously, more experimental data on its productions and decays are still needed. On the theoretical side, we can study the two-body strong decays Pc (4380) → J /ψp with the three-point QCD sum rules or the light-cone QCD sum rules, which maybe give additional support for such assignment.

40

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

32 33 34 35 36 37 38 39

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[m1+; v1.236; Prn:26/09/2016; 12:53] P.14 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

Table 2 The Borel parameters (Borel windows), continuum threshold parameters, ideal energy scales, pole contributions, contributions of the vacuum condensates of dimension 9 and dimension 10 for the hidden-charm pentraquark states. √ T 2 (GeV2 ) s0 (GeV) μ ( eV) Pole D9 D10 10 12  3 −  3.3–3.7 5.1 ± 0.1 2.5 (41–62)% (11–15)% (3–4)% Puuu 2 10 12  3 −  3.5–3.9 5.2 ± 0.1 2.7 (40–60)% (7–9)% (2–3)% Puus 2 10 12  3 −  3.6–4.0 5.3 ± 0.1 2.8 (41–61)% (5–6)% (1–2)% Puss 2 10 12  3 −  3.8–4.2 5.4 ± 0.1 3.0 (40–59)% (3–4)% ∼1% Psss 2 11 12  3 −  3.2–3.6 5.1 ± 0.1 2.5 (42–64)% (12–17)% (3–5)% Puuu 2 11 12  3 −  3.4–3.8 5.2 ± 0.1 2.7 (41–61)% (7–10)% (2–3)% Puus 2 11 12  3 −  3.6–4.0 5.3 ± 0.1 2.9 (40–59)% (4–6)% (1–2)% Puss 2 11 12  3 −  3.7–4.1 5.4 ± 0.1 3.0 (41–60)% (3–4)% (1–2)% Psss 2 11 32  3 −  3.2–3.6 5.1 ± 0.1 2.5 (41–63)% (12–17)% (2–3)% Puuu 2 11 32  3 −  3.4–3.8 5.2 ± 0.1 2.7 (40–60)% (7–10)% (1–2)% Puus 2 11 32  3 −  3.5–3.9 5.3 ± 0.1 2.9 (41–61)% (5–7)% ∼1% Puss 2 11 32  3 −  3.7–4.1 5.4 ± 0.1 3.0 (40–59)% (3–4)% ∼1% Psss 2 10 12  3 +  3.2–3.6 5.2 ± 0.1 2.7 (41–62)% (13–18)% (3–5)% Puuu 2 10 12  3 +  3.3–3.7 5.3 ± 0.1 2.9 (42–63)% (9–12)% (2–4)% Puus 2 10 12  3 +  3.5–3.9 5.4 ± 0.1 3.0 (40–60)% (6–8)% ∼2% Puss 2 10 12  3 +  3.6–4.0 5.5 ± 0.1 3.2 (41–61)% (4–5)% (1–2)% Psss 2 11 12  3 +  3.4–3.8 5.6 ± 0.1 3.3 (50–69)% −(4–6)% (3–4)% Puuu 2 11 12  3 +  3.6–4.0 5.7 ± 0.1 3.5 (48–68)% −(2–4)% (2–3)% Puus 2 11 12  3 +  3.8–4.2 5.8 ± 0.1 3.6 (47–66)% −(1–2)% (1–2)% Puss 2 11 12  3 +  4.0–4.4 5.9 ± 0.1 3.7 (46–64)% −(1–2)% ∼1% Psss 2 11 32  3 +  3.3–3.7 5.8 ± 0.1 3.7 (60–78)% −(3–6)% ∼−1% Puuu 2 11 32  3 +  3.5–3.9 5.9 ± 0.1 3.8 (59–77)% −(2–3)% ∼−1% Puus 2 11 32  3 +  Puss 2 3.7–4.1 6.0 ± 0.1 3.9 (58–75)% −(1–2)% ∼−0% 11 32  3 +  3.9–4.3 6.1 ± 0.1 4.0 (57–74)% −1% ∼−0% Psss 2

41

44 45

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

42

The 0b can be well interpolated by the current J (x) = ε ij k uTi (x)Cγ5 dj (x)bk (x) [27], the u and d quarks in the 0b form a scalar diquark [ud] (or ε ij k uTi Cγ5 dj ) in color antitriplet, the decays 0b → J /ψpK − take place through the following mechanism,

46 47

2

41

42 43

1

43 44 45 46

0b ([ud]b) → [ud]ccs ¯ → [ud]ccu ¯ us ¯ → Pc+ ([ud][uc]c)K ¯ − (us) ¯ → J /ψpK − ,

(36)

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Table 3 The masses and pole residues of the hidden-charm pentaquark states.

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15

10 1 P 2 uuu

 − 3 2

10 1  −  Puus2 32 10 1  −  Puss2 32 10 1  −  Psss 2 32 11 1  −  Puuu2 32 11 1  −  Puus2 32 11 1  −  Puss2 32 11 1  −  Psss 2 32 11 3  −  Puuu2 32 11 3  −  Puus2 32 11 3  −  Puss2 32 11 3  −  Psss 2 32 10 1  +  Puuu2 32 10 1  +  Puus2 32 10 1  +  Puss2 32 10 1  +  Psss 2 32 11 1  +  Puuu2 32 11 1  +  Puus2 32 11 1  +  Puss2 32 11 1  +  Psss 2 32 11 3  +  Puuu2 32 11 3  +  Puus2 32 11 3  +  Puss2 32 11 3  +  Psss 2 32

1 2

MP ( GeV)

λP (10−3 GeV6 )

4.39 ± 0.13

2.25 ± 0.40

4.51 ± 0.12

2.75 ± 0.45

4.60 ± 0.11

3.19 ± 0.50

4.70 ± 0.11

3.78 ± 0.57

4.39 ± 0.14

3.75 ± 0.68

4.51 ± 0.12

4.64 ± 0.77

4.62 ± 0.11

5.60 ± 0.88

4.71 ± 0.11

6.47 ± 1.00

4.39 ± 0.14

3.74 ± 0.70

4.52 ± 0.12

4.64 ± 0.79

4.61 ± 0.12

5.52 ± 0.90

4.72 ± 0.11

6.52 ± 1.02

4.51 ± 0.13

0.99 ± 0.19

4.60 ± 0.12

1.21 ± 0.22

4.72 ± 0.11

1.45 ± 0.25

4.81 ± 0.12

1.71 ± 0.29

27

4.91 ± 0.09

5.04 ± 0.68

28

4.99 ± 0.09

5.83 ± 0.82

30

5.08 ± 0.09

6.72 ± 0.98

31

5.18 ± 0.09

7.73 ± 1.10

5.14 ± 0.08

7.86 ± 0.91

5.21 ± 0.08

8.79 ± 1.07

5.28 ± 0.08

9.91 ± 1.25

5.36 ± 0.08

11.20 ± 1.42

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

29

32 33 34 35 36 37 38 39

40

40

41

41

42 43 44 45 46 47

42

at the quark level. In the decays Pc+ ([ud][uc]c) ¯ → J /ψp, the scalar diquark [ud] survives in the decay chains, the decays are greatly facilitated and can take place easily. On the other hand, if there exists a light axialvector diquark {ud} (or εij k uTi Cγμ dj ), which has to dissolve to form a scalar diquark [ud], the decays are not facilitated and cannot take place easily, but the decays are not forbidden. It is also possible to assign the Pc (4380) to be the axialvector-

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Fig. 1. The masses of the pentaquark states with variations of the Borel parameters T 2 , where the A, B, C, D, E and 10 1  −  10 1  +  11 1  −  11 1  +  11 3  −  11 3  +  and Puud2 32 F denote the pentaquark states Puud2 32 , Puud2 32 , Puud2 32 , Puud2 32 , Puud2 32 respectively, the horizontal lines denote the experimental value of the mass of the Pc (4380).

40 41 42 43 44 45 46 47

37 38 39 40





10 1 − diquark–scalar-diquark–antiquark type pentaquark state Puud2 32 or the axialvector-diquark–   − 1  11 11 3 − axialvector-diquark–antiquark type pentaquark states Puud2 32 and Puud2 32 .  −  10 1  +  11 1  −  10 1 In Fig. 1, we plot the masses of the pentaquark states Puud2 32 , Puud2 32 , Puud2 32 ,  +  11 3  −   + 11 1 11 3 Puud2 32 , Puud2 32 and Puud2 32 with variations of the Borel parameters T 2 as an exam-

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17

  jL jH j 3 − ple. In this article, we take the isospin limit, the masses of the pentaquark states Puud and 2  − jL jH j 3 degenerate. From the figure, we can see that the platforms in the Borel windows Puuu 2 shown in Table 2 appear at the minimum values for the axialvector-diquark–axialvector-diquark– antiquark type pentaquark states with positive parity, and the Borel platforms are very flat; other P platforms in the Borel windows shown Table 2 are not as flat, the uncertainties δM MP induced by the Borel parameters in the Borel windows are about 1%. In Fig. 1, we plot the masses with variations of the Borel parameters at a large interval T 2 = (2.6–4.6) GeV2 , which is much larger than the interval of the Borel windows shown in Table 2. Compared to the values shown in Table 3, the masses shown in Fig. 1 have larger uncertainties due to the larger interval of the Borel parameters. Furthermore, from the values in Table 3, we can see that the mass gaps between the pentaquark  +  −  +  − 11 1 11 1 11 3 11 3 states with opposite parity Pq1 q22 q3 32 − Pq1 q22 q3 32 and Pq1 q22 q3 32 − Pq1 q22 q3 32 are about 0.5 GeV and 0.7 GeV respectively, which is much larger than the mass gap 0.1 GeV be +  − 10 1 10 1 tween the pentaquark states with opposite parity Pq1 q22 q3 32 − Pq1 q22 q3 32 . An additional P-wave costs about 0.5 GeV for the conventional baryon states, the mass spectra of the pentaquark states have new feature.

18 19

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

4. Conclusion In this article, we construct the axialvector-diquark–scalar-diquark–antiquark type and the ± axialvector-diquark–axialvector-diquark–antiquark type currents to interpolate the J P = 32 hidden-charm pentaquark states, study the masses and pole residues with the QCD sum rules systematically by calculating the contributions of the vacuum condensates up to dimension 10 in the operator product expansion. We obtain the masses of the hidden-charm pentaquark states with the strangeness S = 0, −1, −2, −3, respectively, which can be confronted to the experimental data in the future. As a byproduct, the predicted pole residues can be taken as basic input parameters in studying the strong decays of the hidden-charm pentaquark states with the threepoint QCD sum rules or the  light-cone QCD sum rules. In calculations, we use the empirical

2 − (2Mc )2 with the effective heavy quark mass Mc to deenergy scale formula μ = MX/Y/Z/P termine the ideal energy scales of the QCD spectral densities, which works well in studying the hidden-charm tetraquark states and hidden-charm pentaquark states. The predicted masses of the  −  11 1  −   − 10 1 11 3 and Puud2 32 are compatible with the experimental pentaquark states Puud2 32 , Puud2 32 value MPc (4380) = 4380 ± 8 ± 29 MeV from the LHCb collaboration. We can draw the conclusion  −  11 1  −  10 1 tentatively that the Pc (4380) can be assigned to be the pentaquark state Puud2 32 , Puud2 32  −  −  11 1  −  11 3 10 1 or Puud2 32 , or the Pc (4380) has some pentaquark components Puud2 32 , Puud2 32 and  3  11 − Puud2 32 . More experimental data on its productions and decays are still needed to identify the Pc (4380) unambiguously.

47

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Acknowledgements

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This work is supported by National Natural Science Foundation of China, Grant Number 11375063, and Natural Science Foundation of Hebei Province, Grant Number A2014502017.

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Appendix A

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

2

11 3 ,1 11 3 ,0 11 3 ,1 11 3 ,0 11 3 ,1 11 3 ,0 The QCD spectral densities ρsss2 (s), ρ

sss2 (s), ρuss2 (s), ρ

uss2 (s), ρuus2 (s), ρ

uus2 (s), 11 3 ,1 11 3 ,0 11 1 ,1 11 1 ,0 11 1 ,1 11 1 ,0 11 1 ,1 11 1 ,0 11 1 ,1 ρuuu2 (s), ρ

uuu2 (s), ρsss2 (s), ρ

sss2 (s), ρuss2 (s), ρ

uss2 (s), ρuus2 (s), ρ

uus2 (s), ρuuu2 (s), 11 1 ,0 10 1 ,1 10 1 ,0 10 1 ,1 10 1 ,0 10 1 ,1 10 1 ,0 10 1 ,1

sss2 (s), ρuss2 (s), ρ

uss2 (s), ρuus2 (s), ρ

uus2 (s), ρuuu2 (s) and ρ

uuu2 (s), ρsss2 (s), ρ 10 1 ,0 ρ

uuu2 (s) of the hidden-charm pentaquark states, 11 32 ,1

ρsss



(s) =



4 



1 dydz yz(1 − y − z)4 s − m2c 7s − 2m2c 122880π 8   3   1 5 2 2 2 4 dydz yz(1 − y − z) + m − 51sm + 8m s − 63s c c c 614400π 8   3 mc ¯s s 2 2 dydz (y + z)(1 − y − z) m s − − c 768π 6   2   mc ¯s s 3 2 2 dydz (y + z)(1 − y − z) − m m s − 2s − c c 768π 6    2 11mc ¯s gs σ Gs dydz (y + z)(1 − y − z) s − m2c + 6 4096π     11mc ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) + m s − 5s − 3m c c 8192π 6     ¯s s2 2 2 dydz yz(1 − y − z) s − m m 2s − + c c 24π 4    19¯s s¯s gs σ Gs dydz yz 3s − 2m2c − 768π 4    ¯s s¯s gs σ Gs 2 dydz (y + z)(1 − y − z) 5s − 4m + c 1536π 4  mc ¯s s3 dy − 36π 2    s  13¯s gs σ Gs2 2 dydz (y + z) 1 + − m δ s − c 4 13824π 4    s  11¯s gs σ Gs2 dy y(1 − y) 1 + δ s − m

2c + 4 2 1536π   4 ms mc 3 2 dydz (y + z)(1 − y − z) m s − + c 12288π 8   3   ms mc 4 2 2 dydz (y + z)(1 − y − z) + m s − 7s − 3m c c 49152π 8      2 ms ¯s s dydz yz(1 − y − z)2 s − m2c 5s − 2m2c − 768π 6     ms ¯s s 3 2 2 2 4 dydz yz(1 − y − z) + m − 37sm + 8m s − 35s c c c 768π 6     11ms ¯s gs σ Gs dydz yz(1 − y − z) s − m2c 2s − m2c + 6 1024π

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−

1 2 3

−

4 5

+

6 7 8

−

9 10

+

11 12

+

13 14 15

−

16 17

+

18 19

−

20 21 22

+

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

11 3 ,0 ρ

sss2 (s) =

   ms ¯s gs σ Gs 2 2 2 4 dydz yz(1 − y − z) − 15sm + 4m 12s c c 256π 6      ms ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) m s − 3s − 2m c c 4096π 6    ms mc ¯s s2 2 dydz (y + z) s − m c 24π 4    ms mc ¯s s2 dydz (y + z)(1 − y − z) 4s − 3m2c 4 48π    s  89ms mc ¯s s¯s gs σ Gs 2 dydz (y + z) 1 + m δ s − c 3 3072π 4

  y ms mc ¯s s¯s gs σ Gs z dydz + z y 768π 4  199ms mc ¯s s¯s gs σ Gs dy 9216π 4    ms ¯s s3 s  2 dy y(1 − y) 1 + δ s − m

c 2 18π 2

    2 1−y ms mc ¯s gs σ Gs y 2 dy + δ s − m

c y 1−y 3072π 4    s   17ms mc ¯s gs σ Gs2 2 dy 1 + δ s − m

, c 4608π 4 2T 2 

 4   1 4 2 2 dydz (y + z)(1 − y − z) m m s − 6s − c c 122880π 8   3 1 5 2 dydz (y + z)(1 − y − z) m s − − c 1228800π 8   × 48s 2 − 31sm2c + 3m4c   3 mc ¯s s 2 2 dydz (1 − y − z) − m s − c 192π 6   2   mc ¯s s 3 2 2 dydz (1 − y − z) + m s − 5s − 2m c c 1152π 6    2 11mc ¯s gs σ Gs dydz (1 − y − z) s − m2c + 6 1024π     11mc ¯s gs σ Gs 2 2 2 dydz (1 − y − z) − m m s − 2s − c c 2048π 6     ¯s s2 2 2 dydz (y + z)(1 − y − z) s − + m m 3s − c c 48π 4    19¯s s¯s gs σ Gs dydz (y + z) 2s − m2c − 768π 4  mc ¯s s3 dy − 9π 2

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

(37)

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20

11 3 ,1 ρuss2 (s) =

    11¯s gs σ Gs2 dy 1 + s δ s − m

2c 4 3072π   4 ms mc 3 2 dydz (1 − y − z) m s − c 3072π 8   3   ms mc 4 2 2 dydz (1 − y − z) m m s − 3s − c c 12288π 8      2 ms ¯s s 2 2 2 dydz (y + z)(1 − y − z) m m s − 4s − c c 768π 6     ms ¯s s 3 2 2 2 4 dydz (y + z)(1 − y − z) m − 7sm + m s − 8s c c c 512π 6     11ms ¯s gs σ Gs 2 2 dydz (y + z)(1 − y − z) s − m m 3s − c c 2048π 6    ms ¯s gs σ Gs 2 2 2 4 dydz (y + z)(1 − y − z) − 16sm + 3m 15s c c 1024π 6    ms mc ¯s s2 dydz s − m2c 4 6π    ms mc ¯s s2 2 dydz (1 − y − z) 3s − 2m c 24π 4    89ms mc ¯s s¯s gs σ Gs s  dydz 1 + δ s − m2c 4 2 2304π  199ms mc ¯s s¯s gs σ Gs dy 2304π 4     ms ¯s s3 2 dy 1 + s δ s − m

c 36π 2    s   125ms mc ¯s gs σ Gs2 2 dy 1 + δ s − m

, c 9216π 4 T2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

(38)

  4   1 4 2 2 dydz yz(1 − y − z) m s − 7s − 2m c c 122880π 8   3   1 dydz yz(1 − y − z)5 s − m2c + 63s 2 − 51sm2c + 8m4c 8 614400π    3 ¯ + 2¯s s mc qq 2 2 dydz (y + z)(1 − y − z) m s − − c 2304π 6    2   mc qq ¯ + 2¯s s 3 2 2 dydz (y + z)(1 − y − z) − m m s − 2s − c c 2304π 6    2 ¯ s σ Gq + 2¯s gs σ Gs 11mc qg 2 dydz (y + z)(1 − y − z) s − m + c 12288π 6   11mc qg ¯ s σ Gq + 2¯s gs σ Gs + 24576π 6     × dydz (y + z)(1 − y − z)2 s − m2c 5s − 3m2c

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     ¯s s 2qq ¯ + ¯s s 2 2 dydz yz(1 − y − z) s − m m 2s − c c 72π 4   19 ¯s s¯s gs σ Gs + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 2304π 4    × dydz yz 3s − 2m2c +

¯ s gs σ Gs + ¯s sqg ¯ s σ Gq ¯s s¯s gs σ Gs + qq¯ + 4608π 4     m qq¯ s s2 c ¯ dy × dydz (y + z)(1 − y − z) 5s − 4m2c − 36π 2   13¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs − 41472π 4    s  × dydz (y + z) 1 + δ s − m2c 4   ¯ s σ Gq + ¯s gs σ Gs 11¯s gs σ Gs 2qg + 4608π 4    s  × dy y(1 − y) 1 + δ s − m

2c 2   4 ms mc 3 2 dydz (y + z)(1 − y − z) m s − + c 18432π 8   3   ms mc 4 2 2 dydz (y + z)(1 − y − z) + m s − 7s − 3m c c 73728π 8    2   ms 2qq ¯ − ¯s s 2 2 2 dydz yz(1 − y − z) − m s − 5s − 2m c c 1152π 6     ms ¯s s dydz yz(1 − y − z)3 s − m2c 35s 2 − 37sm2c + 8m4c + 6 1152π    ms ¯s gs σ Gs 2 2 2 4 dydz yz(1 − y − z) − 15sm + 4m 12s − c c 384π 6   ms qg ¯ s σ Gq + ¯s gs σ Gs − 12288π 6     × dydz (y + z)(1 − y − z)2 s − m2c 3s − 2m2c   ¯ s σ Gq + 3¯s gs σ Gs ms 19qg + 3072π 6     × dydz yz(1 − y − z) s − m2c 2s − m2c     ms mc ¯s s 5qq ¯ − ¯s s 2 dydz (y + z) s − + m c 144π 4     ms mc ¯s s qq ¯ + ¯s s dydz (y + z)(1 − y − z) 4s − 3m2c − 4 144π   ms mc 89¯s s¯s gs σ Gs + 57¯s sqg ¯ s σ Gq + 32qq¯ ¯ s gs σ Gs + 9216π 4

21

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  s  dydz (y + z) 1 + δ s − m2c 3   ms mc 231¯s sqg ¯ s σ Gq − 89¯s s¯s gs σ Gs + 256qq¯ ¯ s gs σ Gs dy − 27648π 4  

 ms mc qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq y z dydz + + z y 2304π 4     2 s ms qq¯ ¯ s s dy y(1 − y) 1 + δ s − m +

2c 2 2 27π

    1−y ¯ s σ Gq¯s gs σ Gs y ms mc qg 2 dy + δ s − m

− c y 1−y 4608π 4   ms mc ¯s gs σ Gs 19¯s gs σ Gs − 53qg ¯ s σ Gq − 13824π 4      s 2 × dy 1 + δ s − m

, (39) c 2T 2 ×

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11 3 ,0 ρ

uss2 (s) =

  4   1 1 4 2 2 dydz (y + z)(1 − y − z) m m s − 6s − c c − 8 122880π 1228800π 8   3   × dydz (y + z)(1 − y − z)5 s − m2c 48s 2 − 31sm2c + 3m4c    3 mc qq ¯ + 2¯s s 2 2 dydz (1 − y − z) − m s − c 576π 6    2   mc qq ¯ + 2¯s s dydz (1 − y − z)3 s − m2c + 5s − 2m2c 6 3456π    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz (1 − y − z) s − + m c 3072π 6   11mc qg ¯ s σ Gq + 2¯s gs σ Gs − 6144π 6     × dydz (1 − y − z)2 s − m2c 2s − m2c      ¯s s 2qq ¯ + ¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 3s − c c 144π 4   ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq 19 ¯s s¯s gs σ Gs + qq¯ − 2304π 4     m qq¯ s s2 c ¯ 2 dy × dydz (y + z) 2s − mc − 9π 2      11¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs 2 + dy 1 + s δ s − m

c 9216π 4    4 ms mc dydz (1 − y − z)3 s − m2c + 4608π 8   3   ms mc dydz (1 − y − z)4 s − m2c − 3s − m2c 8 18432π

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   2   ms 2qq ¯ − ¯s s 2 2 2 dydz (y + z)(1 − y − z) m m s − 4s − c c 1152π 6     ms ¯s s 3 2 2 2 4 dydz (y + z)(1 − y − z) − m − 7sm + m s − 8s c c c 768π 6    ms ¯s gs σ Gs 2 2 2 4 dydz (y + z)(1 − y − z) + − 16sm + 3m 15s c c 1536π 6   ms 19qg ¯ s σ Gq + 3¯s gs σ Gs + 6144π 6     × dydz (y + z)(1 − y − z) s − m2c 3s − m2c     ms mc ¯s s 5qq ¯ − ¯s s 2 + dydz s − m c 36π 4     ms mc ¯s s qq ¯ + ¯s s 2 + dydz (1 − y − z) 3s − 2m c 72π 4   ms mc 89¯s s¯s gs σ Gs + 57¯s sqg ¯ s σ Gq + 32qq¯ ¯ s gs σ Gs − 6912π 4    s  × dydz 1 + δ s − m2c 2   ms mc 231¯s sqg ¯ s σ Gq − 89¯s s¯s gs σ Gs + 256qq¯ ¯ s gs σ Gs − dy 6912π 4     ms qq¯ ¯ s s2 2 + dy 1 + s δ s − m

c 54π 2   ms mc ¯s gs σ Gs 19¯s gs σ Gs − 269qg ¯ s σ Gq − 27648π 4    s   × dy 1 + 2 δ s − m

2c , (40) T

1

−

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23

11 3 ,1 ρuus2 (s) =

  4   1 4 2 2 dydz yz(1 − y − z) m s − 7s − 2m c c 122880π 8      3 1 5 2 2 2 4 dydz yz(1 − y − z) + m − 51sm + 8m s − 63s c c c 614400π 8    3 mc 2qq ¯ + ¯s s 2 2 − dydz (y + z)(1 − y − z) m s − c 2304π 6    2   ¯ + ¯s s mc 2qq 3 2 2 dydz (y + z)(1 − y − z) − m m s − 2s − c c 2304π 6    2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 + dydz (y + z)(1 − y − z) s − m c 12288π 6   11mc 2qg ¯ s σ Gq + ¯s gs σ Gs + 24576π 6     × dydz (y + z)(1 − y − z)2 s − m2c 5s − 3m2c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.236; Prn:26/09/2016; 12:53] P.24 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

24

     qq ¯ qq ¯ + 2¯s s 2 2 dydz yz(1 − y − z) s − m m 2s − c c 72π 4   19 qq ¯ qg ¯ s σ Gq + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 2304π 4    × dydz yz 3s − 2m2c +

¯ s gs σ Gs + ¯s sqg ¯ s σ Gq qq ¯ qg ¯ s σ Gq + qq¯ + 4 4608π    × dydz (y + z)(1 − y − z) 5s − 4m2c   13qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs − 41472π 4    s  × dydz (y + z) 1 + δ s − m2c 4   11qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs + 4608π 4     m qq 2 ¯ s s s  c ¯ 2 dy × dy y(1 − y) 1 + δ s − m

c − 2 36π 2   4 ms mc + dydz (y + z)(1 − y − z)3 s − m2c 8 36864π   3   ms mc 4 2 2 dydz (y + z)(1 − y − z) + m s − 7s − 3m c c 147456π 8       2 ¯ − 3¯s s ms 4qq 2 2 2 dydz yz(1 − y − z) − m s − 5s − 2m c c 2304π 6      ms ¯s s 3 2 2 2 4 dydz yz(1 − y − z) + m − 37sm + 8m s − 35s c c c 2304π 6   ¯ s σ Gq − 8¯s gs σ Gs ms 19qg + 3072π 6     × dydz yz(1 − y − z) s − m2c 2s − m2c    ms ¯s gs σ Gs 2 2 2 4 dydz yz(1 − y − z) − 15sm + 4m − 12s c c 768π 6      ¯ s σ Gq ms qg 2 2 2 dydz (y + z)(1 − y − z) m − s − 3s − 2m c c 12288π 6     ms mc qq ¯ 3qq ¯ − ¯s s 2 dydz (y + z) s − + m c 144π 4    ms mc qq¯ ¯ s s 2 dydz (y + z)(1 − y − z) 4s − 3m − c 144π 4   ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq ms mc 32qq¯ + 9216π 4    s  × dydz (y + z) 1 + δ s − m2c 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.25 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

  ms mc 288qq ¯ qg ¯ s σ Gq − 57¯s sqg ¯ s σ Gq − 32qq¯ ¯ s gs σ Gs dy 27648π 4

  ms mc qq y ¯ qg ¯ s σ Gq z + dydz + z y 2304π 4    2 s  ¯ ¯s s ms qq 2 dy y(1 − y) 1 + + δ s − m

c 2 54π 2

    2 ms mc qg 1−y ¯ s σ Gq y 2 − dy + δ s − m

c y 1−y 9216π 4   ms mc qg ¯ s σ Gq 36qg ¯ s σ Gq − 19¯s gs σ Gs + 4 13824π    s   2 × dy 1 + δ s − m

(41) c , 2T 2

1

−

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

25

11 3 ,0 ρ

uus2 (s) =

  4   1 4 2 2 dydz (y + z)(1 − y − z) m m s − 6s − c c 122880π 8   3 1 5 2 dydz (y + z)(1 − y − z) − m s − c 1228800π 8   × 48s 2 − 31sm2c + 3m4c    3 mc 2qq ¯ + ¯s s 2 2 dydz (1 − y − z) m s − − c 576π 6    2   mc 2qq ¯ + ¯s s 3 2 2 dydz (1 − y − z) + m s − 5s − 2m c c 3456π 6    2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 dydz (1 − y − z) s − + m c 3072π 6   ¯ s σ Gq + ¯s gs σ Gs 11mc 2qg − 6144π 6     × dydz (1 − y − z)2 s − m2c 2s − m2c      qq ¯ qq ¯ + 2¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 3s − c c 144π 4   19 qq ¯ qg ¯ s σ Gq + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 2304π 4     m qq 2 ¯ s s c ¯ dy × dydz (y + z) 2s − m2c − 9π 2      11qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs 2 + dy 1 + s δ s − m

c 9216π 4   4 ms mc 3 2 dydz (1 − y − z) m s − + c 9216π 8   3   ms mc dydz (1 − y − z)4 s − m2c − 3s − m2c 8 36864π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.26 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

26

   2   ms 4qq ¯ − 3¯s s 2 2 2 dydz (y + z)(1 − y − z) m m s − 4s − c c 2304π 6     ms ¯s s 3 2 2 2 4 dydz (y + z)(1 − y − z) m − 7sm + m s − 8s − c c c 1536π 6   ms 19qg ¯ s σ Gq − 8¯s gs σ Gs + 6144π 6     × dydz (y + z)(1 − y − z) s − m2c 3s − m2c    ms ¯s gs σ Gs 2 2 2 4 dydz (y + z)(1 − y − z) + − 16sm + 3m 15s c c 3072π 6     ms mc qq ¯ 3qq ¯ − ¯s s dydz s − m2c + 36π 4    ¯ s s ms mc qq¯ 2 dydz (1 − y − z) 3s − 2m + c 72π 4   ms mc 32qq¯ ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq − 6912π 4    s  × dydz 1 + δ s − m2c 2   ms mc 288qq ¯ qg ¯ s σ Gq − 57¯s sqg ¯ s σ Gq − 32qq¯ ¯ s gs σ Gs dy − 6912π 4     ms qq ¯ 2 ¯s s 2 + dy 1 + s δ s − m

c 108π 2   ms mc qg ¯ s σ Gq 144qg ¯ s σ Gq − 19¯s gs σ Gs + 27648π 4      s × dy 1 + 2 δ s − m

2c , (42) T

1

−

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

11 3 ,1 11 3 ,1 ρuuu2 (s) = ρsss2 (s) |ms →0, ¯s s→qq, ¯ ¯s gs σ Gs→qg ¯ s σ Gq ,

(43)

11 3 ,0 11 3 ,0 ρ

uuu2 (s) = ρ

sss2 (s) |ms →0, ¯s s→qq, ¯ ¯s gs σ Gs→qg ¯ s σ Gq ,

(44)

11 12 ,1

ρsss



(s) =



4 

31 32 33 34



1 dydz yz(1 − y − z)4 s − m2c 8s − 3m2c 61440π 8   4   1 5 2 2 dydz yz(1 − y − z) − m s − 9s − 4m c c 307200π 8    3 mc ¯s s dydz (y + z)(1 − y − z)2 s − m2c − 768π 6   3 mc ¯s s 3 2 dydz (y + z)(1 − y − z) + m s − c 2304π 6   2 11mc ¯s gs σ Gs 2 dydz (y + z)(1 − y − z) s − + m c 4096π 6   2 11mc ¯s gs σ Gs dydz (y + z)(1 − y − z)2 s − m2c − 6 8192π

35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.27 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–••• 1

+

2 3

−

4 5

+

6 7 8

+

9 10

−

11 12

−

13 14

−

15 16 17

+

18 19

+

20 21

+

22 23 24

+

25 26

−

27 28

−

29 30 31

+

32 33

+

34 35

+

36 37 38

−

39 40

−

41 42 43 44 45 46 47

11 1 ,0 ρ

sss2 (s) =

    ¯s s2 dydz yz(1 − y − z) s − m2c 5s − 3m2c 4 24π    m ¯s s3  19¯s s¯s gs σ Gs c 2 dydz yz 4s − 3mc − dy 384π 4 36π 2    s  11¯s gs σ Gs2 2 dy y(1 − y) 1 + δ s − m

c 3 512π 4   4 ms mc 3 2 dydz (y + z)(1 − y − z) m s − c 12288π 8   4 ms mc dydz (y + z)(1 − y − z)4 s − m2c 8 49152π   2   ms ¯s s 2 2 2 dydz yz(1 − y − z) m m s − 2s − c c 128π 6      2 ms ¯s s 3 2 2 dydz yz(1 − y − z) m s − 7s − 4m c c 384π 6     11ms ¯s gs σ Gs 2 2 dydz yz(1 − y − z) s − m 5s − 3m c c 1024π 6     ms ¯s gs σ Gs dydz yz(1 − y − z)2 s − m2c 3s − 2m2c 6 128π    ms mc ¯s s2 2 dydz (y + z) s − m c 24π 4    ms mc ¯s s2 2 dydz (y + z)(1 − y − z) s − m c 48π 4  89ms mc ¯s s¯s gs σ Gs dydz (y + z) 9216π 4  199ms mc ¯s s¯s gs σ Gs dy 9216π 4

  y ms mc ¯s s¯s gs σ Gs z dydz + z y 768π 4    3 s  ms ¯s s 2 dy y(1 − y) 1 + δ s − m

c 3 6π 2

    2 7s ms mc ¯s gs σ Gs 2 dy 16 + δ s − m

c 2304π 4 T2

    1−y ms mc ¯s gs σ Gs2 y dy + δ s −m

2c 4 y 1−y 3072π      2 s ms mc ¯s gs σ Gs 2 dy 1 + δ s − m

, c 6144π 4 T2

  4   1 4 2 2 dydz (y + z)(1 − y − z) m s − 7s − 2m c c 245760π 8   4   1 dydz (y + z)(1 − y − z)5 s − m2c − 8s − 3m2c 8 1228800π

27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

(45)

41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

1

−

2 3 4 5 6

+ +

7 8

−

9 10

+

11 12 13 14 15

− +

16 17

+

18 19 20 21 22

− +

23 24

−

25 26

−

27 28 29 30 31 32 33

− + +

34 35

−

36 37 38 39 40

+ +

41 42

−

43 44 45 46 47

[m1+; v1.236; Prn:26/09/2016; 12:53] P.28 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

28

+ +

  3 mc ¯s s 2 2 dydz (1 − y − z) m s − c 192π 6   3 mc ¯s s 3 2 dydz (1 − y − z) m s − c 576π 6   2 11mc ¯s gs σ Gs 2 dydz (1 − y − z) s − m c 1024π 6   2 11mc ¯s gs σ Gs 2 2 dydz (1 − y − z) m s − c 2048π 6     ¯s s2 2 2 dydz (y + z)(1 − y − z) s − m m 2s − c c 48π 4    19¯s s¯s gs σ Gs dydz (y + z) 3s − 2m2c 4 1536π   mc ¯s s3 ¯s s¯s gs σ Gs dydz (1 − y − z) s − dy 768π 4 9π 2    s  11¯s gs σ Gs2 2 dy 1 + δ s − m

c 2 3072π 4    2 13¯s gs σ Gs 2 dydz s δ s − m c 27648π 4   4 ms mc dydz (1 − y − z)3 s − m2c 8 3072π   4 ms mc 4 2 dydz (1 − y − z) m s − c 12288π 8   2   ms ¯s s 2 2 2 dydz (y + z)(1 − y − z) m s − 5s − 2m c c 1536π 6   2   ms ¯s s 3 2 2 dydz (y + z)(1 − y − z) m m s − 2s − c c 512π 6      11ms ¯s gs σ Gs 2 2 dydz (y + z)(1 − y − z) s − m m 2s − c c 2048π 6     ms ¯s gs σ Gs dydz (y + z)(1 − y − z)2 s − m2c 5s − 3m2c 6 1024π    ms ¯s gs σ Gs 2 2 dydz (1 − y − z) s s − m c 2048π 6    2 ms mc ¯s s dydz s − m2c 6π 4    ms mc ¯s s2 2 dydz (1 − y − z) s − m c 12π 4   199ms mc ¯s s¯s gs σ Gs 89ms mc ¯s s¯s gs σ Gs dydz − dy 2304π 4 2304π 4    s  ms ¯s s3 2 dy 1 + δ s − m

c 2 36π 2

    2 7s ms mc ¯s gs σ Gs 2 dy 1 + δ s − m

c 64π 4 9T 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.29 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–••• 1

−

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

11 1 ,1 ρuss2 (s) =

ms mc ¯s gs σ Gs2 1536π 4 T 2



  dy s δ s − m

2c ,

29

(46)

  4   1 4 2 2 dydz yz(1 − y − z) m s − 8s − 3m c c 61440π 8      4 1 5 2 2 dydz yz(1 − y − z) − m s − 9s − 4m c c 307200π 8    3 mc qq ¯ + 2¯s s 2 2 dydz (y + z)(1 − y − z) − m s − c 2304π 6    3 ¯ + 2¯s s mc qq dydz (y + z)(1 − y − z)3 s − m2c + 6 6912π    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz (y + z)(1 − y − z) s − + m c 12288π 6    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 2 dydz (y + z)(1 − y − z) − m s − c 24576π 6      ¯s s 2qq ¯ + ¯s s dydz yz(1 − y − z) s − m2c 5s − 3m2c + 4 72π   ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq 19 ¯s s¯s gs σ Gs + qq¯ − 1152π 4     m qq¯ s s2 c ¯ dy × dydz yz 4s − 3m2c − 36π 2   11¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs + 1536π 4    s  × dy y(1 − y) 1 + δ s − m

2c 3   4 ms mc 3 2 dydz (y + z)(1 − y − z) m s − + c 18432π 8   4 ms mc 4 2 dydz (y + z)(1 − y − z) − m s − c 73728π 8    2   ms 2qq ¯ − ¯s s 2 2 2 dydz yz(1 − y − z) − m m s − 2s − c c 192π 6      2 ms ¯s s dydz yz(1 − y − z)3 s − m2c 7s − 4m2c − 576π 6     ms ¯s gs σ Gs dydz yz(1 − y − z)2 s − m2c 3s − 2m2c + 6 192π   ms 19qg ¯ s σ Gq + 3¯s gs σ Gs + 3072π 6     × dydz yz(1 − y − z) s − m2c 5s − 3m2c     ¯ − ¯s s ms mc ¯s s 5qq dydz (y + z) s − m2c + 4 144π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.30 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

30

5

    ms mc ¯s s qq ¯ + ¯s s 2 dydz (y + z)(1 − y − z) s − m c 144π 4   ms mc 89¯s s¯s gs σ Gs + 57¯s sqg ¯ s σ Gq + 32qq¯ ¯ s gs σ Gs − 27648π 4 

6

×

1

+

2 3 4

dydz (y + z)   ms mc 231¯s sqg ¯ s σ Gq − 89¯s s¯s gs σ Gs + 256qq¯ ¯ s gs σ Gs − dy 27648π 4  

 ms mc qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq y z dydz + + z y 2304π 4    s  ms qq¯ ¯ s s2 2 dy y(1 − y) 1 + + δ s − m

c 3 9π 2

    ms mc qg 1−y ¯ s σ Gq¯s gs σ Gs y 2 − dy + δ s − m

c y 1−y 4608π 4

    8s 17ms mc qg ¯ s σ Gq¯s gs σ Gs 2 dy 1 + + δ s − m

c 3456π 4 17T 2   ms mc ¯s gs σ Gs 3qg ¯ s σ Gq + 19¯s gs σ Gs − 18432π 4      s × dy 1 + 2 δ s − m

2c , (47) T

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

11 1 ,0 ρ

uss2 (s) =

  4   1 4 2 2 dydz (y + z)(1 − y − z) m s − 7s − 2m c c 245760π 8      4 1 5 2 2 dydz (y + z)(1 − y − z) − m s − 8s − 3m c c 1228800π 8    3 mc qq ¯ + 2¯s s 2 2 dydz (1 − y − z) − m s − c 576π 6    3 mc qq ¯ + 2¯s s dydz (1 − y − z)3 s − m2c + 6 1728π    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz (1 − y − z) s − + m c 3072π 6    2 ¯ s σ Gq + 2¯s gs σ Gs 11mc qg 2 2 dydz (1 − y − z) − m s − c 6144π 6      ¯s s 2qq ¯ + ¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 2s − c c 144π 4   19 ¯s s¯s gs σ Gs + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 4 4608π    × dydz (y + z) 3s − 2m2c  ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq ¯s s¯s gs σ Gs + qq¯ dydz (1 − y − z) s + 2304π 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.31 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

 ¯ s s2 mc qq¯ dy 9π 2     11¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs s  2 dy 1 + + δ s − m

c 2 9216π 4     13¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs 2 dydz s δ s − − m c 82944π 4    4 ms mc dydz (1 − y − z)3 s − m2c + 4608π 8   4 ms mc 4 2 dydz (1 − y − z) − m s − c 18432π 8    2   ms 2qq ¯ − ¯s s 2 2 2 dydz (y + z)(1 − y − z) − m s − 5s − 2m c c 2304π 6   2   ms ¯s s 3 2 2 dydz (y + z)(1 − y − z) m m − s − 2s − c c 768π 6      ms ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) m + s − 5s − 3m c c 1536π 6     ms qg ¯ s σ Gq + ¯s gs σ Gs 2 2 dydz (1 − y − z) − s s − m c 6144π 6   ms 19qg ¯ s σ Gq + 3¯s gs σ Gs + 6144π 6     × dydz (y + z)(1 − y − z) s − m2c 2s − m2c     ms mc ¯s s 5qq ¯ − ¯s s 2 dydz s − + m c 36π 4     ¯ + ¯s s ms mc ¯s s qq 2 dydz (1 − y − z) s − + m c 36π 4   ms mc 89¯s s¯s gs σ Gs + 57¯s sqg ¯ s σ Gq + 32qq¯ ¯ s gs σ Gs dydz − 6912π 4   ms mc 231¯s sqg ¯ s σ Gq − 89¯s s¯s gs σ Gs + 256qq¯ ¯ s gs σ Gs − dy 6912π 4    s  ms qq¯ ¯ s s2 2 dy 1 + + δ s − m

c 2 54π 2

    8s ¯ s σ Gq¯s gs σ Gs 9ms mc qg 2 dy 1 + δ s − m

+ c 864π 4 9T 2     ms mc ¯s gs σ Gs 3qg ¯ s σ Gq + 19¯s gs σ Gs 2 dy s δ s − m

(48) − c , 13824π 4 T 2

1

−

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

31

11 1 ,1 ρuus2 (s) =

  4   1 4 2 2 dydz yz(1 − y − z) m s − 8s − 3m c c 61440π 8   4   1 dydz yz(1 − y − z)5 s − m2c − 9s − 4m2c 8 307200π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.236; Prn:26/09/2016; 12:53] P.32 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

32

   3 mc 2qq ¯ + ¯s s 2 2 dydz (y + z)(1 − y − z) m s − c 2304π 6    3 mc 2qq ¯ + ¯s s 3 2 dydz (y + z)(1 − y − z) + m s − c 6912π 6    2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs dydz (y + z)(1 − y − z) s − m2c + 6 12288π    2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 2 dydz (y + z)(1 − y − z) − m s − c 24576π 6      qq ¯ qq ¯ + 2¯s s 2 2 dydz yz(1 − y − z) s − + m 5s − 3m c c 72π 4   19 qq ¯ qg ¯ s σ Gq + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 1152π 4     m qq 2 ¯ s s c ¯ 2 dy × dydz yz 4s − 3mc − 36π 2   ¯ s σ Gq + 2¯s gs σ Gs 11qg ¯ s σ Gq qg + 1536π 4    s  × dy y(1 − y) 1 + δ s − m

2c 3   4 ms mc dydz (y + z)(1 − y − z)3 s − m2c + 8 36864π   4 ms mc 4 2 dydz (y + z)(1 − y − z) − m s − c 147456π 8    2   ¯ − 3¯s s ms 4qq 2 2 2 dydz yz(1 − y − z) − m m s − 2s − c c 384π 6   2   ms ¯s s dydz yz(1 − y − z)3 s − m2c 7s − 4m2c − 6 1152π     ms ¯s gs σ Gs dydz yz(1 − y − z)2 s − m2c 3s − 2m2c + 6 384π   ms 19qg ¯ s σ Gq − 8¯s gs σ Gs + 3072π 6     × dydz yz(1 − y − z) s − m2c 5s − 3m2c     ms mc qq ¯ 3qq ¯ − ¯s s 2 dydz (y + z) s − + m c 144π 4    ¯ s s ms mc qq¯ dydz (y + z)(1 − y − z) s − m2c + 4 144π   ms mc 32qq¯ ¯ s σ Gq ¯ s gs σ Gs + 57¯s sqg dydz (y + z) − 27648π 4   ¯ s σ Gq − 32qq¯ ¯ s gs σ Gs ¯ qg ¯ s σ Gq − 57¯s sqg ms mc 288qq − dy 27648π 4 −

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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  y ¯ qg ¯ s σ Gq ms mc qq z dydz + z y 2304π 4    2 s  ¯ ¯s s ms qq 2 dy y(1 − y) 1 + δ s − m

+ c 3 18π 2

    2 1−y ms mc qg ¯ s σ Gq y 2 dy − + δ s − m

c y 1−y 9216π 4   ms mc qg ¯ s σ Gq 144qg ¯ s σ Gq − 19¯s gs σ Gs + 55296π 4      s × dy 1 + 2 δ s − m

2c , T

33

+



1 2 3 4 5 6 7 8 9 10

(49)

 4   1 11 1 ,0 4 2 2 dydz (y + z)(1 − y − z) m ρ

uus2 (s) = s − 7s − 2m c c 245760π 8      4 1 5 2 2 dydz (y + z)(1 − y − z) m s − 8s − 3m − c c 1228800π 8    3 mc 2qq ¯ + ¯s s 2 2 dydz (1 − y − z) − m s − c 576π 6    3 mc 2qq ¯ + ¯s s 3 2 dydz (1 − y − z) + m s − c 1728π 6    2 ¯ s σ Gq + ¯s gs σ Gs 11mc 2qg dydz (1 − y − z) s − m2c + 6 3072π    2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 2 dydz (1 − y − z) − m s − c 6144π 6      qq ¯ 2qq ¯ + ¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 2s − c c 144π 4   ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq 19 qq ¯ qg ¯ s σ Gq + qq¯ − 4 4608π    × dydz (y + z) 3s − 2m2c  ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq qq ¯ qg ¯ s σ Gq + qq¯ dydz (1 − y − z) s + 2304π 4  mc qq ¯ 2 ¯s s − dy 9π 2     11qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs s  2 dy 1 + + δ s − m

c 2 9216π 4     13qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz s δ s − m − c 82944π 4    4 ms mc dydz (1 − y − z)3 s − m2c + 9216π 8   4 ms mc dydz (1 − y − z)4 s − m2c − 8 36864π

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

34

   2   ms 4qq ¯ − 3¯s s 2 2 2 dydz (y + z)(1 − y − z) m s − 5s − 2m c c 4608π 6   2   ms ¯s s 3 2 2 dydz (y + z)(1 − y − z) − m m s − 2s − c c 1536π 6      ms ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) m s − 5s − 3m + c c 3072π 6    ms qg ¯ s σ Gq 2 2 dydz (1 − y − z) − s s − m c 6144π 6   ms 19qg ¯ s σ Gq − 8¯s gs σ Gs + 6144π 6     × dydz (y + z)(1 − y − z) s − m2c 2s − m2c     ms mc qq ¯ 3qq ¯ − ¯s s 2 dydz s − m + c 36π 4    ¯ s s ms mc qq¯ 2 dydz (1 − y − z) s − m + c 36π 4   ms mc 32qq¯ ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq dydz − 6912π 4   ¯ qg ¯ s σ Gq − 57¯s sqg ¯ s σ Gq − 32qq¯ ¯ s gs σ Gs ms mc 288qq dy − 6912π 4    ms qq s  ¯ 2 ¯s s 2 + dy 1 + δ s − m

c 2 108π 2    2 s   ms mc qg ¯ s σ Gq 2 dy 1 + + δ s − m

c 192π 4 T2    ¯ s σ Gq¯s gs σ Gs 19ms mc qg 2 dy s δ s − m

(50) − c , 13824π 4 T 2

1

−

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

11 1 ,1 11 1 ,1 ρuuu2 (s) = ρsss2 (s) |ms →0, ¯s s→qq, ¯ ¯s gs σ Gs→qg ¯ s σ Gq ,

(51)

11 1 ,0 11 1 ,0 ρ

uuu2 (s) = ρ

sss2 (s) |ms →0, ¯s s→qq, ¯ ¯s gs σ Gs→qg ¯ s σ Gq ,

(52)

10 12 ,1

ρsss



(s) =



4 

31



1 dydz yz(1 − y − z)4 s − m2c 8s − 3m2c 245760π 8   4   1 5 2 2 − dydz yz(1 − y − z) m s − 9s − 4m c c 1228800π 8    3 mc ¯s s dydz (y + z)(1 − y − z)2 s − m2c − 1536π 6   3 mc ¯s s 3 2 dydz (y + z)(1 − y − z) + m s − c 4608π 6   2 11mc ¯s gs σ Gs 2 dydz (y + z)(1 − y − z) s − + m c 8192π 6   2 11mc ¯s gs σ Gs dydz (y + z)(1 − y − z)2 s − m2c − 6 16384π

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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[m1+; v1.236; Prn:26/09/2016; 12:53] P.35 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–••• 1 2 3 4

− +

5 6

+

7 8 9 10 11

− −

12 13

−

14 15

+

16 17 18 19 20

+ +

21 22

−

23 24

−

25 26 27 28 29 30 31

− + +

32 33

+

34 35 36 37 38

+ +

39 40

−

41 42

−

43 44 45 46 47

+ +

   2 y 7mc ¯s gs σ Gs z 2 2 dydz m + (1 − y − z) s − c z y 16384π 6

    2 y mc ¯s gs σ Gs z dydz + (1 − y − z)3 s − m2c 6 z y 8192π     2 ¯s s dydz yz(1 − y − z) s − m2c 5s − 3m2c 4 96π    19¯s s¯s gs σ Gs 2 dydz yz 4s − 3m c 1536π 4    ¯s s¯s gs σ Gs 2 dydz (y + z)(1 − y − z) 2s − m c 3072π 4  mc ¯s s3 dy 72π 2     13¯s gs σ Gs2 2 dydz (y + z) 1 + s δ s − m c 110592π 4    s  11¯s gs σ Gs2 2 dy y(1 − y) 1 + δ s − m

c 3 2048π 4   4 ms mc dydz (y + z)(1 − y − z)3 s − m2c 8 24576π   4 ms mc 4 2 dydz (y + z)(1 − y − z) m s − c 98304π 8      2 ms ¯s s 2 2 2 dydz yz(1 − y − z) m m s − 2s − c c 512π 6   2   ms ¯s s 3 2 2 dydz yz(1 − y − z) m s − 7s − 4m c c 1536π 6      11ms ¯s gs σ Gs 2 2 dydz yz(1 − y − z) s − m 5s − 3m c c 4096π 6     ms ¯s gs σ Gs dydz yz(1 − y − z)2 s − m2c 3s − 2m2c 6 512π     ms ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) m m s − 3s − c c 16384π 6    2 ms mc ¯s s dydz (y + z) s − m2c 48π 4    ms mc ¯s s2 2 dydz (y + z)(1 − y − z) s − m c 96π 4  89ms mc ¯s s¯s gs σ Gs dydz (y + z) 18432π 4  199ms mc ¯s s¯s gs σ Gs dy 18432π 4

  y 3ms mc ¯s s¯s gs σ Gs z dydz + z y 1024π 4

  ms mc ¯s s¯s gs σ Gs y z dydz + (1 − y − z) z y 512π 4

35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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1

+

2 3

+

4 5

−

6 7 8

−

9 10

−

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.236; Prn:26/09/2016; 12:53] P.36 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

36

10 1 ,0 ρ

sss2 (s) =

   s  ms ¯s s3 dy y(1 − y) 1 + δ s − m

2c 2 3 24π

    2 7s ms mc ¯s gs σ Gs 2 dy 16 + δ s − m

c 4608π 4 T2

    2 ms mc ¯s gs σ Gs z y 2 dydz m + δ s − c y z 3072π 4

    17ms mc ¯s gs σ Gs2 1−y y 2 dy + δ s − m

c y 1−y 18432π 4      2 ms mc ¯s gs σ Gs s 2 dy 1 + δ s − m

c , 12288π 4 T2

  4   1 4 2 2 dydz (y + z)(1 − y − z) m s − 7s − 2m c c 491520π 8   4   1 5 2 2 dydz (y + z)(1 − y − z) − m s − 8s − 3m c c 2457600π 8    3 mc ¯s s dydz (1 − y − z)2 s − m2c − 6 768π   3 mc ¯s s 3 2 dydz (1 − y − z) + m s − c 2304π 6   2 11mc ¯s gs σ Gs dydz (1 − y − z) s − m2c + 6 4096π   2 11mc ¯s gs σ Gs 2 2 dydz (1 − y − z) − m s − c 8192π 6

   2 1 1 7mc ¯s gs σ Gs 2 2 dydz − m + (1 − y − z) s − c y z 16384π 6

    2 1 1 mc ¯s gs σ Gs dydz + + (1 − y − z)3 s − m2c 6 y z 8192π     2 ¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 2s − c c 96π 4    19¯s s¯s gs σ Gs 2 − dydz (y + z) 3s − 2m c 3072π 4   mc ¯s s3 ¯s s¯s gs σ Gs dydz (1 − y − z)s − dy − 1536π 4 36π 2    11¯s gs σ Gs2 s  2 + dy 1 + δ s − m

c 2 6144π 4    2 13¯s gs σ Gs 2 dydz s δ s − m + c 55296π 4   4 ms mc 3 2 dydz (1 − y − z) + m s − c 12288π 8   4 ms mc dydz (1 − y − z)4 s − m2c − 8 49152π

1 2 3 4 5 6 7 8 9 10

(53)

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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[m1+; v1.236; Prn:26/09/2016; 12:53] P.37 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

−

1 2 3

−

4 5

+

6 7

+

8 9

+

10 11 12

+

13 14

+

15 16

−

17 18 19

−

20 21

+

22 23

+

24 25 26

+

27 28

+

29 30 31

−

32 33

−

34 35 36

−

37 38 39 40 41 42 43 44 45 46 47

10 1 ,1 ρuss2 (s) =

  2   ms ¯s s 2 2 2 dydz (y + z)(1 − y − z) m s − 5s − 2m c c 3072π 6      2 ms ¯s s dydz (y + z)(1 − y − z)3 s − m2c 2s − m2c 1024π 6     11ms ¯s gs σ Gs dydz (y + z)(1 − y − z) s − m2c 2s − m2c 6 4096π     ms ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) m s − 5s − 3m c c 2048π 6    ms ¯s gs σ Gs dydz (1 − y − z)2 s s − m2c 4096π 6    ms mc ¯s s2 2 dydz s − m c 24π 4    2 ms mc ¯s s 2 dydz (1 − y − z) s − m c 48π 4  89ms mc ¯s s¯s gs σ Gs dydz 9216π 4  199ms mc ¯s s¯s gs σ Gs dy 9216π 4

  3ms mc ¯s s¯s gs σ Gs 1 1 dydz + y z 1024π 4

  1 1 ms mc ¯s s¯s gs σ Gs dydz + (1 − y − z) y z 512π 4    s  ms ¯s s3 2 dy 1 + δ s − m

c 2 72π 2

    2 7s ms mc ¯s gs σ Gs 2 dy 1 + δ s − m

c 256π 4 9T 2

    1 1 ms mc ¯s gs σ Gs2 2 dydz m + δ s − c y z 3072π 4

    2 1 1 17ms mc ¯s gs σ Gs 2 dy + δ s − m

c y 1−y 18432π 4    ms mc ¯s gs σ Gs2 dy s δ s − m

2c , 4 2 6144π T

  4   1 4 2 2 dydz yz(1 − y − z) m s − 8s − 3m c c 245760π 8      4 1 5 2 2 dydz yz(1 − y − z) m s − 9s − 4m − c c 1228800π 8    3 mc qq ¯ + 2¯s s 2 2 dydz (y + z)(1 − y − z) m s − − c 4608π 6    3 ¯ + 2¯s s mc qq dydz (y + z)(1 − y − z)3 s − m2c + 6 13824π

37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

(54)

36 37 38 39 40 41 42 43 44 45 46 47

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[m1+; v1.236; Prn:26/09/2016; 12:53] P.38 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

38

   2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz (y + z)(1 − y − z) s − m c 24576π 6    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 2 dydz (y + z)(1 − y − z) + m s − c 49152π 6   7mc qg ¯ s σ Gq + 2¯s gs σ Gs − 6

49152π    2 y z × dydz + (1 − y − z)2 s − m2c z y  

  2 mc qg ¯ s σ Gq + 2¯s gs σ Gs y z 3 2 dydz + m + (1 − y − z) s − c z y 24576π 6      ¯s s 2qq ¯ + ¯s s 2 2 dydz yz(1 − y − z) s − + m 5s − 3m c c 288π 4   19 ¯s s¯s gs σ Gs + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 4 4608π    2 × dydz yz 4s − 3mc +

¯ s gs σ Gs + ¯s sqg ¯ s σ Gq ¯s s¯s gs σ Gs + qq¯ 4 9216π    × dydz (y + z)(1 − y − z) 2s − m2c  ¯ s s2 mc qq¯ dy − 72π 2   11¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs + 6144π 4    s  × dy y(1 − y) 1 + δ s − m

2c 3   13¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs + 331776π 4     × dydz (y + z) 1 + s δ s − m2c   4 ms mc 3 2 dydz (y + z)(1 − y − z) m s − + c 36864π 8   4 ms mc 4 2 dydz (y + z)(1 − y − z) − m s − c 147456π 8    2   ms 2qq ¯ − ¯s s dydz yz(1 − y − z)2 s − m2c − 2s − m2c 6 768π   2   ms ¯s s 3 2 2 dydz yz(1 − y − z) m s − 7s − 4m − c c 2304π 6      ms ¯s gs σ Gs dydz yz(1 − y − z)2 s − m2c 3s − 2m2c + 6 768π   ms 19qg ¯ s σ Gq + 3¯s gs σ Gs + 12288π 6

−

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.39 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

39



   dydz yz(1 − y − z) s − m2c 5s − 3m2c   ms qg ¯ s σ Gq + ¯s gs σ Gs + 49152π 6     × dydz (y + z)(1 − y − z)2 s − m2c 3s − m2c     ¯ − ¯s s ms mc ¯s s 5qq 2 dydz (y + z) s − + m c 288π 4     ¯ + ¯s s ms mc ¯s s qq 2 dydz (y + z)(1 − y − z) s − + m c 288π 4   ms mc 89¯s s¯s gs σ Gs + 32qq¯ ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq − 55296π 4  ×

1 2 3 4 5 6 7 8 9 10 11 12 13 14

×

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

10 1 ,0 ρ

uss2 (s) =

1 2 3 4 5 6 7 8 9 10 11 12 13 14

dydz (y + z)   ms mc 231¯s sqg ¯ s σ Gq + 256qq¯ ¯ s gs σ Gs − 89¯s s¯s gs σ Gs − dy 55296π 4  

 ms mc ¯s s qg ¯ s σ Gq + ¯s gs σ Gs y z dydz + + (1 − y − z) z y 1536π 4   ms mc 9¯s sqg ¯ s σ Gq − 7¯s s¯s gs σ Gs + 16qq¯ ¯ s gs σ Gs + 9216π 4

  y z × dydz + z y    s  ms qq¯ ¯ s s2 2 dy y(1 − y) 1 + + δ s − m

c 3 36π 2  

   ms mc ¯s gs σ Gs qg ¯ s σ Gq + ¯s gs σ Gs z y 2 dydz − m + δ s − c y z 9216π 4   ms mc ¯s gs σ Gs 41qg ¯ s σ Gq − 7¯s gs σ Gs − 55296π 4

    1−y y × dy + δ s −m

2c y 1−y   ms mc ¯s gs σ Gs 19¯s gs σ Gs − 125qg ¯ s σ Gq − 55296π 4      s 2 × dy 1 + δ s − m

c 2T 2     ¯ s σ Gq + ¯s gs σ Gs 19ms mc ¯s gs σ Gs qg 2 dy δ s − m

+ (55) c , 110592π 4

15

  4   1 4 2 2 dydz (y + z)(1 − y − z) m s − 7s − 2m c c 491520π 8   4   1 dydz (y + z)(1 − y − z)5 s − m2c − 8s − 3m2c 8 2457600π

43

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

44 45 46 47

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[m1+; v1.236; Prn:26/09/2016; 12:53] P.40 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

40

   3 mc qq ¯ + 2¯s s 2 2 dydz (1 − y − z) m s − c 2304π 6    3 mc qq ¯ + 2¯s s 3 2 dydz (1 − y − z) + m s − c 6912π 6    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz (1 − y − z) s − + m c 12288π 6    2 11mc qg ¯ s σ Gq + 2¯s gs σ Gs 2 2 dydz (1 − y − z) − m s − c 24576π 6   7mc qg ¯ s σ Gq + 2¯s gs σ Gs − 49152π 6

   2 1 1 × dydz + (1 − y − z)2 s − m2c y z  

  2 mc qg ¯ s σ Gq + 2¯s gs σ Gs 1 1 3 2 dydz + m + (1 − y − z) s − c y z 24576π 6      ¯s s 2qq ¯ + ¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 2s − c c 288π 4   ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq 19 ¯s s¯s gs σ Gs + qq¯ − 9216π 4    × dydz (y + z) 3s − 2m2c  ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq ¯s s¯s gs σ Gs + qq¯ dydz (1 − y − z)s − 4608π 4  mc qq¯ ¯ s s2 dy − 36π 2     13¯s gs σ Gs 2qg ¯ s σ Gq + ¯s gs σ Gs 2 + dydz s δ s − m c 165888π 4     ¯ s σ Gq + ¯s gs σ Gs 11¯s gs σ Gs 2qg s  dy 1 + δ s − m +

2c 4 2 18432π   4 ms mc dydz (1 − y − z)3 s − m2c + 18432π 8   4 ms mc 4 2 dydz (1 − y − z) − m s − c 73728π 8    2   ms 2qq ¯ − ¯s s dydz (y + z)(1 − y − z)2 s − m2c − 5s − 2m2c 6 4608π   2   ms ¯s s 3 2 2 dydz (y + z)(1 − y − z) m m s − 2s − − c c 1536π 6      ms ¯s gs σ Gs 2 2 2 dydz (y + z)(1 − y − z) m s − 5s − 3m + c c 3072π 6     ¯ s σ Gq + ¯s gs σ Gs ms qg dydz (1 − y − z)2 s s − m2c + 6 12288π −

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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[m1+; v1.236; Prn:26/09/2016; 12:53] P.41 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

  ms 19qg ¯ s σ Gq + 3¯s gs σ Gs 12288π 6     × dydz (y + z)(1 − y − z) s − m2c 2s − m2c     ¯ − ¯s s ms mc ¯s s 5qq 2 dydz s − m + c 144π 4     ¯ + ¯s s ms mc ¯s s qq 2 dydz (1 − y − z) s − m + c 144π 4   ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq ms mc 89¯s s¯s gs σ Gs + 32qq¯ dydz − 27648π 4   ms mc 231¯s sqg ¯ s σ Gq + 256qq¯ ¯ s gs σ Gs − 89¯s s¯s gs σ Gs dy − 27648π 4  

 ms mc ¯s s qg ¯ s σ Gq + ¯s gs σ Gs 1 1 dydz + + (1 − y − z) 4 y z 1536π   ¯ s σ Gq − 7¯s s¯s gs σ Gs + 16qq¯ ¯ s gs σ Gs ms mc 9¯s sqg + 9216π 4

  1 1 × dydz + y z    ms qq¯ s  ¯ s s2 2 + dy 1 + δ s − m

c 2 108π 2  

   ¯ s σ Gq + ¯s gs σ Gs ms mc ¯s gs σ Gs qg 1 1 2 dydz m + δ s − − c y z 9216π 4   ms mc ¯s gs σ Gs 41qg ¯ s σ Gq − 7¯s gs σ Gs − 4 55296π

    1 1 × dy + δ s −m

2c y 1−y   ¯ s σ Gq ms mc ¯s gs σ Gs 19¯s gs σ Gs − 125qg − 55296π 4    s   × dy 1 + 2 δ s − m

2c T     ¯ s σ Gq + ¯s gs σ Gs 19ms mc ¯s gs σ Gs qg 2 dy δ s − m

(56) + c , 55296π 4

1

+

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

41

10 12 ,1

ρuus

  4   1 4 2 2 dydz yz(1 − y − z) (s) = m s − 8s − 3m c c 245760π 8      4 1 5 2 2 dydz yz(1 − y − z) − m s − 9s − 4m c c 1228800π 8     3 ¯ + ¯s s mc 2qq dydz (y + z)(1 − y − z)2 s − m2c − 6 4608π    3 ¯ + ¯s s mc 2qq dydz (y + z)(1 − y − z)3 s − m2c + 6 13824π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.236; Prn:26/09/2016; 12:53] P.42 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

42

   2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 dydz (y + z)(1 − y − z) s − m c 24576π 6    2 ¯ s σ Gq + ¯s gs σ Gs 11mc 2qg 2 2 dydz (y + z)(1 − y − z) − m s − c 49152π 6   7mc 2qg ¯ s σ Gq + ¯s gs σ Gs − 6

49152π    2 y z × dydz + (1 − y − z)2 s − m2c z y  

  2 mc 2qg ¯ s σ Gq + ¯s gs σ Gs y z 3 2 dydz + m + (1 − y − z) s − c z y 24576π 6      qq ¯ qq ¯ + 2¯s s 2 2 dydz yz(1 − y − z) s − + m 5s − 3m c c 288π 4 ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq qq ¯ qg ¯ s σ Gq + qq¯ − 4 9216π    × dydz (y + z)(1 − y − z) 2s − m2c   ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq 19 qq ¯ qg ¯ s σ Gq + qq¯ − 4608π 4    × dydz yz 4s − 3m2c  ¯ 2 ¯s s mc qq dy − 72π 2   11qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs + 6144π 4    s  × dy y(1 − y) 1 + δ s − m

2c 3   ¯ s σ Gq + 2¯s gs σ Gs 13qg ¯ s σ Gq qg + 331776π 4     × dydz (y + z) 1 + s δ s − m2c   4 ms mc 3 2 dydz (y + z)(1 − y − z) m s − + c 73728π 8   4 ms mc 4 2 dydz (y + z)(1 − y − z) − m s − c 294912π 8    2   ms 4qq ¯ − 3¯s s dydz yz(1 − y − z)2 s − m2c − 2s − m2c 6 1536π   2   ms ¯s s 3 2 2 dydz yz(1 − y − z) m s − 7s − 4m − c c 4608π 6   ms 19qg ¯ s σ Gq − 8¯s gs σ Gs + 12288π 6     × dydz yz(1 − y − z) s − m2c 5s − 3m2c +

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.43 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

    ms ¯s gs σ Gs 2 2 2 dydz yz(1 − y − z) + m s − 3s − 2m c c 1536π 6      ¯ s σ Gq ms qg 2 2 2 dydz (y + z)(1 − y − z) + m m s − 3s − c c 49152π 6     ms mc qq ¯ 3qq ¯ − ¯s s 2 dydz (y + z) s − + m c 288π 4    ms mc qq¯ ¯ s s 2 dydz (y + z)(1 − y − z) s − + m c 288π 4   ms mc 32qq¯ ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq dydz (y + z) − 55296π 4   ms mc 288qq ¯ qg ¯ s σ Gq − 32qq¯ ¯ s gs σ Gs − 57¯s sqg ¯ s σ Gq − dy 55296π 4

  y ms mc ¯s sqg ¯ s σ Gq z dydz + + (1 − y − z) z y 1536π 4  

  ¯ − 7¯s s qg ¯ s σ Gq ms mc 16qq y z dydz + + z y 9216π 4    s  ms qq ¯ 2 ¯s s 2 dy y(1 − y) 1 + + δ s − m

c 3 72π 2      s ¯ s σ Gq¯s gs σ Gs ms mc qg 2 dy 1 + − δ s − m

c 6912π 4 T2

    z y ¯ s σ Gq¯s gs σ Gs ms mc qg 2 dydz m + δ s − − c y z 9216π 4   ms mc qg ¯ s σ Gq 24qg ¯ s σ Gq − 7¯s gs σ Gs − 55296π 4

    1−y y × dy + δ s −m

2c y 1−y   ms mc qg ¯ s σ Gq 24qg ¯ s σ Gq − ¯s gs σ Gs + 18432π 4      s 2 × dy 1 + δ s − m

c 2T 2    ¯ s σ Gq¯s gs σ Gs ms mc qg dy δ s − m

2c , + (57) 4 36864π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

43

10 1 ,0 ρ

uus2 (s) =

  4   1 4 2 2 dydz (y + z)(1 − y − z) m s − 7s − 2m c c 491520π 8      4 1 5 2 2 dydz (y + z)(1 − y − z) − m s − 8s − 3m c c 2457600π 8    3 mc 2qq ¯ + ¯s s 2 2 dydz (1 − y − z) − m s − c 2304π 6    3 ¯ + ¯s s mc 2qq dydz (1 − y − z)3 s − m2c + 6 6912π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.236; Prn:26/09/2016; 12:53] P.44 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

44

   2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 dydz (1 − y − z) s − m c 12288π 6    2 11mc 2qg ¯ s σ Gq + ¯s gs σ Gs 2 2 dydz (1 − y − z) − m s − c 24576π 6   7mc 2qg ¯ s σ Gq + ¯s gs σ Gs − 49152π 6

   2 1 1 × dydz + (1 − y − z)2 s − m2c y z  

  2 mc 2qg ¯ s σ Gq + ¯s gs σ Gs 1 1 3 2 dydz + m + (1 − y − z) s − c y z 24576π 6      qq ¯ qq ¯ + 2¯s s 2 2 dydz (y + z)(1 − y − z) s − + m m 2s − c c 288π 4   19 qq ¯ qg ¯ s σ Gq + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − 9216π 4    × dydz (y + z) 3s − 2m2c  qq ¯ qg ¯ s σ Gq + qq¯ ¯ s gs σ Gs + ¯s sqg ¯ s σ Gq − dydz (1 − y − z)s 4608π 4  mc qq ¯ 2 ¯s s − dy 36π 2     ¯ s σ Gq + 2¯s gs σ Gs 11qg ¯ s σ Gq qg s  2 dy 1 + + δ s − m

c 2 18432π 4     13qg ¯ s σ Gq qg ¯ s σ Gq + 2¯s gs σ Gs 2 dydz s δ s − m + c 165888π 4   4 ms mc dydz (1 − y − z)3 s − m2c + 8 36864π   4 ms mc dydz (1 − y − z)4 s − m2c − 8 147456π    2   ms 4qq ¯ − 3¯s s 2 2 2 dydz (y + z)(1 − y − z) − m s − 5s − 2m c c 9216π 6   2   ms ¯s s 3 2 2 dydz (y + z)(1 − y − z) − m m s − 2s − c c 3072π 6   ms 19qg ¯ s σ Gq − 8¯s gs σ Gs + 12288π 6     × dydz (y + z)(1 − y − z) s − m2c 2s − m2c     ms ¯s gs σ Gs + dydz (y + z)(1 − y − z)2 s − m2c 5s − 3m2c 6 6144π    ms qg ¯ s σ Gq + dydz (1 − y − z)2 s s − m2c 6 12288π +

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:13871 /FLA

[m1+; v1.236; Prn:26/09/2016; 12:53] P.45 (1-46)

Z.-G. Wang / Nuclear Physics B ••• (••••) •••–•••

45

    ms mc qq ¯ 3qq ¯ − ¯s s 2 dydz s − m c 144π 4    ¯ s s ms mc qq¯ 2 dydz (1 − y − z) s − m + c 144π 4   ms mc 32qq¯ ¯ s gs σ Gs + 57¯s sqg ¯ s σ Gq dydz − 27648π 4   ms mc 288qq ¯ qg ¯ s σ Gq − 32qq¯ ¯ s gs σ Gs − 57¯s sqg ¯ s σ Gq dy − 27648π 4

  1 1 ms mc ¯s sqg ¯ s σ Gq dydz + + (1 − y − z) y z 1536π 4  

  ms mc 16qq ¯ − 7¯s s qg ¯ s σ Gq 1 1 dydz + + y z 9216π 4     2 s ¯ ¯s s ms qq dy 1 + δ s − m

2c + 2 2 216π    ms mc qg ¯ s σ Gq¯s gs σ Gs dy s δ s − m

2c − 4 2 3456π T

    1 1 ¯ s σ Gq¯s gs σ Gs ms mc qg 2 dydz m + δ s − − c y z 9216π 4   ms mc qg ¯ s σ Gq 24qg ¯ s σ Gq − 7¯s gs σ Gs − 4 55296π

    1 1 × dy + δ s −m

2c y 1−y     ms mc qg ¯ s σ Gq 24qg ¯ s σ Gq − ¯s gs σ Gs s   2 dy 1 + + δ s − m

c 18432π 4 T2    ms mc qg ¯ s σ Gq¯s gs σ Gs dy δ s − m

2c , + (58) 18432π 4

1

+

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

33

10 12 ,0

35

38 39

10 12 ,0

ρ

uuu (s) = ρ

sss

34

37



 yf

(s) |ms →0, ¯s s→qq, ¯ ¯s gs σ Gs→qg ¯ s σ Gq ,  1−y



 yf



44 45 46 47

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

32 33

(60) 

34 35 36 37 38 39 40

References

42 43

(59)

1+ 1−4m2c /s 1− 1−4m2c /s where dydz = yi dy zi dz, dy = yi dy, yf = , yi = , zi = 2 2  yf 1  1−y  1−y ym2c (y+z)m2c m2c 2 2 , mc = yz , m

c = y(1−y) , yi dy → 0 dy, zi dz → 0 dz, when the δ funcys−m2c     2

2c appear. tions δ s − mc and δ s − m

40 41

2

31

10 1 ,1 10 1 ,1 ρuuu2 (s) = ρsss2 (s) |ms →0, ¯s s→qq, ¯ ¯s gs σ Gs→qg ¯ s σ Gq ,

32

36

1

41 42

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