arXiv:1802.07711v1 [hep-ph] 21 Feb 2018
K. Azizi1, 2,∗ and U. ¨Ozdem1,†
1
Department of Physics, Dogus University, Acibadem-Kadikoy, 34722 Istanbul, Turkey 2
School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran (Dated: November 9, 2018)
The electromagnetic multipole moments of the open-flavor Zcq¯ states are investigated by assuming a diquark-antidiquark picture for their internal structure and quantum numbers JP C = 1+−for their spin-parity. In particular, their magnetic and quadrupole moments are extracted in the framework of light-cone QCD sum rule by the help of the photon distribution amplitudes. The electromagnetic multipole moments of the open-flavor Z¯cqstates are important dynamical observables, which encode valuable information on their underlying structure. The results obtained for the magnetic moments of different structures are considerably large and can be measured in future experiments. We obtain very small values for the quadrupole moments of Zcq¯ states indicating a nonspherical charge distribution.
Keywords: Tetraquarks, Electromagnetic form factors, Multipole moments, Open-flavor states
I. INTRODUCTION
Since 2003, there are many non-conventional hadrons discovered experimentally, such as many XYZ tetraquarks, P+
c (4380) and Pc+(4450) pentaquarks etc., which could not be described as the conventional hadrons composed of two or three valence quark/antiquarks. They are called exotic hadrons. For some reviews on the theoretical and experimental progress on the properties of these new states see Refs. [1–11]. The greatest achievement with regard to the exotic states was the discovery of the charged multiquark states. The charged states with a hidden pair of heavy quark and antiquark such as the Z±
c (3900) [12], Zc±(4020) [13], Zc±(4430) [14], Zb±(10610, 10650) [15], would be undoubtedly considered as the exotic resonances, because these charged states cannot be explained as excited charmonium-like or bottomonium-like states.
Most of the discovered exotic states up to now share a common properties: they contain a hidden heavy quark-antiquark pair, ¯cc or ¯bb. However existence of the multiquarks, which do not contain ¯cc or ¯bb pairs is also possible, because fundamental laws of QCD do not prohibit existence of such open-flavor multiquark states. It should be noted that they have not been discovered experimentally, and to our best knowledge, there are not any candidates to be considered for these states. They may be seen in the exclusive reactions as the open-charm and open-bottom resonances. In 2003, the two narrow charm-strange mesons Ds0(2317) and Ds1(2460) were observed in the D+
sπ0and D∗+
s π0invariant mass distributions by the BABAR [16] and CLEO [17]] collaborations, are now being considered as candidates to open-charm tetraquark states. In 2016, the D0 Collaboration reported the observation of a state with four different quark flavors, the X(5568), and assigned the quantum numbers JP = 0+for it, but they did not exclude the possibility of JP = 1+[18]. Reported in the B0
sπ±final states, the X(5568) meson, if exist, cannot be categorized into the conventional meson family, and is a good candidate of exotic tetraquark state with valence quarks of four different flavors such as su ¯d¯b or sd¯u¯b. The observation of these states have immediately inspired extensive discussions on the possibility of their internal structure. For more information see for instance Refs. [19–21] and references therein. In 2017, the D0 Collaboration repeated their analysis when the Bsis reconstructed semileptonically. They reported evidence for a narrow structure, which was consistent with their previous measurement in the hadronic decay mode [22]. However, other experimental groups, namely the LHCb [23], CDF [24], CMS [25] and ATLAS [26] collaborations could not find this resonance from analysis of their experimental data.
In order to understand the inner structure of the hadrons in the nonperturbative regime of QCD, the main challenges are the determination of the dynamical and statical features of hadrons such as their decay form factors, masses, electromagnetic multipole moments and so on, both experimentally and theoretically. Many theoretical models accurately estimate the mass and decay width of the discovered exotic states, but the inner structure of these states is still unclear. In other words, the mass and decay width alone can not distinguish the inner structure of the exotic states. Recall that the electromagnetic multipole moments are equally important dynamical observables of the exotic
∗kazizi@dogus.edu.tr †uozdem@dogus.edu.tr
states. The electromagnetic multipole moments include the spatial distributions of the charge and magnetization in the hadrons and these parameters are directly related to the spatial distributions of quarks and gluons inside the hadrons. There are many studies in the literature devoted to the investigation of the electromagnetic multipole moments of the standard hadrons, but unfortunately relatively little are known the electromagnetic multipole moments of the exotic hadrons. There are a few studies in the literature where the magnetic dipole and quadrupole moments of the exotic states are studied: see [27–29] for tetraquarks and [30–35] for pentaquarks. More detailed analyses are needed in order to get useful knowledge on the charge distribution, electromagnetic multipole moments and geometric shapes of the non-conventional hadrons. In this study, we are going to concentrate on the charged open-flavor [qq][qc] states (hereafter we will denote these states as Z¯cq) with spin-parity JP C= 1+−, and calculate their electromagnetic multipole moments in the framework of light-cone QCD sum rule (LCSR). In LCSR, the hadronic parameters are expressed in terms of the vacuum condensates and the light cone distribution amplitudes (DAs) of the on-shell particles (for more about this method see, e.g., [36–38] and references therein).
The rest of the paper is organized as follows: In section II, the calculation of the sum rules in LCSR will be presented. In the last section, we numerically analyze the sum rules obtained for the multipole moments and discuss the obtained results. The explicit expressions of the magnetic and quadrupole moments are moved to the Appendix A.
II. FORMALISM
In this section we derive the LCSR for the magnetic and quadrupole moments of the Z¯cq states. For this aim, we consider a correlation function in the presence of the external electromagnetic field (γ),
Πµν(q) = i Z
d4xeip·x
h0|T {JZ¯cq
µ (x)JνZ¯cq†(0)}|0iγ, (1)
where Jµ is the interpolating current of Z¯cq state with quantum numbers JP C = 1+− in the diquark-antidiquark picture. It is given in terms of three light quark and one heavy quark fields as [39]:
JZ¯cq µ (x) = h q1Ta(x)Cγµq2b(x) ih ¯ q3a(x)γ5C¯c T b(x) − ¯q3b(x)γ5C¯c T a(x) i , (2)
where q1 is u, d and/or s-quark, q2 and q3 are u and/or d-quark, C is the charge conjugation matrix; and a and b are color indices.
In order to acquire sum rules for the magnetic and quadrupole moments, we need to represent the correlation function in two different forms: (1) in terms of the quark-gluon parameters and distribution amplitudes (DAs) of the photon in the deep Euclidean region, so called the QCD representation, and (2) in terms of hadronic properties, so called the hadronic representation.
We start our analysis by calculating the correlation function on Eq. (1) in terms of quarks and gluon properties in deep Euclidean region. For this purpose, the interpolating current is inserted into the correlation function and after the contracting of light and heavy quark pairs using the Wick theorem the following result is obtained:
ΠQCDµν (q) = i Z d4xeipxh0| ( Trhγ5Secb′b(−x)γ5Sa ′a q3 (−x) i TrhγνSeqaa1′(x)γµS bb′ q2 (x) i −Trhγ5Sea′b c (−x)γ5Sb ′a q3 (−x) i TrhγνSeaa′ q1 (x)γµS bb′ q2 (x) i −Trhγ5Secb′a(−x)γ5Sa ′b q3 (−x) i TrhγνSeqaa1′(x)γµS bb′ q2 (x) i +Trhγ5Seca′a(−x)γ5Sb ′b q3 (−x) i TrhγνSeqaa1′(x)γµS bb′ q2 (x) i) |0iγ, (3) where e S(x) = CST(x)C,
with Sq,c(x) being the quark propagators. The light and heavy propagators are given as [40] Sq(x) = Sf ree−h¯qqi 12 1 − imqx/ 4 −h¯qqi 192m 2 0x 2 1 − imqx/ 6 − igs 32π2x2 G µν(x) " / xσµν+ σµν/x # , (4)
and Sc(x) = Sf ree−gsmc16π2 Z 1 0 dv Gµν(vx) " (σµνx/ + x/σµν)K1(mc √ −x2) √ −x2 + 2σ µνK0(mcp −x2) # , (5) where Sqf ree(x) = i x/ 2π2x4 − mq 4π2x2, Sf ree c (x) = m2 c 4π2 " K1(mc√−x2) √ −x2 + i x/ K2(mc√−x2) (√−x2)2 # . (6)
Here K1,2are Bessel functions of the second kind.
The correlation function contains different types of contributions. In first part, one of the free quark propagators in Eq. (3) is replaced by
Sf ree(x) → Z
d4y Sf ree(x − y) /A(y) Sf ree(y) , (7)
and the remaining three propagators are taken as the full quark propagators. In the second case one of the light quark propagators in Eq. (3) is replaced by
Sαβab → − 1 4(¯q
aΓiqb)(Γi)αβ, (8)
and the remaining propagators are taken as the full quark propagators, as well including the perturbative and the nonperturbative contributions. Once Eq. (8) is plugged into Eq. (3), there appear matrix elements such as hγ(q) |¯q(x)Γiq(0)| 0i and hγ(q) |¯q(x)ΓiGαβq(0)| 0i, representing the nonperturbative contributions. The reader can find some details about the transformations of Eqs. (7) and (8) in Ref. [29]. These matrix elements can be written in terms of the photon DAs with definite twists, whose expressions all can be found in Ref. [41]. The QCD side of the correlation function can be acquired in terms of QCD parameters using the Eqs. (3)-(8) and after applying the Fourier transformation to transfer the calculations to the momentum space.
The next step is to calculate the correlation function in terms of the hadronic parameters. To this end we insert intermediate states of Z¯cq with the same quantum numbers as the interpolating current into Eq. (1). As a result, in zero-width approximation, we get
ΠHadµν (p, q) = h0 | J Z¯cq µ | Zcq(p)i¯ [p2− m2 Z¯cq] hZ¯cq(p) | Z¯cq(p + q)iγhZ ¯ cq(p + q) | J†Z¯cq ν | 0i [(p + q)2− m2 Zcq¯ ] + · · · , (9)
where dots stand for the contributions of the higher and continuum states and q is the momentum of the photon. The matrix element h0 | JZ¯cq
µ | Zcqi is determined as¯ h0 | JZ¯cq
µ | Z¯cqi = λZ¯cqε θ
µ, (10)
with λZ¯cq being the residue of the Z¯cq state.
The matrix element hZcq(p, ε¯ θ) | Zcq(p + q, ε¯ δ)iγ can be written in terms of three Lorentz invariant form factors as follows [42]: hZ¯cq(p, εθ) | Z¯cq(p + q, εδ)iγ = −ετ(εθ)α(εδ)β " G1(Q2) (2p + q)τ gαβ+ G2(Q2) (gτ βqα− gτ αqβ) − 1 2m2 Z¯cq G3(Q2) (2p + q)τ qαqβ # , (11)
where εθ and εδ are the polarization vectors of the initial and final Z¯
cq states and ετ is the polarization vector of the photon. The Lorentz invariant form factors G1(Q2), G2(Q2) and G3(Q2) are related to the charge, magnetic and
quadrupole form factors through the relations FC(Q2) = G1(Q2) +2 3ηFD(Q 2 ) , FM(Q2) = G2(Q2) , FD(Q2) = G1(Q2 ) − G2(Q2) + (1 + η)G3(Q2) , (12) where η = Q2/4m2 Z¯cqwith Q
2= −q2. At Q2= 0, these form factors are corresponding to the electric charge, magnetic moment µ and the quadrupole moment D as:
eFC(0) = e , eFM(0) = 2mZcq¯ µ , eFD(0) = m2
Z¯cqD . (13)
Using Eqs. (10)-(13) and imposing the condition q·ε = 0 the Eq. (9) takes the form, ΠHadµν = λ2 Z¯cq [m2 Zcq¯ − (p + q) 2][m2 Z¯cq− p 2] " 2(p.ε)FC(0) gµν−pµqν− pνqµ m2 Zcq¯ ! + FM(0) qµεν− qνεµ+ 1 m2 Z¯cq (p.ε)(pµqν− pνqµ) ! − FC(0) + FD(0) ! p.ε m2 Z¯cq qµqν # , (14) where we inserted X λ εµ(p, λ)εν(p, λ) = −gµν+ pµpν m2 Z¯cq . (15)
Equating the QCD and hadronic sides of the correlation function, we obtain the expression of the electromagnetic multipole moments in LCSR in terms of the QCD degrees of freedom and the photon DAs. We perform the double Borel transforms with respect to the variables p2 and (p + q)2 on both sides of the correlation function in order to suppress the contributions of the higher states and continuum, and use the quark-hadron duality assumption. By matching the coefficients of the structures qµενand (ε.p)qµqν, respectively for the magnetic and quadrupole moments, we get µ = e m2 Z¯cq/M 2 λ2 Z¯cq ΠQCD1 , D = m2Zcq¯ em 2 Z¯cq/M 2 λ2 Zcq¯ ΠQCD2 , (16)
where explicit expressions of the ΠQCD1 and Π QCD
2 are given in Appendix A.
III. NUMERICAL ANALYSIS
In this section, we numerically analyze the results of calculations for magnetic and quadrupole moments. We use mu = md = 0, ms(2 GeV ) = 0.096+0.008−0.004 GeV , mc(mc) = (1.28 ± 0.03) GeV [43], h¯uui(1 GeV ) = h ¯ddi(1 GeV ) = (−0.24±0.01)3GeV3[44], m2
0= 0.8±0.1 GeV2, hg2sG2i = 0.88 GeV4[1], χ(1 GeV ) = −2.85±0.5 GeV−2[45], λZsq ¯qc = 7.3 ± 1.7 × 10−3GeV5, λZ qq ¯qc = 7.6 ± 1.8 × 10 −3GeV5, mZ sq ¯qc = 2.826 +0.134 −0.157 GeV and mZqq ¯qc = 2.843 +0.115 −0.139GeV [46]. The parameters used in the photon DAs are given in Ref. [41].
The predictions for the electromagnetic multipole moments depend on two auxiliary parameters; the Borel mass parameter M2 and continuum threshold s0. Complying with the standard procedure of the sum rule method the predictions on the electromagnetic multipole moments should not depend on M2and s0, but in real computations one can only decrease their effect to a minimum. The working interval for the continuum threshold is chosen such that the maximum pole contribution is acquired and the results relatively weakly depend on its choices. Our numerical computations lead to the interval [10-12] GeV2for this parameter. The working region for M2is determined requiring that the contributions of the higher states and continuum are effectively suppressed. There are two criteria for
Z¯cq |µZ¯cq|[µN] |DZcq¯ |[f m 2] [sd][uc] 1.12± 0.18 0.0086 ± 0.0015 [sd][dc] 0.90± 0.13 0.0085 ± 0.0015 [su][uc] 0.51± 0.24 0.0070 ± 0.0013 [dd][uc] 1.09± 0.17 0.0082 ± 0.0014 [du][uc] 0.84± 0.31 0.0067 ± 0.0012 [dd][dc] 0.93± 0.13 0.0082 ± 0.0014 [uu][dc] 2.05± 0.30 0.016±0.003
TABLE I: Results of the magnetic and quadrupole moments of Z¯cqstates.
choosing working region for the Borel parameter M2: Convergence of the operator product expansion (OPE) and pole dominance. The requirement of the OPE convergence results in a lower bound, while the constraint of the maximum pole contribution leads to an upper bound on it. Our numerical calculation shows that these requirements are satisfied in the region 3 GeV2 ≤ M2 ≤ 4 GeV2 and, the magnetic and quadrupole moments in this region is practically independent of M2. In Figs. 1-2, we plot the dependencies of the magnetic and quadrupole moments on M2 at several fixed values of the continuum threshold s0. As is seen, the variation of the results with respect to the continuum threshold causes a change on the results on the magnetic and quadrupole moments of about 15% and there is a very less dependence of the quantities under consideration on the Borel parameter in its working interval. Hence, one can say that the results of the magnetic and quadrupole moments are almost insensitive to s0 and M2.
Our final results for the magnetic and quadrupole moments are given in Table I. The errors in the results come from the variations in the calculations of the working regions of M2 and s0 as well as the uncertainties in the values of the input parameters and the photon DAs. We also would like to note that in Table I and Figs. 1-2, the absolute values are given since it is not possible to determine the sign of the residue from the mass sum rules. Therefore, it is not possible to estimate the signs of the magnetic and quadrupole moments.
In summary, the electromagnetic multipole moments of the open-flavor Z¯cq states have been investigated by as-suming that these states are represented as diquark-antidiquark structure with quantum numbers JP C= 1+−. Their magnetic and quadrupole moments have been extracted in the framework of light-cone QCD sum rule. The electro-magnetic multipole moments of the open-flavor Z¯cq states are important dynamical observables, which would encode important information of their underlying structure, charge distribution and geometric shape. The results obtained for the magnetic moments are considerably large and can be measured in future experiments. We obtain very small values for the quadrupole moments of Z¯cq states indicating a nonspherical charge distribution. It is easy to see that [sd][uc] and [dd][uc] states belong to a class of doubly charged tetraquarks that the measurements of their electromag-netic parameters, like those of the ∆++ baryon, are relatively easy compared to other exotic states. These kind of exotic states have not been observed so far. We hope our predictions on the electromagnetic moments of these states together with the results of other theoretical studies on the spectroscopic parameters of these states will be useful for their searches in future experiments and will hep us determine exact internal structures of these non-conventional states.
IV. ACKNOWLEDGEMENT
This work has been supported by the Scientific and Technological Research Council of Turkey (T ¨UB˙ITAK) under the Grant No. 115F183.
3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 | µZsduc | [ µΝ ] s0 = 10 GeV2 s 0 = 11 GeV 2 s0 = 12 GeV2 (a) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 |D Zsduc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (b) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0.6 0.8 1 1.2 1.4 1.6 | µZsddc | [ µΝ ] s0 = 10 GeV2 s0 = 11 GeV2 s 0 = 12 GeV 2 (c) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 |D Zsddc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (d) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 | µZsuuc | [ µΝ ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (e) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 |D Zsuuc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (f)
FIG. 1: The dependence of the magnetic and quadrupole moments on the Borel parameter squared M2at different fixed values of the continuum threshold: (a) and (b) for the Zsd¯u¯c state, (c) and (d) for the Zsd ¯d¯c state and, (e) and
3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 | µZdduc | [ µΝ ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (a) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 |D Zdduc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (b) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 | µZduuc | [ µΝ ] s 0 = 10 GeV 2 s0 = 11 GeV2 s0 = 12 GeV2 (c) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 |D Zduuc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (d) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0.6 0.8 1 1.2 1.4 1.6 1.8 2 | µZdddc | [ µΝ ] s 0 = 10 GeV 2 s 0 = 11 GeV 2 s0 = 12 GeV2 (e) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 |D Zdddc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (f) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 | µZuudc | [ µΝ ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (g) 3 3.2 3.4 3.6 3.8 4 M2[GeV2] 0 0.004 0.008 0.012 0.016 0.02 0.024 0.028 0.032 0.036 |D Zuudc |[fm 2 ] s0 = 10 GeV2 s0 = 11 GeV2 s0 = 12 GeV2 (h)
FIG. 2: The dependence of the magnetic and quadrupole moments on the Borel parameter squared M2at different fixed values of the continuum threshold: (a) and (b) for the Zdd¯u¯c state, (c) and (d) for the Zdu¯u¯c state, (e) and (f)
Appendix A: Explicit forms of the functionsΠQCDi In this appendix, we present the explicit expressions for the functions ΠQCD1 and Π
QCD 2 : ΠQCD1 = 1 442368π2 " 32P1 ( eq3mcP3+ 144eq1π2P2(m2 0− m 2 c) + 3eq2 mq1P3+ 12 m2 cmq1+ 8π 2mcmq 1P1 + 4π2m20P2 − 8π2m2 cP2 !) A(u0) + 64χP1 ( 8eq1π2P2 P3+ 6mc − 3mcm20+ 2m3c+ 16π2P1 ! + eq2 P3(3m2cmq1− 8π 2 P2) + 48π2mc3mq1m 2 0P1− 3mcm20P2+ 2m3cP2+ 16π 2 P1P2 !) ϕγ(u0) + 32f3γ ( − eq3m2 cP3+ 3eq1mc mcP3+ 96π2P1(m2 0− m 2 c) + 3eq2 − m2 cP3− 48π 2m2 0(2mcP1+ mq1P2) − m2 c(2mcP1+ 3mq1P2) !) ψa(u0) − 1536π2f3γ ( 3eq1mcm2 0P1+ eq2 − 3m2 cmq1P1+ m 2 0(3mcP1 + 3mq1P2) ) ψν(u0) + 2I6[ψν] ! + 192P1 ( 24eq1π2P2(m2 0− 2m 2 c) + eq2 mq1P3+ 24π 24mcmq 1P1 + m2 0P2− 2m 2 cP2 !) I6[hγ] + 11eq3m2
cf3γP3I2[A] + (eq1− eq2)mcf3γ 23mcP3+ 576π2P1(m2 0− 2m 2 c) I1[A] + 576eq3π2mcmq1P1P2 2I5[S] + I2[S] − 44eq3mcP1P3I4[T1] + I5[T1] − 240eq3π2mq1f3γP2(m 2 0− 3m 2
c)I2[V] + (eq1+ eq2)mcf3γ − 23mcP3− 576π2P1(m20− 2m 2 c) I1[V] # I7[0] + 1 221184m2 cπ4 " − 16mcP1 (
eq3P3+ 36mc3eq2m2cmq1− 8(eq1+ eq2)π 2 P2 ) A(u0) + 192m2 cχP1 n − eq2mq1P3+ 96eq2π 2mcmq 1P1+ 24(eq1+ eq2)π 2P1(m2 0− 2m 2 c) ) ϕγ(u0) + 32mcf3γ (
(eq3− 3eq1)mcP3− 72eq1π2P1(m2 0− 4m 2 c) + 3eq2 mcP3+ 24π2 m2 0P1− 4mc(mcP1 + mq1P2) !) ψa(u0) + 2304π2mcf3γ ( (eq1+ eq2)m20P1 − 2eq2mcmq1P1 ) ψν(u0) + 2I6[ψν] ! + 96P1 (
96(eq1+ eq2)π2m2cP2+ eq2mq1
P3+ 96π2mcP1 )
I6[hγ] + 22eq3mcP1P3 I5[T1] + I5[T2] !
− 288eq3π2mcmq1P1P2I2[S] − 11eq3m 2
cf3γP3I2[A] + (eq1− eq2)m2cf3γ
− 23P3+ 1152π2mcP1
I1[A] − 432eq3π2m2
cmq1f3γP2I2[V] − (eq1+ eq2)m 2 cf3γ − 23P3+ 1152π2mcP1 I1[V] # I7[1] + 1 442368m2 cπ4 " 3456eq2m2 cmq1P1A(u0) + 192χP1 ( 96(eq1+ eq2)π2m2 cP2+ eq2mq1 P3+ 96π2mcP1 ) ϕγ(u0) − 32f3γ (
eq3P3+ 288eq1π2mcP1+ 3eq2P3+ 48π2(2mcP1+ msP2) )
ψa(u0) + 11eq3f3γP3I2[A] − 9216(eq1+ eq2)π2P1P2I6[hγ] + 2304eq2π2mq1f3γP2I3[ψ
a] + 4608eq2π2
mq1f3γP2
ψν(u0) + 2I6[ψν] + (eq1+ eq2)f3γ23P3+ 1152π2mcP1I1[A] + (eq1+ eq2)f3γ
− 23P3+ 1152π2mcP1 I1[V] + 144eq3π2mq1f3γP2I2[V] # I7[2]
−eq3mcmq1P1P2P3 20736π4 " 2ϕγ(u0) + I3[ϕγ] # I7[−1] −1152mP12 cπ4 "
16(eq1+ eq2)π2χP2ϕγ(u0) + 3eq2mq1A(u0) # I7[3] − m 4 c 442368π4 " 64eq3mcχP1P3ϕγ(u0) + 64f3γ ( − eq3P3+ 3eq1 P3+ 96π2mcP1+ 3eq2 − P3+ 48π2 2mcP1 − mq1P2 ) 2ψa(u0) + I3[ψa]− 192f3γ ( (eq1+ eq2)P3+ 96π2mcP1 − 48eq2π2mq 1P2 ) ψν(u0) + 2I6[ψν]
+ 18432(eq1+ eq2)π2P1P2I6[hγ] + 32eq3mcχP3P1I3[hγ
] − 11eq3f3γP3I2[A] − 288eq3π2mq
1f3γP2I2[V] # I8[−3, 1] + m 2 c 442368π2 " − 384eq2mq1χP1
P3+ 96π2mcP1ϕγ(u0) − 64eq3mcχP1P3I3[ϕγ] − 192(eq1+ eq2)f3γ
P3
+ 96π2mcP1ψν(u0) + 2I6[ψν]+ 32f3γ (
(3eq1− 3eq2− eq3)P3+ 288(eq1− eq2)π2mcP1 )
I3[ψa] − 2(eq1− eq2)f3γ
23P3+ 1152π2mcP1I1
[A] + 2(eq1+ eq2)f3γ
23P3+ 1152π2mcP1I1 [V] − 33eq3f3γP3I2[A]
# I8[−2, 1] + P1 110592m2 cπ4 " 8eq3mcP3A(u0) + 576(eq1− eq2)π2mcf3γm2 0
2ψa(u0) + I3[ψa]+ 144eq3π2mcmq
1P2I2[S]
− 1152(eq1+ eq2)π2mcf3γm20
ψν(u0) + 2I6[ψν]+ 48eq2mq1 − P3+ 96π2mcP1 I6[hγ] − 11eq3mcP3I5[T1] − I5[T2] # I8[0, 0] − 1 221184m2 cπ4 " 64χP1 ( P3(2eq3mc− 3eq2mq1) + 288π 2mcmq 1P1 ) ϕγ(u0) + 32f3γ ( 3eq196π2mcP1 − P3 + eq3P3 + 3eq2P3+ 48π2(2mcP1+ mq 1P2) ) ψa(u0) − 2I3[ψa]− 4608eq2π2mq 1f3γP2 ψν(u0) + 2I6[ψν] − 9216eq2π2P1P2I6[hγ] # I8[0, 1] + 1 1536m2 cπ4 "
32eq2mq1χP1ϕγ(u0) + 16(eq1+ eq2)f3γ
ψν(u0) + 2I6[ψν]− 8(eq1− eq2)f3γI3[ψa]
+ f 3γn2(eq1− eq2)I1[A] − eq3I2[V] − 2(eq1+ eq2)I1[V] o# I8[0, 3] + m 4 c 6144π4 " 64eq2mq1χP1 m2cF [−4, 3] + I8[−3, 3] ϕγ(u0) − 16(eq1− eq2)f3γ m4cF [−5, 3] + I8[−3, 3] ψa(u0) + f3γ ( 4(eq1− eq2)m2 cF [−4, 3] + I8[−3, 3] I4[A] − I4[V] + eq3m4 cF [−5, 3] − 2m2cF [−4, 3] + I8[−3, 3] I2[V] + 16(eq1+ eq2)f3γm4 cF [−5, 3] + 2m2cF [−4, 3] + I8[−3, 3]
ψν(u0) + 2I6[ψν]− 8(eq1− eq2) m4 cI8[−5, 3] + 2m2 cI8[−4, 3] + I8[−3, 3] I3[ψa] )#
−eq3mc1382πχP1P34 " 2ϕγ(u0) + I3[ϕγ] # I8[−1, 1] + eq3mcP1P327648π4 I3[A]I8[1, 0] + eq3mcP1 1024π4 4I5[S] − I2[S] I8[−2, 2] − m 6 cP1 1024π4 "
16eq2mq1I6[hγ] + eq3mcI1[S] #
I8[−4, 2] + P1 384m2
cπ4 "
3eq2mq1A(u0) + 16(eq1+ eq2)π
2χP2ϕγ(u0) # I8[0, 2] −m 6 cmq1P1 512π4 " 4eq2A(u0) − 2I6[hγ] − eq3 I2[S] − 2I5[S] # I8[−3, 2]. (17) and ΠQCD2 = − m3 cP1 55296π4 "
11eq3P3I5[T1] + I5[T2]− 4608(eq1+ eq2)π2f3γm20I6[ψν] #
I7[−2]
− f3γ
55296π4 "
11eq3P3I5[A] + 9216eq2π2mq 1P2I6[ψ
ν] + (eq1+ eq2)23P3+ 1152mcP1I4[A] # I7[0] − m2 cI7[1] − f3γ 55296m4 cπ4 "
11eq3P3I5[A] + (eq1− eq2) 23P3+ 1152mcP1I4[A] + 9216eq2π2mq 1P2I6[ψ ν] # I8[0, 0] +f3γm 4 c 128π4 " (eq1− eq2)m2 cI8[−4, 2] − I8[−3, 2]
I4[A] + 8(eq1+ eq2) m4 cI8[−5, 2] − 2m2cI8[−4, 2] − I8[−3, 2] I6[ψν] # +11eq3P1P3 18432π4 I5[T1] + I5[T2] ! I7[−1], (18)
where the values of eq1, eq2, eq3, mq1, P1, P2 and P3corresponding to different states are given in Table II.
Z¯cq eq1 eq2 eq3 mq1 P1 P2 P3
[sd][uc] es ed eu ms h¯qqi h¯ssi hg2sG
2 i
[sd][dc] es ed ed ms h¯qqi h¯ssi hg2sG2i
[su][uc] es eu eu ms h¯qqi h¯ssi hg2sG
2i
[dd][uc] ed ed eu 0 h¯qqi h¯qqi hg2sG
2 i
[du][uc] ed eu eu 0 h¯qqi h¯qqi hg2sG
2 i
[dd][dc] ed ed ed 0 h¯qqi h¯qqi hg2sG
2 i
[uu][dc] eu eu ed 0 h¯qqi h¯qqi hg2sG2i
TABLE II: The values of eq1, eq2, eq3, mq1, P1, P2and P3 related to the expressions of the magnetic and quadrupole moments in Eqs.(17) and (18).
The functions I1[A], I2[A], I3[A], I4[A], I5[A], I6[A], I7[n] and I8[n, m] are defined as: I1[A] = Z Dαi Z 1 0 dv A(α ¯ q, αq, αg)δ′(αq+ ¯vαg− u0), I2[A] = Z Dαi Z 1 0 dv A(α ¯ q, αq, αg)δ′(αq¯+ vαg− u0), I3[A] = Z 1 0 du A(u)δ′(u − u0), I4[A] = Z Dαi Z 1 0 dv A(α ¯ q, αq, αg)δ(αq + ¯vαg− u0), I5[A] = Z Dαi Z 1 0 dv A(αq¯, αq, αg)δ(α¯q + vαg− u0),
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