Magnetic dipole moment of
Z
bð10610Þ in light-cone QCD
U. Özdem1,* and K. Azizi1,2,†1
Department of Physics, Dogus University, Acibadem-Kadikoy, 34722 Istanbul, Turkey
2School of Physics, Institute for Research in Fundamental Sciences (IPM),
P. O. Box 19395-5531 Tehran, Iran
(Received 1 November 2017; published 22 January 2018)
The magnetic dipole moment of the exotic Zbð10610Þ state is calculated within the light cone QCD sum
rule method using the diquark-antidiquark and molecule interpolating currents. The magnetic dipole moment is obtained asμZb ¼ 1.73 0.63 μNin diquark-antidiquark picture andμZb¼ 1.59 0.58 μNin the molecular case. The obtained results in both pictures together with the results of other theoretical studies on the spectroscopic parameters of the Zbð10610Þ state may be useful in determination of the nature and
quark organization of this state.
DOI:10.1103/PhysRevD.97.014010
I. INTRODUCTION
According to QCD and the conventional quark model, not only the standard hadrons, but also exotic states such as meson-baryon molecules, tetraquarks, pentaquarks, glue-ball, and hybrids can exist. The first theoretical prediction on the existence of the multiquark structures was made by Jaffe in 1976[1]. Although it was predicted in the 1970s, there was not significant experimental evidence of their existence until 2003. The first observation on the exotic states was discovery of Xð3872Þ made by Belle Collaboration[2]in the decay Bþ → KþXð3872ÞJ=ψπþπ−. Subsequently, it was confirmed by BABAR[3], CDF II[4], D0[5], LHCb[6], and CMS[7]Collaborations. The discovery of the Xð3872Þ state turned out to be the forerunner of a new direction in hadron physics. So far, more than twenty exotic states have been observed experimentally [for details, see[8–15]]. The fail-ure of these states to fit the standard particles’ structures and violation of some conservation laws such as isospin sym-metry, make these states suitable tools for studying the nonperturbative nature of QCD.
In 2011, Belle Collaboration discovered two charged bottomonium-like states Zbð10610Þ and Zbð10650Þ (here-after we will denote these states as Zband Z0b, respectively) in the processesϒð5SÞ→ππϒðnSÞ, and ϒð5SÞ → ππhbðkPÞ [16]. Here, n¼ 1, 2, 3 and k ¼ 1, 2. The masses and widths of the two states have been measured as
MZb ¼ 10607.2 2 MeV; ΓZb ¼ 18.4 2.4 MeV; MZ0b ¼ 10652.2 1.5 MeV; ΓZ0b ¼ 11.5 2.2 MeV:
The analysis of the angular distribution shows that the quantum numbers of both states are IGðJPÞ ¼ 1þð1þÞ. Both Zb and Z0b belong to the family of charged hidden-bottom states. Since they are the first observed charged bottomoniumlike states and also very close to the thresh-olds of B ¯Bð10604.6 MeVÞ and B¯Bð10650.2 MeVÞ, Zb and Z0b states have attracted attention of many theoretical groups. The spectroscopic parameters and decays of Zband Z0b states have been studied with different models and approaches. Most of these investigations are based on diquark-antidiquark[17–20]and molecular interpretations
[17,21–41], using the analogy to the charm sector.
Although the spectroscopic features of these states have been studied sufficiently, the inner structure of these states have not exactly enlightened. Different kinds of analyses, such as interaction with the photon can shed light on the internal structure of these multiquark states.
A comprehensive analysis of the electromagnetic pro-perties of hadrons ensures crucial information on the nonperturbative nature of QCD and their geometric shapes. The electromagnetic multipole moments contain the spatial distributions of the charge and magnetization in the particle and therefore, these observables are directly related to the spatial distributions of quarks and gluons in hadrons. In this study, the magnetic dipole moment of the exotic state Zbis extracted by using the diquark-antidiquark and molecule interpolating currents in the framework of the light cone QCD sum rule (LCSR). This method has already been successfully applied to study the dynamical and statical properties of hadrons for decades such as, form factors, coupling constants and multipole moments. In the LCSR,
*uozdem@dogus.edu.tr †kazizi@dogus.edu.tr
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
the properties of the particles are characterized in terms of the light-cone distribution amplitudes (DAs) and the vacuum condensates [for details, see for instance[42–44]]. The rest of the paper is organized as follows: In Sec.II, the light-cone QCD sum rule for the electromagnetic form factors of Zbis applied and its magnetic dipole moment is derived. Section III encompasses our numerical analysis and discussion. The explicit expressions of the photon DAs are moved to the Appendix A.
II. FORMALISM
To obtain the magnetic dipole moment of the Zbstate by using the LCSR approach, we begin with the subsequent correlation function, Πμναðp; qÞ ¼ i2 Z d4x Z d4yeip·xþiq·y ×h0jT fJZb μ ðxÞJαðyÞJZνb†ð0Þgj0i: ð1Þ Here, JμðνÞis the interpolating current of the Zbstate and the electromagnetic current Jα is given as,
Jα¼ X q¼u;d;b
eq¯qγαq; ð2Þ
where eqis the electric charge of the corresponding quark. From technical point of view, it is more convenient to rewrite the correlation function by using the external background electromagnetic (BGEM) field,
Πμνðp; qÞ ¼ i Z
d4xeip·xh0jT fJZb
μ ðxÞJZb†
ν ð0Þgj0iF; ð3Þ where F is the external BGEM field and Fαβ ¼ iðεαqβ− εβqαÞ with qα and εβ being the four-momentum and polarization of the BGEM field. Since the external BGEM field can be made arbitrarily small, the correlation function in Eq.(3)can be acquired by expanding in powers of the BGEM field,
Πμνðp; qÞ ¼ Πð0Þμνðp; qÞ þ Πð1Þμνðp; qÞ þ :; ð4Þ and keeping only termsΠð1Þμνðp; qÞ, which corresponds to the single photon emission [45,46] (the technical details about the external BGEM field method can be found in [47]). The main advantage of using the BGEM field approach relies on the fact that it separates the soft and hard photon emissions in an explicitly gauge invariant way [46]. The Πð0Þμνðp; qÞ is the correlation function in the absence of the BGEM field, and gives rise to the mass sum rules of the hadrons, which is not relevant for our case. After these general remarks, we can now proceed deriving the LCSR for the magnetic dipole moment of the Zb state. The correlation function given in Eq.(3)can
be obtained in terms of hadronic parameters, known as hadronic representation. Additionally it can be calculated in terms of the quark and gluon parameters in the deep Euclidean region, known as QCD representation.
We can insert a complete set of intermediate hadronic states with the same quantum numbers as the interpolating current of the Zb into the correlation func-tion to obtain the hadronic representafunc-tion. Then, by isolating the ground state contributions, we obtain the following expression: ΠHad μν ðp; qÞ ¼h0jJ Zb μ jZbðpÞi p2− m2Z b hZbðpÞjZbðp þ qÞiF ×hZbðp þ qÞjJ †Zb ν j0i ðp þ qÞ2− m2 Zb þ ; ð5Þ
where dots denote the contributions coming from the higher states and continuum.
The matrix element appearing in Eq.(5)can be written in terms of three invariant form factors as follows[48]:
hZbðp;εθÞjZbðpþq;εδÞi F ¼−ετðεθÞαðεδÞβ G1ðQ2Þð2pþqÞτgαβ þG2ðQ2Þðg τβqα−gταqβÞ−2m12 Zb G3ðQ2Þð2pþqÞτqαqβ ; ð6Þ whereετ is the polarization vector of the BGEM field; and εθ andεδare the polarization vectors of the initial and final Zb states.
The remaining matrix element, that of the interpolating current between the vacuum and particle state,h0jJZb
μ jZbi, is parametrized as h0jJZb μ jZbi ¼ λZbε θ μ; ð7Þ
whereλZb is residue of the Zb state.
The form factors G1ðQ2Þ, G2ðQ2Þ and G3ðQ2Þ can be defined in terms of the charge FCðQ2Þ, magnetic FMðQ2Þ and quadrupole FDðQ2Þ form factors as follows
FCðQ2Þ ¼ G1ðQ2Þ þ 2 3ðQ2=4m2ZbÞFDðQ 2Þ; FMðQ2Þ ¼ G2ðQ2Þ; FDðQ2Þ ¼ G1ðQ2Þ − G2ðQ2Þ þ ð1 þ Q2=4m2Z bÞG3ðQ 2Þ; ð8Þ At Q2¼ 0, the form factors FCðQ2¼ 0Þ, FMðQ2¼ 0Þ, and FDðQ2¼ 0Þ are related to the electric charge, magnetic momentμ, and the quadrupole moment D as
eFCð0Þ ¼ e;
eFMð0Þ ¼ 2mZbμ;
eFDð0Þ ¼ m2Z
bD: ð9Þ
Inserting the matrix elements in Eqs.(6)and(7)into the correlation function in Eq.(5)and imposing the condition q ·ε ¼ 0, we obtain the correlation function in terms of the hadronic parameters as ΠHad μν ðp;qÞ ¼ λ2Zb ετ ½m2 Zb−ðpþqÞ 2½m2 Zb−p 2 × 2pτFCð0Þ gμν−pμqν−pνqμ m2Z b þFMð0Þqμgντ−qνgμτþ 1 m2Zbpτðpμqν−pνqμÞ −ðFCð0ÞþFDð0ÞÞ pτ m2Zbqμqν : ð10Þ
To obtain the expression of the correlation function in terms of the quark and gluon parameters, the explicit form for the interpolating current of the Zb state needs to be
chosen. In this study, we consider the Zb state with the quantum numbers JPC¼ 1þ−. Then in the diquark-antidiquark model the interpolating current JZb
μ is defined by the following expression in terms of quark fields:
JZbðDiÞ μ ðxÞ ¼ iϵ~ϵffiffiffi 2 p f½uTaðxÞCγ 5bbðxÞ½¯ddðxÞγμC ¯bTeðxÞ − ½uT aðxÞCγμbbðxÞ½¯ddðxÞγ5C ¯bTeðxÞg; ð11Þ
where C is the charge conjugation matrix, ϵ ¼ ϵabc, ~ϵ ¼ ϵdec; and a; b;… are color indices.
One can also construct the interpolating current by considering the Zb as a molecular form of B ¯B and B¯B state,
JZbðMolÞμ ðxÞ ¼ 1ffiffiffi
2
p f½¯daðxÞiγ5baðxÞ½¯bbðxÞγμubðxÞ
þ ½¯daðxÞγμbaðxÞ½¯bbðxÞiγ5ubðxÞg: ð12Þ After contracting pairs of the light and heavy quark operators, the correlation function becomes:
ΠQCD μν ðp; qÞ ¼ −iϵ~ϵϵ 0~ϵ0 2 Z d4xeipxh0jfTr½γ5~Saau 0ðxÞγ5Sbb 0 c ðxÞTr½γμ~Se 0e c ð−xÞγνSd 0d d ð−xÞ − Tr½γμ~Se0e c ð−xÞγ5Sd 0d d ð−xÞTr½γν~S aa0 u ðxÞγ5Sbb 0 c ðxÞ − Tr½γ5~Sa 0a u ðxÞγμSb 0b c ðxÞTr½γ5~Se 0e c ð−xÞγνSd 0d d ð−xÞ þ Tr½γν~Saa 0 u ðxÞγμSbb 0 c ðxÞTr½γ5~S e0e c ð−xÞγ5Sd 0d d ð−xÞgj0iF; ð13Þ
in the diquark-antidiquark picture, and
ΠQCD μν ðp; qÞ ¼ −2i Z d4xeipxh0jfTr½γ5Saab 0ðxÞγ5Sda0að−xÞTr½γμSbbu 0ðxÞγνSb 0b b ð−xÞ þ Tr½γ5Saa 0 b ðxÞγνSa 0a d ð−xÞTr½γμSbb 0 u ðxÞγ5Sb 0b b ð−xÞ þ Tr½γμSaa 0 b ðxÞγ5Sa 0a d ð−xÞTr½γ5Sbb 0 u ðxÞγνSb 0b b ð−xÞ þ Tr½γμSaa 0 b ðxÞγνSa 0a d ð−xÞTr½γ5Sbb 0 u ðxÞγ5Sb 0b b ð−xÞgj0iF; ð14Þ
in the molecular picture, where
~SbðqÞðxÞ ¼ CSTbðqÞðxÞC;
with SqðxÞ and SbðxÞ being the light and heavy quark propagators, respectively. To calculate the correlation functions in QCD representations, the light and heavy quark propagators are required. Their explicit expressions in the x-space are given as SqðxÞ ¼ i =x 2π2x4− h¯qqi 12 1 þm20x2 16 − igs 32π2x2GμνðxÞ½=xσμνþ σμν=x; ð15Þ and
SbðxÞ ¼m 2 b 4π2 K1ðmbpffiffiffiffiffiffiffiffi−x2Þ ffiffiffiffiffiffiffiffi −x2 p þ i=xK2ðmb ffiffiffiffiffiffiffiffi −x2 p Þ ðpffiffiffiffiffiffiffiffi−x2Þ2 −gsmb 16π2 Z 1 0 dvG μνðvxÞ ðσμν=xþ =xσμνÞK1ðmb ffiffiffiffiffiffiffiffi −x2 p Þ ffiffiffiffiffiffiffiffi −x2 p þ 2σμνK0ðmb ffiffiffiffiffiffiffiffi −x2 p Þ ; ð16Þ where Ki are the second kind Bessel functions, v is line variable, and Gμν is the gluon field strength tensor.
The correlation function includes different types of contributions. In first case, one of the free quark propagators in Eqs. (13)–(14)is replaced by
Sfree→ Z
d4ySfreeðx − yÞ==AðyÞSfreeðyÞ; ð17Þ
where Sfreeis the first term of the light or heavy quark propagators and the remaining three propagators are replaced with the full quark propagators. The LCSR calculations are most conveniently done in the fixed-point gauge. For electromagnetic field, it is defined by xμAμ¼ 0. In this gauge, the electromagnetic potential is given by
Aα¼ −1
2Fαβyβ¼ − 1
2ðεαqβ− εβqαÞyβ: ð18Þ
The Eq. (18)is plugged into Eq.(17), as a result of which we obtain Sfree→ −1
2ðεαqβ−εβqαÞ Z
d4yyβSfreeðx − yÞγαSfreeðyÞ;
ð19Þ After some calculations for Sfree
q and Sfreeb we get
Sfree q ¼ eq 32π2x2ðεαqβ− εβqαÞð=xσαβþ σαβ=xÞ; Sfree b ¼ −i ebmb 32π2ðεαqβ− εβqαÞ 2σαβK0ðmb ffiffiffiffiffiffiffiffi −x2 p Þ þK1ðmb ffiffiffiffiffiffiffiffi −x2 p Þ ffiffiffiffiffiffiffiffi −x2 p ð=xσαβþ σαβ=xÞ : ð20Þ
In second case one of the light quark propagators in Eqs.(13)–(14)are replaced by Sab
αβ → −14ð¯qaΓiqbÞðΓiÞαβ; ð21Þ
and the remaining propagators are full quark propagators including the perturbative as well as the nonperturbative contributions. Here as an example, we give a short detail of the calculations of the QCD representations. In second case for simplicity, we only consider the first term in Eq. (13),
ΠQCD μν ðp; qÞ ¼ −iϵ~ϵϵ 0~ϵ0 2 Z d4xeipxh0jTr½γ5~Saa0 u ðxÞγ5Sbb 0 b ðxÞTr½γμ~S e0e b ð−xÞγνSd 0d d ð−xÞj0iFþ ð22Þ
By replacing one of light propagators with the expressions in Eq. (21)and making use of ¯qaðxÞΓiqa0ð0Þ →1
3δaa 0
¯qðxÞΓiqð0Þ; ð23Þ
the Eq.(22) takes the form ΠQCD μν ðp; qÞ ¼ −iϵ~ϵϵ 0~ϵ0 2 Z d4xeipx Tr½γ5Γiγ5Sbb0 b ðxÞTr½γμ~S e0e b ð−xÞγνSd 0d d ð−xÞ 1 12δaa 0 þ Tr½γ5~Saa 0 u ðxÞγ5Sbb 0 b ðxÞTr½γμ~S e0e b ð−xÞγνΓi 112δdd 0 hγðqÞj¯qðxÞΓiqð0Þj0i þ ; ð24Þ
whereΓi¼ I; γ5;γμ; iγ5γμ;σμν=2. Similarly, when a light propagator interacts with the photon, a gluon may be released from one of the remaining three propagators. The expression obtained in this case is as follows:
ΠQCD μν ðp; qÞ ¼ −iϵ~ϵϵ 0~ϵ0 2 Z d4xeipx Tr½γ5Γiγ5Sbb0 b ðxÞTr½γμ~S e0e b ð−xÞγνSd 0d d ð−xÞ δabδa0b0−1 3δaa 0 δbb0 þ δaeδa0e0 −1 3δaa 0 δee0 þ δadδa0d0 −1 3δaa 0 δdd0 þ Tr½γ5~Saa 0 u ðxÞγ5Sbb 0 b ðxÞTr½γμ~S e0e b ð−xÞγνΓi δdbδd0b0−1 3δdd 0 δbb0 þ δdeδd0e0−1 3δdd 0 δee0 þ δadδa0d0−1 3δaa 0 δdd0 1 32hγðqÞj¯qðxÞΓiGμνðvxÞqð0Þj0i þ ; ð25Þ where we inserted ¯qaðxÞΓiGbb0 μνðvxÞqa0ð0Þ →18 δabδa0b0−1 3δaa 0 δbb0 ¯qðxÞΓiGμνðvxÞqð0Þ: ð26Þ As is seen, there appear matrix elements such as hγðqÞj¯qðxÞΓiqð0Þj0i and hγðqÞj¯qðxÞΓiGμνðvxÞqð0Þj0i, rep-resenting the nonperturbative contributions. These matrix elements can be expressed in terms of photon DAs and wave functions with definite twists, whose expressions are given in Appendix A. The QCD representation of the correlation function is obtained by using Eqs. (13)–(26). Then, the Fourier transformation is applied to transfer expressions in x-space to the momentum space.
The sum rule for the magnetic dipole moment are obtained by matching the expressions of the correlation function in terms of QCD parameters and its expression in
terms of the hadronic parameters, using their spectral representation. To eliminate the contributions of the excited and continuum states in the spectral representation of the correlation function, a double Borel transformation with respect to the variables p2andðp þ qÞ2is applied. After the transformation, these contributions are exponentially sup-pressed. Eventually, we choose the structureðε:pÞðpμqν− qμpνÞ for the magnetic dipole moment and obtain
μDi¼e m2 Zb=M2 λ2 Zbm 2 Zb ½Π1þ Π2; ð27Þ μMol¼e m2Zb=M2 λ2 Zbm 2 Zb ½Π3þ Π4: ð28Þ The explicit forms of the functions that appear in the above sum rules are given as follows:
Π1¼ 3m 4 b 256π6ðeu− edÞf32N½3; 3; 0 − 2M2N½3; 3; 1 − 16mbN½3; 4; 1 þ mbM2N½3; 4; 2g −m2bhg2sG2i 9216π6 ðeu− edÞð−M2N½1; 1; 0 þ 2mbN½1; 2; 0Þ −m2bhg2sG2i 147456π6ð2mbM2N½1; 2; 1 þ π2h¯qqið16N½1; 2; 1 þ 5N½1; 2; 2ÞÞ þm4bhg2sG2i 294912π6ðeu− edÞð16N½1; 3; 1 − M2N½1; 3; 2Þ −m2bhg2sG2i 294912π6ðeu− edÞð128N½2; 2; 0 − 8ð2m2bþ M2ÞN½2; 2; 1 þ m2bM2N½2; 2; 2Þ − m3b
98304π6ðeu− edÞð16ðhg2sG2i − 192π2mbh¯qqiÞN½2; 3; 1 þ M2ð13hg2sG2i − 960π2mbh¯qqiÞN½2; 3; 2 −m3bm20h¯qqi
384M8π4 ðeuþ edÞð64m6bFlP½−3; 4; 0 − 48m4bFlP½−2; 4; 0 þ 12m2bFlP½−1; 4; 0 − FlP½0; 4; 0Þ
−mbm20hg2sG2ih¯qqi
Π2¼ −mbhg2sG2ih¯qqi2
2592M10π2 ðeu− edÞðm20− M2ÞI3½hγð4m2bFlNP½0; 1; 0 − FlNP½−1; 1; 0Þ þmbm20hg2sG2ih¯qqi2
10368M10π2 ðeu− edÞI3½hγð4m2bFlNP½2; 1; 1 − FlNP½3; 1; 1Þ −f3γm20hg2sG2ih¯qqi
110592M10π2 ½−ð4eu− 3edÞψaðu0Þ þ 2edI3½ψνð16m4bFlNP½1; 2; 1 − 8m2bFlNP½2; 2; 1 þ FlNP½3; 2; 1Þ −mbm20hg2sG2ih¯qqi
995328M12π2 ð8eu− 5edÞðAðu0Þ þ 8I3½hγÞð16m4bFlNP½2; 3; 2 − 8m2bFlNP½3; 3; 2 þ FlNP½4; 3; 2Þ þ mbh¯qqi2
497664M12π2ð8eu− 5edÞ½−hg2sG2ið−ð5m20− 2M2ÞAðu0Þ − 2m20χM2φγðu0ÞÞ − 8f−5m20hg2sG2i þ 2ðM2hg2sG2i þ 432m2
bm20M2ÞI3½hγgð16m4bFlNP½0; 3; 1 − 8m2bFlNP½1; 3; 1 þ FlNP½2; 3; 1Þ
þ mb
165888M10π4½ðeu− edÞf3γð7M2hg2sG2i − 576π2mbh¯qqiðm20− M2ÞÞψaðu0Þ þ 72π2h¯qqi2ðm2
0− M2Þð−2euI1½~S þ edð3I2½T1 − 3I2½T2 − 5I2½~SÞÞ
×ð−64m6bFlNP½3; 4; 0 þ 48m4bFlNP½2; 4; 0 − 12m2bFlNP½1; 4; 0 þ FlNP½0; 4; 0Þ − mbm20h¯qqi
9216M10π4½−8ðeu− edÞmbf3γψaðu0Þ − h¯qqið−2euI1½~S þ edð3I2½T1 − 3I2½T2 − 5I2½~SÞÞ ×ð64m6bFlNP½−1; 4; 1 − 48m4bFlNP½0; 4; 1 þ 12m2bFlNP½1; 4; 1 − FlNP½2; 4; 1Þ þ h¯qqi 1990656M12π4½hg2sG2if2euð−3M4ð16m4bFlNP½1; 3; 0 − 8m2bFlNP½0; 3; 0 þ FlNP½−1; 3; 0Þ − 32π2m bh¯qqið5m20− 4M2Þð16m4bFlNP½2; 3; 0 − 8m2bFlNP½1; 3; 0 þ FlNP½0; 3; 0ÞÞ þ edð40π2m bh¯qqið5m20− 4M2Þð16m4bFlNP½2; 3; 0 − 8m2bFlNP½1; 3; 0 þ FlNP½0; 3; 0Þ þ 3M4ð−48m6 bFlNP½2; 3; 0 þ 56m4bFlNP½1; 3; 0 − 19m2bFlNP½0; 3; 0 þ 2FlNP½−1; 3; 0ÞÞgAðu0Þ þ 8mbf−4ð8eu− 5edÞπ2M2χhg2
sG2ih¯qqiðm20− M2Þφγðu0Þ − 3ð40eu− 43edÞmbM4hg2sG2i þ 8ð8eu− 5edÞπ2hg2
sG2ih¯qqið5m20− M2Þ − 432ðeu− edÞπ2mb2M2h¯qqiðm20− 4M2ÞgI3½hγ
×ð16m4bFlNP½2; 3; 0 − 8m2bFlNP½1; 3; 0 þ FlNP½0; 3; 0Þ; ð30Þ Π3¼ 9m 4 b 1024π6ðeu− edÞf32N½3; 3; 0 − 2M2N½3; 3; 1 − 16mbN½3; 4; 1 þ mbM2N½3; 4; 2g − m3b 32768π6ðeu− edÞðhg2sG2i þ 48π2mbh¯qqiÞð16N½2; 3; 1 þ 5M2N½2; 3; 2Þ −m3bm20h¯qqi 512M8π2 ðeu− edÞð64m6b4FlP½−3; 4; 0 − 48m4bFlP½−2; 4; 0 þ 12m2bFlP½−1; 4; 0 þ FlP½0; 4; 0Þ; ð31Þ and Π4¼m2bhg2sG2ih¯qqi
294912π4 ½euð−3I1½S − 2I1½~SÞ þ edð3I2½S þ 2I2½~SÞðM2N½1; 2; 2 − 8N½1; 2; 1Þ þ 3m2bh¯qqi
64π4 ðeu− edÞI3½hγðM2N½2; 3; 2 − 8N½2; 3; 1Þ þ3m4bf3γ
128π4 ðeuþ edÞψaðu0ÞðM2N½3; 3; 2 − 8N½3; 3; 1Þ −m3bm20h¯qqi2
96M10π2 ðeu− edÞI3½hγð64m6bFlNP½0; 3; 1 − 48m4bFlNP½1; 3; 1 þ FlNP½2; 3; 1Þ −m3bm20f3γh¯qqi2
− 48FlNP½0; 4; 1 þ 12m2
bFlNP½1; 4; 1 − FlNP½2; 4; 1Þ þ mbf3γ
18432M10π4ðeuþ edÞ½ð−M2hg2sG2i − 48π2mbh¯qqiðm20− M2ÞÞψaðu0Þ
×ð−64m6bFlNP½3; 4; 0 þ 48m4bFlNP½2; 4; 0 − 12m2bFlNP½1; 4; 0 þ FlNP½0; 4; 0Þ − mbh¯qqi
1152M10π4ðeu− edÞ½ð−M2hg2sG2i − 48π2mbh¯qqiðm20− M2ÞÞI3½hγ
×ð−16m4bFlNP½2; 3; 0 − 8m2bFlNP½1; 3; 0 þ FlNP½0; 3; 0Þ; ð32Þ
where, mb is the mass of the b quark, eq is the corresponding electric charge, χ is the magnetic suscep-tibility of the quark condensate, m20¼ h¯qgσαβGαβqi= h¯qqi, h¯qqi and hg2
sG2i are quark and gluon condensates, respectively.
The functions N½n; m; k, FlP½n; m; k, FlNP½n; m; k, I1½A, I2½A, and I3½A are defined as:
N½n; m; k ¼ Z ∞ 0 dt Z ∞ 0 dt 0 e−mb=2ðtþt 0Þ tnðmb t þ mb t0Þ kt0m; FlP½n; m; k ¼ Z s 0 4m2 b ds Z s 4m2 b dle −l2=ϕ lnðl − sÞm ð4m2 b− lÞ2ϕk ; FlNP½n; m; k ¼ Z s 0 4m2 b ds Z s 4m2 b dle −l2=β lnðl − sÞm ðl − 2m2 bÞβk ; I1½A ¼ Z Dαi Z 1 0 dvAðα¯q;αq;αgÞ ×δðαqþ¯vαg− u0Þ; I2½A ¼ Z Dαi Z 1 0 dvAðα¯q;αq;αgÞ ×δðα¯qþ vαg− u0Þ; I3½A ¼ Z 1 0 duAðuÞ; ð33Þ where β ¼ 4lM2− 16m2 bM2; ϕ ¼ 8lM2− 32m2 bM2:
The functionsΠ1andΠ3indicate the case that one of the quark propagators enters the perturbative interaction with the photon and the remaining three propagators are taken as full propagators. The functions Π2 and Π4 show the contributions that one of the light quark propagators enters the nonperturbative interaction with the photon and the remaining three propagators are taken as full propagators. The reader can find some details about the calculations such as Fourier and Borel transformations as well as continuum subtraction in Appendix C of Ref.[49].
As we already mentioned, the calculations have been done in the fixed-point gauge, xμAμ¼ 0, for simplicity. In order to show whether our results are gauge invariant or not we examine the Lorentz gauge,∂μAμ¼ 0. In this gauge, the electromagnetic vector potential is written as
AμðxÞ ¼ εμe−iq:x; ð34Þ with εμqμ¼ 0. In this gauge, the corresponding gauge invariant electromagnetic field strength tensor is written as Fμν¼ iðεμqν− qμενÞe−iq:x: ð35Þ We repeat all the calculations in this gauge and find the same results for the magnetic dipole moment of the state under consideration. Therefore the results obtained in the present study are gauge invariant.
III. NUMERICAL ANALYSIS AND CONCLUSION
In this section, we numerically analyze the results of calculations for magnetic dipole moment of the Zb state. We use mZb¼10607.22MeV, ¯mbðmbÞ¼ð4.18þ0.04−0.03ÞGeV [50], f3γ ¼ −0.0039 GeV2[46],h¯qqið1 GeVÞ ¼ ð−0.24 0.01Þ3GeV3 [51], m2
0¼ 0.8 0.1 GeV2, hg2sG2i ¼ 0.88 GeV4 [8], and χð1 GeVÞ ¼ −2.85 0.5 GeV−2 [52]. To evaluate a numerical prediction for the magnetic moment, we need also specify the values of the residue of the Zbstate. The residue is obtained from the mass sum rule as λZb ¼ mZbfZbwith fZb ¼ ð2.79
þ0.55
−0.65Þ × 10−2 GeV4[19]for
diquark-antidiquark picture and λZb ¼ 0.27 0.07 GeV5 [23]for molecular picture. The parameters used in the photon DAs are given in Appendix A, as well.
The estimations for the magnetic dipole moment of the Zb state depend on two auxiliary parameters; the continuum threshold s0 and Borel mass parameter M2. The continuum threshold is not completely an arbitrary parameter, and there are some physical restrictions for it. The s0 signals the scale at which, the excited states and continuum start to contribute to the correlation function. The working interval for this parameter is chosen such that the maximum pole contribution is acquired and the results relatively weakly depend on
its choices. Our numerical calculations lead to the interval ½119–128 GeV2 for this parameter. The Borel parameter can vary in the interval that the results weakly depend on it according to the standard prescriptions. The upper bound of it is found demanding the maximum pole contributions and its lower bound is found the conver-gence of the operator product expansion and exceeding of the perturbative part over nonperturbative contribu-tions. Under these constraints, the working region of the Borel parameter is determined as 15 GeV2≤ M2≤ 17 GeV2.
In Fig. 1, we plot the dependency of the magnetic dipole moment of the Zb state on M2 at different fixed values of the continuum threshold. From the figure we observe that the results considerably depend on the variations of the Borel parameter. The magnetic dipole moment is stable under variation of s0 in its working region. In Fig. 2, we show the contributions of Π1, Π2, Π3, and Π4 functions to the results obtained at average value of s0 with respect to the Borel mass parameter. It is clear that Π1is dominant in the results obtained when
using diquark-antidiquark current but Π3 is dominant while using the molecular current. The contribution of the Π2 and Π4 functions seems to be almost zero. When the results are analyzed in detail, almost (95–97)% of the total contribution comes from the perturbative part and the remaining (3–5)% belongs to the nonperturbative contributions.
Our predictions on the numerical value of the magnetic dipole moment in both pictures are presented in Table I. The errors in the results come from the variations in the calculations of the working regions of M2 and from the uncertainties in the values of the input parameters as well as the photon DAs. We shall remark that the main source of
15 15.5 16 16.5 17 M2[GeV2] -0.5 0 0.5 1 1.5 2 2.5 3 3.5 |μZb | [ μΝ ]-Diquark-antidiquark s 0 = 119 GeV 2 s0 = 124 GeV2 s 0 = 128 GeV 2 (a) 15 15.5 16 16.5 17 M2[GeV2] -0.5 0 0.5 1 1.5 2 2.5 3 3.5 |μZb | [ μΝ ]-Molecule s0 = 119 GeV2 s 0 = 124 GeV 2 s0 = 128 GeV2 (b)
FIG. 1. The dependence of the magnetic moment for Zb state;
on the Borel parameter squared M2at different fixed values of the continuum threshold. 15 15.5 16 16.5 17 M2[GeV2] -0.5 0 0.5 1 1.5 2 2.5 3 3.5 μ Zb [μ N ]-Diquark-antidiquark Π1 Π2 Π1 + Π2 (a) 15 15.5 16 16.5 17 M2[GeV2] -0.5 0 0.5 1 1.5 2 2.5 3 3.5 μ Zb [μ N ]-Molecule Π3 Π4 Π3 + Π4 (b)
FIG. 2. Comparison of the contributions to the magnetic moment with respect to M2at average value of s0.
TABLE I. Results of the magnetic moment (in units ofμN) for
Zb state.
Picture jμZbj
Diquark-antidiquark 1.73 0.63
uncertainties is the variations with respect to variations of M2.
In conclusion, we have computed the magnetic dipole moment of the Zbð10610Þ by modeling it as the diquark-antidiquark and molecule states. In our calculations we have employed the light-cone QCD sum rule in electro-magnetic background field. Although the central values of the magnetic dipole moment obtained via two pictures differ slightly from each other but they are consistent within the errors. In Ref. [19], both the spectroscopic parameters and some of the strong decays of the Zb state have been studied using diquark-anti-diquark interpolating current. Although the obtained mass in [19] is in agreement with the experimental data, the result obtained for the width of Zb in the diquark-antidiquark picture in [19] differ considerably from the experimental data. They suggested, as a result, that the Zb state may not have a pure diquark-anti-diquark structure. When we combine the obtained results in the present study with those of the predictions on the mass obtained via both pictures in the literature and those result obtained for the width of Zb in Ref. [19]we conclude that both pictures can be considered for the internal structure of Zb. May be a mixed current will be a better choice for interpolating this particle. More theoretical and experimental studies are still needed to be performed in this respect.
Finally, the magnetic dipole moment encodes impor-tant information about the inner structure of particles
and their geometric shape. The results obtained for the magnetic dipole moment of Zb state in both the diquark-antidiquark and molecule pictures, within a factor 2, are of the same order of magnitude as the proton’s magnetic moment and not such small that it appears hopeless to try to measure the value of the magnetic dipole moment of this state. By the recent progresses in the exper-imental side, we hope that we can measure the multipole moments of the newly founded exotic states, especially the Zb particle in future. Comparison of any exper-imental data on the magnetic dipole moment of Zb will be useful to gain exact knowledge on its quark organ-izations and will help us in the course of undestanding the structures of the newly observed exotic states and their quantum chromodynamics.
ACKNOWLEDGMENTS
This work has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under the Grant No. 115F183.
APPENDIX: PHOTON DAS AND WAVE FUNCTIONS
In this appendix, we present the definitions of the matrix elements of the forms hγðqÞj¯qðxÞΓiqð0Þj0i and hγðqÞj¯qðxÞΓiGμνqð0Þj0i in terms of the photon DAs and wave functions[46],
hγðqÞj¯qðxÞγμqð0Þj0i ¼ eqf3γ εμ−qμεx qx Z 1 0 due i¯uqxψvðuÞ hγðqÞj¯qðxÞγμγ5qð0Þj0i ¼ −1 4eqf3γϵμναβενqαxβ Z 1 0 due i¯uqxψaðuÞ
hγðqÞj¯qðxÞσμνqð0Þj0i ¼ −ieqh¯qqiðεμqν−ενqμÞ Z 1 0 due i¯uqx χφγðuÞþx 2 16AðuÞ −2ðqxÞi eq¯qq xν εμ−qμεx qx −xμ εν−qνεx qx Z 1 0 due i¯uqxh γðuÞ
hγðqÞj¯qðxÞgsGμνðvxÞqð0Þj0i ¼ −ieqh¯qqiðεμqν−ενqμÞ Z
Dαieiðα¯qþvαgÞqxSðαiÞ hγðqÞj¯qðxÞgs~GμνðvxÞiγ5qð0Þj0i ¼ −ieqh¯qqiðεμqν−ενqμÞ
Z
Dαieiðα¯qþvαgÞqx~SðαiÞ hγðqÞj¯qðxÞgs~GμνðvxÞγαγ5qð0Þj0i ¼ eqf3γqαðεμqν−ενqμÞ
Z
Dαieiðα¯qþvαgÞqxAðαiÞ hγðqÞj¯qðxÞgsGμνðvxÞiγαqð0Þj0i ¼ eqf3γqαðεμqν−ενqμÞ
Z
Dαieiðα¯qþvαgÞqxVðαiÞ hγðqÞj¯qðxÞσαβgsGμνðvxÞqð0Þj0i ¼ eqh¯qqi εμ−qμqxεx gαν− 1 qxðqαxνþqνxαÞ qβ − εμ−qμqxεx gβν− 1 qxðqβxνþqνxβÞ qα− εν−qνqxεx gαμ− 1 qxðqαxμþqμxαÞ qβ
þ εν−qνq:xεx gβμ− 1 qxðqβxμþqμxβÞ qα Z Dαieiðα¯qþvαgÞqxT 1ðαiÞ þ εα−qαqxεx gμβ−1 qxðqμxβþqβxμÞ qν− εα−qαqxεx gνβ− 1 qxðqνxβþqβxνÞ qμ − εβ−qβεx qx gμα−1 qxðqμxαþqαxμÞ qν þ εβ−qβεxqx gνα−1 qxðqνxαþqαxνÞ qμ Z Dαieiðα¯qþvαgÞqxT 2ðαiÞ þ 1 qxðqμxν−qνxμÞðεαqβ−εβqαÞ Z
Dαieiðα¯qþvαgÞqxT3ðαiÞ
þ 1 qxðqαxβ−qβxαÞðεμqν−ενqμÞ Z Dαieiðα¯qþvαgÞqxT 4ðαiÞ ;
whereφγðuÞ is the leading twist-2, ψvðuÞ, ψaðuÞ, AðαiÞ, and VðαiÞ, are the twist-3, and h
γðuÞ, AðuÞ, SðαiÞ, ~SðαiÞ, T1ðαiÞ, T2ðαiÞ, T3ðαiÞ, and T4ðαiÞ are the twist-4 photon DAs. The measure Dαiis defined as
Z Dαi¼ Z 1 0 dα¯q Z 1 0 dαq Z 1 0 dαgδð1 − α¯q− αq− αgÞ: The expressions of the DAs entering into the above matrix elements are defined as: φγðuÞ ¼ 6u¯uð1 þ φ2ðμÞC 3 2 2ðu − ¯uÞÞ; ψvðuÞ ¼ 3ð3ð2u − 1Þ2− 1Þ þ 3 64ð15wVγ − 5wAγÞð3 − 30ð2u − 1Þ2þ 35ð2u − 1Þ4Þ; ψaðuÞ ¼ ð1 − ð2u − 1Þ2Þð5ð2u − 1Þ2− 1Þ5
2 1 þ 9 16wVγ − 3 16wAγ ; hγðuÞ ¼ −10ð1 þ 2κþÞC12 2ðu − ¯uÞ; AðuÞ ¼ 40u2¯u2ð3κ − κþþ 1Þ þ 8ðζþ
2 − 3ζ2Þ½u¯uð2 þ 13u¯uÞ þ 2u3ð10 − 15u þ 6u2Þ lnðuÞ þ 2¯u3ð10 − 15¯u þ 6¯u2Þ lnð¯uÞ; AðαiÞ ¼ 360αqα¯qα2 g 1 þ wA γ 12ð7αg− 3Þ ; VðαiÞ ¼ 540wV γðαq− α¯qÞαqα¯qα2g; T1ðαiÞ ¼ −120ð3ζ2þ ζþ2Þðα¯q− αqÞα¯qαqαg; T2ðαiÞ ¼ 30α2gðα¯q− αqÞððκ − κþÞ þ ðζ1− ζþ 1Þð1 − 2αgÞ þ ζ2ð3 − 4αgÞÞ; T3ðαiÞ ¼ −120ð3ζ2− ζþ2Þðα¯q− αqÞα¯qαqαg; T4ðαiÞ ¼ 30α2gðα¯q− αqÞððκ þ κþÞ þ ðζ1þ ζþ1Þð1 − 2αgÞ þ ζ2ð3 − 4αgÞÞ; SðαiÞ ¼ 30α2gfðκ þ κþÞð1 − αgÞ þ ðζ1þ ζþ 1Þð1 − αgÞð1 − 2αgÞ þ ζ2½3ðα¯q− αqÞ2− αgð1 − αgÞg; ~SðαiÞ ¼ −30α2gfðκ − κþÞð1 − αgÞ þ ðζ 1− ζþ1Þð1 − αgÞð1 − 2αgÞ þ ζ2½3ðα¯q− αqÞ2− αgð1 − αgÞg: Numerical values of parameters used in DAs areφ2ð1 GeVÞ ¼ 0, wV
γ ¼ 3.8 1.8, wAγ ¼ −2.1 1.0, κ ¼ 0.2, κþ ¼ 0, ζ1¼ 0.4, ζ2¼ 0.3.
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