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BcB_{c} Meson Spectrum Via Dyson-Schwinger Equation and Bethe-Salpeter Equation Approach

Muyang Chen [email protected] Department of Physics, Hunan Normal University, Changsha 410081, China    Lei Chang [email protected] School of Physics, Nankai University, Tianjin 300071, China    Yu-xin Liu [email protected] Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China
Abstract

We predict the masses of the lowlying BcB_{c} mesons with JP=0, 1, 0+, 1+, 2+J^{P}=0^{-},\,1^{-},\,0^{+},\,1^{+},\,2^{+}, using a flavor dependent interaction pattern which gives an unified successful description of the light, heavy-light and heavy mesons and is also appliable to the radial excited heavy mesons. The errors are controlled carefully. With the errors from the RL approximation subduced, our predictions are consistent with the lQCD and quark model results, which supports strongly that the flavor dependent interaction pattern is reasonable. Our predictions provide significant guides to the experiment search of the BcB_{c} mesons.

I Introduction

The BcB_{c} mesons are the only heavy mesons containing two different flavor valence quarks, one charm quark and one bottom quark. The quark flavor difference forbids their annihilation into gluons or photons, so that the ground state pseudoscalar Bc(1S)B_{c}(1S) can only decay weakly, which makes it particularly interesting for the study of weak interaction. The excited BcB_{c} states lying below the BD production threshold decay into the ground state via hadronic or radiative transitions. As a result, the BcB_{c} mesons are supposed to be much more stable than their counterparts in the cc¯c\bar{c} and bb¯b\bar{b} systems, and spin splittings can be studied in detail. However, in the experimental aspect, the BcB_{c} mesons are much less explored than the charmonium and bottomonium due to the small production rate, as the dominant production mechanism requires the production of both cc¯c\bar{c} and bb¯b\bar{b} pairs. The ground state pseudoscalar Bc(1S)B_{c}(1S) meson is first observed in 1998 by CDF collaboration at the Tevatron collider Abe1998 . The vector meson, Bc(1S)B_{c}^{*}(1S), decays into Bc(1S)B_{c}(1S) via radiative transition, Bc(1S)Bc(1S)γB_{c}^{*}(1S)\to B_{c}(1S)\gamma. It’s challenging to detect the emitted photon due to the low energy. The vector meson mass, MBc(1S)M_{B_{c}^{*}(1S)}, is still not determined. Until recently two excited BcB_{c} states were reported by CMS collaboration Sirunyan2019 and LHCb collaboration Aaij2019 , pushing forward the study of the BcB_{c} mesons. More BcB_{c} mesons are expected to be discovered Chang2015 . In the theoretical aspect, the prediction of the BcB_{c} mesons have existed since 40 years ago Eichten1981 , and vairous model calculations occured in these year Kwong1991 ; Eichten1994 ; Gershtein1995 ; Gupta1996 ; Fulcher1999 ; Ebert2003 ; Godfrey2004 ; Li2019 . However, these predictions vary widely. For example, the 1S-spin splitting, i.e. MBc(1S)MBc(1S)M_{B^{*}_{c}(1S)}-M_{B_{c}(1S)}, spreads over a range of 4090 MeV40-90\textmd{ MeV} Kwong1991 ; Eichten1994 ; Gershtein1995 ; Gupta1996 ; Fulcher1999 ; Ebert2003 ; Godfrey2004 ; Li2019 . The predictions of the excited states are even more scattered. First principle calculations are of cource essential to study the BcB_{c} mesons with containable precision. The first principle lattice QCD (lQCD) studies of the BcB_{c} mesons are still lacking Allison:2004be ; Gregory2010 ; Dowdall2012 ; Wurtz2015 ; Mathur2018 . A recent lQCD calculation predicts the BcB_{c} meson masses with JP=0, 1, 0+, 1+J^{P}=0^{-},\,1^{-},\,0^{+},\,1^{+} with high precision Mathur2018 , where JJ is the angular momentum and PP is the PP-parity (CC in the later text is the CC-parity).

The Dyson-Schwinger equation and Bethe-Salpeter equation (DSBSE) approach, a nonperturbative quantum chromodynamic (QCD) approach solving QCD via the basic degree of freedom, i.e. the quarks and the gluons, is complementary to lQCD. In this approach, a meson corresponds to a pole in the quark-antiquark scattering kernel Itzykson1980 , and the amplitude is solved by the Bethe-Salpeter equation (BSE) Salpeter1951 . The propagators of quarks and gluons in the scattering kernel are then solved by the Dyson-Schwinger equations (DSE) Dyson1949 ; Schwinger1951 . However, the DSEs are an infinite tower of coupled equations and no exact solution has been gained. A practical way to solve the problem of hadron spectrum is building a quark-gluon-vertex, constructing a scattering kernel, the gluon propagator being modelled by an effective interation, then the BSE of the hadron amplitude and DSE of the quark propagator could be solved consistently. While the forms of the quark-gluon-vertex and the scattering kernel have been investigated Chang2009 , the most widely used and technically simple one is the rainbow ladder (RL) approximation. This scheme has gained many phenomenological succsess in the hadron spectrum, the form factors, the distribution functions, etc Roberts1994 ; Alkofer2001 ; Maris2003 . The RL approximation is perfect for the light pseudoscalar mesons Maris1997 ; Maris1998 and fairly well for the light vector mesons Maris1999 . A possible way to solve the light meson with orbit angular moment L1L\geq 1 is given in Ref. Chang2009 . The mass spectra of the charmonium and bottomonium with angular momentum J<3J<3 have been investigated in the RL approximation, with relative errors a few percents Fischer2015 ; Hilger2015 ; Chen2017 . However, the original RL approximation fails to describe the heavy-light mesons due to the lacking of the flavor asymmetry. In Ref. Chen2019 we build an interaction model, which takes into account the quark flavor dependence of the effective quark-antiquark interaction properly and preserves the axial-vector Ward Takahashi identity excellently. Our model not only gives a successful unified description of the light, heavy-light and heavy pseudoscalar and vector mesons, but also reduces the error of the Bc(1S)B_{c}(1S) meson mass from 110 MeV110\textmd{ MeV} Qin2018 to about 15 MeV15\textmd{ MeV}. In Ref. Chang2019 our model is applied to the radial excited Bc(2S)B_{c}(2S) and Bc(2S)B_{c}^{*}(2S) mesons. The mass spectrum is reasonable and we predict the decay constants of the radial excited BcB_{c} mesons for the first time.

In this article, we extend our study to BcB_{c} mesons with JP=0,1,0+,1+J^{P}=0^{-},1^{-},0^{+},1^{+} and 2+2^{+}. In section II we give a complete introduction of our model and the formulas. The results are analyzed in section III. We summarize our investigations in section IV.

II Our model

In this Poincare´\acute{\text{e}} covariant framework the quark propagator is solved by the Gap equation Dyson1949 ; Schwinger1951 ; Itzykson1980

Sf1(k)\displaystyle S_{f}^{-1}(k) =\displaystyle= Z2(iγk+Zmmf)\displaystyle Z_{2}(i\gamma\cdot k+Z_{m}m_{f}) (1)
+43g¯2Z1dqΛDμν(l)γμSf(q)Γνf(k,q),\displaystyle+\frac{4}{3}\bar{g}^{2}Z_{1}\int^{\Lambda}_{dq}D_{\mu\nu}(l)\gamma_{\mu}S_{f}(q)\Gamma^{f}_{\nu}(k,q),

where SfS_{f} is the quark propagator. Z1Z_{1}, Z2Z_{2}, ZmZ_{m} are the renormalisation constants of the quark-gluon-vertex, the quark field and the quark mass respectively. DμνD_{\mu\nu} is the gluon propagator, Γνf\Gamma^{f}_{\nu} the quark-gluon-vertex. mfm_{f} is the current quark mass, g¯\bar{g} the coupling constant. f={u,d,s,c,b,t}f=\{u,d,s,c,b,t\} represents the quark flavor. l=kql=k-q, with ll the momentum of the gluon, kk and qq the momentum of outer and inner quark. dqΛ=Λd4q/(2π)4\int^{\Lambda}_{dq}=\int^{\Lambda}d^{4}q/(2\pi)^{4} stands for a Poincare´\acute{\text{e}} invariant regularized integration, with Λ\Lambda the regularization scale. The Bethe-Salpeter amplitude (BSA) of the meson is solved by the BSE Salpeter1951 ; Itzykson1980 ,

[Γfg(k;P)]βα=dqΛ[Kfg(k,q;P)]σβαδ[χfg(q;P)]δσ,\big{[}\Gamma^{fg}(k;P)\big{]}^{\alpha}_{\beta}=\int^{\Lambda}_{dq}\big{[}K^{fg}(k,q;P)\big{]}^{\alpha\delta}_{\sigma\beta}\big{[}\chi^{fg}(q;P)\big{]}^{\sigma}_{\delta}, (2)

where Γfg\Gamma^{fg} is the BSA, KfgK^{fg} the quark-antiquark scattering kernel. kk and PP are the relative and the total momentum of the meson, with P2=M2P^{2}=-M^{2} and MM the meson mass. χfg(q;P)=Sf(q+)Γfg(q;P)Sg(q)\chi^{fg}(q;P)=S_{f}(q_{+})\Gamma^{fg}(q;P)S_{g}(q_{-}), where χfg\chi^{fg} is the wave function, and q+=q+ιP/2q_{+}=q+\iota P/2, q=q(1ι)P/2q_{-}=q-(1-\iota)P/2. ι\iota is the partitioning parameter describing the momentum partition between the quark and antiquark and dosen’t affect the physical observables. α\alpha, β\beta, σ\sigma and δ\delta are the Dirac indexes.

The RL approximation is making the following replacement in Eq.(1) and Eq.(2) Chen2019 ,

g¯2Z1Dμν(l)Γνf(k,q)[Z2]2D~μνf(l)γν,\displaystyle\bar{g}^{2}Z_{1}D_{\mu\nu}(l)\Gamma^{f}_{\nu}(k,q)\to[Z_{2}]^{2}\tilde{D}^{f}_{\mu\nu}(l)\gamma_{\nu}, (3)
[Kfg(k,q;P)]σβαδ43[Z2]2D~μνfg(l)[γμ]σα[γν]βδ,\displaystyle\big{[}K^{fg}(k,q;P)\big{]}^{\alpha\delta}_{\sigma\beta}\to-\frac{4}{3}[Z_{2}]^{2}\tilde{D}^{fg}_{\mu\nu}(l)[\gamma_{\mu}]^{\alpha}_{\sigma}[\gamma_{\nu}]^{\delta}_{\beta}, (4)

where D~μνfg(l)=(δμνlμlνl2)𝒢fg(l2)\tilde{D}^{fg}_{\mu\nu}(l)=\left(\delta_{\mu\nu}-\frac{l_{\mu}l_{\nu}}{l^{2}}\right)\mathcal{G}^{fg}(l^{2}) and D~μνf(l)=D~μνff(l)\tilde{D}^{f}_{\mu\nu}(l)=\tilde{D}^{ff}_{\mu\nu}(l) are the effective quark-antiquark interactions. The dressed function 𝒢fg(s)\mathcal{G}^{fg}(s) is composed of a flavor dependent infrared part and a flavor independent ultraviolet part Chen2019 ,

𝒢fg(s)\displaystyle\mathcal{G}^{fg}(s) =\displaystyle= 𝒢IRfg(s)+𝒢UV(s),\displaystyle\mathcal{G}^{fg}_{IR}(s)+\mathcal{G}_{UV}(s), (5)
𝒢IRfg(s)\displaystyle\mathcal{G}^{fg}_{IR}(s) =\displaystyle= 8π2Dfωf2Dgωg2es/(ωfωg),\displaystyle 8\pi^{2}\frac{D_{f}}{\omega_{f}^{2}}\frac{D_{g}}{\omega_{g}^{2}}e^{-s/(\omega_{f}\omega_{g})}, (6)
𝒢UV(s)\displaystyle\mathcal{G}_{UV}(s) =\displaystyle= 8π2γm(s)ln[τ+(1+s/ΛQCD2)2],\displaystyle\frac{8\pi^{2}\gamma_{m}\mathcal{F}(s)}{\text{ln}[\tau+(1+s/\Lambda^{2}_{QCD})^{2}]}, (7)

where s=l2s=l^{2}. 𝒢IRfg(s)\mathcal{G}^{fg}_{IR}(s) is the infrared interaction responsible for dynamical chiral symmetry breaking (DCSB), with (Df2ωf)1/3(D_{f}^{2}\omega_{f})^{1/3} expressing the interaction strength and ωf\omega_{f} the interaction width in the momentum space. The gaussian form is used as it enables the natural extraction of a monotonic running-coupling and gluon mass Qin2011 . ff and gg label the quark flavors. The equality in ff and gg means that the quark and antiquark contribute equally to the interaction strength and width. 𝒢UV(s)\mathcal{G}_{UV}(s) keeps the one-loop perturbative QCD limit in the ultraviolet. As we are dealing with 5 active quarks, 𝒢UV(s)\mathcal{G}_{UV}(s) is independent of the quark flavors. (s)=[1exp(s2/[4mt4])]/s\mathcal{F}(s)=[1-\exp(-s^{2}/[4m_{t}^{4}])]/s, γm=12/(332Nf)\gamma_{m}=12/(33-2N_{f}), with mt=1.0 GeVm_{t}=1.0\textmd{ GeV}\,, τ=e101\tau=e^{10}-1, Nf=5N_{f}=5, and ΛQCD=0.21 GeV\Lambda_{\text{QCD}}=0.21\textmd{ GeV}\,. The values of mtm_{t} and τ\tau are chosen so that 𝒢UV(s)\mathcal{G}_{UV}(s) is well suppressed in the infrared and the dressed function 𝒢IRfg(s)\mathcal{G}_{IR}^{fg}(s) is qualitatively right in the limit mfm_{f}\to\infty or mgm_{g}\to\infty.

The paremeters DfD_{f} and ωf\omega_{f} that express the flavor dependent quark-antiquark interaction are fitted by the physical observables. See Ref. Chen2019 for the detail. With the parameters well fitted, the axial-vector Ward-Takahashi identity (av-WTI) is perfectly satisfied Chen2019 , which guarantees the ground state pseudoscalar mesons as Goldstone bosons of DCSB. Our model gives a successful and unified description of the light, heavy and heavy-light ground pseudoscalar and vector mesons Chen2019 .

III Results

Three sets of parameters of the charm and bottom system corresponding to the varying of the interaction width are listed in Table 1. The current quark masses, defined in Ref. Chang2019 , are mˇc=1.31GeV\check{m}_{c}=1.31\,\textmd{GeV} and mˇb=4.27GeV\check{m}_{b}=4.27\,\textmd{GeV}.

Table 1: Three sets of the parameters ωf\omega_{f} and DfD_{f} (in GeV) of the charm and bottom system Chen2019 .
flavor Para-1 Para-2 Para-3
ωf\omega_{f} Df2D_{f}^{2} ωf\omega_{f} Df2D_{f}^{2} wfw_{f} Df2D_{f}^{2}
cc 0.690 0.645 0.730 0.599 0.760 0.570
bb 0.722 0.258 0.766 0.241 0.792 0.231

The masses of the charmonium are listed in Tab. 2. Mcc¯RLM^{\textmd{RL}}_{c\bar{c}} is our RL approximation result. Mcc¯expt.M^{\textmd{expt.}}_{c\bar{c}} is the experiment value Tanabashi2018 . ΔMcc¯RL\Delta M^{\textmd{RL}}_{c\bar{c}} is the deviation of the RL results from the experiment value,

ΔMcc¯RL=Mcc¯RLMcc¯expt..\Delta M^{\textmd{RL}}_{c\bar{c}}=M^{\textmd{RL}}_{c\bar{c}}-M^{\textmd{expt.}}_{c\bar{c}}. (8)

The mass of the pseudoscalar (JPC=0+J^{PC}=0^{-+}) meson is used to fit the parameters, so there is no deviation for it. For all the other mesons, the deviations are less than 3.5%3.5\%. Three sets of parameters are used in our calculation. The uncertainties due to the varying of the parameters are very small. The absolute uncertainty is 4 MeV4\textmd{ MeV} for the vector (JPC=1J^{PC}=1^{--}) meson, and 5 MeV5\textmd{ MeV} for the scalar meson (JPC=0++J^{PC}=0^{++}). For other mesons (JPC=1+, 1++, 2++J^{PC}=1^{+-},\,1^{++},\,2^{++}) the absolute uncertainties are less than 26 MeV26\textmd{ MeV}, with the relative uncertainties less than 0.8%0.8\%.

Table 2: Masses (in MeV) of the charmonium with JPC= 0+, 1, 0++, 1+, 1++, 2++J^{PC}\,=\,0^{-+},\,1^{--},\,0^{++},\,1^{+-},\,1^{++},\,2^{++}, the normal states in the quark model. Mcc¯RLM^{\textmd{RL}}_{c\bar{c}} is our RL approximation result. Mcc¯expt.M^{\textmd{expt.}}_{c\bar{c}} is the experiment value Tanabashi2018 . ΔMcc¯RL=Mcc¯RLMcc¯expt.\Delta M^{\textmd{RL}}_{c\bar{c}}=M^{\textmd{RL}}_{c\bar{c}}-M^{\textmd{expt.}}_{c\bar{c}} is the deviation of our results from the experiment value. Three sets of parameters in Tab. 1 are used in our calculation.
JPCJ^{\textmd{PC}} 0+0^{-+} 11^{--} 0++0^{++} 1+1^{+-} 1++1^{++} 2++2^{++}
Mcc¯RLM^{\textmd{RL}}_{c\bar{c}} Para-1 2984 3134 3327 3400 3417 3497
Para-2 2984 3132 3331 3416 3426 3511
Para-3 2984 3130 3332 3426 3431 3518
ΔMcc¯RL\Delta M^{\textmd{RL}}_{c\bar{c}} Para-1 0 37 -88 -125 -94 -59
Para-2 0 35 -84 -109 -85 -45
Para-3 0 33 -83 -99 -80 -38
Mcc¯expt.M^{\textmd{expt.}}_{c\bar{c}} 2984 3097 3415 3525 3511 3556
Table 3: Masses (in MeV) of the bottomonium. The meanings of the quantities are the same as in Tab. 2.
JPCJ^{\textmd{PC}} 0+0^{-+} 11^{--} 0++0^{++} 1+1^{+-} 1++1^{++} 2++2^{++}
Mbb¯RLM^{\textmd{RL}}_{b\bar{b}} Para-1 9399 9453 9754 9793 9788 9820
Para-2 9399 9453 9762 9805 9799 9833
Para-3 9399 9453 9765 9810 9804 9835
ΔMbb¯RL\Delta M^{\textmd{RL}}_{b\bar{b}} Para-1 0 -7 -105 -106 -106 -92
Para-2 0 -7 -97 -94 -94 -79
Para-3 0 -7 -94 -89 -89 -77
Mbb¯expt.M^{\textmd{expt.}}_{b\bar{b}} 9399 9460 9859 9899 9893 9912

The masses of the bottomonium are listed in Tab. 3. Mbb¯RLM^{\textmd{RL}}_{b\bar{b}} and Mbb¯expt.M^{\textmd{expt.}}_{b\bar{b}} are our RL approximation result and the experiment value Tanabashi2018 respectively. ΔMbb¯RL\Delta M^{\textmd{RL}}_{b\bar{b}} is the deviation, defined by

ΔMbb¯RL=Mbb¯RLMbb¯expt..\Delta M^{\textmd{RL}}_{b\bar{b}}=M^{\textmd{RL}}_{b\bar{b}}-M^{\textmd{expt.}}_{b\bar{b}}. (9)

There is no mass deviation for the pseudoscalar meson as it is used to fit the parameters. For all the other mesons, the deviations are less than 1.1%1.1\%. The uncertainty due to the varying of the parameters is very small for the vector meson (less than 1 MeV1\textmd{ MeV}). For other mesons (JPC=0++, 1+, 1++, 2++J^{PC}=0^{++},\,1^{+-},\,1^{++},\,2^{++}) the absolute uncertainties are less than 17 MeV17\textmd{ MeV}, with the relative uncertainties less than 0.2%0.2\%.

The results in Tab. 2 and Tab. 3 have two-sided meanings. On one hand, the RL approximation is reasonble for the charmonium and bottomonium system. The mass deviations are less than 3.5%3.5\% for the charmonium and less than 1.1%1.1\% for the bottomonium. However, the absolute errors do not decrease for the P-wave states, indicating that in the RL approximation some interactions are still lacking even for the heavy system. On the other hand, the results are stable when the parameters change. The uncertainties due to the varying of the parameters are small, less than 0.8%0.8\% for the charmonium and less than 0.2%0.2\% for the bottomonium. So the errors (defined by Eq.(8) and Eq.(9)) of the RL approximation could be estimated quantitatively. The interaction, Eq.(5) \sim Eq.(7), expresses the flavor dependence properly, so that the errors are of the same order for both the open flavor mesons and the qq¯q\bar{q} mesons Chen2019 . As the masses of the BcB_{c} mesons are approximately Mcb¯,JP(Mcc¯,JP+Mbb¯,JP)/2M_{c\bar{b},J^{P}}\approx(M_{c\bar{c},J^{P}}+M_{b\bar{b},J^{P}})/2, the mass errors of the BcB_{c} mesons are assumed to be:

ΔMcb¯,JPRL=1NCC(ΔMcc¯,JPCRL+ΔMbb¯,JPCRL)/2,\Delta M^{\textmd{RL}}_{c\bar{b},J^{P}}=\frac{1}{N_{C}}\sum_{C}(\Delta M^{\textmd{RL}}_{c\bar{c},J^{PC}}+\Delta M^{\textmd{RL}}_{b\bar{b},J^{PC}})/2, (10)

with JPCJ^{PC} the normal states, NCN_{C} the number of the normal states. For example, if JP=0J^{P}=0^{-}, then the corresponding normal cc¯c\bar{c} or bb¯b\bar{b} state is JPC=0+J^{PC}=0^{-+}, and NC=1N_{C}=1. ΔMcb¯,0RL=(ΔMcc¯,0+RL+ΔMbb¯,0+RL)/2\Delta M^{\textmd{RL}}_{c\bar{b},0^{-}}=(\Delta M^{\textmd{RL}}_{c\bar{c},0^{-+}}+\Delta M^{\textmd{RL}}_{b\bar{b},0^{-+}})/2. It’s the same case for the JP=1, 0+, 2+J^{P}=1^{-},\,0^{+},\,2^{+} mesons. If JP=1+J^{P}=1^{+}, then both JPC=1++J^{PC}=1^{++} and 1+1^{+-} are the normal states and NC=2N_{C}=2. The JP=1+J^{P}=1^{+} BcB_{c} mesons are the mixing states of the P11{}^{1}P_{1} state and P13{}^{3}P_{1} state Li2019 . We do not bother to investigate the mixing. The mass errors of the JP=1+J^{P}=1^{+} mesons are estimated to be ΔMcb¯,1+RL=(ΔMcc¯,1++RL+ΔMcc¯,1+RL+ΔMbb¯,1++RL+ΔMbb¯,1+RL)/4\Delta M^{\textmd{RL}}_{c\bar{b},1^{+}}=(\Delta M^{\textmd{RL}}_{c\bar{c},1^{++}}+\Delta M^{\textmd{RL}}_{c\bar{c},1^{+-}}+\Delta M^{\textmd{RL}}_{b\bar{b},1^{++}}+\Delta M^{\textmd{RL}}_{b\bar{b},1^{+-}})/4, i.e., averaging the CC-parity. In Tab. 2 and Tab. 3, the difference of ΔMcc¯RL\Delta M^{\textmd{RL}}_{c\bar{c}} with JPC=1++J^{PC}=1^{++} and JPC=1+J^{PC}=1^{+-} are less than 31 MeV31\textmd{ MeV}, and there is no difference for ΔMbb¯RL\Delta M^{\textmd{RL}}_{b\bar{b}} for these two states. So taking the CC-parity average leads no more than 8 MeV8\textmd{ MeV} error for ΔMcb¯,1+RL\Delta M^{\textmd{RL}}_{c\bar{b},1^{+}}.

Table 4: The masses (in MeV) of the first radial excited states of the charm-bottom system with JP=0J^{P}=0^{-} (cited from Ref. Chang2019 ). The experiment data for Mηc(2S)M_{\eta_{c}(2S)} and Mηb(2S)M_{\eta_{b}(2S)} are taken from Ref. Tanabashi2018 , and that for MBc+(2S)M_{B^{+}_{c}(2S)} is taken from Ref. Aaij2019 .
state ηc(2S)\eta_{c}(2S) Bc+(2S)B^{+}_{c}(2S) ηb(2S)\eta_{b}(2S)
RL 36063606 68136813 99159915
expt. 36383638 68726872 99999999
ΔMRL\Delta M^{\textmd{RL}} -32 -59 -84

The relation Eq.(10) is also supported by the masses of the radial excited states Chang2019 . The center values of the excited state masses are cited in Tab. 4. (ΔMηc(2S)RL+ΔMηb(2S)RL)/2=58 MeV(\Delta M^{\textmd{RL}}_{\eta_{c}(2S)}+\Delta M^{\textmd{RL}}_{\eta_{b}(2S)})/2=-58\textmd{ MeV}, while ΔMBc+(2S)RL=59 MeV\Delta M^{\textmd{RL}}_{B^{+}_{c}(2S)}=-59\textmd{ MeV}. The direct calculation of the BcB_{c} meson masses in the RL approximation and the mass errors due to the RL approximation are listed in Tab. 5. The modified mass is defined by

M¯cb¯RL=Mcb¯RLΔMcb¯RL.\bar{M}^{\textmd{RL}}_{c\bar{b}}=M^{\textmd{RL}}_{c\bar{b}}-\Delta M^{\textmd{RL}}_{c\bar{b}}. (11)

With the errors from the RL approximation subduced, M¯cb¯RL\bar{M}^{\textmd{RL}}_{c\bar{b}} are our prediction for the BcB_{c} mesons. There are two kinds of other errors for all the BcB_{c} mesons, which are estimated as following. Mcc¯,0+RLM^{\textmd{RL}}_{c\bar{c},0^{-+}} and Mbb¯,0+RLM^{\textmd{RL}}_{b\bar{b},0^{-+}} are used to fit the parameters, so the error of Mcb¯,0RLM^{\textmd{RL}}_{c\bar{b},0^{-}} is totally due to the interaction pattern, Eq.(5) \sim Eq.(7). The error due to this interaction pattern, the first error, is about 15 MeV15\textmd{ MeV}. The results vary by a few MeVs as the parameters change, which is the second error. These errors due to the varying of the paramters are much smaller than those of the charmonium and the bottomonium. The uncertainties of the parameters cancel by using Eq.(10), i.e., inferring the errors of the BcB_{c} mesons as the intermidiate of the charmonium and the bottomonium. For the JP=1+J^{P}=1^{+} BcB_{c} mesons, there is a third error due to the CC-parity average in Eq.(10), which is about 8 MeV8\textmd{ MeV}.

Table 5: Masses of the BcB_{c} Mesons (in MeV). Mcb¯RLM^{\textmd{RL}}_{c\bar{b}} is the direct RL result. ΔMcb¯RL\Delta M^{\textmd{RL}}_{c\bar{b}} is the error of the RL approximation defined by Eq.(10). M¯cb¯RL\bar{M}^{\textmd{RL}}_{c\bar{b}} is the modified mass, defined by Eq.(11).
JPJ^{\textmd{P}} 00^{-} 11^{-} 0+0^{+} 11+1_{1}^{+} 12+1_{2}^{+} 2+2^{+}
Mcb¯RLM^{\textmd{RL}}_{c\bar{b}} Para-1 6293 6360 6608 6642 6677 6721
Para-2 6290 6357 6612 6649 6686 6731
Para-3 6287 6354 6612 6651 6688 6733
ΔMcb¯RL\Delta M^{\textmd{RL}}_{c\bar{b}} Para-1 0 15 -97 -108 -108 -75
Para-2 0 14 -91 -96 -96 -62
Para-3 0 13 -89 -89 -89 -57
M¯cb¯RL\bar{M}^{\textmd{RL}}_{c\bar{b}} Para-1 6293 6345 6705 6750 6785 6796
Para-2 6290 6343 6703 6745 6782 6793
Para-3 6287 6341 6701 6740 6777 6790
Table 6: Masses of the BcB_{c} Mesons (in MeV). M¯cb¯RL\bar{M}^{\textmd{RL}}_{c\bar{b}} is our prediction. The first error is due to the interaction pattern Eq.(5) \sim Eq.(7). The second error is due to the varying of the parameters. For JP=1+J^{P}=1^{+} mesons, the third error is due to the CC-parity average in Eq.(10). Mcb¯QMM^{\textmd{QM}}_{c\bar{b}} is the quark model result Li2019 , and the underlined ones are the input values. Mcb¯LQCDM^{\textmd{LQCD}}_{c\bar{b}} is the lQCD prediction Mathur2018 . Mcb¯expt.M^{\textmd{expt.}}_{c\bar{b}} is the experiment value Tanabashi2018 .
JPJ^{\textmd{P}} M¯cb¯RL\bar{M}^{\textmd{RL}}_{c\bar{b}} Mcb¯QMM^{\textmd{QM}}_{c\bar{b}} Mcb¯LQCDM^{\textmd{LQCD}}_{c\bar{b}} Mcb¯expt.M^{\textmd{expt.}}_{c\bar{b}}
00^{-} 6290(15)(3) 6271 6276(7) 6275(1)
11^{-} 6343(15)(2) 6326 6331(7)
0+0^{+} 6703(15)(2) 6714 6712(19)
11+1_{1}^{+} 6745(15)(5)(8) 6757 6736(18)
12+1_{2}^{+} 6781(15)(4)(8) 6776
2+2^{+} 6793(15)(3) 6787

Our predictions of the BcB_{c} mesons and the estimated errors are listed in the second column of Tab.6. Hitherto the only experiment data for the BcB_{c} spectrum is the pseudoscalar meson mass, as the production rate of the BcB_{c} mesons is much lower than those of the charmonium and the bottomonium. Our results are consistent with the recent lQCD predictions (with JP=0, 1, 0+, 1+J^{P}=0^{-},\,1^{-},\,0^{+},\,1^{+}), which are listed in the forth column. Our results are also consistent with the quark model predictions. The quark model predictions from Ref. Li2019 are listed in the third column, where the masses with JP=0, 1J^{P}=0^{-},\,1^{-} are the input values and others are the outputs. The mass splitting of the 1S1S state of our result is MBc(1)MBc(0)=53 MeVM_{B_{c}(1^{-})}-M_{B_{c}(0^{-})}=53\textmd{ MeV}, consistent with the lQCD result 55 MeV55\textmd{ MeV}. The mass splittings of the 1P1P states of our results are (MBc(11+)+MBc(12+))/2MBc(0+)=60 MeV(M_{B_{c}(1^{+}_{1})}+M_{B_{c}(1^{+}_{2})})/2-M_{B_{c}(0^{+})}=60\textmd{ MeV} and MBc(2+)MBc(0+)=90 MeVM_{B_{c}(2^{+})}-M_{B_{c}(0^{+})}=90\textmd{ MeV}, while the quark model results are 53 MeV53\textmd{ MeV} and 73 MeV73\textmd{ MeV} respectively. Our results of the BcB_{c} meson masses also have two-sided meanings. On one hand, we predict the masses of the BcB_{c} mesons (with JP=0, 1, 0+, 1+, 2+J^{P}=0^{-},\,1^{-},\,0^{+},\,1^{+},\,2^{+}), providing a significant guide to the experimental search for the BcB_{c} mesons. On the other hand, the consistency of our results with other predictions supports that the flavor dependent interaction pattern, Eq.(5) \sim Eq.(7), is reasonable. This pattern leads an error about 15 MeV15\textmd{ MeV} for the BcB_{c} mesons.

IV Summary

In this paper, we predict the masses of the BcB_{c} mesons with JP=0, 1, 0+, 1+, 2+J^{P}=0^{-},\,1^{-},\,0^{+},\,1^{+},\,2^{+} using a flavor dependent interaction pattern via the Dyson-Schwinger equation and Bethe-Salpeter equation approach. This interaction pattern, composed of a flavor dependent infrared part and a flavor independent ultraviolet part, gives an unified successful description of the pseudoscalar and vector light, heavy-light and heavy mesons. This interaction pattern could also be applied to the radial excited heavy mesons. Herein we control the errors carefully. Besides the error from the RL approximation, two other kinds of errors are considered. One is the error from the interaction pattern, the other is the error from the varying of the parameters. For the JP=1+J^{P}=1^{+} BcB_{c} mesons, a third error due to the CC-parity average is also considered. With the errors from the RL approximation subduced, our predictions are consistent with the lQCD and quark model results. Our results have two-sided meanings. On one hand, we predict the BcB_{c} meson masses with errors well controlled, providing a significant guide to the experiment search. On the other hand, the excellent consistency of our predictions with the results from other approaches also supports strongly that the flavor dependent interaction pattern, Eq.(5) \sim Eq.(7), is reasonable.

Acknowledgments

We acknowledge helpful conversations with Xian-Hui Zhong, Qi Li, Ming-Sheng Liu and Long-Cheng Gui. Chen thanks Xian-Hui Zhong for the surport and the encourage in publishing this paper. This work is supported by: the National Natural Science Foundation of China under contracts No. 11947108, the Chinese Government Thousand Talents Plan for Young Professionals and the National Natural Science Foundation of China under contracts No. 11435001, and No. 11775041, the National Key Basic Research Program of China under contract No. 2015CB856900.

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